An Experimental and Numerical Investigation of Heat Transfer Downstream of a Normal Film Cooling Injection Slot By Yuping Sun B. Eng., Nanjing University of Science and Technology, 1983 M. Eng., Nanjing University of Science and Technology, 1986 M.A.Sc., Technical University of Nova Scotia, 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1995 © Yuping Sun, 1995 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of Mechanical Engineering The University of British Columbia Vancouver, Canada Date: /:;73i4f 3 Abstract This thesis presents an experimental and numerical investigation of turbulent heat transfer associated with film cooling from a 2D slot. To determine the film cooling heat transfer coefficient, detailed mass transfer measurements have been carried out downstream of a normal film cooling injection slot. The plate downstream of the injection location is porous and air contaminated with propane or methane is bled through the plate beneath it. By measuring the propane or methane concentration very close to the wall, mass transfer measurements are conducted for film cooling mass flow ratios ranging from zero to 0.5. The mass transfer coefficients are calculated using a wall function correction formula, which corrects the measurements for displacement from the surface, and are then related directly to corresponding heat transfer coefficients using the mass/heat transfer analogy. The validity of the method and the wall function correction formula are checked by examining the case with zero film coolant injection, a situation analogous to the well-known turbulent boundary layer mass/heat transfer with impermeable/unheated starting length. Good agreement with data in the literature is obtained for this experiment. For film cooling with low mass flow ratios (M 0.1), heat transfer coefficients close to those of a conventional turbulent boundary layer are obtained. At high values of mass flow ratio (M > 0.1) heat transfer coefficients similar to those of turbulent separated flows are observed, reflecting the important effect of the separation bubble just downstream of injection. Numerical calculations of turbulent flow and heat transfer associated with film cooling have been carried out to validate the present mass transfer method and to ensure that the heat transfer coefficients obtained using the mass/heat transfer analogy are valid. In the present study, the time-averaged continuity equation, Navier Stokes equations and thermal energy equation, together with the k — a turbulence model, are used to describe the turbulent flow and heat transfer. This system of governing equations is discretized using the control volume method. To obtain the numerical solution of the resulting finite difference equations, a novel multi-grid method with no boundary correction is II proposed. This method differs from the FAS multi-grid method in that it employs a special multi-grid procedure to ensure that the correction on the coarse grids vanishes along the boundary, and it offers an easier procedure to overcome difficulties arising from the boundary treatment in both discretization and the multi-grid algorithm. The performance of the method is examined using the laminar cavity flow and the turbulent boundary layer flow, which indicates that it is efficient and can overcome difficulties resulting from the near-wall treatment and the discrete vorticity boundary condition. Numerical simulation of turbulent flow and heat transfer associated with film cooling are conducted using the proposed multigrid method. Detailed turbulent flow field and heat transfer results are obtained for film cooling mass flow ratios ranging from zero to 0.5. The numerical results show that the standard k — a turbulence model, together with the standard wall function, underpredicts considerably the heat transfer coefficients of film cooling at high mass flow ratio, and therefore, is not appropriate for accurate prediction of turbulent heat transfer in the region of flow separation. However, with a modified near-wall treatment, predictions of heat transfer coefficients compare well with the experimental results obtained using the mass/heat transfer analogy. Numerical calculations also show that the overall effect of the wall transpiration for the blowing rates used for the mass transfer measurements on heat transfer is negligible. The present study finally concludes that both the numerical and mass transfer method can be used to investigate film cooling heat transfer with fair accuracy. III Table of Contents Abstract ii Table of Contents iv List of Tables vii List of Figures viii List of Symbols xi xiv Acknowledgements 1 1 Introduction 1.1 Problem Description 4 1.2 Literature Review 6 1.2.1 Analytic Studies 6 1.2.2 Experimental Studies 8 10 1.2.3 Numerical Studies 1.3 Objectives and Scope of the Present Investigation 2 Experimental Arrangement and Measurement Techniques 2.1 Experimental Apparatus and Equipment 15 17 17 2.1.1 Wind Tunnel Facility 17 2.1.2 Hot Wire Anemometer 19 2.1.3 Heat Transfer Simulating Apparatus 20 2.1.4 Flame Ionization Detector 22 2.2 Measurement and Calculation of Mass Transfer Coefficient 27 2.2.1 Mass Transfer Measurement 27 2.2.2 Wall Function Correction of Stanton Number 28 35 2.3 Experimental Procedure Iv v Content 3 Experimental Results 36 3.1 Test Conditions 36 3.2 Two Dimensionality 37 3.3 Mass/Heat Transfer with Impermeable/Unheated Starting Length 39 3.3.1 Mass Transfer with Impermeable Starting Length 42 3.3.2 Effect of Blowing Rate on Mass Transfer 45 3.3.3 Heat Transfer with Unheated Starting Length 46 3.4 Mass/Heat Transfer with Film Cooling 47 3.4.1 Mass Transfer with Film Cooling 49 3.4.2 Mass/Heat Analogy 53 3.4.3 Heat Transfer with Film Cooling 55 3.5 Separation Reattachment Length 57 4 Mathematical Model 61 4.1 Introduction 61 4.2 Time-Averaged Governing Equations 63 4.3 Turbulence Model Equations 64 4.4 Near Wall Treatment 70 4.4.1 Standard Wall Function 71 4.4.2 Modified Wall Function 73 4.5 General Transport Equation 76 4.6 Boundary Conditions 79 5 Numerical Solution Method 80 5.1 Discretization of Governing Equations 80 5.1.1 Grid Arrangement 81 5.1.2 Finite Difference Approximation 81 5.2 Boundary Conditions 86 5.3 Numerical Solution Procedure 88 5.4 Multi-grid Acceleration 89 5.4.1 The NBCFAS Algorithm 92 5.4.2 The Pressure Correction Scheme 94 Content 5.4.3 Restriction and Prolongation Operators 5.5 Applications 5.5.1 Numerical Simulation of Cavity Flow 5.5.2 Numerical Simulation of Turbulent Boundary Layer 5.6 Conclusions 6 Numerical Results and Comparison with Experiments 6.1 Numerical Simulation of Turbulent Boundary Layer Heat Transfer vi 96 98 98 108 109 113 113 6.1.1 Turbulent Boundary Layer Heat Transfer with Unheated Starting Length 114 6.1.2 Effect of Wall Blowing 117 6.2 Numerical Simulation of Turbulent Flow and Heat Transfer with Film Cooling 117 6.2.1 Grid Structure 119 6.2.2 Convergence Performance 121 6.2.3 Turbulent Flow and Heat Transfer with Film Cooling 122 6.2.4 Effect of Blowing Rate 130 6.2.5 Comparison of Near-Wall Treatments 131 7 Conclusions and Recommendations 134 7.1 Conclusions 134 7.2 Recommendations 136 REFERENCES 137 Appendix A Diffusion Coefficients of Air-Propane Mixture 149 Appendix B Diffusion Coefficients of Air-Methane Mixture 150 Appendix C Experimental Procedures 152 LIST OF TABLES 69 model constants Table 4.1 The k Table 4.2 Diffusion coefficients and source terms 78 Table 5.1 Linearized source terms 87 — E VII LIST OF FIGURES 5 Figure 1.1 Schematic of film cooling with normal injection Figure 2.1 Wind tunnel schematic 18 Figure 2.2 Heat transfer simulating apparatus 21 Figure 2.3 Top view of porous plate 22 Figure 2.4 Schematic diagram of the F1D 24 Figure 2.5a FID calibration with air-propane 26 Figure 2.5b FID calibration with air-methane 26 Figure 2.6 The near wall node 32 Figure 3.1 Velocity profile 10 slot width upstream of the film cooling slot 38 Figure 3.2 Two dimensionality of mass transfer with film cooling at M = 0.05 40 Figure 3.3 Two dimensionality of mass transfer with film cooling at M Figure 3.4 Two dimensionality of mass transfer with film cooling at M Figure 3.5 Mass transfer calculation with and without wall function correction 44 Figure 3.6 44 Figure 3.7 Mass transfer Stanton number with impermeable starting length Effect of V on mass transfer with impermeable starting length Figure 3.8 Heat transfer Stanton number with unheated starting length 48 Figure 3.9a Mass transfer Stanton number of film cooling at low M 50 = 0.1 40 0.2 41 46 50 Mass transfer Stanton number of film cooling at high M Figure 3. lOa Effect of V on mass transfer of film cooling at M = 0.2 51 0.3 51 Figure 3.1 Oc Effect of V, on mass transfer of film cooling at M =0.4 52 Figure 3. lOd Effect of V on mass transfer of film cooling at M = 0.5 52 Figure 3.9b Figure 3. lOb Effect of V, on mass transfer of film cooling at M Figure 3.11 Mass transfer Stanton number obtained with methane as tracer gas 54 Figure 3.12 Ratio of Stanton number with methane to that with propane 54 Figure 3. 13a Heat transfer Stanton number of film cooling at low M VIII 58 List of Figures ix Figure 3. 13b Heat transfer Stanton number of film cooling at high M 59 Figure 3.1 3c Normalized Stanton number of film cooling at high M 59 Figure 3.14 Variation of reattachment length 60 Figure 5.1 Staggered grid arrangement 82 Figure 5.2 Fine-coarse grid arrangement 97 Figure 5.3 Cavity configuration 99 Figure 5.4 Upper wall boundary 100 Figure 5.5 A non-staggered grid 102 Figure 5.6a Streamline contours of cavity flow at Re = 100 104 Figure 5.6b Streamline contours of cavity flow at Re = 200 104 Figure 5.6c Streamline contours of cavity flow at Re = 500 105 Figure 5.6d Streamline contours of cavity flow at Re Figure 5.7a Convergence path for cavity flow at Re = 100 106 Figure 5.7b Convergence path for cavity flow at Re = 200 106 Figure 5.7c Convergence path for cavity flow at Re = 500 107 Figure 5.7d Convergence path for cavity flow at Re = 1000 107 Figure 5.8 Variation of turbulent boundary layer shear stress coefficient 110 Figure 5.9a Convergence path of SGM and NBCFAS for U momentum equation 110 Figure 5.9b Convergence path of SGM and NBCFAS for V momentum equation 111 Figure 5.9c Convergence path of SGM and NBCFAS for continuity equation 111 Figure 5.9d Convergence path of SGM and NBCFAS for equation 112 Figure 5.9e Convergence path of SGM and NBCFAS forE equation 112 Figure 6.1 Calculation domain for boundary layer 115 Figure 6.2 Coarsest grid for boundary layer calculation 116 Figure 6.3 Heat transfer Stanton number of boundary layer with unheated starting = 1000 K 105 length 116 Figure 6.4 Effect of wall blowing rate V on heat transfer Stanton number 118 Figure 6.5a Calculation Domain for Film Cooling 120 Figure 6.5b Coarsest grid for film cooling flow and heat transfer 120 Figure 6.5c Finest grid for film cooling flow and heat transfer 121 Listof Figures x Figure 6.6a Convergence path of SGM and NBCFAS for U momentum equation 123 Figure 6.6b Convergence path of SGM and NBCFAS for V momentum equation 123 Figure 6.6c Convergence path of SGM and NBCFAS for continuity equation 124 Figure 6.6d Convergence path of SGM and NBCFAS for K equation 124 Figure 6.6e Convergence path of SGM and NBCFAS for e equation 125 Figure 6.7a Local flow field for film cooling at M = 0.05 125 Figure 6.7b Local flow field for film cooling at M =0.1 126 Figure 6.8a Local flow field for film cooling at M = 0.3 126 Figure 6.8b Local flow field for film cooling at M = 0.5 127 Figure 6.9 Reattachment length of turbulent flow with film cooling 127 Figure 6. lOa Stanton number of heat transfer with film cooling at M = 0.05 128 Figure 6. lOb Stanton number of heat transfer with film cooling at M = 0.1 128 Figure 6.1 la Stanton number of heat transfer with film cooling at M = 0.3 129 Figure 6.1 lb Stanton number of heat transfer with film cooling at M = 0.5 Figure 6.12 Effect of V on heat transfer of film cooling at M = 0.1 129 Figure 6.13 Effect of V on heat transfer of film cooling at M = 0.3 132 132 Figure 6. 14a Predictions of heat transfer with film cooling at M = 0.3 133 0.5 133 Figure 6. 14b Predictions of heat transfer with film cooling at M LIST OF SYMBOLS a Finite difference coefficient Cf Shear stress coefficient Cp Specific heat at constant pressure Cp total pressure coefficient (=[P —P ]1--pU) m d m Defect on grid G D Diffusive flux B Voltage; integral constant in logarithmic law of the wall F Convective flux G Grid of level n h Heat transfer coefficient redefmed for film cooling h’ Convective heat transfer coefficient h,,, Mass transfer coefficient I,’” Interpolation operator from I Total flux C 1 C 2 C k Turbulence kinetic energy Sc Lewis number (= —) Pr Le , GI m to G turbulence model constants , Im Mixing length tm N B tm ,, m L m Finite difference operator on G M Mass flow rate of film cooling m” Mass flux n An empirical constant XI List of Symbols xii p Static pressure Pr Prandtl number Pressure (= p+-p1c) q Heat flux Q Volume flow rate Re Reynolds number St Stanton number St,, Approximate Stanton number t Time U Velocity vector U, V Mean velocity in x, y directions u Nondimensionalized velocity 1 u Instant velocity in x, direction v Wall blowing rate x, x, y Cartesian coordinate 0 x Unheated starting length Xr Reattachment length Nondimensionalized normal wall distance Boundary layer thickness x, y Computational cell dimensions ox, Oy Grid spacing Dissipation rate of turbulent kinetic energy TI Film cooling effectiveness K Von Karman constant p Density Wall shear stress List of Symbols xiii a Schmidt number 1.) Kinematic viscosity I’ Dynamic viscosity Dissipation function Instant temperature 0 General variable a, GK Turbulence model constants N’ Near wall turbulence intensity N’ E Turbulence intensity of a turbulent boundary layer Diffusion coefficient General diffusion coefficient for 0 F Sub/superscripts 1 Propane, methane 2 Coolant ° o Free stream value aw Adiabatic wall value eff Effective value h Heat transfer i, j Tensor indices; grid points m Mass transfer P,N,S,E,W,NE, NW, SE, SW Grid points n, s, e, w Cell faces p Near wall point Turbulent value w Wall value ACKNOWLEDGEMENTS This work was carried out under the supervision of Dr. Martha Salcudean and Dr. Ian Gartshore. I would like to take this opportunity to express my sincere gratitude to them for their invaluable guidance, understanding and financial support. Thanks also go to Dr. E. Hauptmann. His suggestions and advice regarding the wall function correction fonnulae are deeply appreciated. I greatly appreciate the help from Mr. A. Steeves and G. Rohling during various stages of the thesis. Many friends have provided help. My sincere thanks go to all, with special mention of Mr. K. Zhang, Ms. Z. Xiao, Dr. D. Tse, Mr. M. Findlay, and Mr. M. Savage. xiv Chapter 1 INTRODUCTION Modern gas turbines operate at extremely high temperatures to attain the required high performance. The strength and the life of the materials involved may be reduced at these high temperatures. New materials have been developed to improve the high temperature properties; however, the cost of new materials often makes them impractical. As a result, many of the components still require thermal protection. For example, the turbine blades must be cooled several hundred degrees below the gas temperatures. To protect the surfaces from exposure to the hot gas stream, relatively sophisticated methods of cooling such high performance components are necessary. One of the most effective cooling methods is the mass transfer method, which involves injection of a secondary coolant fluid into the boundary layer on the surface to be protected. Two different approaches can be used to introduce the coolant into the boundary layer, transpiration cooling and film cooling. In transpiration cooling, the surface is porous, and the coolant enters the boundary layer through the permeable surface. This is primarily designed to protect the region where the coolant enters the boundary layer. Transpiration cooling is very effective. In addition, the coolant can effectively thicken the boundary layer and reduce the heat transfer rate. However, the porous structure involves a difficult fabrication process, and porous materials to date have not had the high strength required for some applications (such as turbine blades). In addition, the small pore size of porous materials could lead to clogging and maldistribution of coolant flow along the surface, and variations in the external pressure 1 Chapter 1 2 distribution could also result in a non-optimum secondary flow distribution through the permeable surface. Film cooling involves the injection of coolant into the boundary layer by means of a series of orifices located along the surface. The key difference between film cooling and transpiration cooling is that film cooling is not primarily intended for protection of the surface where the coolant is introduced into the boundary layer, but rather the protection of the region downstream of the injection location. Various injection geometries have been employed in film cooling such as tangential or near tangential slots, angled slots, multiple slots and more recently rows of discrete holes. To design the film cooling arrangements, one must predict or measure the wall temperatures, the stream velocities and temperatures, and the heat transfer rates. Two calculation methods, designated as calculation method A and B, have been proposed to predict the heat transfer with film cooling (Eckert, 1984). Both of these calculation methods are based on the assumption that the fluid properties involved are constant and equal for the main stream fluid and the coolant fluid. Upon this assumption, the thermal energy equation is linear and superposition of the temperature field is possible. For film cooling applications, calculation method A, which is widely used, redefines the heat transfer coefficient using an adiabatic wall temperature, namely: h=T w (1.1) Taw Equation (1.1) uses an adiabatic wall temperature T, as the “reference” temperature. , 4 This definition of h has the advantage that q =0 when T, = Taw as it should, so that h is well behaved (no zero or infinite values). The adiabatic wall temperature is defined in a dimensionless form as the film cooling effectiveness 7ZWTO 1 (1.2) , and the mainstream, T, are assumed to be 2 where the temperatures of coolant, T constant. Chapter 1 3 In calculation method A, the required calculations of q and T are made possible by a knowledge of h and ri. This method appears to have been introduced by Scesa (1954), and a complete description of its use is given later by Eckert (1984). The heat transfer coefficient (h), measured when “coolant” and free stream temperatures are identical, measures the effect of heat transfer near the wall without the complication of heat transfer between coolant and free stream gases. The film cooling effectiveness (rj), measured with no heat transfer at the wall, represents the external effects of coolant and free stream mixing without complications near the wall. As Eckert (1984) pointed out, these two effects can be combined into a general case provided all temperature variations are small, so that the fluid equations are independent of the thermal equation. The above approach has been widely accepted in the study of heat transfer associated with film cooling. At low values of mass flow ratio (M), the values of h are often reported to be similar to those with no coolant injection. However, at high values of mass flow ratio (M) this similarity is not valid. Heat transfer rates in the region immediately downstream of the injection location, which is the region of interest in many practical applications, cannot be satisfactorily predicted with only the primary stream heat transfer coefficients and adiabatic wall temperature distributions. In this region the coolant injection significantly alters the flow pattern from that of the primary fluid alone, and the effective heat transfer coefficient may be changed considerably. To accurately predict the heat transfer rate, the adiabatic wall temperature measurements must be supplemented with isoenergetic heat transfer measurements, which must be carried out for each case with close attention being given to the region immediately downstream of the film cooling injection location. By defining the heat transfer rate using the adiabatic wall temperature, the adiabatic wall condition becomes an important consideration in the design of the film cooling heat transfer experiment. Two expansive test plates may then be required for both the direct heat transfer measurement and the indirect measurement using the mass/heat transfer analogy. Chapter 1 4 For these reasons, calculation method B was adopted by Metzger (1968) and later by Moffat (1980) to describe the film cooling heat transfer. In calculation method B, the film cooling heat transfer coefficient is defined by the following equation: h’ q (1.3) and a dimensionless parameter, 9, is used, defined as: (1.4) All the effects of film cooling are now carried in h as a function of the dimensionless parameter 9. This method is especially convenient for situations where the wall temperature is uniform. However, the heat transfer coefficient can go to infinity when it is defined by Equation (1.3). Both calculation A and B are equivalent, and results obtained in one method can be transferred to parameters used in the other. The choice of a method for calculation and presentation of heat transfer with film cooling is largely a matter of convenience. Because it is easier to use the method of Scesa (1954) for determining the heat transfer rate using the mass/heat transfer analogy, this method is used in the present investigation of film cooling heat transfer, as described in the following section. 1.1 Problem Description Figure 1.1 depicts a schematic of the turbulent flow field with coolant injection from a normal film cooling injection slot into a main stream. The main stream flow has a known turbulence intensity, and a boundary layer exists upstream of the film cooling injection location. The main stream flow starts to mix with the coolant flow from the leading edge of the film cooling slot. For film cooling with low mass flow injection, the effect of the coolant injection on flow field is not significant, and the heat transfer rate downstream of the film cooling slot can be predicted with the primary stream heat transfer coefficients obtained without coolant. If the film cooling mass flow rate is high, separation and Chapter 1 5 Mainstream Flow Shear layer Turbulent Mixing I Separation Region Coolant Flow Figure 1.1 Schematic of Film Cooling with Normal Injection subsequent reattachment will occur downstream of the film cooling injection location. In the reattachment zone, the flow is characterized by large pressure gradients, low mean velocities, very large local turbulent intensities and unsteady flow reversals, which lead to the failure of the heat/friction analogy and the logarithmic wall function for temperature. An important length scale of this flow is the reattachment length Xr• This length, which measures the extent of the separation bubble, is defined as the distance from the leading edge of the separation bubble to the point where the mean wall shear stress vanishes. The size of the separation bubble or the reattachment length increases with the film cooling mass flow rate. The heat transfer rate downstream of the film cooling slot will be altered by the separation and reattachment. It does not vanish at the point of reattachment, but rather reaches its local maximum value near the reattachment and then gradually decreases to the values typical of a turbulent boundary layer further downstream. In the present film cooling problem, the following assumptions are made: • the main stream flow has a low turbulence intensity and a boundary layer exists upstream of the film cooling slot. Ier1 6 • the separation bubble at high mass flow ratios is stable, and the flow field is steady. • the temperature of the main stream fluid is equal to that of the coolant. • the temperature variations are small, so that buoyancy is negligible and the energy equation is linear. A ieview of the previous studies relevant to the heat transfer with film cooling injection sw presented. LI Literature Review rally, three types of tools can be used for the prediction of fluid flow and heat fer phenomena: 1) analytical solutions of the Navier-Stokes equations or other matical models, 2) numerical solutions using CFD techniques to solve the Linatical models of the fluid flow and heat transfer, and 3) experimental vations of real fluid phenomena as they occur in practice or in a laboratory where arements are easier to obtain. Previous studies relevant to turbulent flow and heat fer with film cooling using analytical, experimental and numerical techniques are &wed in the following sections. 