The effect of nonvertical shear on turbulence in a stably stratified mediumFrank G. Jacobitz and Sutanu Sarkar Citation: Physics of Fluids (1994-present) 10, 1158 (1998); doi: 10.1063/1.869640 View online: http://dx.doi.org/10.1063/1.869640 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/10/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Relevance of the Thorpe length scale in stably stratified turbulence Phys. Fluids 25, 076604 (2013); 10.1063/1.4813809 Buoyancy generated turbulence in stably stratified flow with shear Phys. Fluids 18, 045104 (2006); 10.1063/1.2193472 On the equilibrium states predicted by second moment models in rotating, stably stratified homogeneous shearflow Phys. Fluids 16, 3540 (2004); 10.1063/1.1775806 Anisotropy of turbulence in stably stratified mixing layers Phys. Fluids 12, 1343 (2000); 10.1063/1.870386 Length scales of turbulence in stably stratified mixing layers Phys. Fluids 12, 1327 (2000); 10.1063/1.870385

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The effect of nonvertical shear on turbulence in a stably stratified mediumFrank G. Jacobitz and Sutanu SarkarDepartment of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla,California 92093-0411

~Received 2 January 1997; accepted 5 January 1998!

Direct numerical simulations were performed in order to investigate the evolution of turbulence ina stably stratified fluid forced by nonvertical shear. Past research has been focused on vertical shearflow, and the present work is the first systematic study with vertical and horizontal components ofshear. The primary objective of this work was to study the effects of a variation of the angleubetween the direction of stratification and the gradient of the mean streamwise velocity fromu50, corresponding to the well-studied case of purely vertical shear, tou5p/2, corresponding topurely horizontal shear. It was observed that the turbulent kinetic energyK evolves approximatelyexponentially after an initial phase. The exponential growth rateg of the turbulent kinetic energyKwas found to increase nonlinearly, with a strong increase for small deviations from the vertical,when the inclination angleu was increased. The increased growth rate is due to a strongly increasedturbulence production caused by the horizontal component of the shear. The sensitivity of the flowto the shear inclination angleu was observed for both low and high values of the gradientRichardson number Ri, which is based on the magnitude of the shear rate. The effect of a variationof the inclination angleu on the turbulence evolution was compared with the effect of a variationof the gradient Richardson number Ri in the case of purely vertical shear. An effective Richardsonnumber Rieff was introduced in order to parametrize the dependence of the turbulence evolution onthe inclination angleu with a simple model based on mean quantities only. It was observed that theflux Richardson number Rif depends on the gradient Richardson number Ri but not on theinclination angleu. © 1998 American Institute of Physics.@S1070-6631~98!00905-2#

I. INTRODUCTION

Stably stratified shear flow is an ubiquitous feature offluid motion in the geophysical environment. Consequentlymuch attention has been drawn to the turbulence evolution invertically stably stratified and vertically sheared flow moti-vated by oceanic and atmospheric applications.1,2 The impor-tance of the gradient Richardson number Ri5N2/S2, whereN is the Brunt–Va¨isälä frequency, andS is the shear rate,was discovered early by energy arguments.1,3 The applica-tion of linear inviscid stability theory by Miles4 and Howard5

established Ri.1/4 as the sufficient condition for stability ina stratified shear flow. More recently, laboratory experimentsand direct numerical simulations have been performed in or-der to study many aspects of the turbulence evolution invertically stratified and vertically sheared flow. The evolu-tion of turbulence in vertically stably stratified andnonverti-cally sheared flow has received considerably less attention.The present paper appears to be the first systematic study ofnonvertical shear flow.

A number of laboratory experiments on vertically strati-fied and vertically sheared flow have been performed. Ko-mori et al.6 investigated the turbulence structure in a strati-fied open-channel flow. Experiments on the evolution ofhomogeneous turbulence in stratified shear flow were per-formed by Rohret al.7 using a salt-stratified water channel,and by Piccirillo and Van Atta8 using a thermally stratifiedwind tunnel. Itsweireet al. confirmed the importance of thegradient Richardson number Ri and Piccirillo and Van Atta

addressed the Reynolds number dependence of the turbu-lence evolution. The experimental investigations werecomplemented by direct numerical simulations by Gerzet al.,9 Holt et al.,10 and Jacobitzet al.11 In these simula-tions, the dependence of the turbulence evolution on a widerange of parameters was addressed. Gerzet al. studied theoccurrence of counter gradient buoyancy fluxes at high Ri-chardson numbers. Holtet al. addressed the Reynolds num-ber dependence of the turbulence evolution. Jacobitzet al.investigated the possibility of Reynolds number indepen-dence at high Reynolds numbers and the influence of theshear numberSK/e, which is the ratio of a time scale ofturbulenceK/e to the time scale imposed by the shear 1/S.Here K is the turbulent kinetic energy ande the turbulencedissipation rate. Kaltenbachet al.12 performed large eddysimulations of vertically stratified and vertically shearedflow. In addition, passive scalars with linear gradients in alldirections were introduced. It was observed that these scalarsmix more efficiently in the horizontal than in the vertical.

