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Evolution and mixing of asymmetric Holmboe instabilities Carpenter, Jeffrey Richard 2005

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E v o l u t i o n a n d M i x i n g of A s y m m e t r i c H o l m b o e Instabilities by Jeffrey Richard Carpenter  B.Sc.(Eng.), University of Guelph, 2002 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F THE REQUIREMENTS FOR T H E DEGREE OF Master of A p p l i e d Science in The Faculty of Graduate Studies ( C i v i l Engineering)  The University Of B r i t i s h C o l u m b i a September 2005 © Jeffrey Richard Carpenter 2005  11  Abstract W h e n a s t a b l y stratified density interface is embedded i n a region of strong velocity shear, h y d r o d y n a m i c instabilities result. These instabilities lead to the development of a t u r b u l e n t flow i n w h i c h v e r t i c a l m i x i n g of the density field takes place. M u c h previous research i n the field of stratified shear i n s t a b i l i t y has concentrated on w h a t has become the c a n o n i c a l mode i n such flows - the K e l v i n - H e l m h o l t z ( K H ) i n s t a b i l ity. T h i s is one of two instabilities t h a t are present i n w h a t is termed the s y m m e t r i c case, where the centres of the shear layer and density interface coincide. T h e other mode is the H o l m b o e instability, and relies o n the presence of a t h i n density interface centred w i t h i n the shear layer. In the present s t u d y the stratified shear layer is generalized to allow an offset between the centres of the shear layer and the density interface. B y i n c l u d i n g this asymmetry, and keeping the density interface t h i n w i t h respect to the shear layer, the a s y m m e t r i c H o l m b o e i n s t a b i l i t y is found to emerge. T h e objective of the present s t u d y is to examine the e v o l u t i o n a n d m i x i n g behavior of a s y m m e t r i c H o l m b o e ( A H ) instabilities, and to compare the results to the well k n o w n K H a n d H o l m b o e instabilities. T h i s is done by p e r f o r m i n g a series of direct n u m e r i c a l simulations ( D N S ) . D N S has the advantage of d i r e c t l y resolving the smallest scales of v a r i a b i l i t y present i n the flow such t h a t the turbulence and m i x i n g characteristics do not require parameterization. I n this way the m i x i n g behavior is modeled w i t h o u t r e l y i n g on a turbulence closure scheme.  Ill  T h e s i m u l a t i o n results show t h a t there are two different m i x i n g mechanisms present. T h e first is a feature of K H instabilities a n d is characterized b y a significant o v e r t u r n i n g of the density interface.  T h i s leads to the m i x i n g a n d p r o d u c t i o n of  intermediate density fluid causing a final density profile that is layered. T h e second m i x i n g m e c h a n i s m is found i n A H and H o l m b o e instabilities a n d consists of regions of m i x i n g and turbulence p r o d u c t i o n t h a t are located on one or b o t h sides of the density interface. It is comprised of a cusp-like wave t h a t p e r i o d i c a l l y ejects p a r t i a l l y m i x e d fluid from the top or b o t t o m of the interface. Since the i n s t a b i l i t y does not generate o v e r t u r n i n g the density interface is able to ' r e t a i n its i d e n t i t y ' throughout the m i x i n g event. T h e amount of m i x i n g t h a t takes place is found to be strongly dependent on the degree of a s y m m e t r y i n the flow. A s the a s y m m e t r y is increased the a m o u n t of m i x i n g also increases, however, this is not necessarily a n accurate representation of n a t u r a l conditions as the p a i r i n g m e c h a n i s m is expected to play a role i n the d y n a m i c s of the flow. T h e development of three-dimensional secondary structure appears to agree w i t h previous studies (e.g.:  Caulfield and Peltier (2000), Peltier and Caulfield (2003),  Schowalter et al. (1994)), and consists of the f o r m a t i o n of streamwise vortices, part i c u l a r l y i n the g r a v i t a t i o n a l l y unstable regions. T h e presence of the density interface a n d the p e r i o d i c ejection of interfacial fluid were also found t o influence the development of these vortices. T h e formation and b r e a k d o w n of streamwise vortices appears to be a n i m p o r t a n t step i n the t r a n s i t i o n to turbulence. Since n u m e r i c a l models are h a m p e r e d by difficulties i n s i m u l a t i n g the h i g h R e y n o l d s a n d P r a n d t l numbers found i n nature, the geophysical relevance of the present w o r k is also discussed i n this context.  iv  Contents Abstract  ii  Contents  iv  List of Tables  vi  List of Figures List of Symbols Acknowledgements  vii x xiii  1  Introduction 1.1 General Introduction 1.2 Overview  1 1 3  2  Literature Review . . . 2.1 Theoretical Work 2.1.1 Introduction to the Taylor-Goldstein E q u a t i o n and N o r m a l Modes 2.1.2 Review of Linear Stability i n Stratified Shear Flows 2.2 E x p e r i m e n t a l Studies 2.2.1 Laboratory Experiments and 'One-sidedness' 2.2.2 T h e M i x i n g Transition 2.3 Numerical Studies  4 4 4 6 16 16 20 23  3  Background 3.1 Relevant Parameters 3.2 T h e P a r t i t i o n and Transfer of Energy 3.3 M i x i n g Efficiency  25 28 32 37  4  Numerical Solution Method 4.1 Direct Numerical Simulations 4.2 Initial and Boundary Conditions  39 39 41  V  5  Results 44 5.1 T h e Symmetric Case: K H and Holmboe Instabilities 44 5.2 T h e A s y m m e t r i c Case: A H Instabilities and the Effects of A s y m m e t r y 50  6  Discussion 6.1 M i x i n g Behavior 6.2 Relevance to Geophysical Flows  64 64 74  7  Conclusions and Future Work  76  Bibliography  79  List of Tables 4.1  Summary of numerical simulations performed  vii  List of Figures 2.1 2.2 2.3 2.4 2.5  2.6 2.7  3.1 3.2  5.1  Roll-up of a vortex sheet (from Rosenhead (1931)) Stability diagram for profiles examined by Taylor (1931). Contours represent lines of constant nondimensional growth rate Stability diagrams showing contours of dimensionless growth rate ac for the symmetric case Stability diagrams showing contours of dimensionless growth rate aci for an asymmetric case Instabilities observed in the tilting-tube experiments of Thorpe (1968). A one-sided flow is observed in (a), while K H instabilities are shown in (b) Illustration of a mixing layer (from Breidenthal (1981)) T h e m i x i n g transition i n a homogenous liquid jet experiment (from Dimotakis (2000)). T h e change i n mixing behavior is due to the m i x i n g transition. T h i s is achieved by the larger Re flow i n (b). . . .  8 10  t  Schematic of velocity and density distribution parameters Illustration of the relationships between the various energy reservoirs (after Winters et al. (1995)) Energy reservoirs and transfers for the K H simulation. T h e plot in (a) includes the potential energy reservoirs P (solid line), PB (dashed line), and <3> (dotted line). In (b) b o t h irreversible transfers D (solid line), and M (dashed line) are shown along w i t h Ei i n (c). T h e K d (solid line) and K d (dashed line) are plotted i n (d)  12 14  17 18  21 29 35  2  3  5.2  45  Plots of the density field in the K H simulation showing its time evolution. Slices of the xz-pl&ne are taken at y — L /2 w i t h times shown y  5.3  in each plot • • • 46 Plots of the density field in the symmetric Holmboe simulation showing its time evolution. Slices of the x,z-plane are taken at y = L /2 w i t h times shown in each plot 48 y  viii 5.4  Energy reservoirs and transfers for the Holmboe simulation. The plot in (a) includes the potential energy reservoirs P (solid line), PB (dashed line), and $ (dotted line). In (b) b o t h irreversible transfers D (solid line), and M (dashed line) are shown along w i t h Ei i n (c).  T h e K d (solid line) and Ku (dashed line) are plotted i n (d) 5.5 Energy reservoirs and transfers for the e = 0.25 simulation. The plot in (a) includes the potential energy reservoirs P (solid line), PB (dashed line), and $ (dotted line). In (b) both irreversible transfers D (solid line), and M (dashed line) are shown along w i t h Ei i n (c). T h e Kid (solid line) and K (dashed line) are plotted i n (d) 5.6 Plots of the density field in the e — 0.25 simulation showing its time evolution. Slices of the xz-plane are taken at y = L /2 w i t h times shown i n each plot 5.7 Cross-section in the 't/2-plane of the density field w i t h u- vectors overlaid for the e = 0.25 simulation. T h e streamwise location is taken w i t h i n the primary vortex for x = Ah at t — 89 5.8 Energy reservoirs and transfers for the e = 0.50 simulation. The plot i n (a) includes the potential energy reservoirs P (solid line), PB (dashed line), and <I> (dotted line). In (b) both irreversible transfers D (solid line), and M (dashed line) are shown along w i t h Ei in (c). T h e Kid (solid line) and K (dashed line) are plotted i n (d) 5.9 Plots of the density field i n the e = 0.50 simulation showing its time evolution. Slices of the xz-plane are taken at y = L /2 w i t h times shown in each plot 5.10 Cross-section in the yz-plane of the density field w i t h u vectors overlain for the e = 0.50 simulation. T h e streamwise location is taken w i t h i n the primary vortex for x — 4h at t = 89 5.11 Energy reservoirs and transfers for the K H simulation. The plot i n (a) includes the potential energy reservoirs P (solid line), P g (dashed line), and $ (dotted line). In (b) both irreversible transfers D (solid line), and M (dashed line) are shown along w i t h Ei i n (c). The Kid (solid line) and Kj, (dashed line) are plotted in (d) 2  3d  49  51  y  52  id  2d  54  56  y  57  3d  d  5.12 P l o t s of the density field in the e = 1.0 simulation showing its time evolution. Slices of the rrz-plane are taken at y = L /2 with times shown i n each plot  59  61  y  6.1 6.2 6.3  E v o l u t i o n of the Kzd reservoir for each simulation Comparison of the total amount of m i x i n g i n each simulation E v o l u t i o n of K as a fraction of the initial kinetic energy KQ = for each simulation  62 66 67  K(0) 68  ix 6.4 6.5  6.6  Changes i n (a) the total mixing and (b) E for the various asymmetries. 69 Initial and final density characteristics of the asymmetric simulations for e = 0.25 in (a) and (b), e = 0.50 in (c) and (d). In (a) and (c) the distribution of density elements is shown for the initial time (dark bars) and the final time (light bars). In (b) and (d) the initial (dashed line) and final (solid line) density profiles are given. A l l density values c  are given as the difference from the initial mean density Initial and final density characteristics of the symmetric simulations. In (a) and (c) the distribution of density elements is shown for the initial time (dark bars) and the final time (light bars). T h e initial (dashed line) and final (solid line) density profiles are given in (b) and (d). T h e K H simulation corresponds to (a) and (b), whereas the Holmboe is depicted i n (c) and (d). A l l density values are given as the difference from the initial mean density  71  73  List of Symbols R o m a n Characters: c c  C o m p l e x phase speed R e a l component of the complex phase speed  Cj D D  I m a g i n a r y component of the complex phase speed V o l u m e averaged viscous dissipation rate R a t e of energy gain due yo diffusion of the i n i t i a l density profile  d E E{  D i m e n s i o n a l a s y m m e t r y offset C u m u l a t i v e m i x i n g efficiency Instantaneous m i x i n g efficiency  g g'  G r a v i t a t i o n a l acceleration R e d u c e d g r a v i t a t i o n a l acceleration  H  B u o y a n c y flux  r  p  c  h  Shear layer thickness  i  Imaginary unit, yj — 1  J  B u l k R i c h a r d s o n number  K  V o l u m e averaged kinetic energy  K  I n i t i a l kinetic energy  K  K i n e t i c energy of the mean flow  Kid  T w o - d i m e n s i o n a l kinetic energy  K%d k k  T h r e e - d i m e n s i o n a l kinetic energy D i m e n s i o n a l wavenumber U n i t vector i n the vertical (z) direction  L  D o m a i n length i n the streamwise d i r e c t i o n  L  D o m a i n length i n the spanwise d i r e c t i o n  0  x  y  L  D o m a i n length i n the vertical d i r e c t i o n  z  L*  Dimensionless vertical d o m a i n length  M  R a t e of m i x i n g  N  N u m b e r of g r i d points i n streamwise d i r e c t i o n (course grid)  N  N u m b e r of g r i d points i n spanwise d i r e c t i o n (course grid)  N  N u m b e r of g r i d points i n vertical d i r e c t i o n (course grid)  P  T o t a l p o t e n t i a l energy  P  A v a i l a b l e potential energy  x  y  z  A  PB  B a c k g r o u n d potential energy  PT  P r a n d t l number  V  Pressure  R Scale ratio Re R e y n o l d s number ReL . . . . . . . L o c a l R e y n o l d s number t Dimensionless time t* Time U(z) B a c k g r o u n d velocity profile U~o, Ui,U~2 • • L a y e r velocities u V e l o c i t y component i n the streamwise d i r e c t i o n u Three-dimensional velocity field vector u* Dimensionless three-dimensional velocity field vector Au V e l o c i t y difference between layers u V e l o c i t y of the mean flow ud T w o - d i m e n s i o n a l velocity ud Three-dimensional velocity V V o l u m e of c o m p u t a t i o n a l d o m a i n v V e l o c i t y component i n the spanwise d i r e c t i o n w V e l o c i t y component i n the vertical d i r e c t i o n w* Dimensionless vertical velocity component x Streamwise component of C a r t e s i a n coordinate system y Spanwise component of C a r t e s i a n coordinate system z V e r t i c a l component of C a r t e s i a n coordinate system 2  3  Greek Characters: a e e r\ K A v p p*  Dimensionless wavenumber Dimensionless asymmetry factor L o c a l viscous dissipation rate D e n s i t y interface thickness Diffusion coefficient of density Wavelength Coefficient of k i n e m a t i c viscosity Density Dimensionless density  p Ap  Reference density D e n s i t y difference between layers  0  Pi,p2,P3 • • • p(z)  L a y e r densities I n i t i a l density profile  PB(Z) Ptop Pbottom a  B a c k g r o u n d density profile A v e r a g e density o n top b o u n d a r y of d o m a i n A v e r a g e density o n b o t t o m b o u n d a r y of d o m a i n Dimensionless growth rate  r  Arbitrary time scale  L i s t of A b b r e v i a t i o n s : AH  Asymmetric Holmboe  DNS  D i r e c t numerical s i m u l a t i o n  KH SW03 TG  Kelvin-Helmholtz S m y t h a n d W i n t e r s (2003) Taylor-Goldstein  Xlll  Acknowledgements T h i s thesis w o u l d not be anywhere near the p r o d u c t it is w i t h o u t the help of m a n y people. I ' d first like to acknowledge not only the guidance and insight p r o v i d e d b y my supervisor G r e g Lawrence b u t also his generosity i n p r o v i d i n g the freedom for me to pursue any a n d a l l opportunities as they arose. T h e help of B i l l S m y t h i n b o t h g e t t i n g the m o d e l w o r k i n g and p r o v i d i n g i n p u t on the results has been invaluable. It is m y hope t h a t this correspondence w i l l continue a n d grow i n the future. I have also benefited from the p r o g r a m m i n g prowess and d e t e r m i n a t i o n of Sheng L i i n the dreaded debugging stages of this project. I've h a d m a n y entertaining and helpful discussions w i t h T e d Tedford on topics closely related to this thesis. F i n a l l y , thank you to m y f a m i l y a n d friends.  