12.1 Analytical Studies ytica1 investigations of thermo-fluid systems often start with the Navier-Stokes —’ons and the thermal energy equation or related simplified mathematical models. In 1 “ n cases, elegant closed form solutions can be obtained, which may contain infinite rs or special functions. In principle, closed form solutions can explicitly give the ion in results with changes in controlling parameters. Thus, optimization of thermo I systems can be accomplished directly with a minimum of effort. However, ytical solutions are generally not available owing to the complex nature of the real cooling heat transfer problems. Only a few limited results obtained using simplified hmaticaI models are available. Ehrich (1953) outlined a Helmholtz-Kirchhoff solution for a generalized slot figuration and obtained analytical solutions to the ideal flow equations pertaining to fr normal injection slot. To eliminate the infinite velocity point at the downstream slot Chapter 1 7 lip, he included, as an option, a free streamline leaving smoothly from the downstream slot wall at the lip. However, this resulted in an unrealistic constant pressure separation region of ultimately infinite length. Fitt et. al. (1985) proposed a simple model of irrotational inviscid flow from a vertical slot into a free stream. Their model assumes that the total pressure in the slot exceeds the free stream static pressure by a small amount, implying Cp1 (see list of symbols for detail) and M<<1. Analyzing the injection process as a small disturbance to the free stream, they used a perturbation potential scaled on M and the Hubert transform to express the solution in terms of a single integral equation. For the range of mass flows addressed by their theory, they obtained good agreement with experiment. Stropky (1993) applied an inviscid analysis to the problem of steady incompressible flow from an inclined slot into a main stream of different total pressure. In his study, a dividing streamline was used to separate the flow field into two zones; an internal flow region containing the slot fluid and an external region containing the free stream fluid. The dividing streamline was found approximately. For a normal injection slot, mass flow rates and discharge coefficients obtained compared well with the experimental results measured at moderate Reynolds number provided the separation was not large. The effect of separation could be included in his method by assuming the approximate shape of the separation flow region, and the results showed significant improvement when this was done. On the other hand, a number of analytical results have also been obtained for heat transfer with film cooling. Goldstein (1971) presented several theoretical correlations for film cooling effectiveness. These correlations were based on the assumption that the secondary injection stream does not alter the flow of the primary stream. Therefore, the results are only valid for film cooling with low mass flow ratios. Based on experimental observation, Sørensen (1969) obtained an analytical formula predicting mass transfer coefficients downstream of a separation bubble for low Reynolds number laminar flow around a blunt flat plate, which might be useful for analyzing the heat transfer coefficient of film cooling at low Reynolds numbers. Chapter 1 8 1.2.2 Experimental Studies The first direct experimental study of the heat transfer coefficient downstream of a normal film cooling slot appears have been carried out by Scesa (1954) who held the downstream surface at a constant temperature. He found that for low M the isothermal heat transfer coefficients were reduced, but not significantly altered by the injection of coolant. He concluded that the usual solid wall turbulent heat transfer relation can be used for heat transfer with film cooling if the heat transfer coefficients are redefined with the adiabatic wall temperature, namely, Equation (1.1). Seban et. al. (1957) and Seban (1960) also performed experiments for film cooling with tangentially injected air. Their heat transfer results confirm that for low M film cooling the effect of injection on heat transfer coefficient is not significant. Hartnett et. al. (1961) also investigated the heat transfer for air injected through a tangential slot. The same effect of injection on heat transfer coefficient was observed. As a result, most of the film cooling investigations focused mainly on finding the film cooling effectiveness. Unfortunately, heat transfer rates with film cooling can not always be satisfactorily predicted with the primary stream heat transfer coefficients and the adiabatic wall temperature. Metzger et. al. (1968) studied the film cooling heat transfer with nontangential injection slots and found that the injection of coolant can have a significant effect on the average surface heat transfer coefficient. Later Foster and Haji-Sheikh (1975) conducted an experimental study of heat transfer in the region downstream of normal injection slots. They found that the heat transfer coefficient can be appreciably increased as a result of flow separation produced by the film cooling injection. In three dimensional film cooling, the effect of injection on the heat transfer coefficient has been observed by Metzger et. al. (1973), Erikson and Goldstein (1974), Mayle and Camarata (1975), Jabbari and Goldstein (1978), Crawford et. al. (1980), and Ligrani et. al. (1992), among others. However, the effect of injection on flow separation and accordingly on heat transfer coefficient is not clear, and still needs to be further investigated. Chapter 1 9 To avoid errors resulting from heat conduction and radiation and to obtain more detailed values of the heat transfer coefficient near film cooling slots or film cooling holes, the mass/heat transfer analogy can be used for determination of heat transfer coefficient with film cooling. With this analogy, a mass flux of contaminated gas is used, rather than a heat flux. Based upon the mass/heat transfer analogy, the heat transfer coefficient is obtained from the measured mass concentration. The mass/heat transfer analogy has been extensively used in the experimental study of film cooling effectiveness and convective heat transfer coefficients. Salcudean et. al. (1994) successfully used propane as a tracer gas for the determination of film cooling effectiveness in the leading edge region of a simplified turbine blade model. Recent experimental studies of the mass/heat transfer coefficients have been carried out by Chen (1988) and Goldstein and Chen (1985) using the naphthalene sublimation technique. They obtained heat transfer coefficients over a turbine blade model through the measurement of naphthalene concentration distribution. The naphthalene sublimation technique was also used by Goldstein and Taylor (1982) in the investigation of mass/heat transfer with injection through a row of discrete holes, and the effect of injection on mass/heat transfer coefficients was determined. The effect on heat transfer coefficient due to injection through a row of inclined, or normal holes has also been studied by Hay et. al. (1985) for a range of mass flow ratios using the mass/heat transfer analogy. Laser holography was used to measure the thickness changes of a swollen polymer surface, which was used to determine the mass transfer coefficient. These two different mass transfer experiments indicate that significant effects of injection on heat transfer coefficient can be expected near the film cooling holes particularly at high M. On the other hand, a number of experimental studies were reported on heat transfer from turbulent separated flows, which might shed light on the understanding of turbulent heat transfer with film cooling. Ota and Kon (1974) carried out direct heat transfer measurements in separated and reattached flow regions on a blunt flat plate. They found Chapter 1 10 that the heat transfer coefficient increased sharply from the leading edge of the plate and became maximum at the reattachment locations for Reynolds number ranging from 2720 to 17900. Zemanick and Dougall (1970) performed an experimental study of local heat transfer downstream of an abrupt circular expansion with three expansion geometries and a range of Reynolds numbers. Similar experimental studies were later carried out to determine the local heat transfer coefficient downstream of an abrupt circular expansion by Amano et. al. (1983), Baugim et. al. (1984) and Baughn et. al. (1989), among others. From these investigations, extensive experimental results were obtained, the main features about heat transfer in the region of flow separation were revealed, and several correlations for maximum heat transfer coefficients were proposed. Detailed heat transfer measurements were also made downstream of a backward-facing step by Vogel and Eaton (1985). Stanton number profiles were obtained for a range of upstream boundary layer thickness at a Reynolds number of 28,000. The heat transfer measurements were shown to be in essential agreement with previous experiments for heat transfer with flow separation. However, the maximum heat transfer coefficient was found to occur slightly upstream of the reattachment point, where the peak values of the turbulence intensity were measured. 1.2.3 Numerical Studies Predictions of turbulent flow and heat transfer can be attempted by solving time-averaged Navier-Stokes equations and the thermal energy equation, in conjunction with a suitable turbulence model. However, the numerical investigation of turbulent flows with film cooling injection poses difficulties caused by the film cooling injection and flow separation downstream of the film cooling injection location. For the prediction of heat transfer with film cooling, an additional difficulty is that the standard wall function for temperature is no longer entirely appropriate in the region of flow separation. Early numerical studies of film cooling were mainly carried out using fully or partially-parabolic models. Pai and Whitelaw (1971) developed a procedure for the 11 Chapter 1 prediction of film cooling effectiveness and heat transfer coefficient downstream of twodimensional film cooling slots. Their proposed procedure is based on a modified form of the parabolic procedure of Patankar and Spalding (1967), which solves the twodimensional, parabolic boundary layer equations. Both the turbulence Reynolds shear stress and heat flux are modeled using Prandtl’s mixing length hypothesis and an effective Prandtl number. The numerical results reported compare well with the measured film cooling effectiveness and heat transfer coefficient only in the region downstream of the separation bubble, as expected. Patankar et. al. (1973) reported predictions of film cooling effectiveness of a three dimensional film cooling arrangement, carried out using the three-dimensional parabolic procedure proposed by Patankar and Spalding (1972). The turbulence model used for the predictions was also based on Prandtl’s mixing length hypothesis. The numerical procedure was based on the solution of a finite difference representation of the timeaveraged, three dimensional boundary layer form of the Navier-Stokes equations. For the problem concerned, the numerical predictions compared very well with the corresponding experimental results. The parabolic procedure has also been used, together with the k — turbulence model, for prediction of film cooling by Bergeles et. al. (1976, 1978). They carried out comparisons between the parabolic and locally-elliptic procedures and found that the fully-parabolic model is as good as the partially-parabolic or locally-effiptic models, while requiring much less computer time and memory. However, numerical procedures based on the parabolic models cannot be used to predict film cooling effectiveness and heat transfer coefficient in the region dominated by the flow separation. In this region, the parabolic boundary layer equations are not valid, and the elliptic time-averaged Navier-Stokes equations must be employed. Recently Sinitsin (1989) performed numerical simulations of turbulent flow and heat transfer with film cooling injection from a 200 and a 400 angled film cooling slot using the elliptic time-averaged Navier-Stokes equations and thermal energy equation, together with the k — a turbulence model. Based on experimental observations, he derived possible distributions of velocity across the film cooling slot exit, and then imposed the 12 Chapter 1 resulting velocity distributions as the inlet boundary condition for the numerical simulation. This procedure avoids a ‘staircased’ representation of the slot that would otherwise result from using a coarse Cartesian grid arrangement, and was thought to be a significant improvement over the uniform distribution generally used. The subsequent numerical calculations carried out by Sinitsin (1989) on a coarse grid indicated, for three velocity distribution classes, that the flow direction did not have a large effect on the flow field downstream of the film cooling injection slot and away from the wall. The flow direction can, however, have a significant effect on the shear stress coefficient and the heat transfer rate downstream of the film cooling injection slot. The reason could be that a small change in velocity at the near-wall points leads to a large variation in its gradient, namely, shear stress or heat transfer coefficient. He found good agreement between predicted and measured velocity profiles downstream of the slot for the 200 case, but only fair agreement for the 40° case. Preliminary calculations of heat transfer rate downstream of the film cooling injection slot were also obtained, but the heat transfer results did not capture the main features of turbulent heat transfer with film cooling. One reason for the above inaccurate heat transfer results is that his results, based on a numerical simulation performed on a single coarse grid, were almost certainly grid dependent, and were probably seriously affected by artificial or false diffusion. Realizing the problems of numerical simulation based on a coarse grid arrangement, Zhou (1990) presented a numerical study of film cooling effectiveness for a normal injection slot using a much finer grid arrangement. He used the time averaged Navier Stokes equations and thermal energy equation, together with the k — E turbulence model, to describe the turbulent flow and heat transfer of film cooling with normal injection. The governing equations were discretized on a staggered grid arrangement using the hybrid differencing scheme. The resulting finite difference representation of the Navier Stokes equations and continuity equation was then solved using a multi-grid technique with a pressure correction procedure as a relaxation smoother. The finite difference representations of the turbulence transport equations and the thermal energy equation were solved only on the finest grid. The multi-grid technique was used to accelerate the convergence rate, and numerical experiments indicated that the multi-grid technique led 13 Chapter 1 to faster convergence, which often cannot be achieved by the single grid method. Flow separation was observed from the numerical simulation, and film cooling effectiveness at various film cooling flow rate was obtained. Extensive investigation of film cooling effectiveness downstream of a normal injection slot were carried out later by Zhou (1994). Fair agreement between numerical predictions and measurements were observed. No measurements of heat transfer coefficients were made. So far, no other finite difference predictions of turbulent heat transfer of film cooling with injection from normal slots have been reported in the literature. Many relevant predictions, however, have been carried out for various other geometries, such as the channel with backward facing step and the blunt flat plate, and various near-wall models have been developed for the prediction of turbulent flow and heat transfer with flow separation. Based on the observation that the near-wall treatment could have a significant effect on the local heat transfer predictions, Chieng and Launder (1980) proposed a two-layer near-wall model, which took into account the variations of k and C to evaluate the generation and destruction terms of the k equation near the wall. This two layer nearwall model was applied to the test case studied experimentally by Zemanick and Dougall (1970). Experimental profiles of Nusselt number were under-predicted at low Reynolds number, and over-predicted at high Reynolds number; the general agreement, however, was good. In both cases, as discussed by Johnson and Launder (1982), the results were affected by coding errors. Johnson and Launder (1982) corrected the results presented by Chieng and Launder (1980) and then proposed an important improvement to the twolayer near-wall model, which allows the dimensionless viscous sublayer thickness to become smaller in regions where a significant flux of turbulent kinetic energy from the core to the near-wall region existed (as in the reattachment and redevelopment regions of the separated shear layer). The original idea to allow the non-dimensional sublayer thickness to vary was proposed by Spalding (1967). The new near-wall model of Johnson and Launder (1982) yielded numerical results in good agreement with appropriate experimental data. 14 Chapter 1 Recognizing the importance of the near-wall treatment, Amano et. al. (1983) introduced a three-layer near-wall model. By assuming proper distributions for velocity, turbulence quantities and temperature in each near-wall layer, transport-productiondestruction terms in the k and E transport equations were derived. The authors applied the proposed three-layer near-wall model to a heat transfer test case studied experimentally by themselves. Good agreement between the predictions and experimental results was obtained. Using a different test case, namely, the turbulent flow and heat transfer about a twodimensional blunt flat plate, Djilali et. al. (1989) performed heat transfer predictions using various existing near-wall turbulence models. By comparing the resulting Nusselt number distributions with available experimental data, the authors found that the threelayer near-wall treatment gave good agreement with experimental data; the standard wall function treatment was not satisfactory at all. Based on recent near-wall turbulence measurements in the region of flow separation, Ciofalo and Collins (1989) proposed a modified near-wall treatment, which formally retains classic wall functions and scaling based on the near-wall turbulent kinetic energy, but allows the non-dimensional viscous sublayer thickness to vary as a function of the local turbulence intensity. This new near-wall treatment further extended the idea of Spalding (1967) and the near-wall model of Johnson and Launder (1982), which also permits the non-dimensional sublayer thickness to vary. However, this treatment is simpler and physically more reasonable. Heat transfer predictions with the modified near-wall treatment were compared with those from a standard wall treatment and experimental data, for different geometries including single and double backward facing steps. Significant improvement in heat transfer predictions was demonstrated. The above heat transfer predictions are often accomplished through the numerical solution of the time-averaged Navier-Stokes equations and thermal energy equation, in conjunction with the k — turbulence model and various near-wall models using a finite difference scheme. The hybrid differencing scheme is usually employed. Thus, the numerical predictions could be adversely affected by false or artificial diffusion. A remedy is to refine the grid, which in turn leads to a very slow convergence process. To 15 Chapter 1 accelerate the convergence rate of the iteration process, the multi-grid method is often used. The application of the multi-grid method often results in faster and more accurate predictions, which will be explored in the present study. 1.3 Objectives and Scope of the Present Investigation The forgoing literature survey has indicated that coolant injection will have a significant effect on heat transfer particularly with normal injection at high M. But there are no detailed experimental measurements available on the heat transfer immediately downstream of the film cooling injection location. Despite the extensive information available on the heat transfer of recirculating flows and the development of various nearwall models for the prediction of heat transfer of recirculating flows, there are still no numerical predictions published on the film cooling heat transfer with normal injection at high M and for other injection geometries in the region dominated by flow recirculation or flow separation. Both the experimental and numerical investigations of film cooling heat transfer pose many challenges and are worth exploring. The objectives of the present experimental investigation are the following: • To establish a mass transfer method for determination of the heat transfer with film cooling using the mass/heat transfer analogy, not for determination of the heat transfer with transpiration cooling, • To obtain the heat transfer coefficient downstream of a normal injection slot at various film cooling mass flow ratios, • To provide data for comparison and evaluation of the numerical heat transfer predictions. The present numerical investigation has three objectives: • To predict the turbulent flow and heat transfer of film cooling with normal injection at various film cooling mass flow ratios, • To establish an efficient multi-grid procedure for numerical simulation of turbulent flow and heat transfer with film cooling using the k model and modified near-wail treatment, — E turbulence 16 Chapter 1 • To compare the heat transfer predictions with the experimental data obtained using the mass transfer method, and verify that the transpiration through the porous plate does not affect the flow significantly. In the following chapters, a detailed experimental and numerical investigation of film cooling heat transfer with normal injection will be presented. In Chapter 2, the apparatus and experimental arrangement are described; the measurement of the mass transfer coefficient and modifications of the mass transfer coefficient through a new wall function correction are presented. Chapter 3 presents the validation of the wall function correction formula and the mass transfer measurements downstream of the film cooling injection slot. The heat transfer coefficients downstream of a normal film cooling injection slot at various film cooling mass flow ratios are also obtained using the mass/heat transfer analogy. In Chapter 4, the mathematical model for turbulent flow and heat transfer with film cooling is formulated using the time-averaged Navier-Stokes equations and thermal energy equation, in conjunction with the k — E turbulence model. In Chapter 5, a multi grid procedure is proposed for numerical simulation of turbulent flow and heat transfer and tested using the laminar cavity flow and turbulent boundary layer flow. Finally, in Chapter 6, predictions of heat transfer with normal film cooling injection using the k — £ turbulence model and a modified near-wall model are presented and compared with experimental data. Conclusions are given in Chapter 7. Chapter 2 Experimental Arrangement and Measurement Techniques In this chapter, the experimental apparatus and the wind tunnel model used for the mass transfer measurement are described. The calibration of the Flame Ionization Detector (F]])) and the experimental procedure are discussed. The calculation of mass transfer coefficients and the wall function correction for mass transfer Stanton number are also presented. 2.1 Experimental Apparatus and Equipment The present experimental investigation of heat transfer with normal film cooling injection is conducted at the University of British Columbia in a boundary layer wind tunnel equipped with a specifically designed heat transfer simulating apparatus. The equipment involved in the velocity and mass transfer measurement includes the wind tunnel, the standard DISA constant temperature hot wire anemometer, the flame ionization detector and a specifically designed heat transfer simulating apparatus. A description of the equipment is now presented. 2.1.1 Wind Tunnel Facility The wind tunnel used in the experiment is a small, low speed, blower type wind tunnel. A schematic diagram is given in Figure 2.1. This wind tunnel has a test section of 0.41 m 17 Inlet Fan Test section Porous Plenum for tracer injection Figure 2.1 Wind Tunnel Schematic Coolant Plenum Inlet Cylindrical plenum for coolant Slot Contraction Section Outlet I’) m -I C) 19 Chapter 2 wide by 0.25 m high by 4.0 m long. It is lengthened by 0.98 m in order to accommodate the heat transfer simulating apparatus. The wind tunnel can produce air-flow in the range of 1 rn/s to 20 mIs. In the present experiment, the free stream air-flow speed is kept at 10 rn/s. The secondary or film cooling air-flow is supplied by a compressor and introduced into the mainstream through a cylindrical plenum and a normal film cooling slot. At the entrance of the wind tunnel test section, a trip wire is placed to ensure that the boundary layer flow is turbulent across the test section before reaching the film cooling injection location. A porous plate fed from a plenum directly beneath it, is located immediately downstream of the slot, as shown in Figure 2.1. Two lateral false wails (like fences which extend from floor to ceiling) are installed along the edges of the porous plate to keep the flow two dimensional, since the width of the porous plate is about 60% of that of the wind tunnel test section. This technique was used by Kiebanoff and Diehl (1951) to keep their turbulent boundary layer flow two dimensional. 