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Experimental investigation of the tolerant wind tunnel for unsteady airfoil motion testing Kong, Lingzhe 1991

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E X P E R I M E N T A L I N V E S T I G A T I O N O F T H E T O L E R A N T W I N D T U N N E L F O R U N S T E A D Y A I R F O I L M O T I O N T E S T I N G By LINGZHE K O N G B. Sc. (Mechanical Engineering) East China Institute of Technology M . Eng. (Flight Mechanics & Engine) East China Institute of Technology A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S M E C H A N I C A L E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A April 1991 © LINGZHE K O N G , 1991 In presenting this thesis in partial fulfilment 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. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract Previously, the concept of the tolerant wind tunnel, developed in the Department of Mechanical Engineering, U . B. C , was tested only for stationary models. In the present study, the concept is investigated for unsteady airfoil motion. The new wind tunnel test section, using the opposite effects of solid and open boundaries, is a new approach to reduce wall blockage effects. Consisting of vertical airfoil slats uniformly spaced on both side walls in the test section, it is designed to produce a nearly free-air test environment for the test model, which leads to negligible or small corrections to the experimental results. The performance of this wind tunnel for unsteady model testing is examined ex-perimentally with a two-dimensional N A C A 0015 airfoil in a simple plunging sinusoidal motion. The airfoil is mounted vertically in the center plane of the test section between solid ceiling and floor. An oscillating table is designed to give the airfoil an accurate plunging sinusoidal motion. A full range of open area ratio is tested by varying the num-ber of slats mounted inside the side walls. Pressure distribution along the airfoil surface and displacement of the airfoil are measured as functions of time by a data acquisition system designed for this research. Lift and moment are obtained by integration of the pressure distribution at every time increment. Using a numerical model based on the singularity distribution method, the free air case results for a N A C A 0015 airfoil in the same unsteady motion are obtained. Comparison with the linear theory results by Sears' 1 ' are discussed. Comparing with the numerical and the linear theory results, the experimental investigation shows that the new test section produces low-correction data. ii Table of Contents Abstract ii List of Figures v List of Symbols viii Acknowledgement x 1 Preliminary Remarks 1 2 Introduction of the Theory 4 3 Potential Flow Theory 9 3.1 Assembling Basic Equations of the Theory 9 3.2 Procedures to Solve the Set of Equations 27 3.3 Method of Tracking the Wake Vortex Shape 28 3.4 Pressure, Force and Moment on the Airfoil 33 4 Results of the Theory 37 4.1 Wake Vortex Shedding Shape 37 4.2 Pressure, Force and Moment on N A C A 0015 Airfoil 38 5 Experiment Arrangement 44 5.1 Introduction to the Experiment 44 5.2 Experiment Apparatus and Equipment 45 iii 6 Experiment Results T Concluding Comments Bibliography Appendices A Instrumentation B Digital Low-Pass Filter C Linearized Theory List of Figures 4.1 Shape of the vortex wake generated by N A C A 0015 airfoil oscillating in simple harmonic manner, numerical results (with vortex decay) compared with the experimental results 40 4.2 Shape of the vortex wake generated by N A C A 0015 airfoil oscillating in simple harmonic manner, without vortex decay 41 4.3 Shape of the vortex wake generated by N A C A 0015 airfoil oscillating in simple harmonic manner, with 10% vortex decay 42 4.4 Lift and moment coefficients of oscillating N A C A 0015 airfoil as function of the reduced frequency /x, compared with the linear theory results for V0 = 0.01 43 5.5 U . B . C . Mechanical Engineering low speed closed circuit 'green' wind tunnel 47 5.6 Double slotted wall test section designed for the 'green' wind tunnel . . . 47 5.7 Photograph and the basic working principle of the oscillating table . . . . 48 5.8 Displacement and velocity of the oscillating table as function of time . . 49 5.9 Pressure attenuation as function of tube length and frequency (tube di-ameter = 0.066 in.) 50 5.10 Calibration plots for Barocel pressure transducer 51 6.11 Pressure tap positions on N A C A 0015 airfoil surface 56 6.12 Pressure coeff. of oscillating N A C A 0015 airfoil as function of time at different tap positions, experimental results for fi — 0.52, V0 = 0.0054 and OAR=0.526 57 v 6.13 Continued: Pressure coeff. of oscillating NACA 0015 airfoil as function of time at different tap positions, experimental results for \i = 0.52, V0 = 0.0054 and OAR=0.526 58 6.14 Continued: Pressure coeff. of oscillating NACA 0015 airfoil as function of time at different tap positions, experimental results for fi = 0.52, V0 = 0.0054 and OAR=0.526 59 6.15 Pressure coeff. of oscillating NACA 0015 airfoil as function of time at different tap positions, experimental results for /z = 0.52, V0 = 0.0054 and OAR=0.708 60 6.16 Pressure coeff. of oscillating NACA 0015 airfoil as function of time at different tap positions, experimental results for fi = 0.52, V0 = 0.0054 and OAR=0.000 61 6.17 Lift coeff. of oscillating NACA 0015 airfoil as function of time, experimen-tal results for fi = 0.52, V0 = 0.0054 at OAR=0.0 and 0.526 62 6.18 Lift coeff. magnitude of oscillating NACA 0015 airfoil as function of OAR, experimental results for fi = 0.52 and V0 = 0.0054 63 6.19 Moment coeff. magnitude of oscillating NACA 0015 airfoil as function of OAR, experimental results for fi = 0.52 and V0 = 0.0054 64 6.20 Cptiower-CplUppeT as function of X along the airfoil chord, experimental re-sults for fi = 0.52 and V0 = 0.0054, compared with the linear theory result 65 6.21 Lift coeff. of oscillating NACA 0015 airfoil as function of time, experimen-tal results for added mass term, circulation part at OAR=0.0 and 0.526 for n = 0.52 and V0 = 0.0054 66 6.22 Lift coeff. magnitude of oscillating NACA 0015 airfoil as function of OAR, experimental results for fi = 0.52 and V0 = 0.0054, circulation part . . . . 67 vi 6.23 Moment coeff. of oscillating N A C A 0015 airfoil as function of time, ex-perimental results for added mass term, total moment coeff. at OAR=0.0 and 0.526 for /x = 0.52 and V0 = 0.0054 68 6.24 Lift coeff. magnitude of oscillating N A C A 0015 airfoil as function of OAR, experimental results for // = 0.345 and V0 = 0.0075 69 6.25 Moment coeff. magnitude of oscillating N A C A 0015 airfoil as function of O A R , experimental results for fi = 0.345 and V0 — 0.0075 70 6.26 CptioweT-CPiupper as function of X along the airfoil chord, experimental re-sults for fi = 0.345 and V0 = 0.0075, compared with the linear theory result 71 6.27 Lift coeff. magnitude of oscillating N A C A 0015 airfoil as function of O A R , experimental results for \i = 0.52 and VQ = 0.01082 72 6.28 Moment coeff. magnitude of oscillating N A C A 0015 airfoil as function of O A R , experimental results for /z = 0.52 and V0 = 0.01082 73 6.29 Lift coeff. magnitude of oscillating N A C A 0015 airfoil as function of OAR, experimental results for ft = 0.345 and V0 = 0.0037 74 6.30 Moment coeff. magnitude of oscillating N A C A 0015 airfoil as function of O A R , experimental results for ft — 0.345 and V0 = 0.0037 75 vii List of Symbols n+l O A R Open Area Ratio : OAR = J { X a i ) = 1 ~ T T c Chord of the airfoil-shaped slats mounted on side walls n Number of airfoil slats a, Size of the slots in tunnel test section side walls L Length of the tunnel test section C Chord of the test airfoil model Uoo Uniform freestream velocity Vy unsteady motion of the airfoil, Vv = A sin(u t) A Amplitude of Vv u Circular frequency of airfoil oscillation Tjt Circulation about the airfoil at time tk (cr,)jt Source (or sink) strength at ith element on the airfoil surface at time tk 7/t Vorticity on the airfoil surface at time tk 6k Wake vortex element inclined angle at time tk Ajt Wake vortex element length at time tk (lw)k Vorticity on the wake vortex element at time viii (Vani)k Normal component of the airfoil unsteady motion at iih control point (Vni)k Normal velocity at ith element control point at time tk (Vti)k Tangential velocity at ith element control point at time tk (V^)jt Tangential velocity at ith element control point, excluding its own effect (<f>i)k Velocity potential at ith control point at time tk (Cpi)k Pressure coeff. at ith control point at time tk (Ci)k Lift coeff. at time tk (Cmo)k Moment coeff. to the center of the airfoil at time tk ' fi Non-dimensional frequency, ft — JJJ^ V0 Non-dimensional amplitude of the forced vibration, V0 = Asi Element length on the airfoil surface X{, j/j Coordinates of element end points xpi, ypi Coordinates of element control point Cis quasi-steady lift coefficient Cmos quasi-steady moment coefficient to the center of the airfoil ix Acknowledgement I would like to thank Dr. G.V. Parkinson