Work on nonvertically sheared flow in a vertically stablystratified fluid is restricted to Blumen’s13 application ofHoward’s semicircle theorem to such a flow. Nonverticalshear flow in a stratified fluid occurs and has been studiedindirectly in experimental investigations of wakes14,15 andjets16 in a stratified medium. No previous laboratory experi-ments or numerical investigations of the evolution of homo-geneous turbulence in a nonvertical shear flow with verticalstratification are known to the authors. This is surprising as

PHYSICS OF FLUIDS VOLUME 10, NUMBER 5 MAY 1998

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this type of flow occurs frequently in environmental and en-gineering applications. Examples are flow over topography,river inflow into the ocean, or effluent discharge by powerplants. The absence of previous work together with the widerange of applications are the primary motivation for the cur-rent study.

In vertical shear flow, the gradient Richardson number isusually defined with the vertical shear rate. In the case ofmore complex shear flows with additional nonvertical shearcomponents, the definition of Ri may need to be modified. Innonvertical shear flow, we choose to define the gradient Ri-chardson number as Ri5N2/S2, whereS is the magnitude ofthe shear rate. This selection is motivated by the observationthat N is an external frequency scale imposed by the gravityacceleration and the mean stratification, whileS is a distor-tion scale imposed by the mean velocity gradients. There-fore, the Richardson number Ri5N2/S2, the square of theratio of the two time scales, is a measure for the competingeffects of mean stratification and mean shear.

In Sec. II the equations of motion are presented. In Sec.III the transport equations for second-order moments are dis-cussed. The numerical method is summarized in Sec. IV.The results of direct numerical simulations are presented inSec. V, and in Sec. VI the effective Richardson number isintroduced. In Sec. VII the influence of additional parameterson the turbulence evolution is discussed. Section VIII con-tains a summary of the work presented here.

II. EQUATIONS OF MOTION

This study is based on the continuity equation of an in-compressible fluid, the three-dimensional unsteady Navier–Stokes equation in the Boussinesq approximation, and atransport equation for the density. In the following,xi de-notes thei th component of an orthonormal Cartesian coordi-nate system,Ui the i th component of the total velocity,% thetotal density, andP the total pressure. The dependent vari-ablesUi , %, and P are decomposed into a mean part~de-noted by an overbar! and a fluctuating part~denoted by smallletters!:

Ui5Ū i1ui , %5%̄1r, P5 P̄1p. ~1!

The mean streamwise velocityŪ5(Ū1,0,0) is unidirectionaland has constant horizontal and vertical shear rates]Ū1 /]x25S25S sinu and ]Ū1 /]x35S35S cosu, respec-tively. The mean density has a constant vertical stratificationrate]%̄/]x35Sr . Thus,

Ū i5~S sin ux21S cosux3!d i1 , %̄5r01Srx3 . ~2!

Therefore u50 corresponds to the well-studied case ofpurely vertical shear shown in Fig. 1, andu5p/2 corre-sponds to the case of purely horizontal shear shown in Fig. 2.It is assumed that a mean pressure gradient balances themean buoyancy force:

052] P̄

]x32g~r01Srx3!. ~3!

The decomposition of the dependent variables is intro-duced into the equations of motion, and the following evo-lution equations for the fluctuating parts are obtained:

]uj]xj

50, ~4!

]ui]t

1uj]ui]xj

1~S sin ux21S cosux3!]ui]x1

1~S sin uu21S cosuu3!d i1

521

r0

]p

]xi1n

]2ui]xj]xj

2g

r0rd i3 , ~5!

]r

]t1uj

]r

]xj1~S sin ux21S cosux3!

]r

]x11Sru3

5a]2r

]xj]xj. ~6!

Hereg is the gravity acceleration,n the kinematic viscosity,anda the scalar diffusivity.