2.1.2 Hot Wire Anemometer Upstream of the film cooling injection location, mean velocity profiles in the boundary layer are measured using a hot-wire probe and a DISA constant temperature hot wire anemometer system. The hot-wire probe is the standard DISA single wire probe, which uses a 5j.tm diameter, 1.25 mm long platinum-coated tungsten wire. The DISA constant temperature anemometer bridge is operated at an overheat ratio of 1.6 and the voltage signal is low-pass filtered through a 10 kHz filter. The 10 kHz frequency is chosen to eliminate high frequency noise without affecting the low frequency signal components. The principle of calibration of a hot wire anemometer is based on King’s Law, namely: A+Bu E = 2 (2.1) where E is the output voltage, n is an empirical constant which takes a value of 0.45. A and B are constants, and they are determined by the calibration curve. The calibration of a hot wire anemometer is performed against the velocity measurement of a Pitot-static Chapter 2 20 tube in the free stream under low turbulence conditions. By measuring different free stream velocities using the Pitot-static probe and recording voltage readings from the hot wire anemometer, values of A and B can be obtained using a least squares method. The whole experimental process is carried out using a computer data acquisition system. The system contains an amplifier, an A/D converter and a 386 PC computer equipped with a data acquisition program in C. During the calibration, the free stream velocity values obtained from the Pitot-static tube are entered into the data acquisition program through the keyboard while the voltage signals generated by the hot wire probe are amplified and sent to an A/D converter. The data acquisition program communicates with the A/D board and processes the digital signal from the A/D converter. To facilitate the data processing of velocity measurements, a look-up table is constructed during the calibration using King’s law. The look-up table relates the values of velocity to the values of voltage signals from the hot wire anemometer. 2.1.3 Heat Transfer Simulating Apparatus The specifically designed heat transfer simulating apparatus is shown in Figure 2.2. It consists of a porous plate shown in Figure 2.3, a film cooling slot 0.0635 m in streamwise width, 0.254 m in spanwise width and 0.254 m high, a rectangular transpiration plenum or box 0.304 m long by 0.254 m wide by 0.254 m high and a cylindrical plenum. This apparatus facilitates a controlled coolant injection from a normal film cooling slot and the introduction of a contaminated tracer gas into the cross flow through the porous plate. In the experiment, the air compressor delivers a measured amount of air-flow to the rectangular box, where the air mixes with a given amount of propane or methane (less than 1%) and is then ejected at very low flow rates into the cross flow through the porous plate. The objective of the air-propane or air-methane mass flux introduced into the main stream via the porous plate is to simulate the heat flux so that the mass transfer measurements can be made, and the heat transfer data can be inferred using the mass/heat transfer analogy. However, the amount of the mass flux introduced through the porous plate should be small so that its effect on the flow field is negligible. 21 Chapter 2 The compressor delivers a measured amount of air-flow into the cylindrical plenum, through which air enters the film cooling slot and then into the cross flow. The mass flow rates are read from two float flow meters. The mass concentration distributions are measured using a flame ionization detector, which is discussed below. 2. Porous plate 1. Rake of tubes 4. Transpiration Plenum 3. Wind tunnel bottom plate 6. Coolant injection slot 5. Mobile supporting table 7. Cylindrical plenum for coolant 0 U 1 XX)C) 6 7 Figure 2.2 Heat Transfer Simulating Apparatus 22 Chapter 2 0.305m 0.254m mean flow direction Figure 2.3 Top View of Porous Plate 2.1.4 Flame Ionization Detector On-line measurements of mass concentration distribution downstream of the film cooling slot are made using a flame ionization detector. The flame ionization detector is an instrument which can be used for measuring the concentration of a hydrocarbon contaminant, such as propane. A detailed description of the FID system for measuring mass concentration can be found in the papers by Fackrell (1978, 1980). The FID consists of a base body, a detector cell and a hydrogen-in-air flame burning in an insulated flame chamber across which a voltage is applied. A schematic presentation of FID is shown in Figure 2.4. The inter-changeable flame jet is at the center of the stainless steel base body. At the center of the combustion chamber lies the concentric collector, which consists of a 10 mm diameter cylinder, while the metal jet, electrically connected by a fork contact, acts as polarization electrode. Inside the detector cell is the flame 23 Chapter 2 igniter. The flame igniter and the electrodes are electrically connected to the power supply unit. The introduction of hydrocarbon contaminant into the flame leads to the production of ions and hence of a small current between the two polarization electrodes. This small current is amplified, and the output voltage signals are used to provide a precise measurement of the mass concentration of the contaminated gas. In the present experiment, mass concentration measurements are made through a specifically designed rake of fine tubes. The rake of tubes contains eleven fine sampling tubes 0.0 15 m apart and can be placed anywhere on the wall downstream of the film cooling slot. The contaminated gas is sampled through each tube in turn, and sent through a suitable fluid wafer switch, to the FID. As the rake is placed in a spanwise position, and the flow is expected to be two dimensional, the results from all eleven tubes are averaged. This is necessary to obtain a result independent of the precise position of any one individual sampling tube with respect to the small porous holes in the plate. Each tube, together with the RID, has to be calibrated individually before beginning the test. The calibration of the RD is now discussed. The FED generates a certain amount of ions and consequently a small current between two polarization electrodes after the contaminated hydrocarbon gas enters the flame chamber. Theoretically, the output voltage for a constant mass concentration (mass fraction) C entering the flame should be E = ApCQ, where E is the output voltage, A a constant, p the density of the gas mixture, and Q the total volume flow rate. However, the efficiency of the ionization process varies with different flow rates so that the FID output voltage is a more complicated function of the flow rate, namely, E=ApCf(Q) (2.2) It can be seen from Equation (2.2) that the FID voltage output is a linear function of the hydrocarbon gas concentration. Tests carried out by Fackrell (1978) with calibration Chapter 2 24 SUCTION EXHAUST \ COLLECTOR —ELECTRODE ION CURRE IGNITION COIL___ H HIGH INPUT 300 V AMPLIFIER POLARIZING ELECTRODE 2 H AIR COLUMN op TUBE Figure 2.4 Schematic Diagram of the Fil) Chapter 2 25 gases from 0 to 2000 ppm indicated that the FID response is linear with contaminated gas concentrations as expected from Equation (2.2). Therefore, to calibrate the FID system only two reference points of the calibration line need to be measured. These two reference points are the 0% reference point and the 100% reference point. The 0% reference point corresponds to the FID measurements taken with no hydrocarbon in the gas sample, and the 100% or “plenum” reference point is taken when the fine tubes are put inside the rectangular box beneath the porous plate in which there is a given amount of hydrocarbon contamination. During the calibration, the flow rate through each fine tube is kept constant by a suction pump associated with the FID. The F voltage outputs during the calibration are recorded for the above two reference conditions. Using the voltage output from these two set reference points a linear relationship is assumed between the FID output voltage and the concentration of contaminated gas. To check the linear response of our FID, two mass transfer tests were conducted by Zhang (1993) using propane and methane. The normalized Fifi output voltages were plotted against the content of contaminated gas for each case. These results are shown in Figures 2.5 (a) and (b). The FID voltage outputs are normalized by subtracting the constant voltage corresponding to the 0% concentration, namely, V = V — V,. These results show an essentially linear relationship between the Fifi output voltage and the propane/methane concentration, confirming the observation of Fackrell (1978) and the validity of Equation (2.2). In the present mass transfer experiment, each fine tube is calibrated individually with a known concentration, and the calibration is recorded in the computer. During the mass transfer experiment, the concentration inside the rectangular box is checked before each measurement is taken to ensure that the concentration inside the box is unchanged within ± 2%. All the measurements are facilitated with a data acquisition program, which processes the FID voltage signals and outputs the relative concentration distributions. Chapter 2 26 0.4 0 0 0.3 0 > > 0) c,) as 0.2 0 > .1 0. 0 0.1 100 200 300 400 Propane Content [ppm] Figure 2.5 (a) FID Calibration with Air-Propane 0.20 — 0.15 > > ci c,) (‘5 0 0.10 > D 0 0 0.05 100 200 300 400 Methane Content [ppm] Figure 2.5 (b) FID Calibration with Air-Methane Chapter 2 27 2.2 Measurement and Calculation of Mass Transfer Coefficient 2.2.1 Mass Transfer Measurement In the present experimental investigation, the mass transfer coefficient h is defined as follows: h = m 1 p(c —c) (2.3) where m’ is the mass flux of propane, p is the air-propane mixture density, c mass concentration (mass fraction) of propane at the wall, and c is the is the mass concentration of propane in the free stream, which is very close to zero in the present experiment. Note that =mc where Cb lb is the mass concentration of propane inside the box, and rn’ is the mixture mass flux. Thus we have: QIA h ,,—c 1 c 10 where C * c1w — 1w —, Clb * 0 Cl C lo ——, Clb Q is the mixture flow rate entering the box, A is the surface area of the porous plate. The mass transfer Stanton number is defined as: h St = _!L U 0 (2.4) Chapter 2 28 where U 0 is the free stream velocity (U 0 St= 4 U 1 (c , , — = 10 m I s). Therefore it can be rewritten as (2.5) * ) 1 c 0 Using Equations (3.2) and (3.3), the measurement of mass transfer coefficient h mass transfer Stanton number can be accomplished through the measurement of c and and , c Q. In the present mass transfer experiment, propane was used as a contaminant gas or tracer gas. Measurements of propane concentration along the porous plate were made with the rake of very fine tubes already described, placed directly on the porous plate. Since the sampling tubes take fluid from a point P near the wall, but not the wall, measurements of c, instead of c, are made in the experiment. The mass transfer Stanton number based on c can be expressed as: St = “ “ ) 10 U(c—c (2.6) It will be shown later that St is only an approximation to the correct Stanton number St. and that the correct mass Stanton number must be obtained using the wall function correction formula presented below. 2.2.2 Wall Function Correction for Stanton Number As mentioned above, measurements of propane concentration are made at a near-wall point P. In previous measurements of film cooling effectiveness made by others using the mass/heat transfer analogy, the assumption that the propane mass concentration at point P is equal to the propane mass concentration at the wall has been made. This may lead to an underestimation of the film cooling effectiveness. The error is difficult to quantify, but is generally small because of the adiabatic wall condition, which implies zero gradient at the wall. However, the gradient of the mass concentration is fl zero at the wall when the contaminated gas is injected through the porous wall; by analogy the Chapter 2 29 temperature gradient is not zero in the presence of a heat flux from the wall. The mass transfer Stanton number St becomes considerably higher than the Stanton number based p on the mass concentration of contaminated gas at the wall. To obtain the mass concentration at the wall and accordingly the correct Stanton number, a wall function correction formula must be established to relate St to St. A brief description of this a wall function correction is now presented. Near the wall, the local mean distribution of velocity or temperature can be described by the so called “wall functions”. A list of various wall functions is given by Bejan (1984). The most common form of wall function employs a near-wall viscous sublayer, where the momentum transfer is dominated by molecular viscosity and the heat transfer by molecular conduction, followed by a logarithmic region. In the so called near-wall viscous sublayer, we have: au tpl)— ‘t Integrating the above equation, we get U t w =pl)— y or U+=y+ where U + U =—, U t Uy = 1) U VP In the logarithmic region, we have: tt W =I_t au tay — + + y y 30 Chapter 2 According to the mixing length argument (Hinze 1959), the turbulent viscosity can be expressed as: pKUy J.t where K is the Von Karman constant (K 0.42). Thus we have: PKU y--— t W ciy or + du Ky —=1 dy Integrating the above equation, we obtain the following velocity profile: ’) -1n(Ey1 u= - where E=9.23 Constant E is obtained by fitting experimental data (Kays and Crawford, 1980). The wall function for temperature in the near-wall region can be obtained in a similar way. In the linear viscous sublayer, we have: =—pC q —— PcYay The above equation can be rewritten as: dT dy =c with + T = (T —T)pC p u W c Integration of this equation yields T=ay where a is the Prandtl number, assumed to be constant. y+y 31 Chapter 2 In the logarithmic region, we have: aT q=—pC ——=q aay where a is the turbulent Prandtl number, assumed to be constant as well. Note that, as before, JI =plcuy then we have: q =—pC KUY T — P G or icy + dT =a dy Integrating the above equation, we obtain T =a[!ln(Ey)+P] + >± where P = 9.24[(—--)° at — 1j[1+O.28exp(—O.OO7---)J at is an empirical function given by Jayetilleke (1969). The velocity and temperature distributions in the near-wall region can now be summarized in the following: Profiles of u: u=y yy (2.7) u=---ln(Ey) y>y (2.8) y (2.9) where ic=O.42,E=9.23,y =11. Profiles of T T = ÷ ay + 32 Chapter 2 T =a[--1n(Ey)+P] Y>YT (2.10) with )° —lJ[l+0.28exp(—O.007----)J 75 P=9.24[(_— where y can be found as the intersection of the linear and logarithmic curves. With the above wall functions, a relationship between (Tw , St) and (Tp , St p ) can be established. The heat transfer Stanton numbers St and St are defined in the following q St= pCU(T—T) p o w q St= ‘ pCU(T-T) o p p0 where St and St are two Stanton numbers based on temperatures at the wall and the p near-wall point P respectively. The near-wall point P is shown in Figure 2.6. The Stanton number St is required while St is measured and must be corrected using the p wall function correction formula. UT -oo y o P(y, U,T) Figure 2.6 The Near Wall Point 33 Chapter 2 If the near-wall point P is located in the logarithmic region, namely: y >max(y,y) then we have: u =--ln(Ey) + T (T—T)pCu W = P P i = [—ln(Ey)+P] Using the definition for St and St, we get 1 1 1 (___)!_0 [—ln(Ey)+P] St St U a p The above equation can be rewritten to yield 1 St= U U —+0 u—-+a pe.. t U St (2.11) p Equation (2.11) can be further reduced to 1 St= U U U (2.12) ?‘ St p u U Either Equations (2.11) or (2.12) can be used to obtain the correct Stanton number St when point P is located in the logarithmic region. If point P is located in the linear viscous sublayer, i.e., y satisfies the following <min(y,y) then we have T= (Tw —T)pC u p p In a similar way, we get 34 Chapter 2 St= 1 1 St (2.13) u Pu p and St= 1 1 St (2.14) + a Re P where Re 1) Equations (2.13) and (2.14) can be used to obtain the correct Stanton number when point P is located in the linear viscous sublayer. For the present mass transfer experiment, the wall function correction formulae are the following: If point P is located in the linear viscous sublayer, the correction formula is (2.15) St= St + a Re p If point P is located in the logarithmic region, the correction formula reads —+a Re St p + U a P—°-t U (2.16) where a is the molecular mass transfer Schmidt number, and a is the turbulent Schmidt number. The mass transfer Schmidt number a is determined by the air-propane, or airmethane binary mixture diffusion coefficient. Equations (2.15) and (2.16) are called the wall function correction formulae, which provide a correction to the approximate Stanton number St. In the present mass transfer experiment, very fine sampling tubes are placed in the linear viscous sublayer. This assumption can be verified by calculating the local Reynolds number defined in Chapter 4. (Note that it is difficult to validate the correction procedure for the region with flow Chapter 2 35 separation without comparing the correct Stanton number with the numerical data). Therefore, the correct Stanton number can be obtained using Equation (2.15), which does not require any information about the wall shear stress or the friction velocity. The validity of the wall function correction formula is examined in the next chapter. Its application to the present mass transfer experiment is also discussed in the next chapter. 2.3 Experimental Procedure In the present experiment, certain experimental procedures are followed to ensure the accuracy and reliability of the measurements. A complete description can be found in Appendix C. Chapter 3 Experimental Results In this chapter, the experimental results of mass transfer measurements are presented and discussed. First, the validity of our proposed mass transfer method and the wall function correction formula is examined using the case with zero film cooling injection, a situation analogous to the well-known turbulent boundary layer mass/heat transfer with an impermeable/unheated starting length. Secondly, mass transfer measurements downstream of a normal film cooling injection slot with various film cooling mass flow ratios are presented. The effects of both the wall blowing rate and the Schmidt number on the mass transfer coefficient are investigated, and the relation between mass and heat transfer is obtained. Finally, the heat transfer coefficients for various film cooling mass flow ratios are obtained using the mass/heat transfer analogy, and the reattacbment length for film cooling flow at high M is inferred. 3.1 Test Conditions The mass transfer experiments presented below are conducted under the following operating conditions: Main stream velocity, 10 m / s, Slot widths = 6.35 mm, Boundary layer thickness at 10 s upstream of the film cooling slot (obtained from velocity measurement), 6 3 s, 36 37 Chapter 3 Mainstream pressure gradient, dx 0, Film cooling mass flow ratios, M =0 up to M 0.5, 3 m / s to 9.14 X i0 mis, Transpiration velocity, or blowing speed, v = 6.85 x i0 Temperature of air, T = 15° C —20° C. The boundary layer thickness at lOs upstream of the film cooling injection location is found from the corresponding velocity measurements conducted by hot wire anemometry; these are plotted in Figure 3.1. For 95 percent confidence levels, an uncertainty analysis based on the method of Kline and Mcljntock (1953) has been carried out for mass transfer coefficients. The measurement uncertainty of transpiration velocity is ± 2%, the measurement uncertainty of free stream velocity is ±0.75%, and the measurement uncertainty of concentration is within ±2%. Using the second-power equation of Kline and Mclintock (1953), the uncertainty of mass transfer Stanton number is within ±2.9%. Two dimensionality is also examined by measuring mass concentration at sixteen streamwise locations on two lines, one at the center, the other 0.04m from the center. The maximum variation in concentration is within ± 2.5% found for low mass flow ratios (M 0.1). A more detailed description of the two dimensionality test is presented below. 3.2 Two Dimensionality To ensure the two dimensionality of the flow and mass transfer, two lateral false walls are installed along the edge of the porous plate as part of the experimental set-up. The two dimensionality assumption, however, still needs to be verified, since the experimental results will be compared with the numerical results based on a two dimensional model. A two dimensionality test of the flow field was carried out by Sinitsin (1989) for a similar film cooling set-up. The velocity measurements were obtained by him at two streamwise and three spanwise locations. The mean velocity profiles indicated that the Chapter 3 38 mean flow upstream of the film cooling injection slot was two dimensional within ±2.5%. However, it is difficult to assume two-dimensionality in the separated flow downstream of the film cooling injection slot. Generally, the false walls, the size of the film cooling injection slot, and the distance between the two false walls can affect the two dimensionality of the flow field and mass transfer. Fortunately, Sinitsin’s mean velocity measurements in the region close to the film cooling injection slot indicated only a small degree of three dimensionality. The mean velocities at the centerline were 5% higher than those at the two locations 0.1 m from the centerline. Since the differences were small across the spanwise direction, Sinitsin (1989) concluded that the flow is essentially two dimensional within the range of the spanwise measurements. Therefore, measurements taken close to the centerline may be considered to represent the entire slot. Figure 3.1 Velocity Profile 10 Slot Width Upstream of the Film Cooling Slot Chapter 3 39 To investigate the two dimensionality of the mass transfer measurement as a part of the present research, a series of mass transfer measurements were made downstream of the film cooling injection slot using the FID. The mass transfer measurements were taken at sixteen streamwise locations along two lines: one at the center of the wind tunnel, the other 0.04 m from the centerline. Figure 3.2 shows the variation of the mass transfer Stanton number along these lines for film cooling at a mass flow ratio of M = 0.05. The spanwise variation in the mass transfer Stanton number is within 5 %. The Stanton number presented in Figure 3.2 is obtained using the wall function correction formula Equation (3.2) described in Chapter 2. The value of y., assumed here is the outside radius of the fine sampling tube, namely, 0.25 mm. The value of a is taken as 1.4. Figure 3.3 depicts the variation in mass transfer Stanton number for film cooling at a mass flow ratio of M = 0.1. Here, the maximum spanwise variation in the mass transfer Stanton number is 2.5%. The mass transfer for film cooling at low M can therefore be considered to be two dimensional. For high mass flow ratio film cooling, the two dimensionality of the mass transfer Stanton number may be affected by the flow separation, the size of the film cooling slot, and the width of the porous plate. Figure 3.4 presents the variation in mass transfer Stanton number for film cooling at a mass flow ratio of M = 0.2. This figure indicates that the variation of the mass transfer Stanton number is very small. Therefore, mass transfer coefficients of film cooling at mass flow ratio M = 0.2 can be considered to be two dimensional as well. Two dimensionality is also assumed for mass transfer coefficients of film cooling at higher mass flow ratio (M > 0.2) although this was not explicitly checked experimentally. 