1  Chapter 1 Introduction 1.1  General Introduction  T h i s thesis studies the process of turbulence generation a n d m i x i n g of a stably stratified density interface. There are m a n y different mechanisms b y w h i c h turbulent m i x i n g is accomplished, a n d a focus here is placed o n the case i n w h i c h m i x i n g results from the presence of a velocity shear. T h i s shear allows h y d r o d y n a m i c instabilities to form on the density interface w h i c h lead the flow to a t u r b u l e n t state whereby irreversible m i x i n g of the density profile is accomplished.  T h e character of this  t r a n s i t i o n process as well as the resulting turbulence is h i g h l y dependent on the type of i n s t a b i l i t y t h a t forms. T h i s is i n t u r n dependent o n b o t h the v e l o c i t y a n d density d i s t r i b u t i o n s . Since density stratification is v i r t u a l l y u b i q u i t o u s i n the n a t u r a l environment the suppression of turbulence i n regions of stable stratification plays an i m p o r t a n t role i n the physics of m a n y geophysical systems. It reduces v e r t i c a l transport rates to near-molecular levels and serves to isolate different fluid layers from one another. W h e n enough energy c a n be supplied by a velocity shear to generate instabilities, the result is enhanced rates of v e r t i c a l transport (i.e., a m i x i n g of the density interface). T h i s scenario has major environmental i m p l i c a t i o n s r a n g i n g f r o m the dispersion of atmospheric p o l l u t a n t s , to the s u p p l y of nutrients to the p h o t i c zone i n lakes a n d  2 oceans. T h e r e are a n u m b e r of documented processes by w h i c h i n s t a b i l i t y results from the shearing of a stable stratified layer. In oceanographic a n d l i m n o l o g i c a l scenarios the m i x i n g is often forced to occur across relatively sharp density gradients.  The  formation of sharp density interfaces can be due to a n u m b e r of different  fluid  mechanical processes i n c l u d i n g shear i n s t a b i l i t y itself (see Turner and Caulfield  (1973);  Peltier  (2003)). T h e maintenance of these sharp gradients is due p r i m a r i l y to  their h i g h levels of static s t a b i l i t y a n d the h i g h P r a n d t l n u m b e r environments they are found i n (i.e., temperature and salt stratified water bodies). T h e s t a b i l i t y characteristics of sharp density gradient shear layers p e r m i t a departure from the c a n o n i c a l K e l v i n - H e l m h o l t z ( K H ) m o d e of i n s t a b i l i t y and allows for the f o r m a t i o n of H o l m b o e modes. T h e m i x i n g behavior of H o l m b o e modes is a relatively u n e x p l o r e d area of research w i t h the o n l y direct q u a n t i t a t i v e study being t h a t of Smyth and Winters  (2003) ( S W 0 3 ) . T h i s w o r k .focused on the s y m m e t r i c  H o l m b o e i n s t a b i l i t y (simply referred to herein as the H o l m b o e i n s t a b i l i t y ) , a case t h a t results w h e n the centre of the shear layer a n d the centre of the density interface coincide. S W 0 3 found t h a t for the single point i n parameter space studied, the H o l m b o e i n s t a b i l i t y was able to a c c o m p l i s h a greater degree of m i x i n g t h a n the K H i n s t a b i l i t y . T h i s highlights the possible i m p o r t a n c e of H o l m b o e modes i n c o n t r i b u t i n g to the m i x i n g of sharp density interfaces. In this s t u d y the H o l m b o e mode is generalized to the a s y m m e t r i c case, where a n offset exists between the centre of the shear layer a n d the density interface. T h e r e s u l t i n g instabilities are referred to as a s y m m e t r i c H o l m b o e ( A H ) i n s t a b i l i ties. T h e m i x i n g behavior of A H instabilities has not been s t u d i e d previously. In  3 fact, the A H i n s t a b i l i t y has thus far existed p r i m a r i l y i n theoretical circles, w i t h only a handful of observations i n l a b o r a t o r y experiments. Therefore, the following chapters are devoted to s t u d y i n g the e v o l u t i o n a n d m i x i n g behavior of A H i n s t a b i l i ties, a n d c o m p a r i n g the findings to previous l a b o r a t o r y experiments and theoretical predictions.  1.2  Overview  T h i s s t u d y constitutes a first step at evaluating the i m p o r t a n c e of a s y m m e t r y i n the development of instabilities and m i x i n g i n stratified shear flows. In the course of this study, w h i c h includes a comparison between K H , H o l m b o e , a n d a s y m m e t r i c H o l m b o e instabilities, I have relied on the d i r e c t i o n set out i n S W 0 3 . In S W 0 3 a c o m p a r i s o n is made between the K H a n d H o l m b o e instabilities i n terms of m i x i n g a n d turbulence generation i n the same region of parameter space. A s the present s t u d y c a n be viewed as an extension of the results found i n S W 0 3 , it is i m p o r t a n t to give proper acknowledgment. Interpretation of the s i m u l a t i o n results was done i n c o l l a b o r a t i o n w i t h my supervisor G r e g Lawrence, a n d this work s h o u l d not be viewed solely as the work of myself as a n exclusive author.  4  Chapter 2 Literature Review 2.1 2.1.1  Theoretical Work Introduction to the Taylor-Goldstein Equation and Normal Modes  M u c h early w o r k on density stratified shear instabilities focused on the theoretical p r e d i c t i o n of unstable flow configurations. T h i s led to the development of what is now k n o w n as the T a y l o r - G o l d s t e i n ( T G ) equation. B a s e d o n the pioneering work of  Taylor (1931) a n d Goldstein (1931), the T G equation describes the fate of linear twod i m e n s i o n a l , m o n o c h r o m a t i c , sinusoidal disturbances i n a density stratified fluid. T h e p r i n c i p l e b e h i n d the T G equation is t h a t i f an infinitesimally s m a l l disturbance applied to given basic velocity and density configurations is found to grow i n time, then the flow is deemed unstable, and results i n the g r o w t h of a n instability. T h i s i n s t a b i l i t y does not necessarily lead to turbulence, m e r e l y a change i n the basic flow p a t t e r n . T h i s d e s c r i p t i o n w i l l become more clear i n the following paragraphs where the m e t h o d is described i n more detail. T o determine the s t a b i l i t y of a given flow, the variables of interest are decomposed into m e a n a n d fluctuating components, where the m e a n components become representative of the basic flow. Once the decompositions are s u b s t i t u t e d into the  5  equations of m o t i o n , t h e resulting equations are linearized b y neglecting the p r o d ucts of t h e p e r t u r b a t i o n terms.  Just as neglecting t h e higher order terms i n a  T a y l o r ' s series expansion, the resulting s o l u t i o n is o n l y v a l i d for s m a l l deviations from the basic state. T h i s requires that the p e r t u r b a t i o n quantities do not become too large.  L i n e a r theory is thus only usually assumed to a p p l y to t h e evolution  of infinitesimal disturbances.  Since - even after l i n e a r i z i n g - t h e solutions of t h e  resulting equations are often extremely difficult to solve a n a l y t i c a l l y , a number of s i m p l i f y i n g assumptions are made.  These most often include s o l v i n g for the two-  dimensional flow of a n incompressible, inviscid fluid w i t h negligible diffusion of the scalar field.  It is also convenient to make the Boussinesq a p p r o x i m a t i o n , w h i c h  states t h a t differences i n density can be neglected i n a l l i n e r t i a l terms, b u t i n c l u d e d i n the b u o y a n c y t e r m . J u s t i f i c a t i o n for the use of t h e a p p r o x i m a t i o n i n this s t u d y can b e seen b y the s m a l l density differences t h a t are t y p i c a l l y observed i n flows of p r a c t i c a l i m p o r t a n c e , p a r t i c u l a r l y Ap/p = O ( 1 0 ~ ) i n t y p i c a l ocean environments 3  (Turner, 1973). O n c e t h e a p p r o x i m a t i o n s a n d procedures o u t l i n e d above have been made the result is a single linear homogeneous p a r t i a l differential equation describing the evolution of s m a l l perturbations from the basic flow. T h e most c o m m o n m e t h o d of o b t a i n i n g solutions for this equation is by the m e t h o d of n o r m a l modes. T h i s m e t h o d searches for wavelike solutions i n x a n d t whose a m p l i t u d e varies as a function of z. Specifically, solutions are assumed to be of the form  4(x,z,t)  = 4>(z)e ^ \ ik  ct  (2.1)  where k is t h e wavenumber, c is the phase speed o f t h e wave, a n d 0 represents the  6 p e r t u r b a t i o n q u a n t i t y of interest (e.g.: vertical velocity, density). T h i s leads to the T G equation N  2  i(U-c)  2  U" (U-c)  4> =  (2.2)  0,  where the primes denote differentiation w i t h respect to z, a n d the basic flow is represented b y the velocity profile u = (U(z),0) a n d the density profile p = T h e buoyancy frequency is represented by N(z) (2.2)  = (gp'/po) ^ 1  2  piz).  T h e solutions to  constitute a n eigen-problem i n v o l v i n g the u n k n o w n eigenfunction, 4>(z), a n d  eigenvalues, c = c + iCi, w h i c h m a y be complex. E a c h s o l u t i o n pair {4>,c} is referred r  to as a n o r m a l mode (or s i m p l y just a mode), a n d correspond to specific values of the wavenumber a n d b u l k R i c h a r d s o n number, J = g'h/(Au) , 2  where g' =  (Ap/p )g 0  is the reduced g r a v i t a t i o n a l constant. Here Ap, Au are representative density and velocity scales respectively, h is a length scale, a n d po is a reference density.  The  basic flow is deemed to be unstable when the eigenvalue has a positive i m a g i n a r y component, i.e., Cj > 0. T h i s can be seen by s u b s t i t u t i n g into (2.1)  and noting that  the s o l u t i o n is p r o p o r t i o n a l to a factor growing e x p o n e n t i a l l y i n t i m e (i.e., (j) oc  e ). kCit  For this reason, the growth rate of the mode is defined b y kci a n d the phase speed is given b y c . r  2.1.2  Review of Linear Stability in Stratified Shear Flows  In general, solutions to the T a y l o r - G o l d s t e i n equation (2.2)  must be obtained by  c o m p l i c a t e d methods i n v o l v i n g n u m e r i c a l schemes for a l l but the simplest cases. T h e difficulty arises from the fact that the coefficients are not constant. However, if piecewise-linear profiles are chosen to represent v e l o c i t y a n d density then two of  7 the coefficients v a n i s h a n d (2.2) simplifies to  0" - k j> = 0 2  w h i c h allows for a n a l y t i c a l solutions i n most cases.  (2.3)  D u e to this simplification, a  great m a j o r i t y of w o r k i n developing solutions has been performed on profiles t h a t are piecewise-linear (see for example Howard (1991); Haigh and Lawrence  and Maslowe  (1973); Lawrence  et al.  (1999)).  T h e simplest scenario of shear flow i n s t a b i l i t y t h a t c a n be analysed using linear theory is the homogenous case where the velocity profile is represented b y  if z > 0  (2.4)  if z < 0  B y a p p l y i n g the b o u n d a r y c o n d i t i o n t h a t the v e l o c i t y must vanish as z —> ± o o , and the m a t c h i n g c o n d i t i o n that the pressure is continuous at the discontinuity, a dispersion r e l a t i o n is obtained for the c o m p l e x phase speed (see Drazin (1982, §28) for details). T h e results, w h i c h were first developed b y Helmholtz  and  Reid  (1868),  show t h a t the g r o w t h rate is d i r e c t l y p r o p o r t i o n a l to the wavenumber, and the phase speed s t a t i o n a r y w i t h respect to the m e a n flow. T h i s corresponds to the i n s t a b i l i t y of a v o r t e x sheet, a n d is characterized i n the early stages of nonlinear growth by a rolling-up of the sheet into a periodic t r a i n of billows, as s h o w n i n figure 2.1 below. T h i s is a special case of the density stratified flow s t u d i e d b y Kelvin  (1871), w h i c h  consists of two layers of velocity U\ a n d U~2, each w i t h density pi a n d p , respectively. 2  In this case the results of linear s t a b i l i t y predict t h a t the flow is unstable for a range  8  Figure 2.1: Roll-up of a vortex sheet (from Rosenhead  (1931)).  of wavenumbers satisfying the relation kpip (U~i — U2) > g(p\ — p )2  2  2  T h i s shows  t h a t the presence of stable stratification is responsible for s t a b i l i z i n g the flow for a certain range of wavenumbers. T h e next major developments came from Taylor  (1931) a n d Goldstein  (1931)  i n their analysis of piecewise-linear profiles t h a t i n c o r p o r a t e d shear layers of finite thickness over layers of stable density stratification. T h e m a i n result of these works was the development of the T a y l o r - G o l d s t e i n equation (2.2), but they also revealed part of the subtle relationship that density stratification c a n p l a y i n d e t e r m i n i n g the s t a b i l i t y of a flow.  Stable stratification is generally considered to exhert a  s t a b i l i z i n g influence on a given flow. T h i s can be seen i n the example of the vortex sheet just disscussed, where the a d d i t i o n of stable density stratification stabilized the flow for longer wavelengths, namely A > •K(A.U) / g'. It c a n also be thought of 2  i n t u i t i v e l y b y n o t i n g t h a t it takes a d d i t i o n a l p o t e n t i a l energy i n m o v i n g a fluid parcel across the density interface.  Taylor  (1931) showed t h a t to t h i n k of stratification  solely as a s t a b i l i z i n g influence was an oversimplification; t h a t it is possible for stable stratification to render a previously stable flow unstable.  