III. TRANSPORT EQUATIONS

In this section the transport equations for second-ordermoments are introduced. The overbarā denotes the volumeaverage ofa. The transport equation for the velocity corre-lation Ri j 5uiuj is derived from Eq.~5!,

d

dtRi j 5Pi j 2Bi j 1P i j 2e i j , ~7!

Pi j 52S sin uuju2d i12S cosuuju3d i1

2S sin uuiu2d j 12S cosuuiu3d j 1 , ~8!

FIG. 1. Sketch of the mean velocity with vertical shear and the mean densitywith vertical stratification. This case corresponds tou50.

FIG. 2. Sketch of the mean velocity with horizontal shear and the meandensity with vertical stratification. This case corresponds tou5p/2.

1159Phys. Fluids, Vol. 10, No. 5, May 1998 F. G. Jacobitz and S. Sarkar

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Bi j 5g

r0~uird j 31ujrd i3!, ~9!

P i j 51

r0pS ]ui]xj 1 ]uj]xi D , ~10!

e i j 52n]ui]xk

]uj]xk

. ~11!

Here Pi j denotes the turbulence production term,Bi j thebuoyancy term,P i j the pressure–strain term, ande i j theturbulence dissipation term. Note that the turbulence produc-tion term appears only in equations for velocity correlationsthat contain the streamwise velocity component, and that thebuoyancy term appears only in equations for velocity corre-lations that contain the vertical velocity component. Theequations for the components of the velocity correlation ten-sor are:

d

dtu1u1522S sin uu1u222S cosuu1u31P112e11,

~12!

d

dtu2u25P222e22, ~13!

d

dtu3u3522

g

r0u3r1P332e33, ~14!

d

dtu1u252S sin uu2u22S cosuu2u31P122e12,

~15!

d

dtu1u352S sin uu2u32S cosuu3u32

g

r0u1r1P13

2e13, ~16!

d

dtu2u352

g

r0u2r1P232e23. ~17!

The transport equations for the density fluxesuir arederived from Eqs.~5! and ~6!:

d

dtuir52S sin uu2rd i12S cosuu3rd i12Sruiu3

2g

r0rrd i31

1

r0p

]r

]xi2

11Pr

Prn

]r

]xk

]ui]xk

.

~18!

The components of the above equation are:

d

dtu1r52S sin uu2r2S cosuu3r2Sru1u3

11

r0p

]r

]x12

11Pr

Prn

]r

]xk

]u1]xk

, ~19!

d

dtu2r52Sru2u31

1

r0p

]r

]x22

11Pr

Prn

]r

]xk

]u2]xk

, ~20!

d

dtu3r52Sru3u32

g

r0rr1

1

r0p

]r

]x3

211Pr

Prn

]r

]xk

]u3]xk

. ~21!

Finally a transport equation for the density fluctuations canby derived from Eq.~6!:

d

dtrr522Srru322a

]r

]xk

]r

]xk. ~22!

The transport equation for the turbulent kinetic energyK5uiui /2 is:

d

dtK5P21P32B2e, ~23!

P252S sin uu1u2, ~24!

P352S cosuu1u3, ~25!

B5g

r0u3r, ~26!

e5n]ui]xk

]ui]xk

. ~27!

HereP2 is the turbulence production term due to horizontalshear]Ū1 /]x2 , P3 the turbulence production term due tovertical shear]Ū1 /]x3 , B the buoyancy term, ande thedissipation term. The total turbulence production is given byP5P21P3 . The potential energyKr is computed from thedensity fluctuations:

Kr51

2

g

r0uSrurr. ~28!

IV. NUMERICAL APPROACH

The equations of motion are solved using a direct nu-merical approach. All dynamically important scales of thevelocity and density fields are fully resolved. A method oflines approach is used where a spatial discretization is firstperformed in order to obtain a semidiscrete system of ordi-nary differential equations. Then the system of equations isintegrated to advance the solution in time. The spatial dis-cretization is accomplished by a Fourier collocation method,which yields high accuracy but can be used only for prob-lems with periodic boundary conditions. Following a methodoriginally used by Rogallo,17 the equations of motion aretransformed into a frame of reference moving with the meanflow in order to allow periodic boundary conditions on thefluctuating parts of the dependent variables. The temporaladvancement is accomplished by a low-storage, third-orderRunge–Kutta scheme. The initial conditions are taken from asimulation of decaying isotropic turbulence and allow forlarge scale growth. A computational grid with 1443 pointswas used for all simulations. The evolution of the largest andsmallest turbulence scales was monitored to ensure a properresolution of the simulations.