for his strict supervision and guidance as well as financial support to me throughout this research. I would also like to thank Dr. I.S. Gartshore for his constant help and encouragement. Special thanks to T.F . Besic, technician in the Mechanical Engineering Workshop for his technical assistance in the design and construction of the oscillating table. x Chapter 1 Preliminary Remarks In recent years, due to the fast development of digital computers, computational aero-dynamics has become a main tool for aircraft design. However, the wind tunnel is still the primary means for verification and validation of the results before the first flight test of a prototype. Unfortunately, the presence of solid walls in conventional wind tunnels imposes a flow boundary condition not existing in real unbounded flow. Due to the wall blockage effect, flow speed changes around the model. This gives the model a loading generally higher than the one measured in an unconstrained flow. On the other hand, for an open jet test section, the flow has a sudden expansion to the static atmosphere from upstream. Because of this, the flow also experiences some distortions in the test section. Usually, the model will experience a lower loading than it would in the free-air condition. Although the problem of wall constraint was recognized a very long time ago and some correction theories have been developed, these theories are usually only satisfactory when the test model is relatively small compared with the test section. Today's aircraft design-ers and industrial aerodynamicists are demanding that wind tunnels should accommodate testing of large models. However, the use of large models (or unsteady model testing) in wind tunnels may make the wall corrections become unacceptably large unless the wind tunnel has a very large test section. But this kind of over-size wind tunnel usually is too expensive to build and operate. An alternative is to use some less expensive way to create 1 Chapter 1. Preliminary Remarks 2 an environment in the test section in which boundary corrections are kept small and may be negligible. Basically, two techniques have been used in modifying the conventional wind tunnel, the active and passive wall test sections. The former uses continuously adjusted boundary conditions at the wall to make the test section streamlines approach the free-air pattern. Practically, this is achieved by either deflecting solid flexible walls or using suction and blowing through porous walls. This will require costly equipment. The passive wall idea is a concept of using ventilated walls. The concept arises from the fact that the boundary corrections are of opposite sign in solid wall and open jet test sections. So, a test section with partly solid, partly open boundaries could in principle be designed to give free-air conditions and correction-free data. So far, three kinds of walls have been tried: porous, longitudinal slots and transversely slotted walls. Parkinson and Lim^ attributed the lack of success of longitudinal slots and porous walls in low-speed tunnel tests to the occurrence of separated flows in the slots and holes. Williams^, using a transversely slotted wall opposite to the suction side of the test airfoil and a solid wall opposite to the pressure side, developed another low-correction test section. Known as the tolerant wind tunnel, the slotted wall consists of a row of uniformly spaced symmetrical airfoil slats at zero incidence. No separation flow would occur at the slats. Using two-dimensional airfoils as the test models, Williams^ proved that if the slats were properly spaced, the overall streamline pattern can closely approximate the corresponding free-air flow. Following Williams'^ work, Hameury^ used this concept for two-dimensional bluff Chapter 1. Preliminary Remarks 3 body testing by employing two airfoil-slatted walls opposite the test model. Satisfactory results for the unsteady wake flow behind bluff bodies were obtained at some open area ratios. The purpose of this work is to investigate the possibility of using Hameury's^ con-figuration of the tolerant wind tunnel for unsteady airfoil motion testing, to create a free-air condition in which the unsteady airfoil would experience the corresponding free-air loading. For comparing with the experimental work, the following chapter uses a numerical model based on the singularity distribution method to obtain the free-air loading for a plunging N A C A 0015 airfoil. Comparisons with the linear theory results by Sears^ are discussed. The outline of the linear theory is given in Appendix C. Chapter 2 Introduction of the Theory For calculating the pressure distribution on an oscillating airfoil, a surface singularity distribution method can be easily used. The ones which are most widely used employ sources and sinks on the airfoil surface combined with a vorticity distribution to generate circulation. Such a theory is used here to calculate the loading on the airfoil when it executes an unsteady motion. The theory is a modified version of a method proposed by Basu and Hancock^. For this theory to be valid, it must be assumed that the amplitude of the airfoil unsteady velocity is small compared with the mean flow Uoo. One particular problem with the calculation of unsteady airfoil motion is the Kutta-Joukowski condition. According to Maskelfl6\ this can be expressed as the condition of both finite velocity and zero loading about the trailing edge. The appropriate solution should satisfy the condition in the neighbourhood of the trailing edge. In potential flow theory, the total circulation around the airfoil and wake must be al-ways zero. Let Tfc to be the airfoil circulation at time tk- When the airfoil has an unsteady motion relative to the steady incompressible inviscid flow, circulation of strength — ^ 6t is shed after the airfoil in time 6t and this shed vorticity is then convected downstream. T(t) circulation about airfoil 4 Chapter 2. Introduction of the Theory 5 For an airfoil undergoing time-dependent motion started at t = 0, its solution is calculated at successive time intervals, tk (k = 1,2,3,...). At time tk, the numerical model is shown as: element i source distribution (<Ti)k vortidty distribution (fi)k Wake vortex element: At time tk, the airfoil contour is replaced by N straight line elements. The control point was chosen at the middle point of each element. A uniform source distribution (o~i)k and a uniform vorticity distribution {?fi)h are placed on the ith element (t = 1,2,..., N). (o~i)k varies from element to element and [ji)k is the same for all elements of the airfoil. Subscript k refers to time tk-Tk = 7* x (air foil perimeter) A small straight line wake element of length A* and inclined at an angle 8k to the OX Chapter 2. Introduction of the Theory 6 axis (i.e. free stream direction) is attached to the trailing edge. A * and 6\ are arbitrary at this stage. Their values are determined as part of the solution. The vorticity on the trailing edge wake element is (iw)k : A f c (7„) f c = - ( r f c - r f c _ 1 ) Tk-i is known at time tk-A downstream wake of concentrated vortices is formed from the vorticity shed at earlier times, which is assumed to be concentrated into discrete vortices and convected according to the resultant velocity calculated at the center of each vortex at each suc-cessive time interval. A l l these concentrated vortices are regarded as known at time tk-At time tk, there are N + Z unknowns (<x;)fc(i = 1,2,..., N), 7^ , A ^ and Ok- The basic set of N + 3 equations are formulated. Refer to Basu and Hancock^. (1) N conditions of the normal velocity component at each control point equals to the normal component of the airfoil unsteady motion at that point: (Vnj)k = (Vanj)k (2.1) (i = 1,2,...,//) {Vanj)k is the normal component of the airfoil unsteady motion at ith control point at time tk. (2) Kutta-Joukowski condition: It can be expressed as equal pressures at the control points of the two elements on either side of the trailing edge. Using the unsteady Bernoulli equation, it gives: (Vtz+1)l = (Vtz)l + 2 { l k - T k ^ (2.2) 2 2 (tk — tk-i) Chapter 2. Introduction of the Theory 7 (Vtiz+1)k and (VtE.)k axe the resultant velocities at elements y and y + 1 at time (3) Length and orientation of the trailing edge wake element, A * and 0* are determined from the condition that the element is tangential to the local resultant velocity and its length is proportional to the local resultant velocity (V?)