T h i s can be seen  by e x a m i n i n g the following profiles  U(z) = z,  p(z)={  Pi  for z > d  p  for d < z < -d  P3  for z < —d  2  (2-5)  where p = pi + A p a n d pz = pi + 2 A p . T h e basic flow defined above corresponds to 2  plane C o u e t t e flow w i t h three homogenous layers of different density superimposed. P e r f o r m i n g a linear s t a b i l i t y analysis on these profiles yields the s t a b i l i t y d i a g r a m  10  a Figure 2.2: Stability diagram for profiles examined by Taylor (1931). Contours represent lines of constant nondimensional growth rate.  in figure 2.2. From the stability diagram it can be seen that the flow is stable for J = 0, corresponding to the homogenous case. It should be noted however, that Couette flow is only stable in the inviscid limit - meaning that for large enough (but finite) values of the Reynolds number, viscosity will destabilize the flow (see Drazin and Reid (1982)). In this respect, Taylor found that the flow was destabilized by the addition of stratification. It can also be seen in figure 2.2 that the maximum growth rate is attained at a value of Jft*0.52. This indicates that the mechanism for instability relies on a certain amount of stratification to grow. Howard and Maslowe  11 (1973) give the following explanation for the mechanism: The motion consists approximately of two internal waves on the two density interfaces, the upper one propagating to the left relative to the local flow speed at such a velocity as to stand still relative to the origin, and the lower one similarly standing still relative to the origin.  The  pressure fields of each wave perturb the other a little, and in such a phase that they amplify each other. Holmboe  (1962) developed further the ideas of both stratification as exerting  a possibly destabilizing influence, and the interaction of waves as a mechanism of linear growth. Using a method similar to normal modes, Holmboe examined a case where the density interface is much smaller than, and centred within, a larger shear layer. He represented this flow with the following profiles:  UQ for z > d U(z)  =1  ( pi  ^z  for d<z<  Uo for z < -d  -d  for z > 0  p(z) = <  I  (2.6)  p2 for z < 0  ^  The results of Holmboe's analysis showed that the flow was unstable to two different types of instabilities: (i) the Kelvin-Helmholtz instability, characterized by a stationary phase speed with respect to the mean flow also present in the unstratified case, and (ii) Holmboe's instability, an oscillatory instability with nonzero phase speed that is the only instability to occur for J > 0.07. These instabilities occupy two different regions of the J,o>plane as seen in the stability diagram of figure 2.3. Holmboe's instability was predicted to consist of two simultaneously unstable modes that had equal growth rates and equal but opposite phase speeds.  12  13 T h e m e c h a n i s m responsible for the linear g r o w t h of H o l m b o e ' s i n s t a b i l i t y was first discussed b y Holmboe (1994).  (1962) and further elaborated u p o n b y Baines  and  Mitsudera  S i m i l a r to the T a y l o r case e x a m i n e d above, it consists of the interaction  of disturbances on the v o r t i c i t y interfaces (the locations where the v o r t i c i t y of the shear layer, dU/dz, interface.  changes a b r u p t l y ) w i t h i n t e r n a l g r a v i t y waves o n the density  I n this respect, two H o l m b o e modes are formed t h a t travel at speeds  comparable to i n t e r n a l gravity waves, and are centred above a n d below the density interface. M o t i v a t e d by e x p e r i m e n t a l results (cf. §2.2.1), Lawrence  et al. (1991) modified  H o l m b o e ' s original m o d e l (2.6) by i n c l u d i n g a density interface t h a t is offset from the centre of the shear layer. T h e profiles now take the form  (2.7)  UQ for  z  < —d  T h e s t a b i l i t y d i a g r a m for one such a s y m m e t r i c case is s h o w n i n figure 2.4. F r o m this it c a n be seen t h a t by i n c l u d i n g this offset the g r o w t h rate of one of the two modes of i n s t a b i l i t y is preferentially destabilized w i t h respect to the other. T h e stronger mode b e i n g t h a t o n the upper side of the density interface where a greater amount of shear layer v o r t i c i t y resides, and the weaker m o d e o c c u p y i n g the lower v o r t i c i t y side of the interface. O t h e r interesting features t h a t result i n c l u d e the vanishing of the region of zero phase speed that is associated w i t h the K H instability, as well as the different wavenumbers of m a x i m u m g r o w t h rate for each of the two modes. T h e s p l i t t i n g of the regions of i n s t a b i l i t y into two separate modes w i t h a nonzero  14  15 phase speed indicates that, according to the framework of Baines  and  Mitsudera  (1994) , the r e s u l t i n g modes of i n s t a b i l i t y are H o l m b o e - l i k e modes resulting from the i n t e r a c t i o n of v o r t i c i t y disturbances a n d internal waves. T h e linear s t a b i l i t y results of Lawrence  et al. (1991) was extended by  Haigh  (1995) to i n c l u d e b o t h an offset i n the density interface and a s y m m e t r i c a l l y placed h o r i z o n t a l (free-slip) boundaries. L i n e a r s t a b i l i t y analysis was performed on piecewiselinear a n d h y p e r b o l i c tangent velocity and density profiles t h a t i n c l u d e d the effects of scalar diffusion a n d viscosity. T h e h y p e r b o l i c tangent profiles b e i n g found to more accurately characterize the d i s t r i b u t i o n s found i n n a t u r a l settings (for example i n the saltwedge flows of Yonemitsu  et al. (1996)), a n d to be a close a p p r o x i m a t i o n  to the error function profiles that result from molecular diffusion. L i n e a r s t a b i l i t y analysis of s m o o t h profiles accounting for viscosity h a d been accomplished earlier by Maslowe  and Thompson  (1971) where it was found t h a t viscosity is o n l y effective  i n d a m p i n g w e a k l y amplified waves near the s t a b i l i t y boundaries. T h i s agrees w i t h the results of Haigh (1995) who found t h a t decreasing the R e y n o l d s number led to a shift to lower wavenumbers of the most amplified mode, as well as a n a r r o w i n g of the s t a b i l i t y boundaries.  However, the R i c h a r d s o n number is the d o m i n a n t parameter  governing the s t a b i l i t y of u n b o u n d e d shear flows (Maslowe  and Thompson,  1971).  T h e i n t r o d u c t i o n of continuous h y p e r b o l i c tangent profiles d i d not greatly change the results found i n the piecewise-linear case of Lawrence  et al. (1991) from (2.7).  16  2.2 2.2.1  Experimental Studies Laboratory Experiments and 'One-sidedness'  A t t e m p t s to s t u d y H o l m b o e ' s i n s t a b i l i t y a n d the theoretical predictions of Holmboe (1962) i n the l a b o r a t o r y were often found to result i n a 'one-sidedness'. T h i s t e r m is now used i n the literature to describe flows i n w h i c h the disturbances a n d m i x i n g generated as a result of i n s t a b i l i t y appear i n only one of the two layers (see worthy and Browand  (1975); Lawrence  et al. (1991); Yonemitsu  Max-  et al. (1996)). It  has been identified as a c o m m o n feature of m i x i n g layer facilities, a n d i s due to the a s y m m e t r y i n t r o d u c e d between the centre of the shear layer a n d the centre of the density interface (Lawrence  et al. (1991, 1998)).  T h e first l a b o r a t o r y experiments t h a t appeared to confirm the existence of instabilities other t h a n K H i n a stratified shear flow were those of Thorpe  (1968). U s i n g  a t i l t i n g tube, T h o r p e was able to create a nearly parallel flow for a short p e r i o d of t i m e u n t i l surges formed at the channel ends reached the centre of the tube. Some of the best k n o w n photographs of K H billows were t a k e n from these experiments. These are s h o w n i n figure 2.5b along w i t h some of the observed one-sided instabilities (figure 2.5a), where the disturbances a n d m i x i n g were confined to the upper side of the density interface. These one-sided instabilities appeared as cusp-like waves t h a t travelled w i t h respect to the mean flow. Since H o l m b o e i n s t a b i l i t i e s are predicted to comprise of b o t h left a n d right p r o p a g a t i n g modes, these i n s t a b i l i t i e s do not appear to be of the t y p e H o l m b o e predicted (though they resemble his predictions i n m a n y other respects).  I n this s t u d y Thorpe  (1968) notes t h a t t h r o u g h o u t the evolution  and m i x i n g i n these one-sided experiments the density interface 'retains its i d e n t i t y ' .  17  Figure 2.5: Instabilities observed in the tilting-tube experiments of Thorpe (1968). A one-sided flow is observed in (a), while K H instabilities are shown in (b).  18 In this respect the m i x i n g is fundamentally different t h a n t h a t present i n K H instabilities, where it is accomplished by a complete o v e r t u r n i n g a n d subsequent m i x i n g of the interface. T h e o n l y l a b o r a t o r y investigation t h a t has been carried out w i t h the intent of s t u d y i n g a s y m m e t r i c stratified shear layers is t h a t of Lawrence  et al. (1991). T h e  experiments were performed i n m i x i n g layer facilities, a n d consist of two streams of fluid w i t h densities p\ a n d P2, and velocities U\ a n d U2 separated b y a splitter plate as s h o w n i n figure 2.6. B y adjusting the v e l o c i t y difference between the two  F i g u r e 2.6: I l l u s t r a t i o n of a m i x i n g layer (from Breidenthal  (1981)).  streams, the s t a b i l i t y characteristics of the flow can be e x a m i n e d . A s y m m e t r y results i n m i x i n g layer experiments due to the f o r m a t i o n of differing b o u n d a r y layers on the h i g h - a n d low-speed side of the splitter plate l e a d i n g to a greater amount of v o r t i c i t y i n the high-speed layer (as described i n Lawrence a s y m m e t r y of the shear layer i n the experiments of Lawrence  et al. (1998)).  The  et al. (1991) was found  to produce instabilities w i t h 'one-sided' behavior, a n d were identified as a s y m m e t r i c instabilities.  19 Since the lower layer was salt stratified, the thickness of the density interface is expected to be m u c h less t h a n the v o r t i c i t y layer, a n d H o l m b o e - l i k e instabilities are expected for sufficiently h i g h levels of stratification. T h i s appears to be the case as the instabilities t o o k the form of cusp-like waves w h i c h o c c a s i o n a l l y ejected dense fluid into the d o m i n a n t layer. T h e m i x i n g characteristics of the r e s u l t i n g instabilities were not quantified however. S i m i l a r experiments were conducted by Schowalter  et al. (1994) w h i c h focused  on the f o r m a t i o n of secondary instabilities i n homogenous a n d stratified m i x i n g layers.  I n this s t u d y a m i x i n g layer facility was used w i t h a serrated splitter plate  edge to inject spanwise perturbations, while the p r i m a r y streamwise i n s t a b i l i t y was m e c h a n i c a l l y forced by m o v i n g the splitter plate at a c e r t a i n frequency. T h e experiments were also noted to have a degree of a s y m m e t r y associated w i t h t h e m , t h o u g h the a m o u n t of a s y m m e t r y is not quantified. T h e secondary instabilities took the form of streamwise vortices, as have been observed i n a n u m b e r of other studies (e.g.: Breidenthal  (1981); Bernal  and Roshko (1986); Caulfield  and Peltier  (2000)).  In the more h e a v i l y stratified experiments, streamwise vortices were enhanced b y the unstable stratification i n the overturned regions of the r e s u l t i n g billows. These streamwise vortices are expected to lead the flow to a t u r b u l e n t t r a n s i t i o n .  This  has been confirmed i n the case of stratified K H i n s t a b i l i t i e s i n the n u m e r i c a l simulations of Caulfield  and Peltier  (2000) where the t r a n s i t i o n to turbulence led to a  r a p i d increase i n the rate of m i x i n g of the scalar field.  20 2.2.2  T h e M i x i n g Transition  The role that secondary structures and the onset of small-scale three-dimensional motions have on the mixing of fluid in homogenous shear layers was studied experimentally by Breidenthal (1981). Here it was found that the amount of mixing between the two streams increased by an order of magnitude once small-scale threedimensional motion set in. This phenomenon has been termed the 'mixing transition', and is found to be a function of the local Reynolds number, Rei, (Dimotakis, 2000). In fact, Dimotakis (2000) predicts that the mixing transition occurs in all flows for which an appropriately defined RCL > 1 — 2 x 10 . The significance of 4  this result, Dimotakis (2000) states, is that Re^ becomes large enough such that a fully-developed turbulent flow is achieved. It is further conjectured that "a necessary condition for fully-developed turbulence . . . is the existence of a range of scales that are uncoupled from the large scales, on the one hand, and free from the effects of viscosity, on the other" (Dimotakis, 2000). Therefore, the mixing transition is not the result of small-scale three-dimensional motions alone, but the generation of turbulence at a sufficiently high Rer, so that it may become fully-developed. This transition denotes a significant change in the dependence of the mixing rate on Re and Pr, where Pr = V/K is the ratio of the two diffusivities of momentum (or vorticity), v, and that of the scalar quantity, K. The study of Breidenthal (1981) concluded that the effect of Pr on the mixing at low Re is pronounced, while at high Re the mixing changes by only a factor of 2 or less for a three orders of magnitude variation in Pr. Stated another way, above the mixing transition the generation of mixed fluid is essentially independent of Pr. There are numerous examples of flows for which this mixing transition exists,  21 and can be recognized by a qualitative change in the composition of the scalar field. This is illustrated in figure 2.7 for the case of a homogenous liquid jet. Here figure (a)  (b)  Figure 2.7: The mixing transition in a homogenous liquid jet experiment (from Dimotakis (2000)). The change in mixing behavior is due to the mixing transition. This is achieved by the larger Re flow in (b).  2.7a corresponds to the scalar field for a flow that has developed instabilities and turbulence, but has not undergone the mixing transition. The effect of the mixing transition on the scalar field for a similar experiment at a larger Re is shown in figure 2.7b, where large amounts of mixed fluid and the presence of small-scale motion can be seen.  22 A l t h o u g h the m i x i n g t r a n s i t i o n has been well d o c u m e n t e d i n homogenous  flows,  it appears t h a t l i t t l e w o r k has been undertaken to s t u d y this phenomenon i n stratified shear layers. A qualitative change i n the m i x i n g behavior of the scalar was noted i n the experiments of Pawlak  and Armi  (1998).  field  These consisted of a  spatially-accelerating stratified shear generated b y - t h e down-slope flow of a dense layer over a t o p o g r a p h i c sill. E n t r a i n m e n t between the two layers occurs t h r o u g h the K H i n s t a b i l i t y w h i l e m i x i n g is observed downstream, after the f o r m a t i o n of turbulence on the interface. A t the final downstream p o s i t i o n of the developing m i x i n g layer a ReL ~ 5 x 1 0 is attained, a n d the beginnings of the m i x i n g t r a n s i t i o n are 3  observed. E v i d e n c e of a m i x i n g t r a n s i t i o n is present i n the n u m e r i c a l simulations of K H instabilities i n Caulfield  and Peltier  (2000). These results showed heightened levels  of m i x i n g t h a t accompanied the onset of small-scale t u r b u l e n t motions after  the  b r e a k d o w n of b o t h the p r i m a r y b i l l o w and the secondary streamwise vortices. T h i s behavior is thought to be linked to the m i x i n g t r a n s i t i o n as the flow is shown to be close to the c r i t i c a l R e y n o l d s number c r i t e r i o n w i t h ReL and Caulfield  (2003)).  ~  10  4  (see  Peltier  However, the sudden j u m p i n m i x i n g associated w i t h the  t r a n s i t i o n can be seen to decrease i n strength as the stratification is increased.  It  is possible t h a t the g r o w t h of ReL w i t h the shear layer thickness is sufficiently suppressed b y the stratification such t h a t the c r i t i c a l m i x i n g t r a n s i t i o n criterion is not reached. I n this sense, the m i x i n g t r a n s i t i o n m a y not play as strong a role i n the d y n a m i c s of n a t u r a l l y developing shear instabilities i n s t r o n g l y stratified fluids as it does w i t h homogenous and weakly stratified flows.  23  2.3  N u m e r i c a l Studies  A s the state of computer technology is ever advancing, so too is the a b i l i t y of n u m e r i c a l models i n accurately s i m u l a t i n g fluid m e c h a n i c a l processes.  However, it  is o n l y relatively recently that simulations relevant to the present s t u d y have been feasible. T h e first such s t u d y is that of Smyth et al. (1988), where the nonlinear development of the H o l m b o e i n s t a b i l i t y was s i m u l a t e d i n two-dimensions. T h i s confirmed previous predictions t h a t the i n s t a b i l i t y consisted of two cusp-like waves that travel on the density interface.  M i x i n g i n the H o m b o e 'waves' was observed to occur  t h r o u g h t h i n wisps of fluid being d r a w n off the cusp areas a n d subsequently t r a i n e d into the upper a n d lower layers.  en-  However, the use of a two-dimensional  m o d e l was not able to determine whether significant m i x i n g a n d turbulence was generated by the instability. These questions were later answered, i n part, b y the recent findings i n and Winters  Smyth  (2003), where m i x i n g a n d turbulence i n K H a n d H o l m b o e instabilities  were e x a m i n e d at a single point i n parameter space. T h e results showed that i n this case the H o l m b o e i n s t a b i l i t y is responsible for higher levels of m i x i n g a n d turbulence generation t h a n the K H instability. T h i s result was rather unexpected considering the long-standing belief t h a t the absence of significant regions of overturning precluded it from h i g h levels of m i x i n g (Thorpe,  1987). T h e l o w linear g r o w t h rate of  the H o l m b o e i n s t a b i l i t y was also expected to be a n i n d i c a t o r t h a t it d i d not develop levels of turbulence and m i x i n g comparable to the K H instability. However, the linear g r o w t h rate m a y not be a reliable i n d i c a t o r of the m i x i n g behavior of a p a r t i c u l a r i n s t a b i l i t y , as it is o n l y applicable to the linear  g r o w t h of the  unstable  24 mode a n d says n o t h i n g of the nonlinear development. Indeed, m i x i n g is invariably a nonlinear phenomenon. O t h e r i m p o r t a n t advancements i n the s t u d y of m i x i n g a n d turbulence i n stratified shear flows were made by Caulfield  and Peltier  (2000).  T h i s s t u d y focused  on the o r i g i n a n d development of secondary i n s t a b i l i t i e s i n homogenous a n d stratified K H billows.  These were found to take the form of streamwise vortices a n d  are expected to lead d i r e c t l y to the onset of t u r b u l e n t m o t i o n . T h e origin of the streamwise vortices is dependent on the value of the b u l k R i c h a r d s o n number, J , w i t h different g r o w t h mechanisms present for cases of s t r o n g or weak stratification. In the K H flows e x a m i n e d the onset of turbulence also m a r k e d a phase of heightened scalar m i x i n g - a possible i n d i c a t i o n of the m i x i n g t r a n s i t i o n (as discussed i n Dimotakis  (2000)). T h i s behavior became less apparent as J was increased  and Caulfield,  (Peltier  2003), resulting i n a larger percentage of the m i x i n g o c c u r r i n g i n the  preturbulent phase (i.e., the phase of the flow preceding the t r a n s i t i o n to t u r b u lence). T h e i m p o r t a n c e of this preturbulent phase was recognized i n S W 0 3 , where it was found t h a t the m a j o r i t y of the m i x i n g occurred.  25  Chapter 3 Background M a n y flows of a geophysical and engineering n a t u r e involve the h o r i z o n t a l shearing of a s t a b l y stratified density interface. U n d e r certain c o n d i t i o n s this flow w i l l become unstable a n d develop h y d r o d y n a m i c instabilities. These i n s t a b i l i t i e s are responsible for leading the flow to a turbulent state whereby irreversible m i x i n g of the density field is a c c o m p l i s h e d . T h e exact form that these instabilities take - and hence, the evolution of the flow - is dependent on a number of parameters describing the i n i t i a l velocity a n d density distributions. Studies to date have p r e d o m i n a n t l y focused o n the case i n w h i c h the centre of the shear layer a n d the density interface coincide - referred to herein as the s y m metric case.  I n this case two different instabilities are possible depending u p o n  the relative strength of the stratification and the thickness of the density interface. W h e n levels of stratification are sufficiently weak the K e l v i n - H e l m h o l t z ( K H ) i n s t a b i l i t y results.  T h e K H i n s t a b i l i t y is characterized b y a p e r i o d i c r o l l - u p of the  density interface caused b y the concentration of shear layer v o r t i c i t y into discrete billows. T h e i n s t a b i l i t y has the appearance of a s t a t i o n a r y b r e a k i n g wave i n w h i c h the density interface curls up to form s t a t i c a l l y unstable regions t h a t move w i t h the m e a n v e l o c i t y of the flow. T h e unstable regions of the b i l l o w continue to r o l l u p u n t i l a saturated a m p l i t u d e is reached. D u r i n g this phase of essentially two-dimensional development, secondary instabilities begin to grow.  A s the b i l l o w becomes satu-  26 rated the i n t e r a c t i o n of the secondary instabilities is thought to lead the flow into a t u r b u l e n t collapse, after w h i c h a r e l a m i n a r i z a t i o n occurs (Caulfield 2000; Peltier  and Caulfield,  and  Peltier,  2003).  A s the level of stratification is increased, and the density interface is kept relatively t h i n i n c o m p a r i s o n to the shear layer thickness, the shear is no longer able to o v e r t u r n the density interface and H o l m b o e i n s t a b i l i t y develops (Holmboe,  1962).  H o l m b o e i n s t a b i l i t y is thought to be the result of a resonant i n t e r a c t i o n between disturbances i n the shear layer v o r t i c i t y and i n t e r n a l g r a v i t y waves on the density interface (Baines  and Mitsudera,  1994). Since the density gradient is sufficiently  s t r o n g a n d t h i n , it can be thought of as d i v i d i n g the shear layer v o r t i c i t y into two equal segments (since we are dealing w i t h the s y m m e t r i c case) t h a t have o n l y a l i m i t e d i n t e r a c t i o n w i t h one another.  F o r this reason, i n s t a b i l i t y develops on each  side of the density interface, and is characterized by cusp-like waves t h a t travel at equal and opposite speeds w i t h respect to the m e a n flow. E a c h of these unstable waves is referred to as a H o l m b o e mode. W h e n a s y m m e t r y is i n t r o d u c e d (i.e., the velocity a n d density profiles are offset from one another), and the density interface remains t h i n , linear theory suggests t h a t one of the two unstable H o l m b o e modes become preferentially destabilized while the other is more strongly s t a b i l i z e d (Lawrence  et al., 1991).  T h i s causes  the p h e n o m e n o n of 'one-sidedness', where disturbances a n d m i x i n g are found to occur on o n l y one side of the density interface. T h e occurrence of one-sided flows has been noted i n the t i l t i n g tube experiments of Thorpe experiments discussed i n Maxworthy  and Browand  (1968), a n d m i x i n g layer  (1975). I n the case of m i x i n g  layer experiments, the a s y m m e t r y was found to be caused by the different b o u n d a r y  27 layer thicknesses between the high- and low-speed layers (Lawrence  et ai,  1998).  T h i s results i n a greater amount of v o r t i c i t y i n the high-speed layer, and a onesidedness i n the development of the instabilities. In a d d i t i o n to m i x i n g layer facilities, the effects of a s y m m e t r y have been observed i n a number of flows of considerable oceanographic i m p o r t a n c e .  M o s t notably, i n  the exchange flow t h a t occurs over the C a m a r i n a l S i l l i n the S t r a i t of G i b r a l t a r (Farmer  and Armi,  1998).  Here h i g h s a l i n i t y water from the M e d i t e r r a n e a n Sea  meets lower s a l i n i t y A t l a n t i c water, and a b i - d i r e c t i o n a l exchange flow is set up i n w h i c h an a s y m m e t r y is observed i n the measured v e l o c i t y a n d density profiles. U s i n g acoustic i m a g i n g techniques, the presence of a variety of instabilities resembling K H billows were discovered. T h e presence of instabilities i n flows of this nature have been found to significantly alter their d y n a m i c s (e.g.: Farmer  and Armi  Winters  and Seim  (2000);  (1998)).  One-sidedness is also a c o m m o n feature i n the d y n a m i c s of stratified rivers, where dense salt water meets the lighter, fresh water from the river discharge. T h i s is k n o w n to be the case i n the study of salt-wedge intrusions b y Sargent and (1987), Yonemitsu  et al. (1996) and Yoshida  et al. (1998).  Jirka  I n the latter of these  studies a one-sided overturn was observed using acoustic i m a g i n g i n the Ishikari R i v e r . T h e o v e r t u r n is expected to be the result of a s y m m e t r y i n the velocity a n d density d i s t r i b u t i o n s as well as a s y m m e t r i c a l l y placed v e r t i c a l boundaries. A s y m m e t r i c density stratified shear i n s t a b i l i t y constitutes a largely unexplored area i n the s t u d y of m i x i n g i n shear flows. T o date, the m a j o r i t y of studies have focused on homogenous and stratified free shear layers subject to the K H i n s t a b i l i t y (see Thorpe  (1968); Breidenthal  (1981); Koochesfahani  and Dimotakis  (1986);  28 Caulfield  and Peltier  (2000)). T h i s likely stems from the long-standing belief that  the K H i n s t a b i l i t y is the o n l y shear i n s t a b i l i t y to develop a complete overturning of the density interface, p r e c l u d i n g other instabilities as a significant source of m i x i n g (Thorpe,  1987). However, the recent findings of S W 0 3 indicate that i n certain re-  gions of parameter space the H o l m b o e instability, if given enough time, is able to generate levels of m i x i n g a n d turbulence t h a t exceed those of the K H instability. T h e l o n g - t e r m nonlinear evolution and m i x i n g behavior of A H instabilities have not presently been studied. In this chapter, the results of S W 0 3 are extended to the a s y m m e t r i c case, where the e v o l u t i o n of the turbulent t r a n s i t i o n a n d m i x i n g behavior are e x a m i n e d for a variety of asymmetries. T h i s is done t h r o u g h a series of direct n u m e r i c a l simulations. In the remainder of this section an i n t r o d u c t i o n is given to relevant parameters, and the framework b y w h i c h the evolution of the flow is studied. C h a p t e r 4 focuses on details of the n u m e r i c a l s o l u t i o n m e t h o d used. T h i s is followed b y a description of the s i m u l a t i o n results i n chapter 5, a discussion i n chapter 6, a n d conclusions a n d future work i n chapter 7.  3.1  Relevant Parameters  T h e basic components of a density stratified shear layer consist of i n i t i a l velocity a n d density profiles whose v a r i a t i o n i n the v e r t i c a l w i l l be represented by hyperbolic tangent functions.  A schematic of these idealized profiles is s h o w n i n figure 3.1.  T h e stable, layered density d i s t r i b u t i o n has a t o t a l difference i n density of Ap,  and  varies over a length scale rj between the two layers. S i m i l a r l y , the v e l o c i t y difference between the two streams is represented by Au,  w h i c h varies continuously over a  29  Figure 3.1: Schematic of velocity and density distribution parameters.  30 length scale h. The offset between the centre of the shear layer and the centre of the density interface is denoted by d. From these variables it is possible to define two important dimensionless parameters to the evolution of the flow. These consist of the scale ratio R = h/r), and the asymmetry factor e = 2d/h. They measure the relative thickness of the velocity and density variations, and the magnitude of the profile asymmetry, respectively. These profiles are represented mathematically by  l/(z) = ^ t a n h ( | z - e ) ,  (3.1)  and p(z) = ^  2  tanh %Rz, h  (3.2)  where the velocity field in a Cartesian coordinate system given by u = ( i t , v, w) has the initial velocity profile u — (U(z), 0, 0), and the initial density profile p(z). Here (x, y, z) denote the streamwise, spanwise, and vertical directions, respectively. Although these profiles are an idealization, the use of hyperbolic tangent profiles was tested experimentally in the laboratory investigation of Yonemitsu et al. (1996), where a good agreement was found in the case of salt-wedge intrusions; a flow known to exhibit asymmetry (see §3). If we now consider a stratifying agent (usually either heat or salt) with a molecular diffusivity of AC, in a fluid of kinematic viscosity v, and a reference density of po, three additional dimensionless parameters result:  Re =  Auh v  ;  J=  Apgh po{Auy  ;  v Pr = - . K,  . . (3.3)  The Reynolds number, Re, for the simulations discussed in this thesis is taken as  31 1200.  