3.3 Massflleat Transfer with Impermeable/Unheated Starting Length The mass transfer method has been used for simulating heat transfer in many studies, and various mass transfer techniques have been proposed for the study of heat transfer problems. One of the well established mass transfer techniques is that using the naphthalene sublimation, described in more detail later, which has found wide 40 Chapter 3 St 0.01 0.009 0.008 0.007 0.006 0.005 0.004 - 0.003 - - - - - - - - - - - - - - - - - I nfl te - - - - tr - - ::: 0.002 - - x,s 0.001 0 5 10 15 20 25 30 35 Figure 3.2 Two Dimensionality of Mass transfer with Film Cooling at M=O.05 St 0.01 0.009 0.008 e 0.007 t - - - - - - - - - - tin ne 0006 - - 0.005 - 0.004 - 0.002 - - 0.001 0 5 10 15 20 25 30 x/s 35 Figure 3.3 Two Dimensionality of Mass transfer with Film Cooling at M=O.1 41 Chapter 3 St xis 0 5 10 15 20 25 30 35 Figure 3.4 Two Dimensionality of Mass transfer with Film Cooling at M=O.2 application in the study of mass/heat transfer processes. The basic idea of various mass transfer methods, however, is common, namely: the mass flux is introduced instead of the heat flux, and the mass transfer coefficient is measured and related to the heat transfer coefficient through the mass/heat transfer analogy. In the mass transfer method using naphthalene sublimation, the mass flux is introduced through sublimation of naphthalene from a solid naphthalene coated surface. The mass transfer coefficient is obtained by measuring the variation in coating plate thickness and surface temperature, and then related to heat transfer by applying the mass/heat transfer analogy. The validity of this method was discussed by Karni (1985). However, it is difficult to use the naphthalene sublimation technique to study the effect of the Schmidt number on mass transfer, which can be used to derive the relation between the mass and heat transfer coefficients. The relation between mass/heat transfer 42 Chapter 3 coefficients in the form of St h St ()fl (see Section 3.4.2 for details) must be rn assumed in order to relate the mass transfer coefficient to the heat transfer coefficient. In addition, the mass transfer coefficient obtained by naphthalene sublimation is susceptible to any change in free stream temperature. For example, a 10 C variation of room temperature could cause naphthalene vapor pressure, the local naphthalene density on the surface and the local mass transfer coefficient to change by about 10%. In the present mass transfer method, a porous plate is used to introduce the mass flux. The mass concentration of contaminated gas is measured by the FED, and the mass transfer coefficient is calculated by Equation (2.6) and Equation (2.15). The heat transfer coefficient is then derived from the mass transfer coefficient using the relation between the mass and heat transfer coefficients. Using two different tracer gases, the effect of Schmidt number on mass transfer coefficients, or the relation between the mass and heat transfer coefficients can be obtained, and the value of exponent n can then be found (see Section 3.4.2 for details). To ensure that the present mass transfer method is valid, the mass flux introduced through the porous plate must be small, thus minimizing the wall blowing effect on the flow pattern. The validity of the present mass transfer method is examined in the next section. The test case to be used is that of mass/heat transfer in a turbulent boundary layer with an impermeable/unheated starting length. 3.3.1 Mass Transfer with Impermeable Starting Length Turbulent boundary layer mass transfer has been studied extensively. As discussed by Kays (1980), the turbulent boundary layer mass transfer coefficient can be obtained using the mass/heat transfer analogy (if the wall blowing effect can be neglected). Using the boundary layer integral method, the heat transfer Stanton number for a turbulent boundary layer with an unheated starting length can be expressed as St 0.0296[l()10 j9 Re 2 Pr 4 (3.1) where x is measured from the virtual origin of the turbulent boundary layer. Its validity has been confirmed by the experimental measurements of Scesa (1951), among others. 43 Chapter 3 Using the relation between heat and mass transfer coefficients, the mass transfer Stanton number for a turbulent boundary layer with impermeable starting length can then be written as (if the wall blowing effect can be neglected): Stm =0.0296[l—()’° ]9 (3.2) 2 a°’ Re 4 In the case of a turbulent boundary layer in zero pressure gradient, considered here, the value of exponent n in equation St h = St () is known to be 0.4. For other cases, rny the value of n is not known. To test the present mass transfer method and the wall function correction formula, and to validate the proposed mass transfer method, a series of mass concentration measurements were carried out for turbulent boundary layer mass transfer with impermeable starting length. Both the uncorrected mass transfer Stanton number and the mass transfer Stanton number corrected by the wall function correction formula are presented in Figure 3.5. It can be seen that the value of St is considerably larger than St and is not an acceptable approximation to St. The mass transfer Stanton number St obtained using the wall function correction formula Equation (2.15) is also presented in Figure 3.6 for comparison with the corresponding semi-empirical results derived from the heat transfer Stanton number of Kays (1980) and of Scesa (1951). The variation of heat transfer Stanton number was obtained along a constant temperature plate by Kays (1980), whereas the variation of heat transfer Stanton number was obtained by Scesa (1951) along a constant heat flux plate. The mass transfer Stanton number St obtained using the wall function correction formula compares very well with the semi-empirical data derived from Equation (3.2), namely, the data derived from those of Kays (1980), which suggests that 1) the mass flux used in the experiment as transpiration through the porous plate is neglegibly small as far as the fluid mecahnics of the turbulent boundary layer is concerned, 2) the present mass transfer analogy is valid, and 3) the wall function correction formula can be used to obtain mass transfer coefficient accurately. CD z C D 0 ci) - ci) -4’ CD H a, 0 CD C..) •0) -t -I, 01 -‘ 0 01 0 0 !: uN: a, D 0 0 0 0 C.) C -Il C 0 0 0 C 0 C.) 0 1 0 -I’ CD H 0 0 C’) CD C.) 1 C -Il C.) 01 0 01 r’z 0 01 0 01 0 q [H —-- 0 a, . Ca) 0 . C.) Chapter 3 45 3.3.2 Effect of Blowing Rate on Mass Transfer In the present experiment, the mass flux introduced through the porous plate has the unique purpose of simulating the heat flux. However, the mass flux could have adverse effects on the flow and the mass transfer coefficients if the wall blowing rate is too large. Generally, the effect of wall blowing, or transpiration is to reduce the skin friction and local wall heat transfer rate and therefore, by analogy, to reduce the local mass transfer coefficient. This is true for both laminar and turbulent flow and it has been confirmed by experimental studies. An extensive review of boundary layer flow with uniform wall blowing was conducted by Hartnett (1975), which indicated that the effect of wall blowing is small if the non-dimensional blowing rate _!_ is less than 1%. To make sure that the mass transfer coefficients can be used to obtain the heat transfer coefficients for the turbulent boundary layer with unheated starting length, the effects of the wall blowing rate on the mass transfer coefficient need to be investigated. Figure 3.7 shows the mass transfer Stanton number for turbulent boundary layer flow with an impermeable starting length. Wall blowing rates of V = 9.14 x 10 m / s and V v 6.85 x 10 = m/s were used, which correspond to —-- U0 = 9.14 x 10 V —-- = 6.85 x 10 blowing on —-- = U respectively. It can be seen from Figure 3.7 that the effect of the wall Stanton number is less than 5% 6.85 x 10 . for v = 6.85 x 10 m / s or A further reduction in wall blowing rate or mass flux introduced 0 through the porous plate makes the values of mass concentration measurement smaller and v less = accurate. Therefore, the mass flux with wall blowing rate of 6.85 x 1O 3 nz / s is assumed to be appropriate for the mass transfer experiment and is used for the film cooling mass transfer measurement. Note that the wall blowing rate 46 Chapter 3 of v w = 6.85 x 10 3 V m I s represents a non dimensional wall blowing rate —- U of less than 0 0.1%, which confirms the experimental conclusion of Hartnett (1975). St 0.01 0.009 0.008 -— ---=.;irs 0.007 0.006 0.004 I 0.003 0.002 x’s 0.001 5 0 10 15 20 25 30 35 Figure 3.7 Effect of V on Mass Transfer with Impermeable Starting Length 3.3.3 Heat Transfer with Unheated Starting Length Note that the mass/heat transfer analogy for a turbulent boundary layer flow can be expressed in the form proposed by Kays (1980), which can be found directly from Equations (3.1) and (3.2): Sth =Stm()°’ where Pr = 0.7 (3.3) 47 Chapter 3 a = 1.4 The heat transfer Stanton number derived from the mass transfer coefficient using Equation (3.3) the mass/heat transfer analogy is depicted in Figure 3.8, which appears exactly as in Figure 3.6, after a change of all points by a factor of 1.31. One then concludes that the present mass transfer method is reasonably accurate for simulating the heat transfer process provided that the near-wall correction is appropriately applied, and V that the non-dimensional wall blowing rate _!_ U is less than 0.5%. 0 3.4 Mass/Heat Transfer with Film Cooling The purpose of the present mass transfer investigation is to provide detailed measurements of the mass transfer coefficient downstream of the film cooling injection slot, so that the heat transfer coefficient with film cooling can be obtained using the relation between the mass and heat transfer coefficients. To the author’s knowledge, mass transfer measurements with film cooling have not yet been reported in literature, although the naphthalene sublimation technique has been used to study the local mass transfer from a gas turbine blade by Chen (1988). Direct heat transfer measurements downstream of a normal film cooling slot were carried out by Scesa (1951) and Foster (1972), but these did not provide heat transfer coefficients in the region of flow separation. Scesa (1951) studied isothermal heat transfer with normal film cooling injection and concluded that for low mass flow ratio film cooling the effect of film cooling injection on the heat transfer coefficient is negligible. Foster (1972) conducted a series of heat transfer measurements with various film cooling slot widths and film cooling injection rates. Foster observed the effect of film cooling injection on heat transfer in the region close to the injection slot for high mass flow ratio film cooling. However, a limited number of measurements in the region of separation were conducted. For technical reasons, direct heat transfer measurements did not provide sufficient Chapter 3 48 St 1 O1 1 02 XIS 0 5 10 15 20 25 30 35 Figure 3.8 Heat Transfer Stanton Number with Unheated Starting Length information in the region of separation, which is located immediately downstream of the film cooling injection slot. In the present study, detailed mass transfer measurements are carried out immediately downstream of the film cooling injection slot using the FID. Different tracer gases are used to investigate the effect of Schmidt number on mass transfer. The effect of wall blowing rate on mass transfer with film cooling is also examined by introducing mass flux at various blowing rates. Mass transfer experimental results are now presented and discussed in the following sections. Chapter 3 49 3.4.1 Mass Transfer with Film Cooling Detailed mean concentration measurements have been carried out at sixteen streamwise locations along the porous plate with film cooling mass flow ratio (M) ranging from 0 to 0.5. Using Equations (2.6) and (2.14), the mass transfer Stanton number is calculated and presented in Figure 3.9. Figure 3.9 (a) and Figure 3.9 (b) show the variation in mass transfer Stanton number of film cooling at low and high mass flow ratios respectively. The above figures indicate that (1) for the low M film cooling, the secondary injection from the film cooling slot alters the mass transfer Stanton number slightly, but the overall influence of film cooling injection is negligible, and the primary mass transfer Stanton number of a turbulent boundary layer can be used to predict mass transfer with film cooling, as expected; (2) for high M film cooling, the mass transfer is greatly affected by the leading edge separation. The value of Stanton number is low inside the separation region, it increases from the leading edge of the film cooling slot to a local maximum near the reattachment of the separation bubble. After a critical point, which is often assumed to be the reattachment point, the Stanton number decreases to the value typical of turbulent boundary layer flow. In sum, the leading edge flow separation has significant effects on the local Stanton number with film cooling, and the effects are similar to those of turbulent heat transfer downstream of a backward facing step where the reattachment occurs by Vogel and Eaton (1985) and other cases by Sørensen (1969). 50 Chapter 3 St 0.005 0.004 0.003 0.002 xis 0.001 0 5 10 15 20 25 30 35 Figure 3.9 (a) Mass Transfer Stanton Number of Film Cooling at Low M St 0.005 0.004 0.003 0.002 xis 0.001 0 5 10 15 20 25 30 35 Figure 3.9 (b) Mass Transfer Stanton Number of Film Cooling at High M Chapter 3 51 St 0.01 0.009 0.008 T” 0.007 --— 0.006 - --}-- =.4iiI --)-- I=Jr 0.005 0.004 0.003 : --- t:: 0.002 xis 0.001 0 5 10 20 15 25 30 35 Figure 3.10 (a) Effect of V on Mass Transfer of Film Cooling at M=O.2 0.01 0.009 0.008 St 0.007 0.006 0.005 0.004 0.003 - - --:- :-- ::..j 0.002 xis 0.001 0 5 10 15 20 25 30 35 Figure 3.10 (b) Effect of V, on Mass Transfer of Film Cooling at M=O.3 Chapter 3 52 x’s 0 5 10 15 20 25 30 35 Figure 3.10 (c) Effect of V on Mass Transfer of Film Cooling at M=0.4 0.01 St 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 x,s 0.001 0 5 10 15 20 25 30 35 Figure 3.10 (d) Effect of V on Mass Transfer of Film Cooling at M=0.5 Chapter 3 53 The effect of wall blowing rate on mass transfer with film cooling at high mass flow ratio is shown in Figure 3.10 for wall blowing rate v = 9.14 x m / s and v 6.85 x m I s respectively. The effect of wall blowing, as expected, is to reduce the local mass transfer rate along the porous plate. However, the difference between mass transfer Stanton numbers of these two wall blowing rates is small. This confirms the conclusion reached earlier for turbulent boundary layer mass transfer with impermeable starting length, that the effect of wall blowing on mass transfer coefficient is small for wall blowing rate v = 6.85 x 1O 3 mis, and that mass flux with this blowing rate is appropriate for the present mass transfer experiment. 3.4.2 Mass/heat transfer analogy In general, the mass/heat transfer relation must be known in order to relate the mass transfer data to the corresponding heat transfer data. The mass/heat transfer analogy takes the following general form (Chen, 1988): St h = St ()fl = ma St Le m (3.4) where Pr is the heat transfer Prandtl number, a is the mass transfer Schmidt number, and Le is the Lewis number. To obtain the mass/heat transfer analogy in film cooling, namely, the value of n, either direct heat transfer measurements or mass transfer measurements of a different tracer gas with a different diffusion coefficient must be conducted. In the present study, mass transfer measurements with propane and (alternatively) methane as tracer gases have been carried out to determine the effect of Schmidt number on the mass transfer Stanton number, namely, the mass/heat transfer analogy. The Schmidt number of air-methane mixture is 0.71, based on the kinetic gas theory whereas that of propane, used earlier is 1.4. The mass transfer Stanton number obtained with methane as tracer gas is determined in the same way as the mass transfer Stanton number with propane, and it is presented in Figure 3.11. The ratio of the mass Chapter 3 54 St 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.001 .... — — - 0 - - 5 - 10 - - _i_1_1_I_1_1_1_1_1_1_i_1_1_1_1_1_1_I_L_ 15 20 25 - xis - 30 35 Figure 3.11 Mass Transfer Stanton Number Obtained with Methane as Tracer Gas 2.C Stm(M) Stm(P) 1.5 - 1.0 - - -I-1-l-4-4 4. 4. 4 4- IIH::: 0.5 - - — — :::i:i:i:i:i:i:::::z::: — — —4—4—4—4—4—+ 4. 4—4—4— 111111 0.0 0 5 xis III 10 15 20 25 30 35 Figure 3.12 Ratio of Stanton Number with Methane to That with Propane Chapter 3 55 transfer Stanton number with methane to that with propane is shown in Figure 3.12. An average value of the ratio of St (0.71) to Stm (1.4) is then used to determine the mass/heat transfer analogy. The ratio varies from a high of 1.68 near the film cooling slot to 1.48 far from the slot and is affected only slightly by changes in M. The average value of the ratio is 1.57. Thus one concludes that: St (methane) 1•57Stm (propane) (3.5) This relationship indicates the effect of Schmidt number on mass transfer, and it can be used to derive the mass/heat transfer analogy in the following. From Equations (3.2), (3.3) or Equation (3.4), we have at any location and for idential flow conditions: Stm 0” const. or (St) “ (St) mm a (J!L)” a (3.6) p where subscripts p and m indicate propane and methane respectively. Using a and a p =1.4, and the ratio of St m = 0.71 values to be 1.57, the value of n is found to be 0.66. Thus we have the following relation between the mass and heat transfer coefficients for film cooling at high M: Pr —0.66 St (Pr) = St (a)(—) h (3.7) m The empirical constant n takes the value 0.66, or essentially .-. For film cooling at low M, the value of exponent n is assumed to take 0.4, since low M film cooling mass/heat transfer is close to boundary layer mass/heat transfer. Intermediate values for n may be appropriate for other geometries or for intermediate blowing ratios. Chapter 3 56 3.4.3 Heat Transfer with Film Cooling The heat transfer Stanton number with film cooling can be obtained using the relation between the mass and heat transfer coefficients St =St h Pr (—) -n (3.8) where St and St are the heat and mass transfer Stanton numbers, Pr and h m a are the Prandtl number and Schmidt number respectively. The Air-Propane Schmidt number a is calculated as 1.4 using the kinetic gas theory by Perry (1973) since the experimental data is not available. The value of n is an empirical constant ranging from 0.33 to 0.8 (Chen, 1988), and it is dependent on the surface geometry and flow characteristics. In the case of film cooling with high mass flow ratios, the value of n is no longer equal to 0.4 as in Equation (3.3). The value of the empirical constant n is derived empirically in the previous section to be 0.66. For film cooling with low or zero mass flow injections, the flow pattern resembles the standard turbulent boundary layer and the value for n of 0.4 is appropriate (Kays, 1980). The film cooling heat transfer coefficients obtained using the relation between mass and heat transfer coefficients given by Equation (3.8) are presented in Figure 3.13 (a) and Figure 3.13 (b) for both low and high mass flow ratio injections respectively. Note that n = n = 0.4 is used for M =0, M = 0.05, and M = 0.1 assuming no separation, and 0.66 for M = 0.2, M = 0.3, M = 0.4 and M = 0.5. The variations in normalized Stanton number (St / St) of film cooling at high M is also depicted in Figure 3.13 (c) against a nondimensional streamline coordinate (x = (x — x,. ) / Xr), where Xr is the reattachment length (see next section for details). Such a nondimensionalization has been used by Vogel and Eaton (1985) to collapse the heat transfer data for single backwardfacing step. Since the value of heat transfer Stanton number Sth differs from mass transfer Stanton number St by a constant factor, the conclusions reached earlier for St apply to as well, namely: the heat transfer Stanton number for film cooling at low Chapter 3 mass flow ratios (M 57 1) is close to that of a turbulent thermal boundary layer, and at high mass flow ratios, the heat transfer coefficient variation is similar to that required for turbulent separated flows. These conclusions, shown clearly in Figure 3.13, agree with those reached by Scesa (1954) for film cooling heat transfer at low M, and by Ota and Kon (1979) and Vogel and Eaton (1985) for film cooling heat transfer at high M. 3.5 Separation Reattachment Length In an early study of turbulent heat transfer with flow separation by Seban (1964) , it was observed that the analogy between friction and heat transfer does not hold; the heat transfer coefficients often exhibit a maximum value at the location of reattachment while , the mean shear stress is zero. An explanation of this observation was provided by Spalding (1967) using a power-law relation between the Stanton number and the Reynolds number, expressing the law of heat transfer for a wall adjacent to a region of turbulent separated flow. The derivation of the power-law relation is based on Prandt Fs (1945) proposal for the laws of dissipation, diffusion and generation of turbulent kinetic energy. Good agreement was obtained between the power-law predictions and the experimental data available for heat transfer from turbulent separated flows. Based upon the above experimental observation that h attains its local maximum at a reattachment point, the separationlreattachment length can be deduced from the present mass/h eat transfer measurements. An experimental and numerical investigation was conducted later by Sparrow (1987) to determine the relation between the points of flow reattachment and maximum heat transfer for regions of flow separation. He showed that the heat transfe r maximum might occur upstream or downstream of the reattachment point, and the above assumed equality of points of flow reattachment and maximum heat transfer coeffic ient is at best approximate. The reattachment lengths at various film cooling mass flow ratios, obtained from the present mass transfer measurements by assuming equality of points of reattachment and Chapter 3 58 maximum heat/mass transfer coefficient, are presented in Figure 3.14. It can be seen that the apparent reattachment length increases monotonically with the film cooling mass flow ratio. Other factors, such as the upstream boundary layer thickness, could also have some effect on the reattachment length. but this thickness is constant, and given by 6 / s 3, for the present results. St 001 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 XIS 0 5 10 15 20 25 30 35 Figure 3.13 (a) Heat Transfer Stanton Number of Film Cooling at Low M Chapter 3 59 00o:fl 0.002 0.001 - 1_I_i_i — 0 5 1_I_I_i 15 10 20 x’s 25 30 35 Figure 3.13 (b) Heat Transfer Stanton Number of Film Cooling at High M St/StmW( ------------------[- 1.0 0.5 x• 0.0 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Figure 3.13 (c) Normalized Stanton Number of Film Cooling at High M 60 Chapter 3 X/s 5- 0 4- 0 3 0 2 0 o 0.0 0.1 0.2 0.3 .1. I 0.4 0.5 0.6 Figure 3.14 Variation of Separation Reattachment Length Chapter 4 Mathematical Model One of the major objectives of the present investigation is to predict the turbulent flow and heat transfer associated with film cooling. This can be accomplished by solving a set of governing equations using various numerical techniques. In this chapter, the timeaveraged governing equations for turbulent flow and heat transfer are derived, the turbulence model used in the present study is discussed, and the near-wall treatments, namely, the standard wall function approach and the modified wall function method, are reviewed. Next, the general convective diffusion equation of various transport phenomena in fluid flow and heat transfer is presented, and its physical and mathematical properties are then discussed. Finally the boundary conditions are presented. 4.1 Introduction Various flow and heat transfer phenomena in engineering can be accurately described by some fundamental natural laws, which are expressed mathematically as equations of mass conservation, momentum conservation and energy conservation. For the incompressible Newtonian flow with constant properties, these equations have the following form: v•u=o (4.1) DU p--—=—Vp+V•(j.tVU) (4.2) 61 Chapter 4 62 DT pc —=V.(icVT)+ PDt where C1 (4.3) is called the dissipation function because it indicates the energy which is dissipated into heat. For incompressible Newtonian flow it can be written as 1 au. au. au = 1 ax ax, The above dissipation function is negligible for low speed flows with thermal energy transport. Equations (4.1), (4.2) and (4.3) are also called continuity equation, Navier-Stokes equation and energy equation respectively. Using the Cartesian tensor notation, the above equations become (4.4) ax, au. au. at aT at a ap a ax, a 1 ax pc (—+u —)=—(ic 1 ax, ax, au. au. ax, —)+ ax, (4.5) (4.6) where i=1,2,3. Details of derivation of these equations can be found in the book by Schlichting (1979), and the book by Kays and Crawford (1980). Equations (4.4), (4.5) and (4.6) are considered to be valid for both the laminar and turbulent flows. The reason is that the smallest eddy in turbulence still contains a significant number of molecules, and has a much larger scale than the molecular mean free path. Theoretically speaking, all the viscous flow and heat transfer problems can be predicted by solving the above governing equations numerically. However, direct simulation of high Reynolds number flow, namely, turbulent flow, places an overwhelming requirement on computer memory and speed. To resolve the finest eddy scale in a turbulent flow, the grid size required for numerical simulation must be of the Chapter 4 63 3! order of the Kolmogorov scale, ‘q = (p—) E . For instance, the grid size required for air moving at lOm I s should be 1= lO m (Chen, 1987). This very fine resolution requires an enormous number of grid points even for a simple turbulent channel flow at moderate Reynolds number, and therefore lead to computations beyond the capability of available supercomputers. A comprehensive study of the grid requirements for computational aerodynamics was conducted by Chapman (1979), which indicated that direct simulation of turbulent flow and heat transfer using the Navier-Stokes equation appeared very difficult. Therefore, the time-averaged equations, together with turbulence model equations, have to be used for predictions of turbulent flow and heat transfer in most engineering applications. The time-averaged equations for turbulent flow and heat transfer are now presented. 