1160 Phys. Fluids, Vol. 10, No. 5, May 1998 F. G. Jacobitz and S. Sarkar

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The code developed during our previous study of turbu-lence in a stratified fluid with vertical shear only (u50)discussed in Jacobitzet al.11 was modified to account for avariable shear inclination angleu. The governing equationswere written in a frame of reference (x18 ,x28 ,x38) wherex18 iscoincident with thex1 direction. Thex28 andx38 axes lie in thesame plane as thex2 andx3 axes but are rotated around thex1 axis by the inclination angleu with respect to thex2 andx3 axes. The advantage of this new coordinate system is thatthere is a single mean shear componentS5]Ū18/]x38 , andthe standard Rogallo transformation can be applied. In thenew coordinate system, there are two gravity componentsg285g sinu in the spanwise directionx28 andg385g cosu inthe shear directionx38 as well as two stratification compo-nentsSr sinu and Sr cosu, respectively. However, the ad-ditional gravity and stratification components do not requireany additional technique in the simulation.

V. RESULTS

In this section the results of a series of simulations withdifferent shear inclination anglesu are presented. All simu-lations were started from the same initial conditions takenfrom a simulation of decaying isotropic turbulence withoutdensity fluctuations. The Richardson number Ri5N2/S2

50.2 ~where N252gSr /r0 is the Brunt–Va¨isälä fre-quency!, the Prandtl number Pr5n/a50.72, the initial valueof the Taylor microscale Reynolds number Rel5ql/n533.54 ~where q5A2K is the magnitude of the velocity,and l5A5nq2/e is the Taylor microscale!, and the initialvalue of the shear numberSK/e52.0 are fixed. The Rich-ardson number and the shear number are based on the mag-nitude S5A(dŪ1 /dx2)21(dŪ1 /dx3)2 of the shear rates.While the Richardson number and the Prandtl number re-main constant, the Reynolds number and the shear numberevolve as the simulations are advanced in time.

Figure 3 shows the evolution of the turbulent kineticenergyK as a function of the nondimensional timeSt fordifferent inclination anglesu. Initially K decays in all simu-lations due to the isotropic initial conditions. Foru50,

which corresponds to the case of purely vertical shear shownin Fig. 1, the turbulent kinetic energy continues to decaythroughout the simulation. In this case the stratification in-fluences the shear production of turbulence directly. Whenthe angleu is increased, the decay ofK is less strong. Foru5p/8 the turbulent kinetic energy remains constant in time.For larger anglesu the evolution of the turbulent kineticenergy changes from decay to growth. Foru5p/2, whichcorresponds to the case of purely horizontal shear shown inFig. 2, the turbulent kinetic energy grows the strongest.However, the growth ofK is not as strong as in the unstrati-fied case~labeled Ri50 in Fig. 3! suggesting an indirectinfluence of the stratification on the turbulence production asdiscussed below.

In the case of purely vertical shear (u50), the stratifi-cation influences the shear production directly. Buoyancyfluxes decrease the vertical velocity fluctuationsu3u3 @seeEq. ~14!# and the magnitude of the 1–3 velocity correlationu1u3 @see Eq.~16!#. Therefore the vertical shear productionP352S cosuu1u3 is directly reduced by buoyancy fluxes. Itwas shown in previous investigations7,10,11 that the primaryeffect of stable stratification is to decrease the shear produc-tion and, to a smaller extent, act as a sink for turbulent ki-netic energy through the buoyancy flux. In the case of purelyhorizontal shear (u5p/2), this direct mechanism of stabili-zation does not exist. The buoyancy fluxes still decrease thevertical velocity fluctuations and the 1–3 velocity correla-tion. However, there is no direct influence on the 1–2 veloc-ity correlationu1u2 @see Eq.~15!# and the horizontal shearproductionP252S sinuu1u2. Only a redistribution throughthe pressure–strain terms leads to an indirect influence ofstratification on the turbulence evolution in the case of purelyhorizontal shear.

It was found that the asymptotic evolution of the turbu-lent kinetic energy follows approximately an exponentiallaw. In this case the exponential growth rate obtained fromEq. ~23!,

g51

SK

dK

dt5S P2SK1 P3SK2 BSK2 eSKD , ~29!

FIG. 3. Evolution of turbulent kinetic energyK as a function of the incli-nation angleu. The dashed lines are the exponential approximation to thesolution.

FIG. 4. Dependence of the asymptotic value of the growth rateg on theinclination angleu.