*. (V '^)fc is the resultant velocity excluding its own effect, to be distinguished from (V*)*. Ak = (Vt')k x At (2.3) (Vn)k = 0 (2.4) (Vn)k is the total normal velocity induced at the control point of the trailing edge wake element. If Aij and Bij are the appropriate influence coefficients, the tangential and normal velocities at the control point of the ith element due to the distribution of sources and vortices on the airfoil are: (^)i = E ^ - f o ) * + 7 * Z ; * « <2-5) 3=1 3=1 (Vni)l = £ Byte)*-ikj^Aii (2.6) j=l j=l The perturbation velocity components induced at the control point of the ith element by the small vortex element attached to the trailing edge can be expressed: (Vti)2k = MkBitN+1 (2.7) (Vm)l = -MkA,N+1 (2.8) Due to the concentrated wake vortices r* T O: (Vu)l = £ ^ J * m (2.9) m=l (vni)l = £ Btmrm (2.io) m=l Chapter 2. Introduction of the Theory 8 Here A*im and B?m are the appropriate influence coefficients. The total perturbation velocity is the sum of all components given in (2.5)-(2.10). This total perturbation velocity is superimposed on the mainstream component and substituted into the basic equations (2.1)-(2.4). There are altogether N + 3 equations and N + 3 unknowns, so that it is a well posed problem. However, since the basic set of equations is nonlinear, some iterative procedures are needed for its solution. Once the source and vorticity strengths are determined, the velocity distribution on the airfoil surface will be known from (2.5)-(2.10). The pressure coefficient then can be calculated from the unsteady Bernoulli equation: C p ~ 1 vi uidt ( 2 - n ) Vti is the total tangential velocity at the control point of ith element, (f>i is the per-turbation velocity potential. The force and moment coefficients are obtained by direct integration of the pressure distribution along the airfoil surface. Once the solution at time tk is determined, the model is set up for time tk+i, with the wake pattern calculated from the solution at time tk- The small fine vortex element attached to the trailing edge at time tk is now assumed to be concentrated into a vortex of strength (fw)k^k and situated at: X = (X trailing edge)k + O.5AfcCOS0fc + ( V' ( , N + 1 )fc At-COS0fc V = {y'trailing edge)k + O.5Afcsin0fc + ( Vt,N + l )k At-sinflfc The resultant velocity at the center of each concentrated vortex in the wake is calcu-lated from the solution at time tk- Then the position of that vortex at time tk+i can be decided directly. Chapter 3 Potential Flow Theory 3.1 Assembling Basic Equations of the Theory Using this theory, the pressure distribution on a N A C A 0015 airfoil was calculated when it proceeds in a simple plunging motion. Define: Aif. normal velocity at control point t, induced by element j distributed with unit strength source. Biy. tangential velocity at control point t, induced by element j distributed with unit strength source. According to the relation between a source and a vortex, A{j is also the tangential velocity at control point i induced by vortex element j ; — Bij is the normal velocity at control point * induced by vortex element j. ay. source strength at the jth element. 9 Chapter 3. Potential Flow Theory 10 7ji vortex strength at jth element. Ucx,: mainstream velocity. Vy: plunging motion in y direction. 9: inclined angle of the element Refer to thesis by A.F.MalekN LOCATION OF ELEMENTS ON A I R F O I L SURFACE Chapter 3. Potential Flow Theory 11 The airfoil contour is discretized and x, y coordinates for each control point axe found. 2 • — Control point for each element. * — End point for each element. For N A C A 00** symmetrical airfoil, its contour is given by: y = ^ ( 0 . 2 9 6 9 0 ^ - 0 . 1 2 6 0 0 i - 0 . 3 5 1 6 1 1 +0.28430i 3 - 0.10150i4) Here x, y, t are all non-dimensional variables relative to the chord C of the airfoil. t — thickness of the airfoil. x is calculated from the leading edge. From the figure on the last page, the end point coordinates of each element are: x\ — 0.5(1 — cos <j>i) *' = lv" / = 0 ,1 ,2 , . . . , JV Chapter 3. Potential Flow Theory 12 yi = - ^ - ( 0 . 2 9 6 9 ^ - 0.12600x, - 0.35160a:? +0.28430a:? - 0.10150acf) Approximately, the coordinates of the control point are given: xpi — 0.5(a:;_i + Xi) yPi = 0.5(2/,-! + yi) i = l,2,...,N The element length: = yj(xi - aii-i) 2 + (y{ - 2/i_i)2 i = l,2,...,N Now, according to the principles in Chapter 2, the basic equations at every control point on the airfoil surface and at the wake can be set up. (1) Equation at the ith control point on the airfoil surface (Refer to Williams^): a. Effects on the ith control point by jth source and vortex element (j = 1,2,. . . , N and j 7^  i): dy,3 - yPi - VPJ Xij = dxijCosdj + dyijsinOj yij = dyijCosOj — dxijsh\9j Chapter 3. Potential Flow Theory 13 ^ - 1 O S l ( x , , - 0 . 5 A 5 I ) 2 4 . ^ . J V v 0 - = 2 { a r c t a n [ 3 ; ' J + 0 - 5 A ^ ] - arctan[ X , J ; ~ ° - 5 A s > ] } Vii Vii s i n 6 j = yi^it± cos9j = Xj X j _ i As,-j = l,2,...,Nandj ^ i The effects on the ith control point by jih element with unit strength source are: Normal direction (Perpendicular to the ith element): Aij = Vyij { cosOi cos0j + sin^j sin#,} — V^ij{sin#;cos0j — cos0;sin0j} Tangential direction (parallel to the iih element): Bij = V^,j{cos0,cos0j + sin0,-sin0j} + Vyij {s'mOiCosOj — cos0,-sin#j } Then the effects on the ith control point by jth element with unit strength vortex are: Normal direction: —B^ = — Vx;j{cos0jcos#j + sin^sin^j} — Vy,-j{sin#,cos#j — cos#;sin#j} Tangential direction: Aij = Vyij{cos0icos9j -f sin#,sin0j} — \4,j{sin^,cos^j: — cos0,sin0j} j = l , 2 , . . . , » - l , t + l , . . . , J V Chapter 3. Potential Flow Theory 14 So, the effects on the ith control point by all source elements with strength cr, (j = 1,2, ....,N) will be (except the ith element itself): Normal direction: V ^ = £ AijO-j (3.12) Tangential direction: Vt}= £ BijCi (3.13) The effects by all vortex elements with strength 7 j - are (except the ith element itself): Normal direction: V% = ~ £ Bin = - 7 £ Bi5 (3.14) Tangential direction: V* = £ Aim = 7 £ An (3.15) For 7 constant over all airfoil elements, b. Effects of the ith element on itself: When the ith element is distributed with uniform source o-,-, its effects on itself are: Normal direction: V£ = 2™,- (3.16) Tangential direction: Vt3 = 0 (3.17) « When it is distributed with uniform vortex 7, its effects on itself are: Chapter 3. Potential Flow Theory 15 Normal direction: 0 (3.18) Tangential direction: 2TT7 (3.19) c. Effects on the ith control point by the wake vortex element: Assuming the length of the wake vortex element is A , its strength is 7^ : Actually, the concept is the same for calculating the effect on the ith control point by vortex elements on the airfoil and by the wake vortex element, all formulas are the same except the coordinates. From the figure above, the control point for the wake element is: XpN+l = XtraUing e d g e + 0.5 AcOS0tf+i yPN+l = ytrailing edge + 0.5Asinfyv+l COS0 ;v + i = sin0N+i = Following the procedures in part a.: Chapter 3. Potential Flow Theory 16 Xi,N+i = dxi,N+icoaON+i + dyiiN+isinON+i yi,N+i = dyi^+icosOx+i — dxitN+1sm6N+1 (xi,N+i + 0.5A) 2 + VxilN+i = log[ (xi<N+l + 0.5A)2 + yl N+l VyiN+i = 2{arctan X ' ' N + 1 _ a r c t a n X*,N+I— yi,N+\ Vi,N+i This wake element is a vortex element, its effects on the ith control point should be: Normal direction: -Bi,N+i = -Vi,-,jv+i{cos^-cos^jv+i+sin^,-sin^jv+i} —V^,Ar+i{sin0,-cos0jv+i — cos0,sin0jv+i} Tangential direction: A'IN+I = Vyi,N+i{cos9icos9N+i + sin0jsin0;v+i} - Ki,^+i{sin^ tcos^7v+i - cos0,sin07v+i} So, the effects on the ith control point by the wake vortex element with strength *)w are: Normal direction: Vni = - J % r + i 7 « (3-20) Tangential direction: V U = K N ^ (3.21) d. Effects on the ith control point by all discrete vortices in the wake: At time tk, according to the theory, there should be k — 1 concentrated vortices in the wake, they are numbered as N + 2,N + 3,...,N + k respectively. Define N + 2 be the first one shed to the wake and N + k be the latest shedding one. Chapter 3. Potential Flow Theory 17 As usual, clockwise is defined positive for vortex strength. Xth element V, From the figure, the velocity induced on the ith control point by the concentrated vortex j (j = N + 2, N -f 3, ,N + k) can be written easily. For a unit strength concentrated vortex j, its induced velocities on the ith control point are: 2 V •• = vxij = 0 ^(XXJ - xxif + (yyj - yytf 3 j = N + 2,N + Z, ,N+k Its projections on the normal and tangential directions to the ith element are: Normal direction: Aij = Vyij(co&0icos9j + sin0;sin0j) Tangential direction: B{j = V j^j(sin0;cos0j — cos0;sin0j) Chapter 3. Potential Flow Theory 18 sin#j = Vvi ~ Vpi yj(xpj-xpi)2 + (ypj-ypi)2 y/(xP}-xpiy + (ypj-ypi)2 j = N + 2,N + 3, ,N + k It is assumed that at time tk, the circulation around the airfoil is IV At time t = 0, To = 0; At time t\ there is one wake vortex element and no concentrated vortex in the wake. At time t2 this vortex element is taken as a concentrated vortex and its strength is To — T i = — IY The new shed vortex element is T\ — T2, and so on. At time tk, the wake vortex element is Tk-i — Tk and the k — 1 concentrated vortices are Tk-2 — Tk~i, I\-3 — Tk-2, ,T2 - T 3 , T x - T 2 , and - T i . The effects on the ith control point by all these concentrated vortices are: Normal direction: (3.22) Tangential direction: N+k £ BijiTj.N-2-Tj.N.l) (3.23) e. Effects on the ith control point by the uniform flow £/«,: Normal direction: Vni = -t/oosine,- (3.24) Tangential direction: (3.25) Assume airfoil plunges sinusoidally in the y direction. Chapter 3. Potential Flow Theory 19 Take Vy = As'mut Where, A