4.2 Time-Averaged Governing Equations The time-averaging or statistical approach was first proposed by Osborne Reynolds (Chen, 1987) about one hundred years ago. Following Reynolds, we assume that the instantaneous value of any variable in a turbulent flow can be decomposed into two parts, a mean and a fluctuating component. Then we have for a general variable 0 (4.7) The time-averaged value, 0, is defined as :+At p=— So’ (4.8) where z t is the averaging time which is larger than the largest time scale of turbulence. Introducing Equation (4.7) into Equations (4.4), (4.5) and (4.6), we obtain, for a statistically steady turbulent flow and heat transfer, (49) Jxi p au,u p a 1 a.r 1 ax au. au 1 ax 1 ax — 1 (4.10) Chapter 4 64 and aT pc (—+ p aU•T )=—(ic xi aT — ——pc u.O) P x (4.11) where the overbars for mean variables are omitted for convenience. Equation (4.10), called the Reynolds equation, and Equation (4.11) contain some new unknown terms, p u, u and —pc u 1 0, which are known as Reynolds stress and — turbulent heat flux. Obviously, the Reynolds stress and turbulent heat flux are produced by the time-averaging of the non-linear convection terms in the Navier-Stokes equation and energy equation. Equations (4.9), (4.10) and (4.11), combined with boundary conditions, are now no longer a closed system because of the Reynolds stress and turbulent heat flux. Therefore, additional equations, also known as turbulence model equations, must be introduced for these new unknowns in order to obtain a solution for turbulent flow. The theories, developed to close the above system by relating the Reynolds stress and turbulent heat flux to mean flow information, are at the basis of various turbulence models. The turbulence model theory is semi—empirical as experimental information is necessary for its derivation. 4.3 Turbulence Model Equations The role of a turbulence model is to express the unknown correlation of fluctuating components in terms of mean flow variables and characteristic turbulence properties (or scales) in order to close the above time-averaged equations. According to Launder and Spalding (1974), turbulence models can be classified into zero-equation models, oneequation models, two-equation models and multi-equation models. In 1877, Boussinesq (Chen, 1987) presented an assumption for the Reynolds stress which was later revised to take the following form: — au. au. ax ax, 2 3 Similarly, the turbulent heat flux can be expressed as (4.12) Chapter 4 65 (4.13) Pr, ax 1 Equations (4.12) and (4.13) are based on the analogy to viscous diffusion. They are not useful until an expression for turbulence viscosity or eddy viscosity i. is established, and the value of turbulence Prandtl number Pr is found. In 1925, Prandtl developed the first zero equation model, the mixing length theory, for the turbulent Reynolds stress. Prandtl assumed that the eddy viscosity is proportional to a mean fluctuating velocity and mixing length, that is, =i.ajj (4.14) Then the Reynolds stress can be expressed as (4.15) where Im is the mixing length, which is a problem dependent. This mixing length turbulence model is valid for simple turbulent flows, such as the pipe flow and channel flow. However, the mixing length implies that the eddy viscosity vanishes wherever — is zero, which is not true. The mixing length seems invalid for complex turbulent flows. To overcome this difficulty, Prandtl (1945) and Kolmogorov (1942) suggested that the length scale, together with the turbulence kinetic energy, is required to describe the eddy viscosity. This leads to the following expression for eddy viscosity: (4.16) where 1 is the length scale, k is the turbulence kinetic energy and C is an empirical constant. They also suggested that a differential equation with k as dependent variable should be used to account for the influence of local turbulence kinetic energy and the influence of convection and diffusion. As a result, a one equation turbulence model with a differential equation for k was developed for the solution of the turbulent flow. However, the one 66 Chapter 4 equation model still needs experimental input to determine the constants in the k equation, and the length scale has to be prescribed as well. The major shortcoming of the one equation model, such as the k equation model, is that it does not account for the transport of turbulence length scale, and it offers only small advantages over the mixing length model in practice. In attempt to eliminate the need for specifying the turbulence length scale as a function of position through the flow, several workers have explored the use of a second differential equation, which in effect gives the length scale I . The group at Imperial College experimented with ad hoc transport equations for 1, which never became popular. However, success has been achieved by using the k equation and a model equation based on the rate of dissipation of turbulence kinetic energy equation model is the well-known k The k — — E. This two e turbulence model. e model of Launder and Spalding (1974), also referred as the standard k — model, is used in the present study. It contains two additional differential equations, one for the turbulence kinetic energy, k, the other for the rate of dissipation of turbulence kinetic energy E. The standard k — E model has been most widely used in the prediction of turbulent flow in engineering since it provides a good compromise between generality and computational economy. The standard k — e model, like other two equation turbulence models, uses the same Prandtl-Kolomogrov formula for turbulence viscosity, p,, that is: (4.18) where 1 is the turbulence length scale, which is determined by k and using dimensional analysis from: 3 loc— E Combining Equations (4.18) and (4.19), we have: (4.19) 67 Chapter 4 (4.20) where C is an empirical constant. Using Equation (4.20), the turbulent viscosity can now be determined via k and However, the values of k and transport equations for k and E . have to be determined from the solution of turbulence E. The exact transport equation for turbulence kinetic energy k can be easily deduced from the exact turbulence shear stress equation. Using the exact turbulence shear stress equation derived by Chou (1945) a D— b(pu,uj) = -----(--pu,uu, —p( , u, 6 1 _au +6,, —au, ---—)— , --——+puu, —(pu u 1 1 fiX 1 OX au. 1 ax aiit u) +j.t 1 au,Ju t-—-—---— 1 1 cJX fiX au. (4.21) ax, and the definition of turbulence kinetic energy The transport equation for turbulence kinetic energy can be written as: —ati. ak D a p pu , +J.t-——)— u u,u,—pu, 1 1 —(pk) = —(--—u ox, cix, ax, 2 Dt — (4.22) Equation (4.22) contains unknown higher order correlation terms and pressure-strain terms in the diffusion term. Therefore it still can not be used to obtain the solution for the turbulence kinetic energy k. To obtain the solution for the turbulence kinetic energy k, these unknown terms must be modeled or approximated. Using the analogy between turbulent diffusion and viscous diffusion, the turbulent diffusion term in Equation (4.22) can be modeled to yield the standard modeled transport equation for k in the following: pt], ak a — ax, ax, ak ax, au. au. au. ax, (4.23) 68 Chapter 4 where k 0 is the turbulent Prandtl number for the transport of turbulence kinetic energy. Terms of Equation (4.24) from left to right successively represent the convective transport, the diffusive transport, the generation and the dissipation of turbulence kinetic energy. The exact transport equation for the rate of dissipation can be derived by manipulating the fluctuating velocity equation. However, the complete derivation is very time consuming and tedious. Therefore we only present the exact equation in the following (Chen, 1987): D —(pE) Dt = a apau, ax, 1 1 ax ax —(—pEu,—211——)+JI—) au. au, au, 1 1 ax, ax ax au 1 aU. aU ax, , ax, au. au. +_!__) ax, ax, —2p( . a u 2 1 ax,ax U. 2 1 ax,a. ) 2 (4.24) Terms in the right hand side of Equation (4.24) represent the diffusion, production and destruction of the rate of dissipation, and these unknown terms must be modeled in order to solve for the rate of dissipation. The turbulent diffusion of the rate of dissipation can be modeled in a similar way to the modeling of turbulent diffusion of kinetic energy. The production term of the rate of dissipation can be neglected following the dimensional analysis and isotropic assumption. The modeling of the destruction term introduced by Lumley (1972, 1975) is obtained using the assumption of local equilibrium between production and dissipation of the turbulence kinetic energy. Using all the above approximations, we obtain a less accurate modeled equation for the rate of dissipation, which reads D —(pE) Dt ae a € au, au au, 1 j.t — (— =—[(— +i. )—]--C ax ax, )— K ax, ax, a 2 —C For steady turbulent flow, it reduces to (4.25) 69 Chapter 4 Fip Uj £ = Fix, Fix, — a 1 ji )---]—C Fix, )L . K Fix, Fix, Fix (4.26) 2 C Values of the empirical constants appearing in the above modeled equation for k and £ are listed in Table 4.1. These constants can be determined by examining the near-wall turbulence, the decay of turbulence behind a grid in absence of mean velocity gradients and studying the energy equation in the vicinity of a wall. The choice of the values of these constants listed in Table 4.1 were made after extensive examination of turbulent flows by Launder et. a!. (1972). See also Chen (1987). These values have been adopted by most workers as “standard” values in prediction of turbulent flows. Table 4.1. The k — E Model Constants 70 Chapter 4 4.4 Near Wall Treatment The standard k — a turbulence model described above neglects the viscous effects which are important in the vicinity of solid boundaries, and it is valid only for high Reynolds number fully turbulent flows. Close to the solid wall, there are regions where the local turbulence Reynolds number is so small that viscous effects predominate over the turbulent ones. Two methods can be used to account for these regions in numerical calculations of turbulent flows: one is the use of wall function method, the other is the low-Reynolds-number-modeling method. For turbulent flow, gradients of dependent variables, such as velocity and temperature, are steep near the wall. To accurately resolve profiles of velocity and temperature, the use of low-Reynolds-number-modeling method requires a large number of grid points placed in the near-wall region. For many practical purposes, the wall function methods are preferred. The wall function method has two merits: it economizes computer time and storage; and it allows the introduction of additional empirical and physical information in special cases, as when there is separation. Wall functions bridge the viscosity affected layer by matching the dependent variables appearing in the turbulence models to universal values at points in the fully developed turbulent region. In turbulent heat transfer with separation, the modeling of conduction affected nearwall region can have a significant effect on the heat transfer coefficient along the wall. The standard wall function of Launder and Spalding (1974) often under-predicts the heat transfer coefficient for separated turbulent flows. To overcome the difficulty, a modified wall function, which was based on the physical observation of the structure of separation, was proposed by Ciofalo and Collins (1989) for prediction of heat transfer with separation. It has been shown that the modified wall function is able to predict the heat transfer coefficient reasonably. In the present study, both the standard and modified wall functions will be used for prediction of the turbulent heat transfer with film cooling. The performance of these two wall functions will be discussed in Chapter 6. In the following, 71 Chapter 4 a brief description of the standard wall function and the modified wall function is presented. 4.4.1 Standard Wall Function The standard wall function method was proposed by Launder and Spalding (1974), and it has gained wide application in numerical simulation of turbulent flow and heat transfer. It is used to relate the surface boundary conditions to the near-wall points and then provide a set of finite difference equations for the near-wall points. The wall function method implies the existence of a near-wall linear sublayer, where momentum transfer is dominated by molecular viscosity and heat transfer by conduction, by a logarithmic region. In the standard wall function (very similar to the wall function equations in Section 2.2.2), y,u and T are nondimensionalized as 1 I 4 k C (4.27) 1) I u u I 2 4 k C (4.28) tw /P 1 1 (T—T )pC C 4 k T_= (4.29) Profiles of u are then expressed as = y’ u=--ln(Ey) y y (4.30) y>y (4.31) y (4.32) y>y (433) where yD+ = 11 Profiles of T obey T = n(Ey)+P] T=c[ 1 1 + 72 Chapter 4 where a and a are the molecular and turbulent Prandtl numbers respectively, and P is an empirical function described in Chapter 2. The rationale for the above near-wall treatment is based on the fact that there exists a — 60) where the turbulent kinetic energy k attains a nearly near-wall region (- 20 constant value, namely: k t 2 C (4.34) p In numerical computation of turbulent flows, k is assumed to take the value of k at the first near-wall node which must be located in the equilibrium region. In the wall function method, a one-dimensional Couette flow is assumed in the near wall region, and similar conditions to those of an equilibrium turbulent boundary layer flow are assumed to prevail. Because the computational nodes immediately adjacent to the wall are placed in the fully turbulent region, the wall shear stress can be calculated using Equations (4.28) and (4.31), which give !! U 2 t =piC ic ln(Ey) (4.35) The wall shear stress calculated using the above equation is used to formulate the finite difference equations for momentum equations in the near-wall cells. The dissipation rate of turbulence kinetic energy at the near-wall nodes is computed according to the local equilibrium condition as 3 2 k Ky,, (4.36) The value of the turbulent kinetic energy k,,, however, must be determined by solving the turbulent kinetic energy equation in the near-wall cell. To this end, proper forms of the corresponding production, dissipation, and transport terms must be chosen. In most implementations, the diffusion of turbulent kinetic energy to the wall is set to zero and 73 Chapter 4 the average energy dissipation rate over the control volume is evaluated using the following equation: 3 8= k 2 C —-—1n(Ey) (4.37) K Equation (4.37) is used to evaluate the dissipation term —pe in the turbulent kinetic energy equation. 4.4.2 Modified Wall Function The above standard wall function of Launder and Spalding (1974) generally produces satisfactory results for turbulent flows without flow separation. It is based on the analogy between friction and heat transfer, which breaks down in the region of flow separation. Therefore, when applied to separated flows, the standard wall function method, in general, yields disappointing results, characterized by severe under-predictions of heat transfer coefficients in the region of separation, and a general disagreement of heat transfer Stanton number or Nusselt number profiles along the wall. Collins (1983) underpredicts Nusselt numbers for an abrupt circular expansion using the TEACH code. Ciofalo and Collins (1987) also under-predict Nusselt number downstream of backwardfacing steps using FLOW3D. The use of low-Reynolds-number k — E model also fails to produce reliable predictions for heat transfer. Gooray et. a!. (1981) reported strong under-predictions of Nusselt number (an order of magnitude!) using low-Reynolds Number k — e model. On the other hand, the same model is reported by Chieng and Launder (1980) to yield far too high Nusselt number values. To overcome the difficulty, Chieng and Launder (1980) introduced a somewhat different treatment for the near-wall region, which employs the value of k at the edge of the viscous sublayer k for scaling instead of k. Its most important feature is to specify the viscous sublayer thickness according to 74 Chapter 4 Re 2“ y k =20(const) (4.38) 1) In this way, the viscous sublayer thickness is allowed to vary according to the value of turbulent kinetic energy at the edge of the viscous sublayer. Unfortunately, the numerical simulation of the sudden circular expansion test problem by Chieng and Launder (1980) using this model was affected by coding errors, and the use of the above treatment by Collins (1983) was reported to yield values of Nusselt number lower than that reported by Chieng and Launder (1980). Johnson and Launder (1982) proposed an improvement to the near-wall treatment of Chieng and Launder (1980), which extended the concept allowing the nondimensional sublayer thickness to vary according to the following equation: 3 Re 2 yk ‘ “ = Re (4.39) 1) k,, where k is the value of k at the edge of viscous sublayer, k is the value of k at the wall, and =20 0 Re c=3.1 This treatment allows the sublayer thickness to become even smaller than that given by Equation (4.38) in the region of flow separation. When applied properly in the code, this model yields results in good agreement with the experimental data. This suggests that improvements on turbulent flow and heat transfer prediction can be made from the assumption of a varying sublayer thickness. Ciofalo and Collins (1989) realized that the most reasonable way to modify the wall function to account for the turbulence structure near the wall, while fonnally retaining the simple linear-log laws, would be to modify the constants of E and/or k in the law of wall. This means modification of the nondimensional sublayer thickness y and/or the slope of the logarithmic region of the u versus curve. Then Ciofalo and Collins 75 Chapter 4 (1989) proposed a modified near-wall treatment, which relates the nondimensional viscous sublayer thickness to the near-wall turbulence intensity N’ p (4.40) —-- up according to the following equation y 0 y N’ (_L) -c (4.41) where y 0 N’ E are the nondimensional viscous sublayer thickness and the turbulence intensity of a turbulent boundary layer, and c is an empirical constant ranging from! to For the thermal sublayer, we have + + v YT G (—) —0.25 at Equation (4.42) implies that y (4.42) oc y. In the near-wall region of an equilibrium turbulent boundary layer, the turbulence intensity, is a known function of the local Reynolds number, ‘E R=—=uy (4.43) 1) In fact, using the scaling based on the friction velocity, the nondimensional velocity distribution is given by the linear-log law described in Chapter 2, namely: u=--ln(Ey) where Y+ + 1) U U Ut The turbulent kinetic energy can be expressed as Chapter 4 76 2 k÷=C2(4) (4.44) yvo k = C (const) y+ (445) > The parabolic profiles of k are the same as those used by Chieng and Launder (1980). Using the above profiles of velocity and kinetic energy, the turbulence intensity in the equilibrium boundary layer, I I — N’ E k 2 u k 2 u (4.46) —=------ can be expressed as a function of local Reynolds number as follows; WE = = W (const) R y (4.47) R>y (4.48) + 4 C K N’ = C, ln(E4) where satisfies the following —ln(E)=R The ratio (4.49) can now be calculated and then used to obtain the viscous sublayer WE thickness with Equation (4.41). The relation between the viscous sublayer thickness and the local turbulence intensity is based on the physical meaning of the ratio ---, which WE can be interpreted as an index of the “distance from equilibrium” in the near-wall region. Therefore, in the reattachment and redevelopment region, we have --- 1, and WE while --- 1, and y > 0 in the laminarizing boundary layer. y WE 4.5 General Transport Equation For the two-dimensional incompressible turbulent flow and heat transfer, the time averaged governing equations, together with the k can be summarized in the following: — E eddy viscosity turbulence model, Chapter 4 77 mass conservation equation: au ax + (4.50) ay momentum equation in x direction: auu avu ay = ar-p — ax + a 2—{.t au axe a y --—}+--{i au av (—+--—)J eifay (4.51) momentum equation in y direction: auv avv +p ax ay aP* a =—-—+—{j.i. ax ay eff av au a (—+—)}÷2 ax ay ay av eff (4.52) turbulence transport equation for k: auk + av a L.!]+±[fL.!]+G....p )= ay ay Gk ay axak ax (4.53) turbulence transport equation for : auE ax + avE a ..GC 1 _!J+[_LJ+C P 2 )=— ax ay ax ay ay (454) thermal energy equation auav a a a a + )= axeffa ay’effay 2 ax ay (4.55) where p* =P-f-pk au, auf)au G=J.L(----+----— 1 ax eif ji JI+II! Pr Pr 1 ‘eif For convenience, Equations (4.51), (4.52), (4.53), (4.54) and (4.55) are written in the following general form: p( au ax + av ) ay = ax [1’ ax j+ ay [r ay (4.56) Chapter 4 78 where F is a general diffusion coefficient are and S a general source term, and their values listed in Table 4.2 for each of the transport equations. The transport Equation (4.56) is a typical convective-diffusion equation, and it has the following properties: 1. if Equation (4.56) is homogeneous, its non-constant solution can not attain its maximum or minimum values at the interior points of the solution domain. (4.56) 2. if the source term of Equation is positive, and its value along the boundary is larger than zero, its solution will be larger than zero. These two mathematical properties are consistent with the physical properties of the convective diffusion process, and must be satisfied by the numerical scheme to be employed in order to simulate the convective diffusion process properly. Table 4.2. Diffusion Coefficients and Source Terms Dependent variables (1) r x-momentum equation U neff y-momentum equation V 0 S P a Energy equation kequation a T k JU + + a a )+—(p. au ’ff 0 eff ay a ayeff 0 J1 _L GpE a’c t a equation a —f6 a 1 C G —C 2 3V k —) eff 3x av Chapter 4 79 4.6 Boundary Conditions To make the problem well-posed, the corresponding boundary conditions must be specified. The boundary conditions under consideration can be summarized by the following: 1. Inlet boundary conditions: Distribution of all dependent variables, except pressure, are specified in such a way that they are consistent with the measured boundary layer and free stream values. For the uniform flow inlet, uniform profiles for U, k and E are prescribed; V is set to zero. The turbulent kinetic energy is determined using the free stream turbulence intensity measured from the experiment, while the rate of dissipation is estimated from the kinetic energy and a characteristic length scale. 2. Outlet boundary conditions: Theoretically, when the outlet of the flow is located in the region where the flow is fully developed, a zero gradient condition along the streamline can be applied for all variables, except pressure. In the present study, the outlet of the flow is located sufficiently far away from the film cooling injection location where the flow changes slowly enough that the zero gradient condition is acceptable. 3. Symmetry line boundary conditions: The normal velocity across the symmetry line is set to zero, and a zero cross-stream gradient condition is applied to all other variables. 4. Wall boundary conditions: For velocity, the non-slip boundary condition is applied for a solid impermeable wall, while the normal velocity is assigned the value of wall blowing velocity for a porous wall. The heat flux boundary condition, namely, either the adiabatic wall condition or the constant heat flux condition, is imposed for heat transfer. Both the standard wall function and the modified wall function are used to calculate the wall shear stress, the turbulence kinetic energy at the near-wall nodes and the temperature on the wall. Chapter 5 Numerical Solution Method In order to obtain the numerical solution of a turbulent flow, the governing equations presented in the previous chapter are discretized using the finite control volume method. The resulting system of algebraic equations is then solved on a computer by a certain iteration procedure. In this chapter, cliscretization of the governing equations is first described and then the generalized pressure correction method for solving the resulting system of finite difference equations is presented. Next, an overview of the numerical method and implementation of the boundary conditions are described. To increase the convergence rate of the pressure correction iteration procedure, a multi-grid acceleration technique employing no correction along the boundary is proposed. This method is applied to the numerical solution of a laminar cavity flow and turbulent boundary layer flow in order to validate the proposed multi-grid method, the multi-grid code, and examine the turbulence and numerical models. 