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reaches an approximately constant value. Then the equationcan be integrated and the exponential law

K5K0 exp~gSt! ~30!

is obtained. This exponential assumption is shown as dashedlines in Fig. 3. It approximates the asymptotic evolutionwell. The constant of integration is used to fit the graphs. Thedependence of the asymptotic value of the growth rateg onthe inclination angleu is shown in Fig. 4. The growth rategincreases strongly for 0

vertical shear is smaller than the normalized turbulence pro-duction in the case of purely horizontal shear.

The observation of a larger stabilizing effect of stablevertical stratification in the case of vertical shear relative tohorizontal shear is consistent with the transport equations forthe second-order moments. A buoyancy term appears di-rectly in Eq. ~14! for the vertical velocity varianceu3u3,reducingu3u3. A buoyancy term also appears in Eq.~16! forthe 1–3 velocity correlationu1u3, and it reduces the magni-tude of u1u3 in addition to the reduction fromu3u3. Thedecreasedu1u3 reduces the vertical production rateP352S cosuu1u3. A similar direct influence of buoyancy onthe horizontal production rateP252S sinuu1u2 does notexist, because a buoyancy term neither appears in Eq.~13!for u2u2 nor in Eq.~15! for u1u2. The 2–3 velocity correla-tion u2u3 remains small compared tou1u2 andu1u3. How-ever, there is an indirect effect of gravity through thepressure–strain terms on the horizontal turbulence produc-tion as shown in Fig. 8 by the reducedP2 /SK at u5p/2with respect to the unstratified case.

The anisotropic action of buoyancy in the vertical direc-tion is evident in the evolution of the Reynolds stress anisot-ropy tensor. In Fig. 10 the anisotropyb13 is shown, on whichthe normalized vertical production rateP3 /SK depends. Themagnitude of the anisotropyb13 decreases with increasingangleu. On the other hand, the magnitude of the anisotropyb12 increases with increasing angleu as shown in Fig. 11.The anisotropyb12 determines the normalized horizontalproduction rateP2 .

VI. THE EFFECTIVE RICHARDSON NUMBER

In this section, the effective Richardson number Rieff isintroduced in order to parametrize the dependence of theturbulence evolution on the shear inclination angleu. In ad-dition, the relationship between the effective Richardsonnumber Rieff , the flux Richardson number Rif , and the gra-dient Richardson number Ri is discussed.

A series of simulations with different gradient Richard-son numbers Ri but purely vertical shear was performed.These simulations matched the initial conditions of the series

of simulations with constant gradient Richardson numbers Ribut different inclination anglesu discussed in Sec. V. Again,an exponential evolution of the turbulent kinetic energyKwas observed as shown in Fig. 12. The turbulent kineticenergyK decays for large Ri and grows for small Ri. Thevalue of the critical Richardson number, for whichK re-mains constant in time, is about Ricr50.138 in this series ofsimulations. The dependence of the asymptotic value of thegrowth rate g on the gradient Richardson number Ri isshown in Fig. 13. In agreement with Jacobitzet al.11 thegrowth rateg decreases approximately linearly with increas-ing Richardson number Ri. The following relationship wasobtained from linear regression:

g50.156~6!21.13~4!Ri. ~31!

In the oceanic and atmospheric environment, the fluxRichardson number Rif5B/P is frequently used instead ofthe gradient Richardson number Ri. The flux Richardsonnumber Rif can be related to the gradient Richardson numberRi, if an eddy diffusivity model is used:

FIG. 9. Evolution of the normalized turbulence productionP/SK as a func-tion of the inclination angleu.

FIG. 10. Evolution of the Reynolds stress anisotropyb13 as a function of theinclination angleu.

FIG. 11. Evolution of the Reynolds stress anisotropyb12 as a function of theinclination angleu.

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Rif5B

P52

g

r0

u3r

Su18u3852

a t

n t

g

r0

Sr

S25

Ri

Prt. ~32!

Here u185u1 and u385u2 sinu1u3 cosu are the velocitycomponents in the plane of shear,a t52u3r/Sr is the eddydiffusivity of the density field,n t52u18u38/S is the eddy vis-cosity of the velocity field, and Prt5n t /a t is the turbulentPrandtl number. Therefore the flux Richardson number Rifcoincides with the gradient Richardson number Ri, if theturbulent Prandtl number Prt is equal to one.