T h i s value is more representative of l a b o r a t o r y experiments b u t also includes  low-Re m i x i n g events expected i n oceans a n d lakes (Thorpe, 1985). N o t a b l y , this Re is the same order of magnitude as the instabilities observed i n the seasonal thermocline of the M e d i t e r r a n e a n Sea b y Woods (1968). T h e b u l k R i c h a r d s o n number, J , represents the r a t i o of the s t a b i l i z i n g effect of s t r a t i f i c a t i o n t o the destabilizing effect of the v e l o c i t y shear, a n d is chosen to be 0.15. T h i s value of J allows for the g r o w t h of b o t h K H a n d H o l m b o e instabilities t h r o u g h a n adjustment of the scale r a tio R, a n d is representative of conditions found i n b o t h field a n d l a b o r a t o r y studies (Thorpe, 1985). T h e P r a n d t l number, Pr, is the r a t i o of the k i n e m a t i c viscosity t o the molecular diffusivity of the stratifying agent, a n d is therefore a property of the fluid. A value of Pr = 9 was chosen for the current s i m u l a t i o n s w h i c h corresponds to t h e r m a l stratification i n b o t h fresh a n d salt water. It s h o u l d be noted that Pr is also used as a guide i n the choice of the i n i t i a l value of R. T h i s c a n easily be seen by considering o n l y the process of diffusion a c t i n g o n b o t h the v o r t i c i t y of the shear layer, a n d the stratifying agent of the density interface. G i v e n a t i m e scale r , each interface w i l l grow b y diffusion according to h ~ (UT) /  1 2  ratio of these l e n g t h scales R ~ Pr /  1 2  R = Pr '  1 2  a n d r\ ~ (KT) / . 1 2  T a k i n g the  is found. Therefore, i n most circumstances  w i l l be taken.  C h o i c e of the parameters mentioned above was also m o t i v a t e d b y n u m e r i c a l constraints (to be discussed further i n section 4.1) a n d to facilitate better comparison to the results of the simulations performed i n S W 0 3 .  32  3.2  T h e P a r t i t i o n a n d Transfer o f E n e r g y  T o gain insight into the processes involved i n the e v o l u t i o n of the flow through its various stages of development i t is useful to p a r t i t i o n the energy into a number of different reservoirs. A s the flow evolves, the energy transfers between these reservoirs are c a l c u l a t e d based o n the framework of Winters  et al. (1995). T h i s begins w i t h  the definition of the average t o t a l p o t e n t i a l energy, expressed i n dimensionless units as  =(Sfe<^>'"  p  (3  -  4)  where (-)y denotes a volume average over the c o m p u t a t i o n a l d o m a i n , a n d g is the g r a v i t a t i o n a l acceleration.  P c a n n o w be p a r t i t i o n e d into the available potential  energy, PA, a n d b a c k g r o u n d p o t e n t i a l energy, PB, reservoirs. Here PB is defined as  P  b  = (\ \2 9  (PB )V,  (3.5)  Z  where ps = PB{Z) is the background density profile.  T h i s profile represents the  m i n i m u m p o t e n t i a l energy state of the density field i f fluid elements are rearranged and deformed adiabatically, i.e., w i t h o u t altering their density.  S u c h a profile is  a m o n o t o n i c a l l y decreasing function of height t h a t is s t a t i c a l l y stable everywhere. T h e significance of ps lies i n the fact t h a t for a closed d o m a i n a n y changes to this profile are irreversible, and lead to an increased PB as t i m e proceeds ( Winters 1995). These changes are closely linked w i t h the rate of m i x i n g of the density  et al, field.  It is now possible t o define PA by  P  A  =P - P  B  .  (3.6)  33 T h i s q u a n t i t y represents the amount of p o t e n t i a l energy t h a t is readily available to be converted back to kinetic energy to drive fluid m o t i o n s . U n l i k e the PB reservoir, changes i n PA are reversible a n d represent ' s t i r r i n g ' processes (Peltier and Caulfield, 2003).  These relationships c a n be seen more clearly b y l o o k i n g at the evolution  equations for t h e kinetic a n d potential energy reservoirs.  F o r a closed d o m a i n i n  dimensionless units (see Winters et al. (1995) a n d Peltier and Caulfield (2003)):  ^  = -J(p.w„)  at  - Re~ (u  §  •V u,)v = H - D  l  v  2  t  = -J( .v.)v P  +  1  ^  = -H  +  D  r  (3.7)  (3.8)  where the v o l u m e averaged kinetic energy is given b y  K  - V W  a n d L* is the dimensionless d o m a i n height.  ,3,,  T h e variables a n d operators w i t h a n  asterisk as subscript have been ndhdimensionalized b y the velocity, length, density, a n d t i m e scales given b y A n , h, A p , a n d h/Au, the n o t a t i o n of Caulfield and Peltier  respectively. (Here we have chosen  (2000) a n d Peltier and Caulfield (2003) as  m u c h as possible.) T h e b u o y a n c y flux, H, represents the reversible transfer of energy between the k i n e t i c a n d p o t e n t i a l energy reservoirs. T h i s c a n be seen b y its presence i n (3.8), where i t is of opposite sign, i n d i c a t i n g t h a t transfers exist between b o t h reservoirs. Here we have used the convention of Peltier and Caulfield (2003) where H > 0 (H < 0) represents the lifting (sinking) of lighter fluid parcels, a n d leads t o a corresponding decrease (increase) i n P.  T h e second t e r m i n (3.7), D, gives the average loss of  34 kinetic energy due to viscous dissipation. This constitutes an irreversible transfer from K to the internal energy reservoir of the fluid. Since D acts most effectively on the small scales of velocity variation it will be used as an indicator for the level of turbulence present in the flow. The last term in (3.8), D , represents the irreversible p  rate of increase of P due to the molecular diffusion of the initial density profile. In this representation it has been assumed that pbottom — Ptop = Ap, where pbottom and Ptop denote the average density on the bottom and top vertical domain boundaries, respectively. This assumption holds for sufficiently large domain heights such that the average density at the vertical domain boundaries remains constant, and the difference is always Ap. This term plays no active role in the dynamics of the flow but serves to quantify the rate of increase in P that occurs in the absence of fluid motion. In order to quantify the rate of fluid mixing similar equations can be written for PA and PB as follows: =^- = -H-M dt  (3.10)  ^ • = M +D. P  (3.11)  Here M represents the instantaneous mixing rate, defined as the rate of increase in PB  due solely to fluid motions (Peltier and Caulfield, 2003). Since changes in PB can  be computed quite easily by a simple sorting of the density field, this quantification of the mixing rate lends itself well to the numerical simulations performed here. The interaction between various energy reservoirs described by evolution equations (3.7) - (3.11) are shown schematically in figure 3.2. The partition of P into PA and PB allows for the distinction between stirring and mixing processes. The stirring  35  Kinetic EnergyK  D Viscous Dissipation  Buoyancy Flux  Available PE F'A  H  Mixing  M  r  Background PE Internal Energy  Internal Energy Mass Diffusion  PB Total P E  Figure 3.2: Illustration of the relationships between the various energy reservoirs (after Winters et al. (1995)).  36 processes being associated with reversible energy transfers to and from PA, and mixing processes being due exclusively to the irreversible transfer of energy to PB caused by fluid motion. It is also instructive to partition K into components that can be associated with the kinetic energy of motions related to the mean flow, the two-dimensional primary instability, and the three-dimensional secondary instabilities and subsequent turbulence. Following Caulfield and Peltier (2000), this is done as follows:  K = j^((u )/2) ,  (3.12)  2  2  (Au)  2  K 2 d  K  =  1 „ A ( ( ^ ' U d)/2)yz, (A^)  / A  2  2  ™ =7 A 2  ' ^)/2) , xyz  (3-13)  (3.14)  where u(z,t) = (u) ,  (3.15)  xy  u (x, z, t) = (u- u(z)) , 2d  y  u (x,y,z,t) = u-u-u d3d  2  (3.16) (3-17)  From these definitions it can be seen that  K = K +K  2d  +K . 3d  (3.18)  It is important to note that the exchange between these reservoirs is reversible, implying that energy may be transfered directly from PA via H to any of the kinetic energy reservoirs.  37  3.3  M i x i n g Efficiency  In the s t u d y of stratified flows, the efficiency of a m i x i n g event has t r a d i t i o n a l l y been defined as the ratio of the increase i n p o t e n t i a l energy of the system to the work done b y the k i n e t i c energy of the d r i v i n g m e c h a n i s m (Caulfield  and  Peltier,  2000). I n this case, k i n e t i c energy is extracted from the shear layer, a n d work is done b y the i n s t a b i l i t y on the fluid w i t h i n the c o m p u t a t i o n a l d o m a i n t h r o u g h m i x i n g of the density field, a n d i n losses due to viscous d i s s i p a t i o n .  W i t h this i n m i n d , a  suitable measure of the m i x i n g efficiency can be a r r i v e d at b y inspection of the energy e v o l u t i o n equations (3.7) - (3.11). F i r s t , note t h a t if a h y p o t h e t i c a l reservoir is defined t h a t consists of b o t h K and PA then the reversible transfer associated w i t h H is e l i m i n a t e d and a l l transfers between reservoirs are irreversible. Ignoring  D, P  since it plays no active role i n m i x i n g , it c a n be seen t h a t the energy consumed by the m i x i n g event can be used for two different processes: to be lost to the internal energy of the fluid by viscous dissipation, or to be used to m i x the density  field  a n d s u p p l y b a c k g r o u n d p o t e n t i a l energy. So b y defining the instantaneous m i x i n g efficiency, Ei, as  ^wh  < >  E  319  we are t a k i n g the r a t i o of the rate at w h i c h the m i x i n g event is s u p p l y i n g potential energy, to the t o t a l rate at w h i c h energy is b e i n g extracted i n the process. definition is equivalent to t h a t discussed i n Caulfield and Caulfield  and Peltier  (2000) a n d  This Peltier  (2003), a n d has the property 0 < Ei < 1.  O f course Ei alone is not sufficient to describe the m i x i n g behavior i n a p a r t i c u l a r flow - the m a g n i t u d e of the energy transfers are also i m p o r t a n t .  F o r example, i f  38 b o t h M a n d D are s m a l l then Ei m a y be close to u n i t y i n d i c a t i n g h i g h l y efficient m i x i n g , yet the transfers themselves do not represent a significant m i x i n g process. It is also useful to define a c u m u l a t i v e m i x i n g efficiency as  f Mdt T  E =  rp  c  J o ¥  ,  (3.20)  fiMdt-ffDdt' where  T denotes some d u r a t i o n of interest (Peltier and Caulfield, 2003). If T is  t a k e n as the t o t a l d u r a t i o n of a m i x i n g event t h e n E gives the t o t a l p r o p o r t i o n of c  energy t h a t is used to perform m i x i n g to the t o t a l a m o u n t of energy expended i n the process.  T h i s gives an i n d i c a t i o n of the m i x i n g efficiency of the entire event,  rather t h a n a n instantaneous value.  39  Chapter 4 Numerical Solution M e t h o d 4.1  Direct Numerical Simulations  T h e n u m e r i c a l simulations performed i n this s t u d y were carried out using a threed i m e n s i o n a l spectral m o d e l designed for the s t u d y of stratified flows (see  Winters  et al. (2004) for a full discussion). It is used here to solve the incompressible equations of m o t i o n for a Boussinesq fluid i n the absence of external forces a n d r o t a t i o n a l effects described by ^ Dt  = — — V p - —gk + uV u po po  (4.1)  2  and V - u = 0.  Here D/Dt  = d/dt  (4.2)  + u • V denotes a m a t e r i a l derivative, k is the unit vector i n the  vertical d i r e c t i o n (positive upwards), a n d u = (u,v,w)  represents the velocity field  i n the streamwise, spanwise, and v e r t i c a l directions, respectively. T h e pressure field is given b y p, a n d the density field by p. Since the density field is governed by an active scalar it evolves subject to the advection-diffusion equation  40 For the purposes of this s t u d y any nonlinearities i n the equation of state are neglected. Since the viscous a n d diffusive terms from (4.1) and (4.3) are solved d i r e c t l y w i t h o u t any turbulence closure scheme this n u m e r i c a l s o l u t i o n m e t h o d fits into the direct n u m e r i c a l simulations ( D N S ) category.  D u e to the present l i m i t a t i o n s  i n v o l v i n g c o m p u t a t i o n a l resources, D N S is constrained to s t u d y flows w i t h relatively s m a l l d o m a i n sizes, and lower values of Re and Pr t h a n w o u l d often occur i n nature. In the D N S of a homogenous fluid the difficulty of s i m u l a t i n g h i g h Re flows lies i n the large range of length scale variability, where the largest scales are generally given by the d o m a i n size, a n d the smallest scales are O(LK) 1998). Here, L  = (i/ /e) / 3  K  of viscous d i s s i p a t i o n .  1  4  (Moin  and  Mahesh,  is the K o l m o g o r o v length scale, where e is the rate  T h i s represents the scale at w h i c h velocity gradients  diffusively s m o o t h e d b y viscosity.  However, i n a stratified flow where Pr  are  > 1,  as is the case here, the smallest scales are determined by the B a t c h e l o r length L  = Ln/Pr /  1 2  B  (Batchelor,  1959). T h i s expresses the fact t h a t the smallest scales  are now d e t e r m i n e d by scalar gradients smoothed b y molecular diffusion - a relatively slower process.  H i g h Pr fluids are therefore able to generate significantly smaller  scales t h a n realized i n otherwise homogenous  flows.  T h i s l i m i t a t i o n has been p a r t i a l l y alleviated by a m o d i f i c a t i o n of the  Winters  et al. (2004) m o d e l by W . D . S m y t h to aid i n the s i m u l a t i o n of higher Pr  fluids.  T h e m o d i f i c a t i o n includes an a d d i t i o n a l active scalar field t h a t is resolved on a grid twice as fine as the velocity a n d pressure grids. O r i g i n a l l y designed for the s t u d y of differential diffusion, i.e., the m i x i n g of two scalar fields w i t h different Pr Smyth et al. (2004)), it is used here for a single scalar at moderate  Pr.  (see  41  4.2  Initial and Boundary Conditions  In order to reduce the number of time steps required for the i n i t i a l g r o w t h of the p r i m a r y i n s t a b i l i t y a p e r t u r b a t i o n is a p p l i e d to the i n i t i a l velocity and density fields. T h e p e r t u r b a t i o n varies o n l y i n the streamwise d i r e c t i o n (as the p r i m a r y i n s t a b i l i t y is t w o - d i m e n s i o n a l , confirmed by Haigh (1995)) a n d is d e t e r m i n e d from the s o l u t i o n of the T a y l o r - G o l d s t e i n equation i n c o r p o r a t i n g the effects of viscosity and mass diffusion. S o l u t i o n of the T a y l o r - G o l d s t e i n equation predicts the structure of the resulting instability, given by the eigenfunctions, as well as the g r o w t h rate and phase speed for a disturbance of a given wavenumber, as found i n the eigenvalues.  