5.1 Discretization of Governing Equations Turbulent flow and heat transfer can be described by the time-averaged continuity equation, Navier-Stokes equations and thermal energy equation, together with the k — E turbulence model. A description of the governing equations is provided in Chapter 4. Cast in the primitive variable from, the above governing equations do not have an explicit equation for pressure which appears in the Navier-Stokes equations in its gradient form. The pressure field is indirectly specified through the continuity equation. 80 Chapter 5 81 Therefore, numerical solution methods based on governing equatio ns in the primitive variable form must find an efficient procedure to update the pressu re field during each iteration. In the present study, the governing equations are discretized on a staggered grid using the control volume method. The resulting system of finite difference equations are solved using the pressure correction method of Patankar (1980) , in which a pressure correction equation is derived for updating the pressure field. A detaile d description of the discretization and numerical solution procedure is discussed below. 5.1.1 Grid Arrangement To simulate the turbulent flow and heat transfer, the computational domain is divided into a set of non-uniform staggered grids. The staggered grid is used to overcome the numerical instability and difficulties from the pressure bound ary condition. The staggered grid, together with control volumes for velocity and scalar variables, is shown in Figure 5.1. The velocity components u and v are defined at points that lie on the faces of each scalar control volume. In each scalar control volume the pressu re and turbulence quantities axe defined at the grid points. In the above grid arrangement, grid points are placed at the centers of the control volumes so that control volume faces of coarse grids are also the control volume faces of fine grids in case the grid needs to be refined when a multi-grid technique is required to accelerate the convergence rate. 5.1.2 Finite Difference Approximation Momentum equations can be integrated over their control volumes to produce a set of discretized equations for each velocity component. The momentum equations for turbulent flow in 2D rectangular coordinates can be written as follow s: a ox ox a au oy oy (5.1) Chapter 5 82 a) P control volume N ) . 1/ / / / / w’ W 1 P I I ,P/le / I’ / I / () S S b) U control volume c) V control volume Figure 5.1 Staggered Grid Arrangement: — = u, f E / = v, o = p 83 Chapter 5 a a av ax (Pt41iteff )+(PVVIIeff av —)—S (5.2) where a S ax au ax au (Ite a ay ejr ax a av ap (5.3) av ap (5.4) aOteff a +(I.teff )_ Integration of Equation (5.1) for velocity component u yields — a ay au )}dxdYllsudxdY 5 eff)(P lieff ax ax” — (5.5) or + J, — — J= if S dx dy where e =Fe+De J =E+D J, =P+D, 3 J =F+D and 1 =$puu!dy au De = f ‘w SPUUIXXdY (liff -;)I au f P =$puvIyy,d av D J j)i dy D = (lieff dy = (lieff dx F =SPUVty=y,dx D =J(—j.ICff av 2I=A (5.6) Chapter 5 84 The above convective fluxes, F, F,,, F,, F and diffusive fluxes, De D, D, D are exact, and approximations must be made to obtain the finite difference equations for each velocity component. Different schemes have been used to discretize the convective and diffusive fluxes. One method is to use central differencing to approximate the convective and diffusive flux terms, which leads to a second-order accurate central differencing scheme. However, the central differencing scheme can not simulate convective fluxes properly, and is stable only at very low cell Reynolds numbers. A remedy to this difficulty is to employ the socalled upwind differencing scheme for the convective flux terms. However, the stability is achieved at the cost of introducing numerical diffusion or truncation error, which is especially high when the grid is not aligned with the flow. High-order approximations have been used to solve the problems arising from truncation. Leonard (1979), for instance, developed a quadratic upwind interpolation formula. This formulation (QUICK) sometimes leads to negative coefficients, thus violating the physical principles of the convective-diffusion process or the mathematical properties of the governing transport equations. Other approximations have also been introduced to reduce the numerical diffusion or truncation error. Patankar (1980) proposed a power law difference scheme as an alternative to the upwind differencing scheme. This scheme was analyzed by Huang et. al. (1985). They found that it suffers from the same limitations as the upwind differencing scheme, namely, it produces good results only when the flow is aligned with the grid. Raithby (1976) developed the skew-upstream differencing scheme (SUDS), which accounts for the flow skewness with respect to the grid. This scheme is of first order accuracy, but capable of reducing some numerical diffusion at the expense of simplicity, since the influence of neighboring nodes has to be also included in the expressions of the coefficients. Recently, a second order accurate numerical diffusion free piece-wise parabolic finite analytic method has been established by Sun and Militzer (1992) to simulate the convective diffusion process. This scheme (PPFAM) can 85 Chapter 5 automatically simulate the convective-diffusion process. Accuracy and stability are again achieved at the cost of simplicity. In the present study, the hybrid differencing scheme of Patankar (1980) is used. This scheme (HD) is a combination of the central differencing and upwind schemes. It is second order accurate at small cell Reynolds numbers, but first order accurate at high cell Reynolds numbers. It is a reasonable compromise between accuracy, stability and simplicity for prediction of turbulent flow and heat transfer. The hybrid differencing scheme implies that the central differencing scheme is used for low cell Reynolds numbers while for high cell Reynolds numbers it reduces to the upwind differencing scheme, in which the diffusion is set to zero. Using the formula proposed by Patankar (1980), the convective fluxes and the diffusive fluxes can be related as follows ep =a(uP—uE) ‘ ’ e 1 J—Fu =a(u —un) Jfl—IuP =a(u,,—u,) u 3 J—F u) =a(u — 5 For the hybrid differencing scheme, the coefficients read a; =[_FDe_-0] (5.7) a; (5.8) a; =[_1D a’ , = 3 D _.L,0] (59) 4- 0] (5.10) + , The integration of the source term is approximated using the central differencing scheme as follows ffsdy=s;u+s: (5.11) 86 Chapter 5 For velocity component u, we have the following finite difference equations a uA,, = a UN +a 5+ u,, + a u + a u (5.12) where a=a+a+a+a’—S Similarly, we have finite difference equations for v, k, £ and T in the following: a = a VE +a v,,, +a VN +a’ v ÷s: (5.13) ak —akE+akW+akN+akS+S’ (5.14) a (5.15) E, = aEE+aEW +a EN +a:E+S a=a+a1+aI,+a1+S (5.16) where a=a+a+a+a—S a=a+a+a+a—S a=a+a+a+a—S a=a+a.+a+a—S,’, For convenience, expressions of source term discretization coefficients are listed in Table 5.1. Integration of the continuity equation over the scalar control volume yields (5.17) Equations (5.12), ..., (5.17) constitute the system of finite difference equations for velocity, pressure, turbulence quantities and temperature, which can be used to solve for turbulent flow and heat transfer. 5.2 Boundary Conditions Near the boundary, discretization of governing equations can be carried out in a similar manner, but some modifications must be made to account for various boundary conditions. For turbulent flows, the wall shear stress is evaluated using the wall function 87 Chapter 5 of Launder and Spalding (1974), which leads to some modifications of the finite difference coefficients and source term expressions. Discretization of the turbulence kinetic energy equation near a solid wall is performed by evaluating the dissipation of turbulence kinetic energy over the near-wall control volume using the wall function and setting the diffusive flux at the wall to zero. There is no discretization required for the turbulence dissipation equation since the rate of dissipation at the near-wall nodes is calculated using the wall function. With a heat flux specified along the wall, modification of the discretization of the thermal energy equation near the wall is trivial. For other types of boundaries, such as outlet flow and symmetry lines, the modifications to discretized equations are generally simple and straightforward. Detailed description can also be found in the thesis by Djilali (1987). Table 5.1 Linearized Source Terms St Transport equation ‘P x-momentum u o y-momentum v o k-equation k k 2 C p — Ax+(J.L. sx ty Ax y AV AU Ay+(J.t-_-) —)I4’ Ay G zX b.y G Ax ‘If p c p 2 E -equation a — k 1 C Ax Chapter 5 88 5.3 Numerical Solution Procedure Discretization of the governing equations for turbulent flow and heat transfer, as described in the previous section, leads to a set of algebraic equations for all the dependent variables. This set of equations is non-linear and can only be solved on a computer using certain iterative methods. One of the difficulties in using iterative methods is updating the pressure field since no explicit pressure equation is available. The pressure must be determined by solving the entire set of hydrodynamic equations. Two types of iterative methods have been developed for solving the above discrete system, namely, the de-coupled and the coupled method. The most widely used de coupled method to date is the SIMPLE method of Patankar (1980), which is known as the pressure correction method. In this method, a pressure correction equation is derived based on the discrete continuity equation, which is then used for updating the pressure field during the iteration process. The SIMPLE method produces a numerical solution to the discrete system when the iteration process is convergent. However, the SiMPLE method does not ensure the convergence of the iteration process, and theoretically the pressure correction equation can not replace the continuity equation. Modifications to the SIMPLE method lead to the SIMPLER and the SIMPLEC algorithms. The efficiency of these schemes has been examined by Latimer and Pollard (1985), Jang et. al. (1986). They concluded that the improvement in performance is not as good as expected. On the other hand, Vanka (1986) developed a coupled method for obtaining numerical solutions to the discrete system, which is known as the symmetrically coupled Gauss-Seidel (SCGS) relaxation method and seems to be very similar to SIVA of Caretto et. al. (1972). In this coupled method, dependent variables associated with one control volume, such as velocity and pressure, are updated simultaneously. The merits of the SCGS method are that it eliminates the need for a pressure equation or a pressure correction equation, and it is often more efficient in reducing errors simultaneously. Chapter 5 89 5.4 Multi-grid Acceleration It is well known that both the de-coupled and the coupled methods suffer from convergence stalling problems as iterations proceed and often their convergence rates deteriorate as the grid is made fine for higher resolution. This phenomenon can be explained by performing a discrete Fourier analysis on the error, which shows that the iteration scheme is good at reducing the high frequency error components, but not the low frequency error components. Since a slow convergence rate is due to the persistence of relatively low frequency error components, it might be effective to remove these low frequency components on a coarse grid. A systematic exploitation of this finding leads to the development of the multi-grid acceleration technique. The multi-grid method was first introduced into the numerical simulation of partial differential equations by Fadorenko (1962) and Bakhvalov (1966). Brandt (1977) made it practical by developing multi-grid algorithms and demonstrating their advantages. Later Brandt (1980) introduced the multi-grid method into the field of computational fluid dynamics. Following Brandt, attempts have been made to apply and develop multi-grid techniques for numerical simulation of fluid flow. Some notable contributions using the multi-grid method include the articles by Fuchs and Zhao (1984), Ghia et. al. (1982), Vanka (1986), and Sivaloganathan and Shaw (1988). Fuchs and Zhao (1984) obtained solutions to the incompressible Navier-Stokes equations in the primitive variable form with a multi-grid technique. They employed an upwind finite difference scheme with a “distributive Gauss-Seidel” (DGS) as the smoothing operator and found that the multigrid method improved the convergence rate considerably. Ghia et. al. (1982) reported the use of a stream function and vorticity formulation and a coupled strongly implicit multigrid technique (CSI-MG) for numerical solution of a two dimensional driven cavity flow. Good convergence rate was achieved for high Reynolds number cavity flow. The article by Vanka (1986) describes a multi-grid method based on the Navier-Stokes equations in 90 Chapter 5 the form of primitive variables and its application to increase the convergence rate of a symmetric block Gauss-Seidel method. For a typical case, the multi-grid solution procedure was shown to be approximately 25 times as efficient as the single grid procedure. Based upon the SIMPLE algorithm, Sivaloganathan and Shaw (1987, 1988) worked out a multi-grid pressure correction method (MGPCM) for the numerical solutions of the Navier-Stokes equations and investigated its convergence rate. For the two dimensional driven cavity flow, it is found that MGPCM is much faster than the SIMPLE algorithm. Numerical simulation of laminar flows using multi-grid techniques with various smoothing operators, fmite difference schemes and grid arrangements have also been carried out by Lacroix et. al. (1984), Philips and Schmidt (1985), Braaten and Shyy (1987), Mifier and Schmidt (1988), Lonsdale et. al. (1988), Karki et. al. (1989), Thompson and Ferziger (1989), Joshi and Vanka (1991), hang and Chen (1991), Luchini et. al. (1991) and Tzanos (1992), among others. All the above attempts are focused on the numerical solution of laminar flows, namely, the Navier-Stokes equations. Only a few studies have ventured into the field of turbulent flow. Yakota (1990) developed an implicit multi-grid scheme for compressible turbulent flow, which solves the time-averaged Navier-Stokes equations with the k — E model of turbulence. The mean flow equations are solved using a multi-grid algorithm with a four level W-cycle, while the k — e equations are solved only on the fmest grid and uncoupled from the mean flow calculations. Claus and Vanka (1992) extended the block-implicit multi-grid method to numerical solution of a 3D jet in crossflow. Jets in crossflow have also been studied by Demuren (1992) using SIMPLEC with multi-grid acceleration. In these two cases, turbulence transport equations for ic and a are solved only on the finest grid level during the multi-grid process. Values required for diffusion fluxes on coarser grids are simply restricted from those on the finest grid level. The work of Peric et. al. (1989) represents possibly the first application of the multi-grid method to the turbulence transport equations. Peric reported attaining a speed-up factor of the order 91 Chapter 5 of 30 by applying the full approximation storage (FAS) multi-grid method to a turbulent backward-facing step flow. FAS has also been used by Lien and Leschziner (1991) in simulating complex recirculating turbulent flows with the k — £ turbulence model. Their study shows that the convergence acceleration is significant in turbulent conditions, but is not of the same order as that for laminar flows. Recently, Rubini et. al. (1992) applied multi-grid acceleration to a fmite volume method for the calculation of a three dimensional, variable-density turbulent flow. A significant reduction in overall computing time is obtained for both the constant-density and variable-density flow over a three dimensional backward-facing step. Peric (1989), Lien (1991) and Rubuni (1992) all extended the use of multi-grid method to the k — £ equations. However, the boundary treatment of turbulent quantities is not reported. The recent work of Shyy et. al. (1993) gives a detailed description of boundary treatment for both k and e. By retaining the grid line next to the solid wall during the multi-grid restriction procedure, an improved convergence rate is obtained. So far, numerical calculation of turbulent flow using the multi-grid method is still in its infancy. The main difficulties in calculation of turbulent flow lie in the implementation of a multi-grid algorithm for turbulence quantities along the boundary on coarse grids. The reason is that the wall function is employed for the calculation of k and e at the near-wall points. It prevents the direct application of the FAS algorithm and contributes to the slow convergence rate or even divergence of the multi-grid method in numerical simulation of a turbulent flow. To address the difficulties arising from the wall function approach, a no boundary correction full approximation storage (NBCFAS) multi-grid method has been developed. The distinct features of the NBCFAS algorithm are that it does not involve a correction along the boundary and thus eliminates the need for an exact understanding of the boundary conditions for the dependent variables as long as the finite difference equations along the boundary can be obtained on the finest grid level. To determine the efficiency of the proposed multi-grid method, two test problems are studied. Numerical experiments 92 Chapter 5 with the test problems indicate that the present multi-grid method can reduce all the residuals much faster than the single grid method, and that it can produce convergent solutions for both laminar cavity flow and turbulent boundary layer flow. 5.4.1 The NBCFAS Algorithm Let us approximate the calculation domain via a sequence of grids G°, G’,...,G. On the finest grid, G’, we assume that the flow problem of concern can be approximated by finite difference equations in the following general form: LU(x)=F(x) XEG (5.18) BU(x)=T?’(x) xeG (5.19) Equations (5.18) and (5.19) can be derived by discretizing the governing equations and their corresponding boundary conditions on grid G”. The exact form depends on the discretization methods employed. In the NBCFAS mode, full approximation is determined by the following equations: LkUk(x)=Fk(x) xeGk (5.20) EkUk(x)=(x) xeJGk (5.21) x E Gk (5.22) xeaGk (5.23) XEaG’ (5.24) xeG” (5.25) xeJG’ (5.26) xeJG’ (5.27) where fk (x) = Lk 1 u÷ (x)) + k+I (I pk(x)..Jkuk+I(x) (f’ (x) — L’u (x)) k=0,1,...,n—1 and f(x)=F(x) E”(x)=B”(x) 93 Chapter 5 Hence Equations (5.20) and (5.21) reduce to Equations (5.18) and (5.19) respectively for k = n. The interpolation of correction is expressed as: (5.28) u +vk+l < is an approximation to U”, Where v is the correction, and they satisfy the following relations: v+l jk+1(,,k) = jk+1 (Uk = U” = k÷l 1 — k+l 1 ) (u ) 1 ) 1 (u a xe Gk u G” (5.29) xe (5.30) k=0,1,...,n—1 Along the boundary, vk =u” a Gk, the correction satisfies Jk(Uk+l)_Uk uk+1)=O _I+ ( 1 XeaGk This confirms that no correction along the boundary is involved in NBCFAS algorithm. The differences between NBCFAS algorithm and FAS algorithm are clear now, and can be summarized in the following: • NBCFAS algorithm does not make any correction along the boundary, whereas FAS algorithm may make corrections along the boundary. • NBCFAS algorithm and FAS algorithm are identical for Dirichiet boundary value problems. • NBCFAS algorithm does not require a well-posed boundary condition as long as finite difference equations along the boundary on the finest grid can be obtained, whereas FAS algorithm does. • On coarse grids, NBCFAS algorithm always has a system of finite difference equations with known values for dependent variables, whereas FAS algorithm may not. • It is much easier to program with NBCFAS algorithm than FAS algorithm. The above advantages of the NBCFAS algorithm make it very promising for numerical simulation of fluid flow. In fluid mechanics there are cases in which we do not Chapter 5 94 know whether the system of governing equations and corresponding boundary conditions is well-posed or not, but finite difference equations can be obtained for the purpose of numerical simulation. For instance, numerical simulation of laminar flow using the stream function and vorticity approach, and numerical solution of turbulent flow using the k — a turbulence model fall into this category. In the stream function and vorticity formula, we only have finite difference equations to evaluate vorticity along the wall numerically. In the k — a turbulence model, we use wall functions to calculate k and a numerically for points adjacent to the wall. To solve the 2D laminar driven cavity flow with the stream function and vorticity approach on a non-staggered grid and turbulent flow using the k — a turbulence model on a staggered grid, the FAS multi-grid method will encounter some difficulties along the boundary due to the lack of suitable boundary conditions. This problem may lead to a divergent multi-grid process. However, the NBCFAS algorithm can be suitable for such problems without a well-posed mathematical model since the present method does not require a correction scheme on the coarse grid along the boundary. In the NBCFAS algorithm, the boundary condition for dependent variables is specified on the coarse grid, and updating of the boundary values for dependent variables is only made on the finest grid. The details of the implementation of NBCFAS algorithm are given in Section 5.5. Note that a very similar multi-grid algrothm has been developed independently by He (1994) in our CFD group for numerical simulation of laminar and turbulent flows with complex geometries. 5.4.2 The Pressure Correction Scheme To solve the above finite difference equations, we need an efficient procedure to update all the dependent variables. The major difficulty is to update the pressure. The SIMPLE procedure, proposed by Patankar (1980), employs a pressure correction equation for updating the pressure field. In the present NBCFAS multi-grid procedure, the SIMPLE pressure correction method is used as a smoothing operator due to its popularity in Chapter 5 95 engineering calculations and its attractive smoothing rate, as demonstrated by Shaw and Sivaloganathan (1988). To use the SIMPLE procedure in the multi-grid process, a generalized pressure correction method needs to be derived, which is now discussed in the following section (Shaw and Sivaloganathan, 1988). Let us consider the following system: NU=f (5.31) where U,, =(u,,,v,,,p,,)T _çu n 1 Let U — v cT “in ‘in in I = (U:, v:, : )Tbe an approximate solution to Equation (5.31), and U,, be the solution to Equations (5.25) and (5.26). The correction U,, = = (us, v v,, , T can be (u,v,,,v) written as follows: u=u,,—u; (5.32) By substituting Equation (5.32) into Equation (5.31), we have a (u + u,) = a (u ) 4 (4 + 4) + 4(4 + 4) +u + (5.33) a(v +v,)=a(v +v)+a,(v, +v,,)+a,(v +v) (5.34) (pu) —(pu), +(pv) —(pv) +(pu) —(pu), — (pv) =0 (5.35) Following Patankar (1980), we assume that U =44 + 44 + 4, 4, + a u a = 4, v +4 v +4 v +a v (PU): —(pu), +(pv) —(); =0 a Then we have: satisfies: = + h(p — p ) + S, +h(p —p)+S,, Chapter 5 96 = h (p,, a a v h (p — — p’,.) + a uE p. ) + a v +4u +4 +au +4 v,, + aX, v + a v The SIMPLE pressure correction method also neglects the mixing derivatives terms which yields: au,, =h(p—p) (5.36) a v, = h(p —p’,,) (5.37) If we substitute the above correction expressions into Equation (5.35), we obtain the following equation for pressure correction: ap = a p + ap + a,, p,, +a p + S where a,, =aE+aW+aN+aS aE = Pe h aw =ph a%, = p h = Ps h = (pu), — (Pu); + (pv); — (pv) Using the above pressure correction equation, the general pressure correction method for smoothing can now be described in the following way: 1. solve for velocity with a given pressure field. 2. solve the pressure correction using the pressure correction equation. 3. update velocity and pressure using the above correction formula. 4. update the dependent scalar variables. 5.4.3 Restriction and Prolongation Operators In the multi-grid process, the restriction operator is used to transfer fine grid values to a coarse grid, whereas the prolongation operator is used to extrapolate the coarse grid Chapter 5 97 corrections to a fine grid. For the fine-coarse grid arrangement shown in Figure 5.2, the restriction is made by averaging neighboring values, namely: = = + Uff] — 1 [v +v 1I + + P+1,ff+1 1 + P+1,ff + P, 1 — ‘:4 “-,- — . —, — . ,—, — zi__ Figure 5.2 Fine-Coarse Grid Arrangement The prolongation relations are obtained by a bi-linear interpolation. For each coarse grid node, four corresponding fine grid values are derived. For the velocity component u, they are: Chapter 5 Uff 98 1 N [3u, — = -- ff = 1 U U÷ff÷ = N + u, _I 1 N [u ÷ 1 + 3 uLJ 1 1 [u_ + 3u 11 + 3u’+ 1 + u/÷ 1 - N 1 [u,’. + 3u + I +1 .,÷ 3u’+ ] The velocity component v can be prolongated in a similar way. However, prolongation for pressure is different. For each coarse grid cell, four fine grid pressures are obtained, which read P,ff 1 + j [9p + 3p = P,ff+1 = P+1,ff = P’+l,ff+I ÷ 3p÷ 1 + 3pJC + [9p = + + 3p + [9 p + + + I + :+1.1+1 I In the present study, the same prolongation operators are used for both the solution and correction for simplicity, although they may be different. 