amplitude of the motion. u = 2TT/ / frequency of the motion. The normal component of the airfoil unsteady motion on the ith control point is: Summarizing from items a, b, c, d and e, at time tk, there are N source elements on the airfoil surface a ; ( i = 1,2,...,N), vorticity about the airfoil 7^ , vorticity on the trailing edge wake element (7^ )* and k— 1 discrete vortices Tk-2 — Tk-i, Tk-3 — I V - 2 , . • •, T i — T 2 , — I V Tfc_2,. . . , T 2 , T i are all known from previous time. The effects on the ith control point at time i f c by all influences will be: Normal direction: Because flow can not cross the solid airfoil surface, according to the principle in Vani = VyCOsfl,- (3.26) V • = V1 4- V2 4- V3 4- V4- 4- V 5 - 4- V6- 4- V7-y n t — vnt 1 y n t ' y n t ' y n t ' v n t " y n t > y n i Chapter 2: Vni = va a m i.e., N N N+k j=N+2 i = l,2, ,N Chapter 3. Potential Flow Theory 20 Where Oj(j = 1,2, ,N), 7 * , ( 7 u , ) f c , A ^ and (0N+i)k are unknown, other quantities (including r,-_w_2 — T J - J V - I , j = N + 2, ....,JV + k) are all known. Actually, 2iro~i can be put into the first term if lets Aij equal to a special value of 27r when j = i. In this way, the above equation becomes: N N E AHai ~ ^ E B'i ~ ^N+iil^k 3=1 i=i N+k + E ^(lVtf-a - ry.jv.i) - U^smOi = Vycos0,- (3.27) .7=^+2 i = 1,2, , J V Where B,j = 0, when j = i. At time t*, the new shedding vortex element and the circulation on the airfoil have the relation: (lw)k&k = T f c - i - Tk Where, rfc = perX7*. N per = E Put into the above equation: (7i»)*Ajk = rfc_! - perx% (3.28) For convenience in the calculation, equation (3.27) can be written in a different way: £.Vn(r*-i - P e r x 7 f c ) A , B"N+i(lw)k — ^ A f c A , ' 7 f c Chapter 3. Potential Flow Theory 21 Let, B'i,N+l A* B",N+lPer So, Finally, equation (3.27) becomes: N N + l N+k -l-L/oosin^ + V„cos0, (3.29) J = 1,2, ,N Which is a group of TV* linear equations. (2) Equation at the control point (numbered N -f 1) of the wake vortex element: The basic idea and procedures are the same as in part (1). a. Effects on the (N + l ) " 1 control point by all source elements: Normal direction: Chapter 3. Potential Flow Theory 22 Tangential direction: N Vt,N+i = H BN+IJVJ (3.31) Where, AN+U = VyN+1j{cos9N+icos9j + sm9N+ism9j} - VxN+1j{s\n9N+1cos9j - cos0jv+1sin0j} BN+1J = VxN+lij{cos9N+1cos9j + sin^+isin^-} — VyN+ij {sin^yv+icos% — cosfl/v+isinfy} = 2 { a r c t a n f " ^ + ° - 5 A j ' ] - arctan[ X j V + 1 - j " ° - 5 A ^ ] } VN+lj VN+l,j xN+i,j = dxN+ijcos9j + dyN+i,jsin9j VN+I,J = dyN+ltjcos9j - dxN+lijsin9j dyN+i,j = J/pjv+i - J/pi Similarly, the effects on the (N + l)th control point by all vortex elements: Normal direction: K V + i = - 7 i k E f i N + i j (3-32) Tangential direction: ifrr+i = 7fc2>N+lli • (3-33) i=i b. Effects by itself: Chapter 3. Potential Flow Theory 23 Normal direction: VnVn = 0 (3-34) Tangential direction: V&r+i = 27t(lw)k (3.35) c. Effects of all concentrated vortices in the wake: Normal direction: N+k Vn,N+l = E ~ Tj-N-l) (3.36) j=N+2 Tangential direction: Where, N+k Vt4,N+l - YI BN+\A^i-N-2 - Tj-AT-i) (3.37) j=N+2 A-N+ij = VyN+1j(cos0N+1cos6j +sin^AT+isin%) BN+I,J = VyN+ij(sm9N+1cos0j - cos^+isin^) 2 VyN+\j — sinfy = COS0j = • XPN+I)2 + {yPj -yPN+i)2 j = :N + 2,N + 3.... . . , /V + fc yPj - yPN+i \/(XpN+l - Xpj)2 + (yPN+i - y p i ) 2 Xpj — XpN+l \J(xpN+l - Xpj)2 + (yPN+i - yPj)2 j = N + 2,N + 3, ,,N + k d. Effects of the uniform flow [Zoo*. Chapter 3. Potential Flow Theory 24 Normal direction: < N + I = -Uoosm6N+1 (3.38) Tangential direction: KN+I = tfoocosfljv+i (3.39) Because the wake vortex element is tangential to its local resultant velocity, the total normal velocity at this control point should be zero. i.e., N N j'=i i=i JV+fc + £ ^ N + i , j ( r j - 7 v -2 - T j - ^ - i ) - Uoosin0N+l = 0 j=JV+2 For convenience, put the equation in the following form: N N N+k £ AN+itj<Tj - BN+I,J = - E ^+IJ(TJ_7V-2 - Ty_^_i) j=l J=l j=N+2 +U00sin6N+1 (3.40) There have been N+l equations from (3.29) and (3.40), but there are iV+3 unknowns o~j(j = 1 , 2 , N ) , fit, A * and (0N+i)k, another two equations are needed for the closed solution problem. (3). Equation about the wake vortex element length: Approximately, the wake vortex element length is proportional to the local resultant velocity Vt'N+1. (excluding the effect on itself, to be distinguished from V^AT+I). Chapter 3. Potential Flow Theory 25 i.e., at time i * : A * = K N + I ( ' * - ' * - I ) (3-41) Where, i.e., N N N+k Vt,N+\ = Y1BN+Uai+ 7*$34/V+l,j + £ BN+l,j(Tj-N-2 ~ Tj^N_t) j = l i = l js=N+2 - r { / o o C O S ^ + i The other equation needed is from the Kutta-Joukowski condition. (4). Kutta-Joukowski condition equation: According to the theory, the Kutta-Joukowski condition for unsteady airfoil motion can be expressed by: Where, N N j = N + 2 - r C / 0 O c o s ^ + 1 (3.42) Chapter 3. Potential Flow Theory 26 and, N N Vt,% = T,B%jai + 7><12A%,3 + A"%,N+iM>< 3=1 3=1 N+k T1-3=N+2 -(-t/ooCOS^W 2 Take: ( K , f + i ) 2 = (vt *) 2 = z2 (3.42) becomes: By: „ 7 nTk - Tfc-i Z i — z 2 = i— tk — tk-i (iw)kAk = r^_i - r* Z - Z = 2 ^ w ^ k ^ k tk — tk-i Finally, the wake vortex element strength was expressed as: (7w)* = 7^-{Z2 - Zx}(tk - tk.x) (3.43) lt\k Take (3.41) and (3.43) as the final two equations needed for the closed solution prob-lem. For convenience, another equation was used in the computer iteration. It can be deduced from equation (3.40): N N sin 0N+1 = {]T AN+hj<jj - 7fc£ BN+\,j j = l j=l N+k )}/U0O (3.44) j=N+2 Chapter 3. Potential Flow Theory 27 3.2 Procedures to Solve the Set of Equations In section 3.1, N + 3 equations (3.29), (3.40), (3.41) and (3.43) were set up. There are N + 3 unknowns <Tj (j = 1 , 2 , J V ) , 7^ , A* and (#Ar+i)fc. It is a well posed problem. However, (3.41) and (3.43) are not linear, the following steps should be taken at time tk: [1] Guess A f c and (6N+i)k. [2] Solve N + 1 linear equations (3.29) and (3.40) to get <7j (j = 1 , 2 , N ) and 7^ . [3] Use relation (7,^ = (Tk-i - Tk)/Ak to get [fw)k. [4] Use Kutta-Joukowski condition (3.43) to obtain a better result for (7™) .^ [5] Then, use (3.41) equation to obtain the wake vortex element length A*.. [6] Substitute all new results into equation (3.44) to get a new value of the wake vortex element angle (#yv+i)fc-[7] If the new A*., (#jv+i)fc and (^ w ) k do not satisfy the desired accuracy, put them back into equations (3.29) and (3.40) again and repeat all steps until these three values converge to the desired accuracy. [8] Then, calculate the velocity, velocity potential and pressure distribution along the airfoil surface (Vti)k, ((f>i)k and [Cpi)k (i - 1 , 2 , N ) , lift coefficient {Ci)k and moment coefficient (Cmo)k. [9] Go on to next time step tk+i = tk + At and repeat the same process. Note: at time tk+i, there are k discrete vortices and a wake vortex element. The new Chapter 3. Potential Flow Theory 28 position of all these vortices can be found in the following way. 3.3 Method of Tracking the Wake Vortex Shape According to the theory, the wake vortex shedding shape behind the airfoil can be cal-culated as below. a. For the small line wake vortex element: At time ifc+i, the wake vortex element at time tk is assumed to be concentrated into a vortex of strength (7^ ,)^  and situated at: b. For all other discrete vortices in the wake: (a). Effects on the ith discrete vortex by all source elements (i = + iV + 3, , N + XpN+k+l = Xtrailing edge + O.5AfcCOS0fc -f Vr/)jV+1COs(^+l)fc(^+l — tk) VpN+k+1 = Vtrailing edge + 0.5A f csin^ + VttN+1sm(9N+1)k(tk+1 - tk) t,N+l is the same as in equation (3.41). k): Proceeding as before, the effects on the i discrete vortex are: x direction: TV (3.45) i=i y direction: N (3.46) 3=1 Chapter 3. Potential Flow Theory 29 Where, Axij = VxijcosOj - VyijsmOj i = N + 2,N + 3, ,N + k i = 1,2, ,N XH + 0.5 As,- ± Xij-0.5Asj Vyij = 2{arctan-^ ^ - a r c t a n - ^ l } Vij Vij Xij = dxijeosOj + dyijSinOj yij = dyijeosOj — dxijs'mBj dxij — Xpi Xpj dyij = yPi - yPj (b). Effects on the ith discrete vortex by all vortex elements (i - N + 2, iV+3, , iV+ *): x direction: N y direction: ^ = E ^ i 7 , (3-47) i = iV + 2,iV + 3, ,N + k J = 1,2, ,N Chapter 3. Potential Flow Theory 30 Where, Ayij = VyijeosOj + Vxijsindj -Axij = —VxijCo&Bj + Vyijsm0j (c). Effects on the ith discrete vortex by the small wake vortex element: x direction: y direction: V* = AyitN+1(lw)k (3.49) V*- = -AxitN+l(lw)k (3.50) i = N + 2,N + 3, ,N + k Where, —•A.xitN+i = -VxitN+iCos0N+i + Vyi,N+1s'mON+1 , ,(xi,N+i +0.5A) 2 4-y? Vxi,N+l = log 7 A >, 2  (x,Viv+, - 0.5A) 2 + j / / > J V + 1 T , o r . Xi,N+i + 0.5A s.yv+i - 0-5A = 2{arctan arctan ) Xi,N+i = dxi t^+icos0N+i + rfj/i^+isin^yv+i Vi,N+i = dyi tN+icos0N+i — dxi>N+ism0N+i dXi,N+l — Xpi — XpN+l dyi,N+i = Vpi - VpN+i Chapter 3. Potential Flow Theory 31 (d). Effects on the ith discrete vortex by all other discrete vortices: Y ' x direction: y direction: j=N+2 N+k Vy\= YM^i-N^-T^-l) j=N+2 Note An — 0, Bu = 0, i.e., the effects by itself are zero. (3.51) (3.52) Where, Aij = -VyijCosSij Vyxj — ^(xxi - xx,)1 +[yy{ - yy,)7 i = iV + 2,iV + 3, ,JV+fc j = N + 2,N + Z, ,N + k Chapter 3. Potential Flow Theory 32 (e). Effects of the uniform flow U^: x direction: Vx) = U„ (3.53) y direction: Vi = 0 (3.54) The total induced velocity will be the summation of all influences. A l l discrete vortices will move downstream at the total velocity V ,^- and Vy,-: x direction: i.e., Vri = V 1 . + V2- + V3- + V4 + V5-' Xt — ' xt • r Xt ' Xt • XX • r XI N N Vxi = Axij<Tj + 7*53 AiH + Ayi,N+ihw)k i=\ i = i N+k j=N+2 y direction: N N Vyi = Yl AlH^i ~ 7 * £ Aii ~ Ai,N+l(lw)k i=i i = i N+k + £ Aj(Xi-N-2 ~ T j_Ar_i ) 3=N+2 i = N + 2,N + 3, ,N + k So, the new position of the iih discrete vortex will be: (*jn)fc+l = (xpi)k + VxiAt (yPi)k+i = (yPi)k + VyiAt i = JV + 2,JV + 3, ,N + k Chapter 3. Potential Flow Theory 3.4 Pressure, Force and Moment on the Airfoil 33 After = 1,2, , k), 7*, (7^ ,)*, A * and (0yv+i)jt are all known at time i * , the tangential velocity Vu at every control point on the airfoil surface can be found: N N Vti = E B 0- f f i + E^«7fc + <jv+i(7i»)fc i=i j=i N+k + E Bij(Fj-N-2 ~ Tj-N-l) j=/V+2 -rUooCOsBi Once the velocity distribution on the airfoil surface is known, the main problem of calculating the pressure coefficient from the unsteady Bernouli equation is to determine | £ , i.e., the time rate of change of velocity potential <j> at each time step. At the ith control point at time tk, ^ can be evaluated by: The following part decides (<f>i)k- Refer to Williams^. (a). Velocity potential at the ith control point due to all source elements: r (g < i + 0.5Aa J-)2 + y?-1 ^ = l o s [ ( x , , - 0 . 5 A 5 j ) 2 + ^ ] ^ = 2 a r c t a n [ 4 + ^ A ; ^ ) 2 ] fo0- = 0 .5A3 j X log {[(x t i + 0 . 5 A 5 j ) 2 + yDKxn - O.bAsj)2 + y?-]} .7 = 1,2, , iV Here A S J , and y,j are all the same as before. The potential function due to a source element j of width A S J and unit strength: <f>aij = xij<f>A<j + Vij<f>Bij + <t>Dij — 2Asj Chapter 3. Potential Flow Theory 34 So, the potential due to all source elements: N <l>si = £ \xii^Mj + Vij<f>Bij + <f>Dij - 2As i]<r i (3.55) t = 1,2, ,N (b) . Potential at the ith control point due to all vortex elements: to* = A 3 , a r c t a n _ y f _ % A j . ) a l The potential function due to a vortex element j of width A S J and unit strength: <f>vij = -Vij<i>Aij + Xij<f>Bij - 4>Cij The potential due to all vortex elements: N <t>vi ~ £ [-yij<t>Aij + Xij<j>Bii ~ <f>Cij]lk (3.56) 3=1 (c) . Potential due to the small wake vortex element: Similarly as (b): <l>wi = [—yi,N+l<f>Ai,N+l + Xi,N+i(f>Bi,N+l ~ <j>Ci,N+l]{lw)k (3.57) (d) . Potential due to all wake discrete vortices: Potential function at the ith control point due to discrete vortex j of unit strength: fay = 2 arctan (———) The potential due to all discrete vortices: N + k 11-11 • <f>ki = J2 2 a r c t a n ( ^ _ J ^ - ) ( r i _ N _ 2 - r j _ ^ _ 1 ) (3.58) Chapter 3. Potential Flow Theory 35 At time i * , the potential at the ith point due to all influences will be: i.e. N (<f>i)k = E [XiJ$AiJ + yij^Bij + <f>Dij ~ 2ASj]cTj j = l /v + E ["Vy^ii + xij4>Bii - <t>Cij)lk 3=1 + (lw)k[—yi,N+l<j>Ai,N+l + Z . , / V + l < t a t , / V + l — 0C.,/V+l] + £ 2arc tan( i^_Jw)(r j _ A r _ 2 _r j _ / v_ 1 ) (3.59) j=N+2 XP> XP3 (<t>i)k-i is known, ( ^ ) * can be calculated now. Then, from the equation: c -1-(VL.)*-±A p~ {uj Ul}dt] The pressure coefficient at the t** control point at time tk will be: ( ^ . - ( ^ - ^ ( f ) * (3.60) t = 1,2, , iV After {Cpi)k is determined, (Ci)k and (Cmo)k will all be known by integration of the (Cpi)k distribution at time tV Y Chapter 3. Potential Flow Theory Lift coeff.: 1 r 1 N Ci = --j>Cpdx = --J^CpiAxi Mid-chord pitching moment coeff.: Cmo = -^jCpixdx + ydy) 1 N = "F2 D Cpi{xiAxi + y.Ay.) ° «=i X f = X p j X c xc and ?yc are the center coordinates of the airfoil. Chapter 4 Results of the Theory The present method was used to calculate the vortex shedding shape in the wake of an unsteady airfoil, and also the pressure, force and moment on the airfoil. A NACA 0015 airfoil was used in the calculation and comparisons to the linear theory results are given. Part 1 gives the wake vortex shedding shape behind the unsteady airfoil. Part 2 gives the force and moment on the NACA 0015 airfoil. 4.1 Wake Vortex Shedding Shape The wake vortex shedding shape was calculated when the airfoil is in simple plunging si-* nusoidal motion. The unsteady motions Vy = 0.07869sin(4.3r) and Vy — 0.3105sm(17.0£) were chosen for comparison with the experimental results by Bratt®. These values are all non-dimensional with Uoo = 1 and C = 1. For the same unsteady motion, Bratt^ obtained the experimental results by smoke-trace photographs. A qualitative comparison of the results by the present method with those by Bratt^ is shown in Figure 4.1. For W = 4.3, the wake shear layer is a harmonic type with its amplitude slowly built-up. See Figure 4.2 (a). It has the same frequency as the forced vibration. The wake is not stable because the amplitude becomes larger and larger with time. This instability can also be seen from the case when u = 17.0. See Figure 4.2 (b); the wake shear layer rolls 37 Chapter 4. Results of the Theory 38 up to a series of big vortices. At each cycle of the vibration, each pair of vortices sheds to the wake. After about two and one half cycles, the shedding vortices become very irregular and the wake becomes unstable. But the experimental results by Bratt^ does not show this instability. Most possibly, the reason for this is due to the viscosity in the real flow. Because of viscous dissipation, the wake vortex decays with time continuously until it vanishes. Its influence on other vortices will decrease. In potential flow theory, the wake vortex keeps its strength through all the calculations. When two vortices are very close to each other, the induced velocity will be very large. So, with time increasing and without vortex decay, the wake will become unstable. Maull^ suggested the concept of vortex decaying in the wake calculation and Wong^ used this method in his calculation and satisfactory results were obtained. Using the present method, with 10% vortex strength decreasing after one cycle from its shedding, these two cases were calculated again and the results are given in Figure 4.3. Now the wakes for both cases are stable and they are very close to the experimental results by Bratt^. It can be predicted from the results that the wake vortex decay is more impor-tant for the calculation of wake vortex motion at high frequency unsteady airfoil motion. The analysis of the accuracy of wake vortex shape calculations was not undertaken. This problem has been treated by Rosenhead^ and //ama' 1 2 ' . 4.2 Pressure, Force and Moment on N A C A 0015 Airfoil For comparing with the linear theory results, non-dimensional frequency p. = jjj^ was used, where, ui = 2irf, C is the chord of the airfoil. Lift and moment coefficients for p, = 0~1 were calculated by both the linear theory and the present method. In all of the Chapter 4. Results of the Theory 39 calculations, Vy — 0.01 sin u;t (or Vv = V0 sin 2pt*, VQ = 0.01 - non-dimensional velocity amplitude, £* = - non-dimensional time.) was used as the plunging motion in the y direction. Figures 4.4 (a) and (b) show the comparison between the linear theory and the present numerical results for the lift and moment coefficient magnitudes. Figure 4.4 (a) is the ratio of the unsteady lift coefficient magnitude and the quasi-steady lift coefficient vs the nondimensional frequency p; Figure 4.4 (b) is the moment coefficient ratio. Figure 4.4 shows that when p < 1, the lift and moment coefficients from the numerical model are essentially the same as the linear theory results. Both the linear theory and the numerical results can be taken as the theoretical free air case results. Lift and moment coefficients were also calculated with wake vortex decay. Results show that for p < 1, the influence on the force and moment by wake vortex decay is very small. The numerical method outlined here leads to the calculation of the inviscid flow about an airfoil undergoing an unsteady motion. It is assumed that the flow remains attached at the airfoil surface and separates at the trailing edge of the airfoil. The appropriate Kutta-Joukowski condition was assumed, and satisfactory solutions were obtained comparing with the linear theory. Chapter 4. Results of the Theory 40 by Brott by present method (a) Vv = 0.07869 iin(4.31) by Sroit by present method (b) Vv = 0.3105 sin(17.01) Figure 4.1: Shape of the vortex wake generated by N A C A 0015 airfoil oscillating in simple harmonic manner, numerical results (with vortex decay) compared with the experimental results Chapter 4. Results of the Theory 41 (a) Vv = 0.07869 sin(4.31) (b) Vv = 0.3105 sm(17.01) Figure 4.2: Shape of the vortex wake generated by NACA 0015 airfoil oscillating in simple harmonic manner, without vortex decay Chapter 4. Results of the Theory 42 10% decay-no decay (b) Vy = 0.3105 sin(17.01) Figure 4.3: Shape of the vortex wake generated by NACA 0015 airfoil oscillating in simple harmonic maimer, with 10% vortex decay Chapter 4. Results of the Theory 43 0.8 . CilCu 0.7 0.6 — Linear Theory Numerical Model 1 i 1 1 1 1 1 1 1 1 — 0 0.1 0.2 0.3 0 .4 0.5 0.6 0.7 0.8 0.9 1 (a) Lift Coeff. Ratio 0.9 CmojCn 0.6 0.5 Linear Theory Numerical Model 1 1 1 1 1 1 1 1 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (b) Moment Coeff. Ratio Figure 4.4: Lift and moment coefficients of oscillating N A C A 0015 airfoil as function of the reduced frequency p, compared with the linear theory results for V0 = 0.01 Chapter 5 Experiment Arrangement 5.1 Introduction to the Experiment The idea of using the tolerant wind tunnel for unsteady airfoil motion testing is investi-gated here. The experiment is performed in a two-dimensional test-section insert designed and built for an existing low speed closed-circuit wind tunnel (Figure 5.5). This insert is 915mm wide by 388mm deep in cross-section and 2.59m long. The two-dimensional test model is mounted vertically in the center plane of the test section between solid ceiling and floor. Both side walls consist of vertical uniformly spaced airfoil shaped wooden slats of section N A C A 0015 and chord of 89mm at zero incidence. These slotted walls are surrounded by 0.39mxO.30mx2.44m wooden plenum chambers. Figure 5.6 gives a plan view of the new setup. The purpose of this tunnel is to make the flow about the model behave as nearly as possible as in free air, so that the loading measured on the model will not be affected by the tunnel walls. The open area ratio (OAR) of the slotted walls is adjusted to best achieve this condition. A full range of open area ratios was tested by varying the number of slats mounted inside the walls. Several stationary model configurations (airfoil, cylinder, plate, etc.) have been tested in the tunnel in steady flow and results showed that the optimum O A R values he between 0.55 and 0.65. For the unsteady case, Hameury^ tested the unsteady wake behind bluff bodies. His results showed that for one kind of bluff body of different sizes, the Strouhal Numbers collapsed to each other also at this O A R range. This means there is negligible wall interference on 44 Chapter 5. Experiment Arrangement 45 the model at these OAR. The unsteady airfoil motion was tested here. The loading on the airfoil was measured and compared with the theoretical free air results. 1.2 Experiment Apparatus and Equipment A N A C A 0015 airfoil with 1ft chord was tested. Pressure with time was measured along the airfoil surface. Lift and pitching moment were obtained by integration of the pressure coefficients at every time increment. To make the airfoil model as light as possible, there were several holes drilled inside. Al l pressure taps are flush mounted and have 5mm di-ameter orifices. Plastic tubes of 0.066in. inside diameter and approximately 2ft. length transmit the surface pressure to a pressure transducer outside of the test section. Pres-sures were measured via a 48 port scanivalve manual scan pressure transducer, called the Barocel Modular Pressure Transducing System. The signal coming out of the scanivalve was fed to a DT 2801 A-D converter, which was connected to an I B M PC computer. With the help of a computer program, the surface pressure and the displacement of the airfoil were measured and recorded very nearly instantaneously and simultaneously at successive time increments. Displacement was measured using a linear displacement transducer. A l l instrumentation is listed in Appendix A. The floor and ceiling tip clear-ances were 9mm in all tests, giving the airfoil enough clearance to oscillate back and forth inside the tunnel. The clearance is larger than that (2.5mm) of the general two dimensional model testing in the same tunnel. Since measurements were only taken from the pressure taps around the airfoil surface at the middle span of the model, tip influence will not be expected. The test airfoil was mounted vertically on an oscillating table. The table was designed on the basis of generating a plunging sinusoidal motion to the airfoil. A picture of its construction is given on Figure 5.7 (a) and its basic principle is shown on Figure 5.7 (b). Although the slider-crank mechanism does not give the airfoil an exact Chapter 5. Experiment Arrangement 46 sinusoidal plunging motion, since the flywheel offset (amp) is very small relative to the connecting rod (L), the experimental results showed that the airfoil motion is very close to the exact sinusoidal motion. Figures 5.8 (a) and (b) are examples of the displacement and velocity of the oscillating table. A sine curve of the same amplitude and frequency was plotted on the same graphs for comparison. The Barocel transducer is accurately calibrated for static pressure. For unsteady pressure signals transmitted through relatively long tubes, considerable attenuation is expected. The attenuation of the signal depends on such variables as tube length, tube diameter, number and size of pressure taps, frequency and shape of the pressure signal and, in addition, the basic frequency response of the Barocel itself. WilanS13^ calibrated the transducer for unsteady pressure response. The calibration results are given in Figure 5.9 and Figure 5.10. Figure 5.10 demonstrates the linearity of the system. For a tube length of 2ft and frequency around 5Hz, the output is attenuated by about 5%. The experiments were carried out at a Reynolds number of about 2.5 x 105 based on the airfoil chord. The displacements (amp) of the airfoil are 1/16 and 1/8 of an inch. The tunnel wind speed was continuously monitored through a calibrated pitot-static tube, mounted off centerline in the test section. The sample frequencies of the data acquisition system are 1080 and 720 respectively for p — 0.52 and 0.345. The sample time is 1 second. The values of dimensional and non-dimensional airfoil oscillating frequencies and amplitudes at which experiments were performed are given in the Table below. Frequency 6.5 Hz 4.3 Hz plitude i . H = 0.52 K = 0.0054 fi = 0.345 V0 = 0.0037 E < 1 in« ft = 0.52 V0 s 0.01082 ft = 0.345 V0 = 0.0075 Chapter 5. Experiment Arrangement 47 -Tuning 4 * •act ion diflUMT 5* dHhaw i d i f f u H r l Bmthar _ l 1*' tflffuMT 36*«2T Ta»t taction 7:1 Contract on d i t l u M r Selling •action Scraan •n ID sua Figure 5.5: U .B .C . Mechanical Engineering low speed closed circuit 'green' wind tunnel Figure 5.6: Double slotted wall test section designed for the 'green' wind tunnel Chapter 5. Experiment Arrangement 48 (a) Photograph of the Oscillating Table amp/L « 1 (b) The Basic Working Principle Figure 5.7: Photograph and the basic working principle of the oscillating Chapter 5. Experiment Arrangement 49 i n . 0 4.6875 . i n . / s 0 -"i 1 1 1 1 1 1 r 0.4 0.8 second (a) displacement -4.6875 _ -Experiment results -Exact sine curves (b) velocity 0.8 second (a) Displacement (in.): y = — gsin(37.5£) (b) Velocity (in./s): V„ = -4.6875 cos(37.5i) Figure 5.8: Displacement and velocity of the oscillating table as function of time Figure 5.9: Pressure attenuation as function of tube length and frequency (tube diameter = 0.066 in.) Chapter 6 Experimental Results In this chapter, the experimental results for a N A C A 0015 airfoil in small sinusoidal plunging motions are presented. Figure 6.11 shows the pressure tap positions on the airfoil surface. Several O A R were examined between 0 and 1 by changing the number of slats on both slotted side walls. Figures 6.12, 6.13 and 6.14 give the pressure variation with time at some tap points for fi = 0.52 and OAR=0.526 for the case of amp=l/16 inch (non-dimensional unsteady velocity amplitude VD = 0.0054). The original signal is very noisy with many high frequency signals. Filtered by a low-pass digital filter (Appendix B), the pressure signal at the oscillation frequency only attenuated by less than 2%, but the noise was deleted quite completely. The pressure signal around the trailing edge part is still very noisy after being filtered. This is because the boundary layer becomes thicker at this region. The signal becomes dominated by the airfoil oscillation frequency from tap 9 to the leading edge. The leading edge point is the stagnation point if the airfoil stays at rest. When the airfoil plunges in a small sinusoidal motion, the stagnation point changes a little around the leading edge. The pressure frequency at this point should be two times the airfoil oscillating frequency. The reason for this is that when the airfoil proceeds through one complete oscillation, the leading edge becomes stagnated twice. The experimental results showed this, but not very clearly. This is because the real pressure change at this point is very small and possibly the noise (60 Hz electrical noise) is almost at the same scale as the real pressure signal. 