Since it  is generally found t h a t the wavenumber of the fastest g r o w i n g mode dominates the nonlinear development of the flow (Lawrence  et al. (1991)), the p e r t u r b a t i o n is  applied at the wavenumber of m a x i m u m g r o w t h rate. T h i s s o l u t i o n is based on linear theory a n d assumes t h a t the perturbations have infinitesimally s m a l l a m p l i t u d e . F o r this reason, the p e r t u r b a t i o n must be s m a l l enough to ensure it spends some t i m e i n the linear regime and not adversely affect the g r o w t h of the i n s t a b i l i t y . A m a x i m u m displacement a m p l i t u d e of 0.2h was found to be sufficient for these purposes. In a d d i t i o n to the p e r t u r b a t i o n described above, a r a n d o m p e r t u r b a t i o n is applied to the velocity field to stimulate the g r o w t h of three-dimensional secondary instabilities.  T h e a m p l i t u d e of the p e r t u r b a t i o n is evenly d i s t r i b u t e d between  ± 0 . 0 l A u , a n d is centred on the density interface. T h e c o m p u t a t i o n a l d o m a i n consists of lengths i n each of the streamwise, spanwise, and v e r t i c a l directions denoted by  {L ,L ,L }, x  y  z  respectively. T h e b o u n d a r y  conditions are p e r i o d i c i n b o t h the streamwise and spanwise directions so t h a t L  x  is  42 Simulation e  J Re Pr R N Ny N L /h Ly/k L /h a  1  2  3  4  5  6  7  0  0  0.25  0.50  1.0  1.5  2.0  0.15  0.15  0.15  0.15  0.15  0.15  0.15  1200  1200  1200  1200  1200  1200  1200  9 1  9 3  9 3  9 3  9  9  3  9 3  128  128  128  128  128  128  128  96  64  96  96  96  96  96  x  192  128  192  192  192  192  192  6.98  7.85  6.16  5.71  5.71  6.04  6.28  4.50  4.50  4.50  4.50  4.50  4.50  4.50  9.00 0.172  0.176  z  x  9.00  9.00  9.00  9.00  9.00  0.081  0.025  0.090  0.131  0.163  z  3  9.00  T a b l e 4.1: S u m m a r y of n u m e r i c a l s i m u l a t i o n s performed.  given by the fastest g r o w i n g wavelength d e t e r m i n e d by the linear s t a b i l i t y analysis. T h e streamwise extent, therefore, varies for each s i m u l a t i o n . I n order to accommodate a n u m b e r of wavelengths of the resulting secondary i n s t a b i l i t i e s a l l s i m u l a t i o n s have L  y  = 4.5h.  T h i s estimate is based o n the observations of secondary i n s t a b i l i -  ties from K H a n d H o l m b o e simulations i n Caulfield and Peltier (2000) a n d S W 0 3 . T h e no-flux, free-slip b o u n d a r y conditions are e m p l o y e d i n the v e r t i c a l directions r e q u i r i n g the v e r t i c a l velocity a n d v e r t i c a l fluxes to v a n i s h , i.e.,  ™\z=0,L  z  dv  du = ~d~z >,=O,L  z  A value of L  z  dz z=Q,L z  = o, '  &  = 0.  (4.4)  dz z=0,L  z  = 9h was chosen so t h a t the v e r t i c a l boundaries are large enough to  have a negligible influence o n the development of the i n s t a b i l i t y . A table s u m m a r i z i n g the s i m u l a t i o n s discussed i n this chapter is shown i n table 4.1. Here {N , x  N, y  N} z  correspond to the n u m b e r of g r i d points used i n each of the component directions  43  in the coarser velocity fields, and a is the exponential growth rate of the instability determined from linear theory.  44  Chapter 5 Results 5.1  T h e S y m m e t r i c Case: K H a n d H o l m b o e Instabilities  S i m u l a t i o n results are examined first i n the s y m m e t r i c case (e = 0), where K H and H o l m b o e i n s t a b i l i t i e s are found to develop. T h i s gives a basis for c o m p a r i s o n i n the a s y m m e t r i c cases (e ^ 0), and helps illustrate the effects t h a t a s y m m e t r y has on the flow.  Kelvin-Helmholtz case: F i g u r e 5.1 shows the evolution of the p o t e n t i a l energy reservoirs, the irreversible transfers M a n d D, a n d Ei for the K H i n s t a b i l i t y (as performed i n S W 0 3 using a slightly different m i x i n g efficiency and d o m a i n size). I n figure 5.1, the dotted line represents the p o t e n t i a l energy gain resulting from diffusion of the i n i t i a l profile, obtained from (5.1)  T h i s is done for c o m p a r i s o n purposes w i t h P and P , B  given b y the solid and dashed  lines respectively. T h e K H curves show a relatively r a p i d g r o w t h i n P  A  from the difference between P a n d P ) B  (obtained  due to the r o l l - u p of the p r i m a r y billow.  45  F i g u r e 5.1: E n e r g y reservoirs a n d transfers for the K H s i m u l a t i o n . T h e plot i n (a) includes the p o t e n t i a l energy reservoirs P (solid line), PB (dashed line), a n d $ (dotted line). I n (b) b o t h irreversible transfers D (solid line), a n d M (dashed line) are s h o w n a l o n g w i t h Ei i n (c).  The K  2d  (solid line) a n d K  3d  (dashed line) are  p l o t t e d i n (d).  T h i s can be seen i n the two-dimensional x z - s l i c e s of the density field shown i n figure 5.2.  T h e s e slices are taken at y = L /2 y  a n d are s i m i l a r to v i s u a l i z a t i o n s of the  density field encountered i n l a b o r a t o r y investigations where light sheets are used to i l l u m i n a t e the flow (e.g.: Lawrence  et al. (1991); Schowalter  et al. (1994); Hogg and  Ivey (2003)). A t t ~ 45 the flow has reached its state of m a x i m u m p o t e n t i a l energy (and maximum P ) A  w h e n the billow has saturated. S h o r t l y after this t i m e the billow core  begins to collapse o w i n g to the g r o w t h of three-dimensional secondary instabilities associated w i t h the l o c a l l y unstable regions a r o u n d the b i l l o w core. T h u s far the rate of m i x i n g has steadily increased u n t i l its peak at i w 50.  A t this time the  secondary i n s t a b i l i t i e s have evolved into coherent structures w i t h i n the billow core. These structures resemble those observed i n the stratified m i x i n g layer experiments of Schowalter  et al. (1994).  However, the flow w i t h i n the b i l l o w (or elsewhere)  46  Figure 5.2: Plots of the densityfieldin the KH simulation showing its time evolution. Slices of the arz-plane are taken at y = L /2 with times shown in each plot. y  47 cannot be called turbulent; but, on the verge of turbulent collapse.  As can be  seen in figure 5.2, the greater degree of mixing within the core region has produced a 'pocket' of intermediate density fluid that is vertically centred between the two relatively undisturbed layers. Through a combination of both the buoyancy forces and the background shear, this pocket spreads laterally throughout the centre of the domain where the density profile assumes a statically stable configuration as the flow relaminarizes. By inspection of figure 5.1b, it can be seen that the highest levels of viscous dissipation occur well after the highest levels of mixing, and can be identified with the lateral spreading and shearing of the mixed billow core. In this stage of the flow, the surplus of PA gained initially by the billow roll-up has largely been consumed. With no other stirring mechanism present for the formation of PA the small-scale turbulent motions have little effect on mixing. This general trend can be seen in figure 5.1c, where E peaks in the early stages of 'preturbulent mixing', while the T  later stages are marked by a much lower efficiency corresponding to the spreading and shearing of the mixed billow core.  S y m m e t r i c H o l m b o e case:  As the scale ratio R, is increased, the stratification is compressed to a layer thinner than the shear thickness. While J remains unchanged, the density gradient is locally sharpened in the centre of the domain. This results in the formation of Holmboe's instability as seen in figure 5.3. The growth of Holmboe's instability is considerably slower than that of the K H , as is expected from its smaller linear growth rate (see table 4.1 for a listing of the linear growth rate <7, for each simulation). Before the  48  Figure 5.3: Plots of the density field in the symmetric Holmboe simulation showing its time evolution. Slices of the xz-plane are taken at y = L /2 with times shown in each plot. y  49 onset of t u r b u l e n t motions at t « 190 the g r o w t h of PA a n d K d c a n be seen i n figure 2  5.4a a n d 5.4b to be m a r k e d by a pronounced o s c i l l a t i o n . T h i s is due to a standing wave-like m o t i o n t h a t develops d u r i n g the g r o w t h of the instability. A s was shown  0  50  100  150  200  250  300  350  0  400  50  100  150  t  t  200  250  300  350  F i g u r e 5.4: E n e r g y reservoirs and transfers for the H o l m b o e s i m u l a t i o n . T h e plot i n (a) includes the p o t e n t i a l energy reservoirs P (solid line), PB (dashed line), and $ (dotted line). In (b) b o t h irreversible transfers D (solid line), a n d M (dashed line) are shown along w i t h Ei i n (c). T h e K d (solid line) a n d K d (dashed line) are p l o t t e d i n (d). 2  3  i n S W 0 3 , b o t h the m i x i n g a n d viscous dissipation rates (figure 5.4b) a t t a i n higher levels t h a n the K H . Here the rate of m i x i n g i n figure 5.4b exhibits a strong v a r i a b i l i t y t h a t was not present i n a l l other simulations to the same extent. A s suggested by the p r e l i m i n a r y results of an a d d i t i o n a l s i m u l a t i o n at a finer resolution, the density field is not well resolved o n the scale necessary for accurate c o m p u t a t i o n of M. However, b o t h the v e l o c i t y field a n d the b u l k m i x i n g characteristics of the flow (taken over the entire d u r a t i o n of the simulation) appear to be accurate. T h e ill-resolved behavior of M carries over to the c o m p u t a t i o n of Ei i n figure 5.4c. T h e low g r o w t h rate of the H o l m b o e i n s t a b i l i t y c a n be seen to postpone the  50 development of secondary instabilities (indicated by K )  u n t i l the finite a m p l i t u d e  3d  waveforms develop at t w 60. Once the g r o w t h of K  3d  begins, it is at a lower rate  t h a n the K H s i m u l a t i o n a n d is not influenced by the s t a n d i n g wave m o t i o n t h a t develops. A t no p o i n t i n the s i m u l a t i o n are large-scale o v e r t u r n i n g motions present. T h e m i x i n g accomplished by the H o l m b o e i n s t a b i l i t y is due to other processes t h a t leave the density interface relatively intact. These w i l l be discussed i n the following paragraphs i n the context of A H instabilities as well.  5.2  T h e A s y m m e t r i c Case: A H Instabilities a n d the Effects of A s y m m e t r y  A series of simulations are now presented i n w h i c h the a s y m m e t r y factor, e, is gradu a l l y increased w h i l e leaving a l l other parameters constant ( w i t h the exception of L  x  as discussed i n §4.2). N o t e that i n these simulations R = 3, so t h a t we are e x a m i n i n g A H i n s t a b i l i t i e s t h a t result from the relatively t h i n density interface, rather t h a n a s y m m e t r i c K H instabilities. T h i s can also be seen from a linear s t a b i l i t y analysis of the a s y m m e t r i c configurations w h i c h shows the presence of two modes of i n s t a b i l i t y : a d o m i n a n t m o d e w i t h a larger g r o w t h rate t h a t forms on the side of the density interface w i t h greater shear layer vorticity, and a weaker m o d e w i t h s m a l l g r o w t h rate i n the opposite layer (Lawrence  et ai,  1991; Haigh,  1995).  However, for a l l  simulations performed i n this s t u d y the value of J chosen results i n the s t a b i l i z a t i o n of the weaker m o d e according to linear theory. A s c a n be seen i n the evolution of the density fields i n the following simulations, o n l y the d o m i n a n t A H mode develops i n i t i a l l y (due i n part to the i n i t i a l p e r t u r b a t i o n a n d streamwise p e r i o d i c b o u n d a r y  51 conditions) a n d grows more r a p i d l y t h a n b o t h the K H a n d H o l m b o e instabilities, as is expected from its larger linear g r o w t h rate (cf. table 4.1).  e = 0.25 case: V a r i o u s energy reservoirs, selected transfers, a n d m i x i n g efficiency for the e = 0.25 a s y m m e t r y are shown i n figure 5.5. Initially, the p o t e n t i a l energy of the e = 0.25  ]»0.5  r  F i g u r e 5.5: E n e r g y reservoirs and transfers for the e = 0.25 s i m u l a t i o n . T h e plot i n (a) includes the p o t e n t i a l energy reservoirs P (solid line), PB (dashed line), a n d 4> (dotted line).  In (b) b o t h irreversible transfers D (solid line), a n d M  line) are s h o w n along w i t h Ei i n (c). T h e K  2D  (solid line) a n d K  3D  (dashed  (dashed line) are  p l o t t e d i n (d).  A H i n s t a b i l i t y bears resemblance to the K H case, where there is a r a p i d g r o w t h of PA due to the f o r m a t i o n of a billow structure. T h i s b i l l o w is q u a l i t a t i v e l y different from s t a n d a r d K H billows i n that it never accomplishes a complete overturning; leaving the density interface intact, it draws fluid of intermediate density from the upper p o r t i o n s of the interface (figure 5.6). T h e b i l l o w structure thins soon after its i n i t i a l development to more closely resemble a H o l m b o e mode, where a cusp-like  52  Figure 5.6: Plots of the density field in the e = 0.25 simulation showing its time evolution. Slices of the rrz-plane are taken at y = L /2 with times shown in each plot. y  53 wave forms t h a t is c o n t i n u a l l y e n t r a i n i n g a w i s p of fluid into the leading vortex, located i n the d o m i n a n t upper layer. T h i s entrainment process supplies PA w h i c h is then m i x e d w i t h i n the p r i m a r y vortex at a relatively steady rate (figure 5.5b). M i x i n g continues i n this manner for a s u b s t a n t i a l p e r i o d of t i m e (t ~ 45 — 110) d u r i n g w h i c h secondary instabilities grow w i t h i n the p r i m a r y vortex. T h i s ' p r e t u r b u l e n t ' (to use the t e r m i n o l o g y of S W 0 3 ) p e r i o d shows a developing three-dimensional structure that remains coherent.  A s i n the K H s i m u l a t i o n , this  coherent m i x i n g stage is m a r k e d by the highest levels of Ei. T h e secondary i n s t a b i l ities also appear to be associated w i t h the s t a t i c a l l y unstable regions created i n the p r i m a r y vortex, s i m i l a r to the K H case. F i g u r e 5.7 shows the development of this three-dimensional m o t i o n i n the y z - p l a n e taken from the t r a i l i n g edge of the p r i m a r y vortex s l i g h t l y d o w n s t r e a m of the cusp, i n the later stages of the preturbulent p e r i o d (t = 89). D e n s i t y structure similar to this was observed i n the m i x i n g layer experiments of Schowalter  et al. (1994) where the l o c a l l y unstable stratification led to the  amplification of streamwise vortices t h r o u g h the b a r o c l i n i c generation of v o r t i c i t y (see figure 26a i n p a r t i c u l a r ) . Interaction of these streamwise vortices is thought to cause the t r a n s i t i o n to turbulent flow i n K H billows (Caulfield Peltier  and Caulfield,  and Peltier,  2000;  2003) and appears to be i n agreement w i t h the findings here.  