5.5 Applications To determine the efficiency of the proposed multi-grid method, two test problems were studied, namely: the laminar cavity flow and the turbulent boundary layer flow. These test problems have been studied extensively, and comparison of numerical results in the next sections with those in the literature is possible. 5.5.1 Numerical Simulation of Cavity Flow To test the NBCFAS algorithm, we now apply it to the numerical simulation of laminar driven cavity flow using the stream function and vorticity approach. As shown in Figure 5.3, the cavity flow under consideration is induced by the top cover which moves at speed U . The physical constants defining the cavity flow are the following: 0 Chapter 5 99 0 =1.0 mIs U H=l.Om L=1.Om 2 i)=0.1/ s 5.OXlO m 0 U Figure 5.3 Cavity Configuration The above cavity flow is assumed to be a two-dimensional, laminar, steady, incompressible Newtonian flow with constant viscosity. Therefore, the flow obeys the Navier-Stokes equations and satisfies the governing equations with the stream function and vorticity as dependent variables, namely: ax (5.38) ay ax u=-,v=— ax ay (5.39) (5.40) Chapter 5 100 ax (5.41) ay where N is the stream function, is the vorticity function, and u, v are the velocity components in the x, y directions respectively. The boundary conditions for velocit y read: Uab = UbC = UCd = Vab = Uad = = VCd = VOd =0 0 U (5.42) The boundary conditions for the stream function take the following form: N a 1 b ‘Vbc = Wcd = Vad =0 (5.43) However, we do not have a proper boundary condition for vorticity. For the purpose of numerical simulation, discrete boundary conditions are derived using the definition of vorticity and the non-slip boundary conditions. Various discrete vorticity boundary conditions can be found in the book by Roach (1976). As an example, we present the most common vorticity boundary conditions in the following: r An 2 The corresponding solid upper wall boundary is shown in Figure 5.4. w in Figure 5.4 Upper Wall Boundary Chapter 5 101 To obtain the finite difference equations for the governing equations, the central differencing scheme is used to discretize the stream function equation on a non-staggered grid, as shown in Figure 5.5. The finite difference equation obtained for the stream function is written in the following general form: I1 JP c, +Cg +CeWe +C ÷C = (5.45) where CL 2(l+CL) c= S 2(l+CL) CL e 2(l+CL) W 2(l+a) c= “ 2(l-i-x) 2y h The finite difference equations for velocity components are obtained by applying the central differencing scheme to the definition of the stream function, namely: u“ v 2h (5.46) .._‘VeWw 2h (547) The finite difference equation for vorticity is obtained using the upwind differencing scheme which is expressed as follows: Cp=Cnn+CsCs+CeCe+Cwtw where = CL[l+max(—2A h,.,o)] 2(l+CL+IAxhxI+AyhyI) = S CL[l+max(—2Ah.,o)] 5 2(l+ ( 1 h +CL a+IA Ah() (5.48) Chapter 5 102 y n e w S Figure 5.5 A Non-staggered Grid cc[1+max(—2A h,o)J e 2(1+f3+IAxhx+fLIAyhyI) = A [1+max(—2Ah,o)J lv The finite difference equation system for i and on grid G”, for convenience, is written as L”U”(x)=F’(x) B”U’(x)=”(x) (5.49) XEJG (5.50) Chapter 5 103 The NBCFAS multi-grid procedure is now used to obtain the solution of Equations (5.49) and (5.50). As an example, the two grid NBCFAS multi-grid procedure is briefly described in the following: 1. Solve Equations (5.49) and (5.50) approximately, using a Gauss-Seidel iteratio n method to obtain U” and calculate the defect d” on the fine grid G”. 2. Restrict U” and d” to get U” and d” 1 , on the coarse grid G . 1 3. Solve on the coarse grid for U’ and V”’. 4. Prolongate V’ to get V” on the fine grid G” and then U”. 5. Repeat the process until the solution converges. In the above multi-grid procedure, a 9-point full weight restriction operator and a bilinea r interpolation prolongation operator are used. The point Gauss-Seidel iteration is used as a smoothing operator. The above proposed multi-grid solution procedure is then applied to solve the following cavity flow problems: 1. Re= l00(Re= U0L)with65x6sgrids 2. Re = 200 with 65 x 65 grids. 3. Re = 500 with 65 x 65 grids. 4. Re = 1000 with 65 x 65 grids. The results obtained are depicted in Figure 5.6, which agree very well with the results in the literature. The performance of the NBCFAS solution procedure is compared with the single grid solution procedure for each case and is presented in Figure 5.7. The comparison between the NBCFAS multi-grid method and the single grid method shows that the present NBCFAS multi-grid procedure is very much faster. In general, it is 15 to 40 times faster. Therefore it is very promising in the numerical simulation of real flow problems. It is interesting to note that application of the FAS procedure to the numerical solution of cavity flow using stream function and vorticity led to a divergent process. Chapter 5 104 F B 0 C B A 9 8 7 6 5 4 3 2 I -00064 -0.0128 -0.0193 -0.0257 -0.0321 -0.0385 -0.0449 -0.0514 -0.0578 -0.0642 -0.0706 -0.0771 -01)835 -0.0899 -0.0963 Figure 5.6 (a) Streamline Contours of Cavity Flow at Re = 100 F E O C B A 9 8 7 6 5 4 3 2 1 -0.0064 -0.0129 -0.0194 -0.0269 -0.0324 -0.0389 -0.0453 -0.0518 -0.0583 -0.0648 -0.0713 -0.0778 -0.0843 -0.0908 -0.0973 Figure 5.6 (b) Streamline Contours of Cavity Flow at Re = 200 Chapter 5 105 F E D C 8 A 9 8 7 6 5 4 3 2 1 -0.0053 -0.0114 -0.0175 -0.0236 -0.0297 -0.0358 -0.0419 -0.0480 -0.0541 -0.0602 -0.0662 -0.0723 -0.0784 -0.0845 -0.0906 Figure 5.6 (c) Streamline Contou rs of Cavity Flow at Re F E O C B A 9 8 7 6 5 4 3 2 1 Figure 5.6 (d) Streamline Contours of Cavity Flow at = 500 -0.0049 -0.0113 -0.0177 -0.0241 -0.0305 -0.0369 -0.0434 0.0498 -0.0562 -0.0626 -0.0690 -0.0754 -0.0818 -0.0882 -0.0946 Re = 1000 Chapter 5 106 Rm 1 o° I 02 NBCFAS SGM —0-1 o° 102 10’ 100 10•’ 10.2 0 500 1000 1500 wu 2000 Figure 5.7 (a) Convergence Path for Cavity Flow at Re = 100 15 i0 —o-_ NBCFAS SGM —0— 102 102 10’ 10° 1 0’ 10.2 14 1o io 500 1000 wJ 1500 2000 Figure 5.7 (b) Convergence Path for Cavity Flow at Re = 200 107 Chapter 5 R,, —0—- NBCFAS SGM 10 102 10’ 100 10’ wu 0 500 1500 1000 2000 Figure 5.7 (c) Convergence Path for Cavity Flow at Re = 500 102 i0 —0-— NBCFAS SGM 102 OCCD33 102 101 100 10.2 1o 500 1000 1500 2000 Figure 5.7 (d) Convergence Path for Cavity Flow at Re = 1000 108 Chapter 5 5.5.2 Numerical Simulation of a Turbulent Boundary Layer To demonstrate the most significant advantage of the NBCFAS algorithm we now apply it to the numerical solution of a turbulent boundary layer flow. The turbulent boundary layer flow under consideration is assumed to be two-dimensional, steady, and incompressible, flow of a Newtonian Fluid with constant properties, which is described by the time averaged continuity equation and Navier-Stokes equations. The Reynolds stress resulting from the averaging process is modeled using Bousnesq’s eddy viscosity concept, and the turbulent viscosity is expressed in terms of turbulent kinetic energy and its dissipation rate. To close the above problem, the k — E turbulence model is used. The wall function method of Launder and Spalding (1974) is used to relate the surface boundary conditions to the near-wall points and then provide a set of difference equations for the near-wall points. As discussed before, the governing equations are discretized using the control volume method on a staggered grid. The resulting difference equations on grid & are again written as: LU(x)=F(x) xeG (5.51) x)=(I)”(x) B’U’ ( 2 xEaG” (5.52) The NBCFAS multi-grid algorithm is used to perform the numerical simulation. The NBCFAS algorithm for a two grid system is as follows: 1. Solve approximately for procedure on (u,v,p,k,e)’ Ic 2. Calculate the defect dxc, restrict (u,v,p,k,E) with the SIMPLE pressure correction and then calculate 3. Solve approximately for and (u,v,p,k,e)!c to get d’’ and fkl (u, v,p,k,e) with the SIMPLE procedure on 4. Do correction, and repeat the process until it is convergent. Chapter 5 109 For a turbulent flow, however, an additional treatment must be introduced to prevent negative values of k and a. To ensure positive values, any negative coarse grid correction is set to zero before prolongation. The proposed NBCFAS multi-grid solution procedure is then used to solve the L) 0 = U with an turbulent boundary layer flow at Reynolds number Re = io (Re 81 x 81 grid system. The shear stress coefficient obtained is depicted in Figure 5.8, and it compares very well with the results of Schlichting (1979). The performance of the NBCFAS multi-grid solution procedure is again very satisfactory. As shown in Figure 5.9, the NBCFAS multi-grid method reduces the residual of all the above variables to 108 with about 120 multi-grid cycles while the multi-grid method without a special treatment along the boundary falls to converge and the multi-grid method excluding the k and a equation can not decrease the residual down to the required criterion. Therefore, the NBCFAS algorithm appear to be a very suitable candidate for numerical simulation of turbulent flow problems. 5.5 Conclusions The following are the main conclusions of the proposed multi-grid method: 1) The NBCFAS algorithm is capable of producing fast convergent solution for cavity flow and turbulent boundary layer flow. 2) The NBCFAS algorithm can overcome the difficulties resulting from the discrete vorticity boundary treatment and the wall function treatment for turbulent quantities. 3) The NBCFAS algorithm is easy to program and takes the same form for different boundary conditions on the coarse grids. 110 Chapter 5 Cf 0.01 0.009 0.008 Numerical Schlichting 0.007 0.006 0.005 0.004 0.003 0.002 Re 0.001 6.Oxl 0° 4.OxlO° 2.OxlO° 8.OxlO° Figure 5.8 Variation of Turbulent Boundary Layer Shear Stress Coefficient R 102 NBCFAS SGM 101 10° 10-I 102 14 1 .4 io io ioio 0 500 1000 1500 2000 2500 3000 3500 4000 fterations of Finest Grid Figure 5.9 (a) Convergence Path of SGM and NBCFAS for U Momentum Equation •111 Chapter 5 102 101 NBCFAS SGM 100 10.1 0 500 1000 1500 2000 2500 3000 3500 4000 Iterations of Finest Grid Figure 5.9 (b) Convergence Path of SGM and NBCFAS for V Momentum Equation RM 102 NBCFAS SGM 101 100 10.1 10.2 1o io 2000 2500 3000 3500 4000 Iterations of Finest Grid Figure 5.9 (c) Convergence Path of SGM and NBCFAS for Continuity Equation Chapter 5 112 NBCFAS SGM I 01 10.2 1o 1o 4 4 to 7 1o 4 to 0 500 1000 1500 2000 2500 3000 3500 4000 Iterations of Finest Grid Figure 5.9 (d) Convergence Path of SGM and NBCFAS forK Equation 102 101 —a-— 100 NBCFAS SGM io 10.2 4 to 1o 4 0 io 1a 7 io 0 500 1000 1500 2000 2500 3000 3500 4000 Iterations of Finest Grid Figure 5.9 (e) Convergence Path of SGM and NBCFAS fore Equation Chapter 6 Numerical Results and Comparison with Experiments In the previous chapter, a numerical solution procedure has been established and tested by computing a laminar cavity flow and a turbulent boundary layer flow. In this chapter, the numerical solution procedure is used to solve the turbulent flow and heat transfer associated with film cooling. First, the numerical simulation of a turbulent thermal boundary layer with uniform blowing is conducted. The turbulent boundary layer heat transfer coefficients are presented and compared with the well-known correlation available in literature. Effects of wall blowing on turbulent thermal boundary layer are also discussed. Secondly, numerical simulations of turbulent flow and heat transfer downstream of a normal film cooling injection slot at various film cooling mass flow ratios are performed. Local flow fields near the film cooling injection for both low and high mass flow ratio film cooling are obtained. The separation reattachment lengths for high mass flow ratio film cooling are established from the flow field and compared with the experimental results. Heat transfer coefficients at various film cooling mass flow ratios are obtained and compared with the experimental results. Effects of wall blowing and wall functions on film cooling heat transfer are also presented. 6.1 Numerical Simulation of Turbulent Boundary Layer Heat Transfer Turbulent thermal boundary layers have been studied extensively in the past. Both the theoretical and experimental results are available in the literature. In addition, turbulent 113 Chapter 6 114 heat transfer with film cooling is an extension of the turbulent boundary layer heat transfer, and heat transfer coefficients of the turbulent boundary layer were used in literature as an approximation to those downstream of the film cooling injection slot. For all these reasons, the numerical simulation of turbulent thermal boundary layer is carried out in the following section in order to validate the mathematical model, numerical model, and the computer code, and to provide approximate information for heat transfer with film cooling. 6.1.1 Turbulent Boundary Layer Heat Transfer with Unheated Starting Length Turbulent thermal boundary layer with an unheated starting length occurs in a variety of engineering applications. It has been investigated extensively. Both analytical and experimental studies have resulted in an analytical correlation for the heat transfer coefficient given by Equation (3.1). Figure 6.1 shows the geometry of a turbulent thermal boundary layer with an unheated starting length similar to the case reported in Section 3.3.3 The physical constants describing the turbulent thermal boundary layer are as follows: L=0.565m H=0.2m l=0.260m 0 =lOm/s U i=1.745x10 2 m I s V 0 10mm/s The above thermal boundary layer flow is assumed to be a fully turbulent, incompressible flow with zero pressure gradient and constant properties. The turbulent Prandtl number of the flow is assumed to be 0.9, and the buoyancy effects and viscous heating are neglected for simplicity. At inlet, the temperature is assumed to be uniform. A uniform heat flux and a uniform wall blowing rate are specified along the Chapter 6 115 0 U Figure 6.1 Calculation Domain for Boundary Layer porous plate. Boundary conditions for flow parameters are described in the previous chapter. To simulate the turbulent thermal boundary layer with unheated starting length, the standard k — e turbulence model and the standard wall function of Launder and Spalding (1974) are employed, and the numerical solution procedure described in the previous chapter is used. The non-uniform numerical grid used for calculation is shown in Figure 6.2. In the present numerical experiment, the finest grid used is 196 x 98. A four level multi-grid method is used in the present numerical calculation. The numerical results for the heat transfer Stanton number are presented in Figure 6.3, which shows the excellent agreement between the numerical prediction and the well-known empirical correlation. Figure 6.3 indicates that the difference between the present prediction and the correlation of Scesa (1951) is less than 5%. This numerical experiment demonstrates that the present numerical solution procedure is very effective, and the turbulent model and wall function are suitable for numerical calculation of the turbulent thermal boundary layer. ‘1 0 0 0) I’) CO z z. CO ICD Cl) 0. CD CD c D -‘ CD 0 0 C,’ 0 ci I’, 0 -‘ 0 C 0 C) CD I 0. C 0 w -‘ 0 0 0 CD ci -Il CO C G) -I Cl) -‘ q C’) 0 3 cT 0 CD 0 ci 0 z 0 Cl) CD Cl) —I z CD 9) c) Co a) - C) C-) Chapter 6 117 6.1.2 Effect of Wall Blowing As discussed in Chapter 3, the wall blowing generally reduces the mass/heat transfer coefficients along the porous plate. The previous mass transfer measurements also indicate that the effect of wall blowing is small if is less than 1%. In the following numerical experiment, the effect of wall blowing on the heat transfer coefficient along the porous plate is investigated. To study the effect of wall blowing on heat transfer and to verify the experimental findings, numerical simulations are carried out with wall blowing rates of v,, =0 rn/s and v = 6.85 x i0 rn/s. The numerical heat transfer results are depicted in Figure 6.4, which shows the variation of the heat transfer Stanton number along the porous plate for wall blowing rates of v,, =0 rn/s and v,,, = 6.85 x i0 rn/s respectively. It can be seen that the effects of wall blowing rate of v = 6.85 x iO rn /s on the heat transfer Stanton number is less than 5%. This confirms our previous mass transfer experimental results. Therefore, the mass transfer method is essentially correct for the study of a turbulent thermal boundary layer with a wall blowing rate of v,, = 3 m Is. The suitable 6.85 x i0 wall blowing rate, under which the effect on heat transfer is small, is identified. In other words, the present mass transfer method is fully justified numerically. 6.2 Numerical Simulation of Turbulent Flow and Heat Transfer with Film Cooling Having validated the reliability of the numerical solution method for turbulent thermal boundary layer flow, we now apply the numerical procedure to the prediction of the turbulent flow and heat transfer with film cooling injection. Prediction of the turbulent flow and heat transfer at various film cooling injection rates is one of the main objectives of the present study. The purpose of numerical simulation of film cooling turbulent flow and heat transfer is to obtain detailed information on both the flow field and heat transfer coefficient downstream of the film cooling injection slot, to investigate the effect of wall Chapter 6 118 1o St 1 02 0 xis 5 10 15 20 25 30 35 Figure 6.4 Effect Of Wall Blowing Rate V on Heat Transfer Stanton Number blowing rate on the flow field and heat transfer coefficient so as to determine the validity of the proposed mass transfer method and provide detailed data for comparison with experimental measurements. To the author’s knowledge, very few attempts have been made to simulate the turbulent heat transfer downstream of a normal film cooling injection slot. A numerical study was made by Sinitsin (1988). However, his preliminary numerical heat transfer results failed to capture the features of turbulent heat transfer with flow separation, mainly due to the use of standard wall function and a rather coarse grid system in the numerical simulation. Numerical calculation of turbulent flow with film cooling injection from a normal injection slot was also investigated by Zhou (1989) using the multi-grid technique. Improvement of the flow field prediction was achieved. However, no attempts Chapter 6 119 were made to calculate the heat transfer coefficient downstream of the film coolin g injection slot in the numerical investigation. In the present numerical experiment, numerical calculations are carried out for film cooling turbulent flow and heat transfer with mass flow ratios ranging from 0.05 to 0.5. The numerical simulations are based on the time-averaged continuity equation, Navier Stokes equations and thermal energy equation, in conjunction with the k E turbulence model and a modified wall function, which is described in the previous chapter. Numerical results are obtained using the numerical solution procedure describ ed in Chapter 5. Some details of the numerical calculation and numerical results for turbulent flow and heat transfer are discussed in the following sections. In Section 6.2.1, the grid — arrangement for numerical simulation is described. The performance of the solutio n procedure for turbulent flow and heat transfer with film cooling is presented in Section 6.2.2. Numerical results for the flow field and heat transfer coefficients are compa red with experimental data in Section 6.2.3. The effects of wall blowing and near-w all treatments on the heat transfer coefficient are examined in Section 6.2.4 and Sectio n 6.2.5 respectively. 6.2.1 Grid Structure It is well-known that the grid structure or grid distribution over the solution domai n affects not only the convergence of the numerical solution procedure, but also the accuracy of the numerical solution. One of the major difficulties for the predict ion of heat transfer associated with film cooling is to provide a better resolution for the flow separation bubble downstream of the normal film cooling injection slot. To this end, a non-uniform numerical grid is used to discretize the computational domain . The computational domain of the film cooling problem is shown in Figure 6.5 (a). A schematic of the discretized solution domain for the present film cooling turbulent flow and heat transfer problem is presented in Figure 6.5 (b) and Figure 6.5 (c). In the above grid arrangement, dense grids cover the film cooling slot and the possible separa tion bubble zone. Then the grid expands gradually both upstream and downstream to cover the rest of the solution domain where the flow is more simply structured. This grid Chapter 6 120 Uo O.16m vw ttttttttttttttt—— O.04m - O.254m 6.35mm O.304m Figure 6.5 (a) Calculation Domain for Film Cooling EEEFEEEEEE_ Figure 6.5 (b) Coarsest Grid for Film Cooling Flow and Heat Transfer Chapter 6 121 Figure 6.5 (c) Finest Grid for Film Cooling Flow and Heat Transfer arrangement is expected to reduce the truncation errors associated with the discretization and provide a better resolution for the separation bubble. The present grid arrangement is based on many preliminary studies, and places only two cells in the film cooling slot for the coarsest grid system while sixteen cells are employed for the finest grid system. 6.2.2 Convergence Performance The numerical simulation of turbulent flow and heat transfer associated with film cooling is carried out using a four level multi-grid procedure on a series of non-uniform staggered grids. The convergence path for calculation of film cooling heat transfer at M = 0.3 is presented in Figure 6.6, which indicates the comparison of convergence rate between the single grid method and the proposed multi-grid method. It can be seen from Figure 6.6 that the multi-grid acceleration is effective. The multi-grid procedure, in general, is very much faster than the single grid SIMPLE procedure. However, the linear acceleration is still not observed in the numerical simulation of turbulent flow and heat Chapter 6 122 transfer, and the acceleration for turb ulent flow is not as great as for the laminar flow. This phenomenon has also been observ ed by Leschziner (1988). 6.2.3 Turbulent Flow and Heat Transfer with Film Cooling To investigate the turbulent flow and heat transfer associated with film cooling, a series of numerical experiments have bee n carried out for various film cooling mass flow ratios. Detailed flow fields near film injection slot and heat tran sfer coefficients downstream of the film cooling inje ction slot are obtained for film coo ling mass flow ratios ranging from 0.05 to 0.5. These numerical results are now discussed in the following. Figures 6.7 and Figure 6.8 present the local flow field near the film cooling injection slot. Two different flow patterns are found within the limit of observatio n and present numerical resolution: one for film cooling with low injection mass flow ratios, the other for film cooling with high mass flow injection ratios. It can be seen from Figures 6.7 and 6.8 that the tendency towards separat ion increases with increasing mas s flow ratio. The reattachment length variation with mass flow ratio is illustrated in Fig ure 6.9. The length increases with increasing mass flow ratio. The figure shows good agreem ent between computations and experimental data . From the above numerical experim ents, we can conclude that for low M film cooling, the deviation of the flow pattern from that of turbulent bounda ry layer is small while the flow pattern for high M film cooling is totally different from that of a turbulent boundary layer. The change in flow pattern for high mass flow ratio film cooling definitely has an impact on the hea t transfer coefficient downstream of the film cooling injection slot, which is discussed belo w. Figures 6.10 and 6.11 present the variation in heat transfer Sta nton number downstream of the film cooling inje ction slot with M from 0.05 to 0.5. Again, two different patterns are observed wit hin the limit of observation and pre sent numerical resolution: one for the low M film cooling, the other for high M film coo ling. For low M film cooling, the heat transfer Stanto n number is close to that of a turbulen t boundary layer with an unheated star ting length, and the overall agreement Chapter 6 123 R 102 101 NBCFAS SGM 100 10.1 ioio• 106 io• 1o 0 500 1000 1500 2000 2500 3000 3500 4000 Iterations of Finest Grid Figure 6.6 (a) Convergence Path of SGM and NBCFAS for U Momentum Equation R 101 —0-— 100 NBCFAS SGM 1 .1 14 1o I o- io1o io 0 500 1000 1500 2000 2500 3000 3500 4000 Iterations of Finest Grid Figure 6.6 (b) Convergence Path of SGM and NBCFAS for V Momentum Equation 124 Chapter 6 RM 102 101 —°--- NBCFAS SGM 10.1 10.2 io 1 0 500 1000 1500 2000 2500 3000 3500 4000 Iterations of Finest Grid Figure 6.6 (c) Convergence Path of SGM and NBCFAS for Continuity Equation 102 NBCFAS 101 —ci--- SGM 10° 10.1 io 1O.