52 Chapter 6. Experimental Results 53 Pressures for p = 0.52 and OAR=0.708 at some tap positions are also given. See Figure 6.15. The signal at tap 10 is still irregular. Only from tap 11 it becomes dominated by the airfoil oscillating frequency. Al l experimental results show that for larger O A R , the region of irregular signal extends nearer to the leading edge. More evidence can be seen from Figure 6.16 for p = 0.52 and OAR=0 where the region of the irregular signal extends only to tap 5. This may be due to a possible interaction between the sudden expansion flow at the test section entrance and the unsteady airfoil motion. The flow area is suddenly enlarged there, inducing two separation regions at the entrance corners. Abbott and Kline^ investigated this flow field and found that the two unsteady separation regions would give some unsteadiness to the main free stream flow. Though the pressure signals around the airfoil trailing edge are irregular, they have little effect on the lift coefficients. But they do have some effect on the moment coeffi-cients, since the trailing edge part has a big moment arm. By integrating the pressure distribution at each time increment, the variation of lift and moment coefficients with time are calculated. Figure 6.17 shows the results of the total lift coefficient changing with time for p = 0.52, VQ = 0.0054 at OAR=0 and 0.526. For each OAR, the abso-lute magnitude values of the lift and moment coefficients from each cycle are averaged and plotted together against OAR in Figure 6.18 and Figure 6.19. The phases of the lift and moment with respect to the quasi-steady values are not shown, but qualitative observations indicated no effect of OAR on the phases. Both figures clearly show the great blockage effect of the conventional solid wall. The wall blockage effect is reduced by the increasing of the open area ratio. The free air case results are shown on the same graph for comparison. The lift coefficient for the free air case agrees with the experi-mental results at about OAR=0.55. For the moment coefficient, it is about OAR=0.1. This value is much smaller than expected. The reason can be seen from Figure 6.20, the Chapter 6. Experimental Results 54 pressure difference (between the airfoil lower and upper surfaces) distribution along the airfoil chord for both the experimental and the linear theory results at the time when the maximum lift coefficient occurs. The linear theory has an infinite suction at the leading edge point, this would increase the moment value (to the center of the chord) in the linear theory. The experimental results show that the pressure difference has a sudden increase near the trailing edge for both the traditional solid wall test section and the slotted wall test section, this would decrease the moment value for the experiment. Both of the effects make the experimental moment value depart from the free air case result. The total lift and moment coefficients include two parts. One is due to the circulation on the airfoil and in the wake. Another is the added mass term due to the motion of the airfoil. In linear theory, the terms are additive, giving Ci(total) = C ^ circulation) + Ci(added mass). For p < 1, this relation should be close to correct. As an example, the relation is used here for p = 0.52 and V0 = 0.0054 to obtain the lift coefficient for the circulation part. The added mass result is obtained in a wind off situation (no free stream velocity) at the same oscillating frequency using the same procedures. Figure 6.21 shows the experimental results of the lift coefficient for the added mass term and the circulation part for p = 0.52 at OAR=0 and 0.526. Proceeding as before, they are all plotted together in Figure 6.22. The free air result matches the experimental result at about OAR=0.52. In linear theory, due to the symmetrical pressure distribution from the added mass term, the moment from this part is zero. The total moment coefficient is the same as the circulation part. The experimental result (Figure 6.23 (a)) shows that this part is very small. Figures 6.23 (b) and (c) are the total moment coefficient results for p = 0.52 at OAR=0 and 0.526. For p = 0.345 and amp=l/8 inch (non-dimensional velocity amplitude Va = 0.0075), Chapter 6. Experimental Results 55 the lift and moment coefficients from the experiment are plotted against O A R in Figure 6.24 and Figure 6.25. The free air lift coefficient matches the experimental result at about OAR=0.53 and the moment coefficient at about 0.71. As before, Figure 6.26 gives the comparison of the pressure difference distribution results between the experiment and the linear theory. From the graph, it is easy to see that there is no sudden jump of the pressure difference near the trailing edge. This explains why the free air moment value matches the experimental result near the desired value for this case. The other two tests taken here are for fi = 0.52, V0 = 0.01082 and p = 0.345, V0 = 0.0037. Figure 6.27 and Figure 6.28 are the lift and moment coefficient results for p = 0.52 and V0 = 0.01082. Figure 6.29 and Figure 6.30 are the lift and moment coefficient results for p = 0.345 and VQ = 0.0037. For p = 0.52, the free air result matches the experimental result for the lift and moment at about OAR=0.68 and 0.0 respectively. For fi — 0.345, they are about 0.65 and 0.55. The optimum O A R for the moment at p = 0.52 also occurs at much too small a value. The reasons are the same as discussed before. So for airfoil unsteady motion at higher p, the pressure difference between the upper and lower surfaces near the trailing edge is not zero as stated in the Kutta-Joukowski condition. Though this deviation from the zero-loading condition does not affect the lift, the contribution to the moment is considerable. Satyanarayana and Davis^^ discussed this problem in their work. They concluded that the application of the Kutta-Joukowski condition is valid below the reduced frequency p — 0.6 for the pitching airfoil motion. Another thing that needs to be noticed is the scattered experimental data in some cases. Except some unknown reasons, an explanation is that the pressure signals are very small due to the small airfoil motion, errors from all influences might contaminate some measurements. One solution for this is that experiments at more O A R values should be taken. Figure 6.11: Pressure tap positions on N A C A 0015 airfoil surface -0.19 -0.21 -0.23 -0.26 T 0.4 0.8 second at tap5 0.4 0.8 second at tap7 8 •8 S 3 8 -0.26 -0.28 -0.30 -0.40 -0.45 -0.50 0.4 0.8 second at tap9 -i r 0.4 0.8 second at tapl4 Figure 6.12: Pressure coeff. of oscillating N A C A 0015 airfoil as function of time at different tap positions, experimental results for fi = 0.52, Va = 0.0054 and OAR=0.526 an -0.42 -0.47 -0.52 -0.45 0.4 0.8 second at tapl7 -0.55 at tap20 i r-" T - - 1 ' • , 0.4 0.8 second 0.4 0.8 second ^ at tap23 Figure 6.13: Continued: Pressure coeff. of oscillating N A C A 0015 airfoil as function of time at different tap positions, experimental results for // = 0.52, Va = 0.0054 and OAR=0.526 Chapter 6. Experiment Results 59 1.00 0.96 i i i i i 0.4 0.8 second at tap28 at tap30 Figure 6.14: Continued: Pressure coeff. of oscillating N A C A 0015 airfoil as function of time at different tap positions, experimental results for p = 0.52, V0 = 0.0054 and OAR=0.526 -0.22 -0.25 -0.30 -0.32 -0.34 0.4 0.8 second at tap7 0.8 second at taplO 8 8 -0.34 -0.36 -0.38 0.8 second at t ap l l Figure 6.15: Pressure coeff. of oscillating NACA 0015 airfoil , j . fund different tap positions, experimental results for e = 0.52, V„ = 0.0054 and OAR-0.708 o Chapter 6. Experiment Results 61 at tap4 i r— i i i ' 0.4 0.8 second at tap5 Figure 6.16: Pressure coeff. of oscillating N A C A 0015 airfoil as function of time at different tap positions, experimental results for u = 0.52, V0 = 0.0054 and OAR=0.000 Chapter 6. Experiment Results 62 0.04 OAR=0.000 Figure 6.17: Lift coeff. of oscillating N A C A 0015 airfoil as function of time, experimental results for p = 0.52, V0 = 0.0054 at OAR=0.0 and 0.526 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 OAR Figure 6.18: Lift coeff. magnitude of oscillating N A C A 0015 airfoil as function of O A R , experimental results for ft = 0.52 and V0 = 0.0054 Figure 6.20: Cp,iower-Cp,uPper as function of X along the airfoil chord, experimental results for u = 0.52 and V0 = 0.0054, compared with the linear theory result 0.04 0.04 OAR=0.000 Figure 6.21: Lift coeff. of oscillating N A C A 0015 airfoil as function of time, experimental results for added mass term, circulation part at OAR=0.0 and 0.526 for a = 0.52 and V0 = 0.0054 Chapter 6. Experiment Results 67 ( S C M C M C N O ' - t ' - t ' - l ^ V*4 Q) O O O Q O O O O added mass term OAR=0.526 — i 1 • i 1 1 1 1 1 0.4 0.8 second OAR=0.000 Figure 6.23: Moment coeff. of oscillating N A C A 0015 airfoil as function of time, exper-imental results for added mass term, total moment coeff. at OAR=0.0 and 0.526 for H = 0.52 and V0 = 0.0054 w oo Chapter 6. Experiment Results 01 .009 'mo 008 .007 _ 006 . OAR Moment C o e f f . 