T h e b r e a k d o w n of the p r i m a r y vortex to incoherent three-dimensional motions at t « 110 signals a start i n the rise of D w i t h a m a x i m u m at t sa 140, after the m a j o r i t y of the m i x i n g has occurred. T h i s stage also results i n a ceasing of the entrainment as seen i n the loss of PA generation. However, the rate of m i x i n g remains relatively constant, a n d begins to t r a i l off only after the turbulence begins to decay and the upper layer relaminarizes. T h i s indicates t h a t the t u r b u l e n t motions are m i x i n g  54  Figure 5.7: Cross-section in the t/z-plane of the density field with vectors overlaid for the e = 0.25 simxdation. The streamwise location is taken within the primary vortex for x = Ah at t = 89.  55 p r e v i o u s l y entrained - b u t not completely m i x e d - fluid i n the d o m i n a n t layer. It s h o u l d be noted t h a t towards the end of the s i m u l a t i o n the weaker mode has begun to develop i n the lower layer. T h i s can be seen i n later times i n figure 5.6. T h e development of this mode is perhaps not s u r p r i s i n g considering t h a t the density gradient of the weaker layer remains sharp while a n appreceable a m o u n t of shear layer v o r t i c i t y remains. However, it is possible t h a t the g r o w t h of the weaker mode has been c o m p r o m i s e d somewhat due to the p e r i o d i c streamwise b o u n d a r y c o n d i t i o n . T h i s b o u n d a r y c o n d i t i o n forces the mode to develop at o n l y those wavenumbers t h a t are h a r m o n i c s of the streamwise d o m a i n length L . x  I n general, the results of  linear s t a b i l i t y analysis show t h a t the weaker m o d e has m a x i m u m g r o w t h rates at higher wavenumbers t h a n the d o m i n a n t mode (Haigh,  1995). T h e s i m u l a t i o n was  not carried out for a long enough d u r a t i o n to quantify the behavior of the weaker mode. T h e g r o w t h of b o t h p r i m a r y and secondary instabilities can be seen i n the  Kd 2  and K d curves i n figure 5.5d, respectively. Here the k i n e t i c energy associated w i t h 3  the p r i m a r y i n s t a b i l i t y , K d, grows at a near constant e x p o n e n t i a l rate u n t i l t « 25, 2  corresponding to the t i m e at w h i c h the i n i t i a l b i l l o w structure begins to t h i n . T h i s t i m e m a r k s the b e g i n n i n g of secondary g r o w t h given b y Kzd- T h e i n i t i a l g r o w t h rate of secondary instabilities can be inferred from the slope of this curve, a n d is seen to be larger t h a n t h a t of the p r i m a r y instability. T h i s is i n agreement w i t h the results for s y m m e t r i c stratified shear layers i n S W 0 3 a n d Caulfield  and Peltier  (2000). A s  the Kzd reservoir reaches appreciable levels K d begins to decay as t r a n s i t i o n occurs. 2  B y the end of the s i m u l a t i o n the signature of the weaker m o d e can be seen i n the leveling off of the K d curve w i t h a pronounced oscillation. 2  56  Figure 5.8: Energy reservoirs and transfers for the e = 0.50 simulation. The plot in (a) includes the potential energy reservoirs P (solid line), PB (dashed line), and $ (dotted line). In (b) both irreversible transfers D (solid line), and M (dashed line) are shown along with Ei in (c). The K d (solid line) and Ksa (dashed line) are plotted in (d). 2  t = 0.50 case: As the asymmetry is increased to e = 0.50, both similarities and departures from the e = 0.25 case can be seen. In figure 5.8 a sharp rise in PA can be seen as a billow structure develops. The quicker formation of the intial instability is consistent with the high growth rate predicted by linear theory. By examining the density field in figure 5.9, it can be seen that the billow grows to a larger diameter than in the e = 0.25 case. This is not surprising since the dominant layer has a greater amount of shear layer vorticity available. The rapid development of the AH instability leads to a considerably larger PA increase in this case, and appears to have overshot the amount of dense fluid that it can entrain in the primary vortex. This is suggested by the thinning of the billow to produce a cusp structure as a portion of the dense entrained fluid sinks to the level of the interface. In this respect the  57  Figure 5.9: Plots of the density field in the e = 0.50 simulation showing its time evolution. Slices of the xz-plane are taken at y = L /2 with times shown in each plot. y  58 i n i t i a l development of the A H i n s t a b i l i t y resembles a transient ejection event rather t h a n the steady r o l l - u p of a K H - l i k e billow. T h i s d e s c r i p t i o n of the i n i t i a l development becomes clearer as the a s y m m e t r y is increased further, discussed i n following paragraphs. F o l l o w i n g i n i t i a l development, the flow enters a p r e t u r b u l e n t p e r i o d i n w h i c h the m a j o r i t y of the m i x i n g is accomplished. Just as i n the e = 0.25 case, this p e r i o d sees the g r o w t h of secondary instabillities leading to the t u r b u l e n t t r a n s i t i o n i n d i c a t e d by elevated levels of D s t a r t i n g at t w 100. T h r o u g h o u t b o t h the preturbulent a n d turbulent phases the m i x i n g stays relatively constant w i t h s m a l l changes  present  i n the t u r b u l e n t phase a n d i n periods associated w i t h ejection events. A g a i n , the preturbulent phase is found to be the most efficient, w i t h a g r a d u a l decline following the development of a more complex flow structure. D u r i n g the preturbulent phase h i g h levels of m i x i n g are m a i n t a i n e d by the 'scraping' of the density interface and subsequent entrainment of this fluid b y the p r i m a r y vortex.  Since the entrained  fluid is of o n l y intermediate density, there appears to be no s a t u r a t i o n of the p r i m a r y vortex as is found i n K H instabilities. T h i s s c r a p i n g of the interface leads to sharper scalar gradients a n d higher mass fluxes. T h e a m p l i f i c a t i o n of streamwise v o r t i c i t y b y l o c a l l y unstable regions s t i l l appears to be a p r o m i n a n t cause of three-dimensional motions. However, a more complex i n t e r a c t i o n between the streamwise vortices a n d the density interface is observed for e = 0.50.  T h i s consists of a three-dimensional entrainment of fluid from the  density interface w i t h i n the p r i m a r y vortex as s h o w n i n figure 5.10. It is possible t h a t the higher levels of m i x i n g and earlier t r a n s i t i o n to turbulence that occurs for e = 0.50 is a result of this more pronounced entrainment m e c h a n i s m o n the density  59  Figure 5.10: Cross-section in the yz-plane of the density field with % vectors overlain for the e = 0.50 simulation. The streamwise location is taken within the primary vortex for x — Ah at t = 89.  60 interface. B y e x a m i n i n g the g r o w t h of K  3d  i n figure 5.8d it c a n be seen t h a t these two  processes are i n t i m a t e l y linked. T h e e v o l u t i o n of K  2d  and K  3d  appear q u a l i t a t i v e l y  quite s i m i l a r to the e = 0.25 s i m u l a t i o n . However, there is a notable change i n the g r o w t h rate of K  3d  b e g i n n i n g at t RJ 45 and corresponds to the s i n k i n g of the i n i t i a l  ejection to the level of the density interface.  I n this p o s i t i o n the development of  three-dimensional motions is suppressed by the stable stratification. T h e g r o w t h of K  3d  does not resume u n t i l a number of t i m e scales later. T h i s relationship between  the g r o w t h of three-dimensional motions and ejection events c a n be seen more clearly as the a s y m m e t r y is increased.  e = 1.0 case: A further increase i n the a s y m m e t r y to e = 1.0 results i n the energy characteristics shown i n figure 5.11. A g a i n , the higher g r o w t h rate a n d larger shear layer v o r t i c i t y leads to the r a p i d f o r m a t i o n of PA- T h i s i n i t i a l spike drops considerably due to the i m m e d i a t e break-off and vertical settling of the dense fluid at the b o t t o m of the p r i m a r y v o r t e x as seen i n figure 5.12. T h i s supports the v i e w proposed i n the previous paragraphs t h a t the i n i t i a l development of the i n s t a b i l i t y is best t h o u g h of as a n ejection event.  It can be characterized by a rise a n d fall i n PA followed  by slightly higher l o c a l values of D and M.  ( R e c a l l t h a t the transfers D and M  are v o l u m e averaged quantities so t h a t they are relatively insensitive to s p a t i a l l y localized increases.)  T h i s same signature can also be seen at a later t i m e of t w  55 — 75 a n d indeed, is visible i n the density field of figure 5.12. A s the m a g n i t u d e of the ejection events increase w i t h increasing a s y m m e t r y (and a greater a m o u n t of available shear layer v o r t i c i t y ) , so too does the transient devel-  61  0  50  100  150 I  200  250  300  0  50  100  150  200  250  300  t  Figure 5.11: Energy reservoirs and transfers for the K H simulation. The plot in (a) includes the potential energy reservoirs P (solid line), PB (dashed line), and $ (dotted line). In (b) both irreversible transfers D (solid line), and M (dashed line) are shown along with Ei in (c). The Kid (solid line) and K d (dashed line) are plotted in (d). 3  opment of three-dimensional motion. The K$d curve in figure 5.11 shows changes in growth rate corresponding to periods where the ejection has contacted the density interface (i.e., at t m 45 and t « 65). This reveals support for the damping of three-dimensional motions as the density interface is approached.  e > 1 cases: Thus far, as the asymmetry is increased so too are the gains in Pg, as the instabilities extract greater energy from the shear layer they are able to produce greater, more energetic mixing. Intuitively, this relationship cannot hold as the asymmetry is increased indefinitely, since the flow would approach the configuration of a homogenous free shear layer overlying an undisturbed density interface. Also, the pairing of adjacent vortices that is present in homogenous shear layers will also be expected to play a role as e increases. For these reasons the e > 1 simulations serve  Figure 5.12: Plots of the density field in the e = 1.0 simulation showing its time evolution. Slices of the xz-plaae are taken at y —Ly/2 with times shown in each plot.  63  only to a d d a level of completeness to the e x a m i n a t i o n of a s y m m e t r y effects.  64  Chapter 6  Discussion R e s u l t s of the n u m e r i c a l simulations enable a direct c o m p a r i s o n of the evolution of s y m m e t r i c a n d a s y m m e t r i c instabilities. In p a r t i c u l a r , a more general description of the m i x i n g behavior is now made. T h e a p p l i c a b i l i t y of the results to geophysical flows is then discussed.  6.1  M i x i n g Behavior  It has been found t h a t m i x i n g by shear instabilities i n this region of parameter space can be broken d o w n into two fundamentally different processes t h a t are characteristic of K H a n d H o l m b o e - l i k e modes.  I n the K H case, where the thickness of the  density interface is of the same order as the thickness of the shear layer (i.e., R = 1), g r o w t h of the i n s t a b i l i t y is characterized by significant regions of overturning. It is found to have a short, but efficient, preturbulent phase i n w h i c h the highest rates of m i x i n g are observed. I n this state, the saturated K H b i l l o w has developed seco n d a r y instabilities of large a m p l i t u d e w h i c h s u p p l y further energy to the m i x i n g process.  T h i s phase ends w i t h the turbulent collapse of the l o c a l i z e d unstable re-  gions w i t h i n the b i l l o w core. In this sense the i n s t a b i l i t y is self-limiting since the preturbulent phase is relatively short-lived, w i t h greatest m i x i n g o c c u r r i n g just prior to the t u r b u l e n t collapse.  65 In the s t u d y by S W 0 3 it was found t h a t the lower p r i m a r y g r o w t h rate of the H o l m b o e i n s t a b i l i t y led to a longer preturbulent phase where greater m i x i n g was accomplished. Here, a longer preturbulent phase of h i g h efficiency m i x i n g was found to occur even i n the A H instabilities e x h i b i t i n g larger g r o w t h rates. T h e i n s t a b i l i t y can support more energetic three-dimensional m o t i o n s before the t u r b u l e n t t r a n s i t i o n , leading to greater m i x i n g . T h i s is linked to the entrainment processes identified i n A H instabilities, n a m e l y the ejection of intermediate density fluid into the p r i m a r y vortex, a n d the 'scraping' of the density interface i n the region u p s t r e a m of the cusp. B o t h these processes lead to a sharpening of the interface w h i c h enhances m i x i n g and p o s s i b l y helps to m a i n t a i n the preturbulent form of the i n s t a b i l i t y . T o compare the g r o w t h of three-dimensional m o t i o n s i n each s i m u l a t i o n figure 6.1 shows the e v o l u t i o n of the K d reservoir i n each case. It s h o u l d be noted t h a t 3  the i n i t i a l g r o w t h rates of the K H and A H cases are quite similar; a n i n d i c a t i o n t h a t they a l l result from the generation of s t a t i c a l l y unstable regions. T h e K  3d  g r o w t h of  the H o l m b o e s i m u l a t i o n appears to be the result of a different process. It is not clear at this t i m e w h a t m e c h a n i s m is responsible for the g r o w t h of secondary instabilities i n the H o l m b o e case,.(see S W 0 3 for a discussion). T h e a b i l i t y of A H instabilities to support more energetic three-dimensional motions is i n d i c a t e d b y the higher levels of K d a t t a i n e d i n figure 6.1. 3  T h o u g h the properties discussed above a p p l y to a l l A H instabilities i n this study to v a r y i n g degrees, there exist large differences i n the t o t a l a m o u n t of m i x i n g between the various a s y m m e t r i c cases. T h e net amount of m i x i n g - measured as the t o t a l b a c k g r o u n d p o t e n t i a l energy gain due to fluid m o t i o n s - is p l o t t e d for each s i m u l a t i o n i n figure 6.2.  T h e general t r e n d is the larger the degree of asymme-  66  10"  •KH Holmboe e = 0.25 e = 0.50 e= 1.0 e = 1.5 e = 2.0 10  50  100  150  200  250  300  Figure 6.1: Evolution of the Kzd reservoir for each simulation.  350  67  F i g u r e 6.2: C o m p a r i s o n of the t o t a l a m o u n t of m i x i n g i n each s i m u l a t i o n .  68 try becomes, the more the development of the instability is dominated by shear layer vorticity. In this respect, the larger asymmetries are able to extract a greater amount of energy from the initial shear layer. This can be seen by looking at the total average kinetic energy reservoir, K, in each of the asymmetric cases shown in figure 6.3. Since the K reservoir is the energy source of the instability it gives a good  1  0.95  0.9  0.85  €  0.8  sc. 0.75  0.7  0.65  e = 2.0  0.6  0  50  100  150  200  250  300  350  t Figure 6.3: Evolution of K as a fraction of the initial kinetic energy K = K(0) for each simulation. 