° 10.6 10 0 500 1000 1500 2000 2500 3000 3500 4000 Iterations of Finest Grid Figure 6.6 (d) Convergence Path of SGM and NBCFAS forK Equation Chapter 6 125 RE 102 101 NBCFAS SGM 10 1 02 io ioio• io ioiO8 0 500 1000 1500 2000 2500 3000 3500 4000 Iterations of Finest Grid Figure 6.6 (e) Convergence Path of SGM and NBCFAS for E Equation 3 3 3- 3- 3- 3- 3-3-3-3-3-333-3-3-3>33- 3- 3- 3- 3- 3- 3- 3-3-3-3-3-3-3-3-3-3-3-3-3-3- 3- 3- 3- 3- 3- 3- 3-333-3-3-3-3-3-3-3-3->>> 3- 3- 3- 3- 3- 3- 3- 3- 3- 3- 3- 3 3-3-3-3-3-33-3-3-3-3-3->)-)-> 3- 3- 3- 3- 3- 3- 3-3>3-3-3-3-3->>> 3- 3-)- 3- 3- >3-)-3-)-)-)-)-)-3-3->> 3- )_ 3- 3- )-)->3->3-3_>>)-)-)- 3- 3- )- 3- 3- 3- 3-3-3-3---)->> 3- 3 3- 3- 3- 3-> 3- 3- 3- 3- -3-3-3->)-)-> 3- 3- 3- 3- — —3- - Figure 6.7 (a) Local Flow Field for Film Cooling at M=O.05 II 0 ‘3 Co o o 0 0 0 C) I- CD 0) 3 -‘ C cc - 2 II Co 0 0 2 -I 0 - 1 0 0 0 I- CD 0) -I C - - - - WWWIIIH w1mmm 0) Ia) II — 10 I-’ IC, I::r Chapter 6 127 .— .—. .— .—.- .— s—.— - , , , > - - a—.. .__-- . -- -,- — =.—---— ,,).- _- .-------- , U.-. —a.-— - > - -- ). —I-- — • ___. — —. Figure 6.8 (b) Local Flow Field for Film Cooling at M=O.5 Xis 6 —.-—— . (Exp.) 1 X X(Num.) 5 4 3- 2- 0 0.0 .1 0.1 I., 0.2 .1,, I, 0.3 0.4 0.5 0.6 Figure 6.9 Reattachment Length of Turbulent Flow wfth Film Cooling -n Co o - cii——--- —--- ::: ::: ::: 3 --- - : : - :zzz ::: oo -- ,, — —— z ::: 1_fl ——--- I i z C g C’, 0 — ° C) 2 C - 1 - ——--S ———-h ——---i —— ——- :z:: —- —— —--- p Co z = 0 0 3 21 I C Z g SQ a - 9) 2 C Ti — —— - - -- -- -- ——--- I - - -- __z:E:: cii o—_ -‘ —-----J —• —- :z:z:z:::: —- —— —— ———-- I) a) -L m -I a) -C C) Chapter 6 129 St 0.01 0.009 0.008 - - - — 0.007 0.006 - 0.005 0.004 0.003 0.002 0.001 xis . 5 10 15 20 25 30 40 35 45 Figure 6.11(a) Stanton Number of Heat Transfer with Film Cooling at M=O.3 0.01 0.009 0.008 St t 0.007 0.006 0.005 0.004 0.003 0.002 xIs 0.001 0 5 10 15 20 25 30 35 40 45 Figure 6.11(b) Stanton Number of Heat Transfer with Film Cooling at M=O.5 Chapter 6 130 between the numerical and experimental results is satisfactory. Both the numerical and the experimental results confirm that the low M film cooling heat transfer can be predicted approximately by the turbulent boundary layer heat transfer as long as there is no significant flow separation downstream of the film cooling injection slot. Figure 6.10 shows that the agreement between the numerical prediction and experimental results is satisfactory, and the turbulent boundary heat transfer can be used to predict the low M film cooling heat transfer. For high M film cooling, both the numerical and experimental results show that the heat transfer Stanton number downstream of the film cooling injection slot is greatly affected by the presence of the leading edge separation bubble. It can be seen from Figure 6.11 that the heat transfer Stanton number for high M film cooling increases from the leading edge of the film cooling injection slot and reaches its local maximum near the reattachment point due to the local turbulence intensity, and then gradually relaxes to the value of a turbulent thermal boundary layer. The profile of film cooling heat transfer Stanton number is typical to that of turbulent heat transfer with flow separation. However, the satisfactory agreement between the experimental results and numerical predictions is achieved only by using the modified wall function. The use of standard wall function leads to underprediction of the heat transfer rate downstream of the film cooling injection location or the heat transfer coefficient in the region of flow separation. 6.2.4 Effect of Blowing Rate There are two alternatives available to simulate experimentally the film cooling heat transfer with zero blowing velocities. The first group of methods uses an analogous mass transfer process with a fluid flowing over a catalytic surface where the fluid undergoes some chemical reaction so that the mass averaged normal velocity at the wall surface is zero (Eckert and Drake 1972). However, experimental difficulties have prevented researchers from using this group of methods. The second group of methods is based on providing a non-zero, but very small mass transfer rate at the surfaces. The present mass transfer experiment belongs to the second group of methods. However, it is difficult to fully investigate the validity of the mass transfer method with the same mass transfer Chapter 6 131 method alone, since the mass transfer rate introduced through the porous plate can not be reduced infinitely. One of the objectives of the numerical simulation of heat transfer with film cooling is to examine the proposed mass method employed in the experimental study. The validity of the present mass transfer method can be studied via the effect of the wall blowing on heat transfer associated with film cooling, which is similar to the investigation of its validity for boundary layers. The effect of the wail blowing on the heat transfer coefficient can be obtained numerically by changing the value of wall blowing rate and examining the effect on the flow and mass transfer. In the present study, two wall blowing rates are used. The numerical heat transfer coefficients of film cooling for wall blowing rates of v =0 mIs and v = 6.85 xl cr m Is are presented in Figures 3 6.12 and Figure 6.13. It can be seen from these figures that (1) the wall blowing is reducing the heat transfer rate along the wall, as expected; (2) the effect of wall blowing on heat transfer coefficient is small when the blowing rate is less than 0.1%. The above investigation confirms that the present mass transfer method can be used to study the film cooling heat transfer coefficient successfully. 6.2.5 Comparison of Near-wall Treatments It is well known that the standard wall function of Launder and Spalding (1974) is based on the Reynolds analogy. As pointed out by Spalding (1969), the Reynolds analogy is no longer valid in the region of flow separation. Therefore, the standard wail function will experience difficulties when applied in the region of separation. To address the above difficulties, various near-wall treatments have been proposed. An extensive review can be fund in the paper by Ciofilo and Collins (1989). Performance of different near-wall treatments have also been investigated by Djilali et. al. (1989). In the present study, both the standard wall function and the modified wall function of Ciofilo and Collins (1989) are used for the prediction of heat transfer with film cooling. The numerical results are then compared with experimental results in Figure 6.14. It shows that the standard wall function is not suitable for flow with separation while the modified wall function produces very promising results. Chapter 6 132 St 0.01 0.009 0.008 - - 0.007 0.006 - - - 0.005 0.004 0.003 0.002- 0.001 0 5 10 15 20 XIS 25 30 35 40 45 Figure 6.12 Effect of V on Heat Transfer of Film Cooling at M=O.1 0.01 St 0.009 0.008 0.007 0.006 u.uu u.vv.t - 0.003 - 0.002 0.001 5 10 15 20 xIs 25 30 35 40 45 Figure 6.13 Effect of V on Heat Transfer of Film Cooling at M=O.3 Chapter 6 133 0.01 0.009 0.008 St 0.007 0.006 0.005 0.004 A 0.003 0.002 0.001 xis - 5 10 15 20 25 30 35 40 45 Figure 6.14 (a) Predictions of Heat Transfer with Film Cooling at M=03 0.01 0.009 0.008 St - 0.007 0.006 -- 0.005 0.004 -- 0.003 0.002 xIs 0.001 5 10 15 20 25 30 35 40 45 Figure 6.14 (b) Predictions of Heat Transfer with Film Cooling at M=O.5 Chapter 7 Conclusions and Recommendations 7.1 Conclusions This thesis presents an experimental and numerical investigation of turbulent mass/heat transfer downstream of a 2D normal film cooling injection slot. Such studies are useful both for gaining basic understanding about mass/heat transfer from turbulent separated flows, and for numerical simulations and design of film cooling systems. Major contributions and conclusions drawn from the present study are summarized in the following list: 1. A novel experimental mass transfer technique has been developed to study the heat transfer coefficient associated with film cooling using the mass/heat transfer analogy. This technique includes measuring the concentration of a contaminated gas along a porous plate using a Flame Ionization Detector, calculating the mass transfer coefficients using a wall function correction formula and relating the mass transfer coefficients to heat transfer coefficients using the mass/heat transfer analogy, which is problem dependent. 2 The proposed mass transfer method can be used to obtain information on heat transfer coefficients downstream of the film cooling injection slot. 3. The wall function correction formula proposed based on the near-wall turbulence structure can be easily applied to obtain the correct Stanton number provided the probe is placed in the viscous sublayer. 134 135 Chapter 7 4. Using different tracer gases, the relation between the mass/heat transfer coefficients for separated flows can be obtained by the present mass transfer 66 whereas for unseparated flows technique. For separated flows St oc (Pr)° St oc . 4 (Pr)° 5. The heat transfer Stanton number for film cooling at low mass flow ratios (M 0.1) is close to that of a turbulent thermal boundary layer. Therefore, the heat transfer of a turbulent boundary layer can be used to predict the film cooling heat transfer for low mass flow ratio film cooling flow without separation. 6. At higher mass flow ratios (M 0.2), the heat transfer coefficient variation is similar to that obtained with a backward facing step, namely, it starts to increase from the leading edge of the separation bubble, attains its local maximum near the reattachment point, and then gradually relaxes to values typical of a turbulent thermal boundary layer. 7. An efficient multi-grid method (NBCFAS) has been proposed for the numerical solution of incompressible flows. This multi-grid algorithm is able to accelerate the convergence rate, and produces accurate numerical results for both laminar and turbulent flows. Essentially the same multi-grid algorithm has also been developed independently and used by He (1994) for the numerical simulation of turbulent flows with complicated geometries. 8. Using the k a turbulence model and a modified wall function, turbulence flow - and heat transfer associated with film cooling are solved with the above proposed multi-grid technique. Good agreement between numerical predictions and measurements is obtained for turbulent heat transfer downstream of a normal 2D injection slot. 9. The modified wall function proposed can be used to predict heat transfer rates in the region of flow separation, while the standard wall function is clearly not suitable for flow with separation. 136 Chapter 7 10. The standard k e turbulence model, together with the modified wall function, - can satisfactorily predict the turbulent flow and heat transfer downstream of the film cooling injection slot. 11. Numerical calculations confirm that the present mass transfer method is adequate for the study of heat transfer with film cooling provided that the blowing rate is limited to values around 0.1% of the free stream velocity. 12. Both the experimental study and the numerical simulation indicate that a region of separated flow exists downstream of the film cooling injection slot for high mass flow ratio film cooling (M 0.2). The separation length obtained from the mass transfer measurement compares very well with the numerical predictions. The Reynolds analogy is no longer valid in the region of separation. The assumption that mass/heat transfer coefficients reach their maxima near the a reattachment point appears to be approximately correct. 7.2 Recommendations 1. Extend the present experimental mass transfer technique to study the heat transfer associated with other injection geometries in film cooling such as tangential or near tangential slots, angled slots, multiple slots and rows of discrete holes. 2. Extend the proposed numerical procedure to investigate the turbulent flow and heat transfer near various injection geometries in film cooling. 3. Employ the present numerical and experimental method to perform an extensive experimental and numerical study of the mass/heat transfer for separated flows (e.g. for turbulent flow with a backward facing step). 4. 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R., “Heat Transfer with Film Cooling near Nontangential Injection Slots,” Journal of Engineering for Power, Trans. ASME, Series A, Vol. 90, pp. 157-163, 1968. Metzger, D. E., Takeuchi, D. I., and Kuenstler, P. A., “Effectiveness and Heat Transfer with Full-Coverage Film Cooling,” Journal of Engineering for Power, Trans. ASME, pp. 180-184, 1973. References 145 Miller, T. F., and Schmidt, F. W., “Evaluation of a Multilevel Technique Applied to the Poisson and Navier-Stokes Equations,” Numerical Heat Transfer, Vol. 13, pp. 126, 1988. Ota, T., and Kon, N., “Heat Transfer in Separated and Reattached Flow on a Blunt Flat Plate,” Journal of Heat Transfer, Vol. 96, pp. 459-462, 1974. Ota, T., and Kon, N., “Heat Transfer in the Separated and Reattached Flow over Blunt Flat Plates- Effects of Nose Shape,” mt. J. Heat Mass Transfer, Vol. 22, pp. 197- 206, 1979. Pai, B. R., and Whitelaw, I. H., “The Prediction of Wall Temperature in the Presence of Film Cooling,” hit. J. Heat Mass Transfer, Vol. 14, pp. 409-425, 1971. Patankar, S. V., and Spalding, D. B., “Heat and Mass Transfer in Boundary Layer Flows,” Morgan-Grampian Press, 1967. Patankar, S. V., and Spalding, D. B., “A Calculation Procedure for Heat, Mass and Momentum Transfer in Three Dimensional Parabolic Flows,” Tnt. J. Heat Mass Transfer, Vol. 15, pp. 1787-1806, 1972. Patankar, S. V., Rastogi, A. K., and Whitelaw, J. H., “The Effectiveness of ThreeDimensional Film-Cooling Slots-il. Predictions,” Vol. 16. pp. 1673-1681, 1973. Patankar, S.V., “Numerical Heat Transfer and Fluid Flow,” Hemisphere, Washington, D.C., 1980. Peric, M., Ruger, M., and Scheuerer, G., “A Finite-Volume Multigrid Method for Calculating Turbulent Flows”, 7th Symp. Turbulent Shear Flow, Stanford University, California, pp. 7.3.1-7.3.6, 1989. Perry, J., “Chemical Engineers’ Handbook”, 5th edition, New York, McGraw-Hill, 1973. 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M., “Inviscid Flow From a Slot into a Cross Stream”, Ph.D. Thesis, University of British Columbia, 1993. 148 References Sun, Y., and Militzer, J., “The Piecewise Parabolic Finite Analytic Method. Part I: Theory,” Appi. Math. Modelling, Vol. 16, PP. 576-587, 1992. Thompson, M. C., and Ferziger, J. H., “An Adaptive Multigrid Technique for the Incompressible Navier-Stokes Equations”, J. Compt. Phys., Vol. 82, pp. 92-121, 1989. Tse, D. C. M., “Multiple Jet Interactions with Special Relevance to Recovery Boilers,” Ph.D. Thesis, University of British Columbia, 1993. Tzanos, C. P., “Higher-Order Differencing Method with a Multigrid Approach for the Solution of the Incompressible Flow Equations at High Reynolds Numbers,” Numerical Heat Transfer, Vol. 22, pp. 179-198, 1992. Vanka, S. P., “Block-Implicit Multigrid Solution of the Navier-Stokes Equations in Primitive Variables”, J. Compt. Phys., Vol. 65, pp. 138-158, 1986. Yokota, J. W., “Diagonally Inverted Lower-Upper Factored Implicit Multigrid Scheme for the Three-Dimensional Navier-Stokes Equations”, AIAA Journal, Vol. 28, No. 9, pp. 1642-1649, 1990. Vogel, J. C., and Eaton, 3. K., “Combined Heat Transfer and Fluid Dynamic Measurements Downstream of a Backward-facing Step,” Journal of Heat Transfer, Trans. ASME, Vol. 107, pp. 922-929, 1985. Zemanic, P. R., and Dougall, R. S., “Local Heat Transfer Downstream of an Abrupt Circular Channel Expansion,” Journal of Heat Transfer, Trans. ASME, Series C, Vol. 92, pp. 53-62, 1970. Zhang, K. G., “A Private Communication,” 1993. Zhou, J. M., “A Multi-grid Method for Computation of Film Cooling’, M.S. Thesis, University of British Columbia, 1990. Appendix A Diffusion Coefficient of Air-Propane The diffusion coefficient of air-propane mixture could be measured experimentally. In the present study, the diffusivity of air-propane mixture is calculated using Fuller’s Formula (Perry, 1973): 75 0.OOlxT’ I_L+_L VMM G - P[(v) +(v)] where 2 Is [DGJ=m T=293°K 1 = Mair = 28.96 M = Mpropane = 44.09 P=labn 1 (v) =(V)air =20.1 2 =(V)propane = 65.34 (v) Then we have DG =0.1076[cm2/sJ (A.2) The corresponding mass transfer Schmidt number is given by: 1.512 u 1.4 Gair_propane 1.076 iE 149 (A.3) Appendix B Diffusion Coefficient of Air-Methane The diffusion coefficient of air-methane mixture can be calculated using the following formula based on the kinetic gas theory (Perry, 1973): BT 2/3 I 1 1 MM i—+—— ‘12’D where {DG 2 Is j = cm (difusivity) T=293°K 1 =Mair =28.96 M 2 = Mte = 16.04 M P=latm = 0.4972 B=(10.85—2.50 1 1IM 3 =1.075x10 2 P4 2[() Notice that ‘ir =3.617[angstromsj ,) 1 =3.882[angstromsj (,,) 2 =(r propane Then we have = 3.7495[angstroms] 150 APPENDIX B 151 and DG =0.23[cm2ls] (B.2) The corresponding Schmidt number of mass transfer is: —v--- aair_propane 0.66 (B.3) The diffusion coefficient of air-methane mixture can also be calculated using Fuller’s Formula (Perry, 1973), namely: 0.OOlxT Ii 1.75 VMI = 1 2 M ‘—+---— P[(v) +(v)] where 2 Is [DGJ=m T=293°K 1 (v) =(V)ajr = 20.1 2 =(V)tne (v) =23 P=latm Then we have DG =0.206[cm2 is] (B.4) The mass transfer Schmidt number is given by the following: air-methane = 0.73 (B.5) The difference in calculated diffusion coefficients of air-methane mixture is due to the fact that the uncertainty of both Equation (A.1) and Equation (B.1) is about 10%. Therefore, an average value of the above diffusion coefficients or Schmidt numbers is used. Appendix C Experimental Procedures The following procedures are followed in the present experiment to ensure the accuracy and reliability of the measurements. C.1 Velocity Measurement The velocity measurements are carried out using a DISA hot wire anemometer immediately after the calibration. The velocity measurement procedure is outlined in the following: 1. Turn on the wind tunnel and hot wire anemometer. Allow them to warm up for about 1 hour. 2. Calibrate the hot wire system against the Pitot-static tube measurements to create the looking-up table for velocity measurement. 3. Check the calibration of the hot wire system against the standard Pitot-static result and calibrate the system again if the comparison is not satisfied. Repeat until the calibration is satisfied. 4. Check the wind tunnel speed to make sure that the free stream velocity is kept at 10 rn/s. 5. Position the hot wire probe to a near-wall position P, and measure the mean velocity. Move the hot wire probe up to required positions to measure the velocity profile 6. across the whole boundary layer. Check the free stream velocity constantly to make sure it is kept at 10 m/s. 152 APPENDIX C C.2 153 Mass Transfer Measurement The mass transfer measurements are conducted using FID. The following procedure for calibration and measurement is practiced: 1. Turn on the wind tunnel and ignite the FID. 2. Adjust the hydrogen flowrate, and the compressed air flow rate to the required amounts. 3. Let the FID system warm up for about two hours. 4. Check the hydrogen and compressed air flow rates to make sure that they have been stabilized. 5. Calibrate the FID system by taking the 0% concentration measurement for all the sampling fine tubes and repeat the 0% concentration measurement for about 10 to 20 times to get a stabilized result. 6. Add a certain amount of contaminated gas and a given amount air to the rectangular box. 7. Check the contaminated gas and air flow rates, allow them to stabilize. 8. Place the sampling tubes on the porous plate and cover the tubes within a sealed box. 9. Calibrate the FID system by taking the 100% concentration measurement for all sampling tubes and repeat the 100% measurement until the variation is less than 1%. 10. Set the free stream velocity to 10 mIs, the film cooling injection rate at the required value and check the hydrogen, air, compressed air, contaminated gas flow rates to ensure that they have been stabilized. 11. Position the sampling tubes at the required locations. 12. Take the concentration measurement when all the flow rates are stable. 13. Change the film cooling injection rate from low M to high M, repeat step 11 and 12 until all the measurements are completed.
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An experimental and numerical investigation of heat transfer downstream of a normal film cooling injection… Sun, Yuping 1995
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Title | An experimental and numerical investigation of heat transfer downstream of a normal film cooling injection slot |
Creator |
Sun, Yuping |
Date Issued | 1995 |
Description | This thesis presents an experimental and numerical investigation of turbulent heat transfer associated with film cooling from a 2D slot. To determine the film cooling heat transfer coefficient, detailed mass transfer measurements have been carried out downstream of a normal film cooling injection slot. The plate downstream of the injection location is porous and air contaminated with propane or methane is bled through the plate beneath it. By measuring the propane or methane concentration very close to the wall, mass transfer measurements are conducted for film cooling mass flow ratios ranging from zero to 0.5. The mass transfer coefficients are calculated using a wall function correction formula, which corrects the measurements for displacement from the surface, and are then related directly to corresponding heat transfer coefficients using the mass/heat transfer analogy. The validity of the method and the wall function correction formula are checked by examining the case with zero film coolant injection, a situation analogous to the well-known turbulent boundary layer mass/heat transfer with impermeable/unheated starting length. Good agreement with data in the literature is obtained for this experiment. For film cooling with low mass flow ratios (M 0.1), heat transfer coefficients close to those of a conventional turbulent boundary layer are obtained. At high values of mass flow ratio (M > 0.1) heat transfer coefficients similar to those of turbulent separated flows are observed, reflecting the important effect of the separation bubble just downstream of injection. Numerical calculations of turbulent flow and heat transfer associated with film cooling have been carried out to validate the present mass transfer method and to ensure that the heat transfer coefficients obtained using the mass/heat transfer analogy are valid. In the present study, the time-averaged continuity equation, Navier Stokes equations and thermal energy equation, together with the k-ε a turbulence model, are used to describe the turbulent flow and heat transfer. This system of governing equations is discretized using the control volume method. To obtain the numerical solution of the resulting finite difference equations, a novel multi-grid method with no boundary correction is proposed. This method differs from the FAS multi-grid method in that it employs a special multi-grid procedure to ensure that the correction on the coarse grids vanishes along the boundary, and it offers an easier procedure to overcome difficulties arising from the boundary treatment in both discretization and the multi-grid algorithm. The performance of the method is examined using the laminar cavity flow and the turbulent boundary layer flow, which indicates that it is efficient and can overcome difficulties resulting from the near-wall treatment and the discrete vorticity boundary condition. Numerical simulation of turbulent flow and heat transfer associated with film cooling are conducted using the proposed multigrid method. Detailed turbulent flow field and heat transfer results are obtained for film cooling mass flow ratios ranging from zero to 0.5. The numerical results show that the standard k-ε a turbulence model, together with the standard wall function, underpredicts considerably the heat transfer coefficients of film cooling at high mass flow ratio, and therefore, is not appropriate for accurate prediction of turbulent heat transfer in the region of flow separation. However, with a modified near-wall treatment, predictions of heat transfer coefficients compare well with the experimental results obtained using the mass/heat transfer analogy. Numerical calculations also show that the overall effect of the wall transpiration for the blowing rates used for the mass transfer measurements on heat transfer is negligible. The present study finally concludes that both the numerical and mass transfer method can be used to investigate film cooling heat transfer with fair accuracy. |
Extent | 3656008 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-06-11 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0088918 |
URI | http://hdl.handle.net/2429/8977 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1995-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
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