0 .000 0.01034 0.344 0 .00811 0.453 0 .00806 0 .635 0 .00829 0.854 0 .00692 1.000 0 .00603 a A 4 X v Cmo = 0.0081 (free air) X . A T 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 OAR Figure 6.25: Moment coeff. magnitude of oscillating N A C A 0 0 1 5 airfoil as function of O A R , experimental results for /J = 0.345 and V0 = 0 .0075 Figure 6.26: CPliower-CpiUppeT as function of X along the airfoil chord, experimental results for u = 0.345 and Va = 0.0075, compared with the linear theory result 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 OAR Figure 6.27: Lift coeff. magnitude of oscillating N A C A 0015 airfoil as function of O A R , experimental results for \i = 0.52 and Va = 0.01082 0 0.1 0 .2 0 .3 0.4 0 .5 0 .6 0 .7 0 .8 0 .9 1 OAR Figure 6.28: Moment coeff. magnitude of oscillating N A C A 0015 airfoil as function of OAR, experimental results for fi = 0.52 and V„ = 0.01082 -a co 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 OAR Figure 6.29: Lift coeff. magnitude of oscillating NACA 0015 airfoil as function of OAR, experimental results for fi — 0.345 and V0 = 0.0037 -a 005 0 0.1 0 .2 0 .3 0.4 0 .5 0 .6 0 .7 0 .8 0 .9 1 OAR Figure 6.30: Moment coeff. magnitude of oscillating N A C A 0015 airfoil as function OAR, experimental results for ft = 0.345 and Va = 0.0037 Chapter 7 Concluding Comments A two-dimensional potential flow theory, based on the surface singularity distribution method, was used to calculate the loading on a NACA 0015 airfoil in a plunging sinu-soidal motion in the free-air case. Comparison with the linear theory results were given. The above potential flow theory can be extended to include not only the oscillating airfoil, but also the effect of different wind tunnel wall configurations. In this way, it will predict the correction-free wind tunnel test configuration and give more comparison with the experimental results. Also, the non-zero loading trailing edge condition should be con-sidered in the numerical model at higher fi. This will need some preliminary experimental results. Experiments were carried out on a NACA 0015 airfoil in the same unsteady motion. Experimental work showed that the range of optimum OAR values for the plunging airfoil motion lies between 0.5 and 0.7 for the lift coefficients. For the moment coefficients, the free air values also matched the experimental results at this OAR range at /a = 0.345. But for fi = 0.52, the optimum OAR values occured at about 0.1. The reason mainly comes from the experimental results of the pressure distribution near the trailing edge. McCroakey^ pointed out that finite pressure loading and abrupt streamline curvature can exist in the trailing edge region for unsteady model motion, depending on the amplitude and type of the model motion and flow oscillation. The next step of this work would be to see when (at which u) the zero loading condition at the trailing 76 Chapter 7. Concluding Comments 77 edge will breakdown, what the effect will be on the moment and how it will relate to the motion type and amplitude of the airfoil. For further use of the tolerant wind tunnel for unsteady motion testing of other kinds of two dimensional models, such as flow induced vibration, airfoil pitching motion, etc., more work needs to be done to examine the freestream flow quality inside the test section when the test model has an unsteady motion. Also it will be helpful if the practical free air case results are known for the test model and motion. One problem that needs to be clarified is the interaction between the model motion and the two separation regions at both entrance corners. Another important problem is whether the flow near the airfoil slats separates when the test model has an unsteady motion, and how large an unsteady motion might induce flow separation at the slats. Three-dimensional tests for aircraft take-off and landing, buildings and vehicles can be considered by using three slotted walls (two side walls and the ceiling) in the tolerant wind tunnel. Bibliography [1] T H . Von Karman and Sears, W.R., "Airfoil Theory for Non-Uniform Motion", Jour-nal of the Aeronautical Sciences, Vol. 5, No. 10, August 1938, pp. 379-390. [2] Parkinson, G.V. and Lim, A . K . , "On the Use of Slotted Walls in Two-Dimensional Testing of Low-Speed Airfoils", CASI Trans, Sept. 1971. [3] Williams, C D . , "A New Slotted-Wall Method for Producing Low Boundary Cor-rections in Two-Dimensional Airfoil Testing", Ph.D. thesis, Oct. 1975, University of British Columbia. [4] Hameury, M . , "Development of the Tolerant Wind Tunnel for Bluff Body Testing", Ph.D. Thesis, Feb. 1987, University of British Columbia. [5] Basu, B .C. and Hancock, G.J. , "The Unsteady Motion of a Two-Dimensional Aero-foil in Incompressible Inviscid Flow", J . Fluid Mech., 1978, Vol. 87, Part 1, pp. 159-178. [6] Maskell, E .C. , "On the Kutta-Joukowski Condition in Two-Dimensional Unsteady Flow", 1972, Royal Aircraft Establishment, Farnborongh. [7] Malek, A . F . , "An Investigation of the Theoretical and Experimental Aerodynamic Characteristics of A Low-Correction Wind Tunnel Configuration for Airfoil Testing", Ph.D. Thesis, April 1983, University of British Columbia. [8] Bratt, J .B. , "Flow Patterns in the Wake of An Oscillating Airfoil", R & M 2773, 1953, Royal Aeronautical Establishment. 78 Bibliography 79 [9] Maull, D.J . , "An Introduction to the Discrete Vortex Method", I U T A M / I A H R , Karlsruhe, 1979. [10] Wong, L . H . , "A Numerical Model for Vortex Shedding From Sharp Wedges in Os-cillatory Flow", M.A.S . Thesis, April 1990, University of British Columbia. [11] Rosenhead, L. , "Formation of Vortices From A Surface Discontinuity", Philosophical Magazine, Vol. 5, 1928, p. 417. [12] Hama, F.R. and Burke, E.R., "On the Rolling Up of A Vortex Sheet", T N BN-220, 1960, University of Maryland. [13] Wiland, E. , "Unsteady Aerodynamics of Stationary Elliptic Cylinders in Subcritical Flow", M.A.S . Thesis, 1965, University of British Columbia. [14] Abbott, D.E. and Kline, S.J., "Experimental Investigation of Subsonic Turbulent Flow Over Single and Double Backward Facing Steps", Journal of Basic Engineering, September 1962, pp. 317-325. [15] Bodapati Satyanarayana and Sanford Davis, "Experimental Studies of Unsteady Trailing-Edge Conditions", A I A A Journal, Vol. 16, No. 2, February, 1978. [16] W.J . , McCroskey, "Unsteady Airfoils", Ann. Rev. Fluid Mech. 1982. 14: 285-311. Appendix A Instrumentation Pressure Transducer • Barocel Pressure Sensor (DATAMETRICS Inc.) Type 511J-10 • Signal Conditioner (DATAMETAICS Inc.) Type 1015 • Power Supply (DATAMETRICS Inc.) Type 700 Mechanical Pressure Scanner • Scanivalve 48-ports (SCANIVALVE Corp.) Model 48J9-2273 Pressure Lines (INTRAMETRIC) • Inside Diameter: 0.066in.; Outside Diameter: 0.095in.; Length: 2ft. Manometer • Inclined Micro-Manometer (Willi.Lambrecht KG) I.B.M. Personal Computer DT 2801 Analog-Digital Converter Linear Displacement Transducer (Schaevits Model 3000 H.R.) 80 Appendix B Digital Low-Pass Filter A digital low-pass filter was used in processing the data from the experiment. Its expres sion in Laplace form is: ute = 2irfe Where, fc is the cut-off frequency. Let a = j u>, the frequency response of the filter can be shown as: 20 log\G(ju,)\ 81 Appendix B. Digital Low-Pass Filter 82 Assuming X(s) is the original signal, Y(s) is the signal after filtered. Y(s) = G(s) X(s) Y(s) s + u>c Y(s) = ue X(s) Taking the inverse transformation to the above equation, it can be written in the time domain as: Y'(t) + o>c Y(t) = uc X(t) Here, X(t) is filtered to get Y(t) , the filtered data. Take the data sample period in the experiment as T > w c), discretizing the above equation. Y(k + 1)-Y(k) ..... W I X £ ^ + Y(k) = LJC X{k) Y(k + 1) = Y(k) + u>eT [X(k) - Y{k)} (B.63) Equation (B.63) is the final form used in the calculation. Usually, the cut-off frequency w c can be choose in the following way. u>c > 5 U), and, Where, u)t is the useful signal frequency. Appendix C Linearized Theory The linear theory developed by W.R. Sears^ is used to calculate the loading on a small camber, thin airfoil in a small plunging motion. In this theory, the a problem (dealing with the coming flow at a small angle to the airfoil chord), the camber problem, the thickness problem and the plunging motion problem are solved separately and the results are added together. For a plunging N A C A 0015 airfoil (for the case of the coming flow parallel to the airfoil chord), there are no a and camber problems. The thickness effect is an even function (no contribution to the lift and moment), the loading on the airfoil only comes from the motion. So, the problem can be taken as a flat plate airfoil in plunging motion. W.R. Sears^ investigated this problem and gave the results of the pressure distribu-tion, lift and moment to the center of the airfoil. Assume the plunging motion of the flat plate airfoil is: (C.64) x 83 Appendix C. Linearized Theory 84 Where, the airfoil coordinate is: x = —cos0 (C.65) The pressure coefficient along the airfoil surface is: Wh ere, Cp = -{2pV0 sin 6 + ^ tan 6-}e>ui (C.66) y — rt^L 'K^jp)^ Ko{jp)' Ki(jp) and Ko(jp) are the modified Bessel functions of the 2 n d kind. The lift and moment coefficients are: C, = C,.(0.5j> + j r r K ^ r J (C67) C m o = Cmi( „ , • v 1 ^ ' y / . J (C.68) Ki[3P) + A o ( j / i ) Where, the quasi-steady lift and moment coefficients are: Cu = 2TrV0e1"t (C.69) Cmo. = \^V0ejut (C.70) 

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