0  indication of the levels of extraction in each simulation. These plots show a greater extraction of energy as the asymmetry is increased. This result also agrees with the observations of increasing initial billow diameters and an increased presence of  69 ejection events i n the cases of greater asymmetry. A l t h o u g h a n increased e x t r a c t i o n of energy from the shear layer is observed i n cases of greater asymmetry, this does not necessarily lead to a greater m i x i n g of the density field. T h e p r o p o r t i o n of extracted energy t h a t is used to perform m i x i n g is given by the c u m u l a t i v e m i x i n g efficiency, E . c  F i g u r e 6.4 shows b o t h the net energy  gain due to m i x i n g a n d E for various asymmetries. Here, increases i n e i n i t i a l l y lead c  E  E  F i g u r e 6.4: Changes i n (a) the t o t a l m i x i n g a n d (b) E for the various asymmetries. c  to a more efficient m i x i n g process. T h i s efficiency begins to d r o p off for e > 1.5 a n d is most likely due to the greater distance between the centre of i n s t a b i l i t y a n d the density interface. E v e n t u a l l y a point is reached at e « 2 where a l t h o u g h the highest  70 levels of energy are extracted from the shear layer, less m i x i n g is accomplished. T h e i n t u i t i v e n o t i o n espoused i n S W 0 3 - t h a t for sufficiently large J and R, the density interface effectively acts as a flexible barrier to the v o r t i c i t y of the shear layer - c a n be seen to support these results. Since this barrier is sufficiently strong, it prevents c o m m u n i c a t i o n between layers, a n d isolates the v o r t i c i t y i n the weaker layer. B y this process the e x t r a c t i o n of energy i n the A H i n s t a b i l i t y is l i m i t e d to the shear of the d o m i n a n t layer u n t i l the f o r m a t i o n of the weaker mode.  A s the  a s y m m e t r y is increased past e > 1.5 the g r o w t h of the i n s t a b i l i t y is increasingly isolated from the density interface and cannot p e r f o r m m i x i n g as efficiently.  If e  were to continue to increase, the i n s t a b i l i t y w o u l d a p p r o a c h t h a t of a n homogenous shear layer w i t h a v a n i s h i n g m i x i n g efficiency. In accordance w i t h the theoretical and l a b o r a t o r y i n v e s t i g a t i o n of Lawrence et al. (1991) the development of A H instabilities resulted i n a d i s t i n c t one-sidedness. T h e final result of this one-sidedness is a greater m i x i n g of the density profile i n the d o m i n a n t layer. T h i s can be seen i n the final averaged density profiles shown i n figure 6.5 where the i n i t i a l profile is i n c l u d e d for comparison. T h e v e r t i c a l d i s t r i b u t i o n of density elements throughout the c o m p u t a t i o n a l d o m a i n is also s h o w n i n figure 6.5 for b o t h the i n i t i a l a n d final s i m u l a t i o n times. These plots i n d i c a t e t h a t the m i x i n g processes i n A H instabilities are responsible for the p r o d u c t i o n of largely low-density fluid  (since the d o m i n a n t layer coincides w i t h the low-density stream i n this case).  T h i s confirms the observation t h a t it is fluid from the upper portions of the interface t h a t is m i x e d i n the upper layer. T h i s process results i n a slight deepening of the upper layer i n w h i c h the effects can be noticed at a distance of « 2.5 — 3.5 shear layer depths from the new interface p o s i t i o n . T h e weaker layer is left relatively  71  F i g u r e 6.5: I n i t i a l a n d final density characteristics of the a s y m m e t r i c simulations for e = 0.25 i n (a) a n d (b), e — 0.50 i n (c) and (d). I n (a) a n d (c) the d i s t r i b u t i o n of density elements is shown for the i n i t i a l t i m e (dark bars) a n d the final time (light bars). In (b) a n d (d) the i n i t i a l (dashed line) and final (solid line) density profiles are given. A l l density values are given as the difference from the i n i t i a l mean density.  72 untouched w i t h a sharp gradient s t i l l intact. T h i s s h o u l d allow for the formation of the weaker m o d e sometime after the d o m i n a n t m o d e has s t a b i l i z e d , as was observed i n the e = 0.25 a n d e = 0.50 cases. T h e different m i x i n g behavior of the K H i n s t a b i l i t y c a n also be seen to manifest itself i n the final density d i s t r i b u t i o n , shown i n figure 6.6a a n d 6.6b.  Here the  m i x i n g process is responsible for p r o d u c i n g exclusively intermediate density fluid. T h i s observation was noted i n §5.1 as the m i x i n g was concentrated p r i m a r i l y w i t h i n the b i l l o w core a n d originated from the h i g h - a n d low-density fluid of the edges of the interface a n d shear layer. T h e increase i n intermediate density fluid leaves a steplike structure i n the profile w i t h two locations of higher gradients. See Caulfield Peltier  and  (2000) for a discussion of this density profile generation i n K H instabilities.  It s h o u l d also be noted t h a t the v e r t i c a l extent of the m i x i n g is entirely w i t h i n the o r i g i n a l shear layer and density interface thickness - i n m a r k e d contrast the A H cases.  to  T h e H o l m b o e s i m u l a t i o n (figures 6.6c and 6.6d) displays m i x i n g  behavior s i m i l a r to the A H cases i n t h a t the final density d i s t r i b u t i o n shows m i x i n g concentrated above and below the interface. T h e difference between m i x i n g behavior i n A H a n d H o l m b o e instabilities and t h a t of K H instabilities was first noticed b y Thorpe  (1968). Here it was observed  t h a t i n c e r t a i n m i x i n g events, the density interface 'retains its i d e n t i t y ' , m e a n i n g t h a t m i x i n g is accomplished w i t h o u t the collapse of overturned regions. I n this case there is no m i x i n g of fluid across the interface - as is the d o m i n a n t process i n K H instabilities.  73  Q. <  CL <  1  2  3  4  5  F r e q u e n c y (%)  F i g u r e 6.6: Initial a n d final density characteristics of the s y m m e t r i c simulations. In (a) a n d (c) the d i s t r i b u t i o n of density elements is shown for the initial time (dark bars) a n d the final time (light bars). T h e initial (dashed line) a n d final (solid line) density profiles are given in (b) a n d (d). T h e K H s i m u l a t i o n corresponds to (a) and (b) , whereas the H o l m b o e is depicted i n (c) a n d (d). A l l density values are given as the difference from the initial m e a n density.  74  6.2  Relevance to G e o p h y s i c a l F l o w s  T h e objective of the present s t u d y is t o examine the m i x i n g a n d t u r b u l e n t t r a n s i t i o n i n a s y m m e t r i c shear layers. T h i s is m o t i v a t e d i n part by the occurance of a s y m m e t r y i n flows of significant geophysical i m p o r t a n c e , such as exchange flows a n d salt-wedge intrusions, as discussed i n §3. However, n u m e r i c a l l i m i t a t i o n s d u e t o the present state of c o m p u t a t i o n a l technology (see §4.1) requires t h e s i m u l a t i o n s t o be r u n at lower Re a n d Pr t h a n w o u l d n o r m a l l y be expected i n n a t u r a l settings. A l t h o u g h the values used i n this s t u d y are representative of low Re events t h a t are w i t h i n (but at t h e low end of) t h e range observed i n t h e oceanic t h e r m o c l i n e b y Smyth et al. (2001), a n d i n the seasonal M e d i t e r r a n e a n t h e r m o c l i n e b y Woods (1968) (see also Thorpe (1985)). Questions concerning the a p p l i c a b i l i t y of D N S i n shear layers for Re < 1 0 have 4  been voiced i n Moin and Mahesh (1998). T h i s concerns a phenomenon t h a t has been termed t h e ' m i x i n g t r a n s i t i o n ' .  It describes w h a t is thought t o a be a universal  a t t r i b u t e of t u r b u l e n t flows whereby once a c e r t a i n Re ~ O ( 1 0 ) is exceeded, " a 4  q u a l i t a t i v e difference i n the appearance of the scalar field is observed . . . w h i c h results i n a more w e l l - m i x e d state" (Dimotakis, 2000). T h e m i x i n g t r a n s i t i o n has been well d o c u m e n t e d i n homogenous free shear layers (see Dimotakis (2000) a n d references therein) b u t i t s extension t o stratified shear layers remains t o be fully quantified. - T h e n u m e r i c a l s i m u l a t i o n s of Peltier and Caulfield (2003) examine the development of K H billows for a range of J. I n these simulations a q u a l i t a t i v e change i n the behavior of the flow is n o t e d for the l o w - J simulations as three-dimensionality is achieved. Since Re ~ O ( 1 0 ) for these flows, 4  effects of the m i x i n g t r a n s i t i o n can be expected (Peltier and Caulfield, 2003).  75 T h e onset of three-dimensional m o t i o n coincident w i t h a q u a l i t a t i v e change i n the behavior of the flow has been noted previously b y Breidenthal  (1981).  Subsequent  to this t r a n s i t i o n , a p e r i o d of intense turbulence is observed i n w h i c h the m a j o r i t y of the m i x i n g is achieved. However, i n the simulations of Peltier  and Caulfield  (2003)  as the s t r a t i f i c a t i o n is increased such t h a t J > 0.1 the g r o w t h of turbulent kinetic energy becomes i n h i b i t e d by buoyancy forces a n d the m i x i n g i n the turbulent phase becomes less significant. T h i s result certainly agrees w i t h the behavior observed i n the a s y m m e t r i c m i x i n g studied herein. Therefore, the observed i m p o r t a n c e of the preturbulent phase consisting of a coherent m i x i n g behavior, a l t h o u g h influenced b y the low Re, is not necessarily expected to be a result of the low Re. F u r t h e r s t u d y of the role t h a t stratification plays i n the m i x i n g t r a n s i t i o n is required to assess the geophysical significance of low-.Re m i x i n g .  76  Chapter 7 Conclusions and Future W o r k A number of direct n u m e r i c a l simulations have been performed i n w h i c h the effects of a s y m m e t r y on the e v o l u t i o n and m i x i n g behavior of stratified shear instabilities has been e x a m i n e d . T h e results are compared and contrasted to the two s y m m e t r i c instabilities t h a t occur at this point i n parameter space, namely, the K H a n d H o l m boe instabilities. T w o different m i x i n g mechanisms are found to emerge. T h e first is present o n l y i n the K H case, and consists of the f o r m a t i o n of significant regions of overturning. M i x i n g generated i n the collapse of this o v e r t u r n results i n the product i o n of intermediate density fluid, and a layered density profile. T h e second m i x i n g m e c h a n i s m is found i n A H instabilities. It is characterized b y a sharpening of the density interface b y a spanwise vortex, and the entrainment of p a r t i a l l y m i x e d fluid from the edge of the interface. T h e later of these processes is accomplished largely by p e r i o d i c ejection events. B o t h mechanisms were found to have the highest rates of m i x i n g i n the preturbulent phase, while the density structure was still coherent. H i g h levels of m i x i n g observed i n the A H instabilities are the result of a long l a s t i n g preturbulent phase s i m i l a r to the s y m m e t r i c H o l m b o e i n s t a b i l i t y . A s the level of a s y m m e t r y is increased more shear layer v o r t i c i t y becomes i n volved i n the development of the d o m i n a n t A H mode.  T h i s results i n a greater  i n i t i a l e x t r a c t i o n of energy from the shear layer p r o d u c i n g higher levels of b o t h m i x i n g a n d viscous dissipation. In cases e x h i b i t i n g lower asymmetries the weaker  77 A H m o d e is found to develop after the turbulent b r e a k d o w n of the d o m i n a n t mode. T h i s is o n l y possible due to the one-sided m i x i n g behavior of the d o m i n a n t A H mode w h i c h leaves the density interface sharp i n the weaker layer. A s first pointed out by Thorpe (1968), the density interface is able to " r e t a i n its i d e n t i t y " i n the A H m i x i n g process as a result of this one-sided m i x i n g behavior. T h e e v o l u t i o n of the flow a n d a d d i t i o n a l m i x i n g associated w i t h the weaker m o d e has not been studied in detail. T h r e e - d i m e n s i o n a l structure i n the d o m i n a n t A H modes was found to consist of streamwise vortices generated by the l o c a l l y unstable density regions of the A H b i l l o w structure. These structures appear to be influenced b y the p e r i o d i c ejections of fluid from the cusp region a n d are similar to those observed i n the l a b o r a t o r y experiments of Schowalter  et al. (1994). T h e i n t e r a c t i o n of the streamwise vortices  w i t h the density interface - p a r t i c u l a r l y d u r i n g the s i n k i n g of ejections to the interface level - was found to influence the g r o w t h of the k i n e t i c energy of the secondary structures. These processes have further i m p l i c a t i o n s for the t u r b u l e n t t r a n s i t i o n i n A H i n s t a b i l i t i e s t h a t have not been fully quantified. C o n s i d e r a t i o n of the features mentioned above have led to the conclusion that the degree of a s y m m e t r y present i n the velocity a n d density profiles is an i m p o r t a n t factor i n the m i x i n g behavior and turbulent t r a n s i t i o n i n stratified shear flows. However, the s t u d y of a s y m m e t r y i n these flows is far from complete. T h e present s t u d y has focused entirely on a single p o i n t i n parameter space. A more thorough e x a m i n a t i o n w o u l d include a number of points t h r o u g h o u t the b u l k R i c h a r d s o n number d o m a i n as well as e x p l o r i n g R e y n o l d s number effects. T h i s i n e v i t a b l y leads to a c o n s i d e r a t i o n of the m i x i n g t r a n s i t i o n i n stratified shear layers a n d is hampered  78 n u m e r i c a l l y b y the difficulties i n s i m u l a t i n g high-.Re flows. W h i l e future work may look to e x p a n d i n g the d o m a i n of s t u d y there are s t i l l m a n y unanswered questions regarding the present work. T h i s includes consideration of the p a i r i n g m e c h a n i s m i n A H instabilities. If, as e is increased, the flow approaches t h a t of a homogenous shear layer where p a i r i n g is k n o w n to occur, then at what value of e does p a i r i n g become i m p o r t a n t for a s y m m e t r i c flows? A n d w h a t  dependence  does this value have on the R i c h a r d s o n number a n d other parameters? A n o t h e r area t h a t requires further investigation is the f o r m a t i o n of three-dimensional secondary structures a n d their role i n the turbulent t r a n s i t i o n . 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