The evolution of the flux Richardson number Rif isshown in Fig. 14 for different gradient Richardson numbersRi in the case of purely vertical shear. For all gradient Rich-ardson numbers Ri, the flux Richardson number Rif evolvesto a constant asymptotic value that is close to the corre-sponding value of the gradient Richardson number Ri.Therefore, in the parameter range considered here, the turbu-lent Prandtl number Prt remains always close to one. Thisagrees with previous simulations by Schumann and Gerz18

where an increase of the turbulent Prandtl number Prt withincreasing Richardson number Ri is observed only at largerRichardson numbers Ri.0.25, beyond the scope of the cur-rent work. Figure 15 shows the evolution of the flux Rich-ardson number Rif for different inclination anglesu and con-stant gradient Richardson number Ri50.2. The asymptoticvalue of Rif remains very close to the value of Ri50.2.Therefore the turbulent Prandtl number Prt is again close toone.

In the case of nonvertical shear it was shown in Fig. 4that the growth rateg depends on the inclination angleu fora constant gradient Richardson number Ri. Using the linearrelationship~31! between the growth rateg and the Richard-son number Ri, an effective Richardson number Rieff can becomputed from the growth ratesg observed at an angleu.Therefore the effective Richardson number Rieff of a nonver-tical shear flow is defined to be equal to the gradient Rich-ardson number Ri of a purely vertical shear flow with thesame growth rateg. The dependence of the effective Rich-ardson number Rieff on the inclination angleu is shown inFig. 16. The effective Richardson number Rieff decreases

FIG. 12. Evolution of the turbulent kinetic energyK for different gradientRichardson numbers Ri in the case of purely vertical shear (u50). Thedashed lines are the exponential approximation to the solution.

FIG. 13. Dependence of the asymptotic value of the growth rateg on thegradient Richardson number Ri for the case of purely vertical shear (u50).

FIG. 14. Evolution of the flux Richardson number Rif for different gradientRichardson numbers Ri in the case of purely vertical shear (u50).

FIG. 15. Evolution of the flux Richardson number Rif for different inclina-tion anglesu.

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with increasing angleu, because the stabilizing influence ofstable stratification decreases as the shear inclination angleuis increased fromu50 ~vertical shear! to u5p/2 ~horizontalshear!. In the following paragraphs two possible models forthe observed dependence are discussed.

First consider the plane of shear defined by the velocitycomponentsu1 and u2 sinu1u3 cosu. This plane is the(x18 ,x38) plane discussed in Sec. IV on the numerical ap-proach. The mean shearS acts only in this plane. The com-ponentsg cosu and Sr cosu of the gravity constantg andthe stratificationSr act also in this plane. Under the assump-tion that the componentsg sinu and Sr sinu outside theplane of shear do not influence the evolution of turbulence,the effective Richardson number can be written as:

Rieff5Ri cos2 u. ~33!

This relation is shown as a dashed line in Fig. 16. It does notfit the numerical data well and leads to large errors at largevalues of the inclination angleu, because it does not take theeffect of buoyancy outside the plane of shear into account.

In a second approach, the shearS cosu in the directionof gravity andS sinu perpendicular to the direction of grav-ity are weighted differently. Since the horizontal componentS sinu is not directly influenced by gravity, it is given alarger weight than the vertical componentS cosu. Then theeffective Richardson number Rieff can be written as:

Rieff5Ri

cos2 u1a sin2 u. ~34!

Herea52.7 is the weight given to the horizontal componentof shear. Equation~34! is shown as a solid line in Fig. 16. Itfits the numerical data relatively well for a simple modelbased on mean parameters of the flow.

VII. INFLUENCE OF ADDITIONAL PARAMETERS

The primary aim of this work was to study the influenceof the shear inclination angleu on the dynamics of turbu-lence in stratified shear flow, keeping other parameters con-stant. However, more simulations have been performed to

study the influence of additional parameters. First, the per-sistence of the shear inclination angle effect in stronglystratified, high Richardson number flow is studied. Then theinfluence of a high initial shear number is discussed.

In the first series of simulations discussed in Sec. V, theshear inclination angleu was varied fromu50 to u5p/2 tostudy the influence of the inclination angle on the turbulenceevolution. All other parameters were fixed in these simula-tions. In the second series of simulations discussed in Sec.VI, the gradient Richardson number Ri was varied from Ri50 to Ri50.2 in order to compare its influence on the tur-bulence evolution in vertically stratified and verticallysheared flow with the effects of a variation of the inclinationangleu in vertically stratified and nonvertically sheared flow.Both parameters influence the growth rateg of the turbulentkinetic energyK. Since stratification effects are often param-etrized by the gradient Richardson number Ri, the shear in-clination angle effect was parametrized by the introductionof the effective Richardson number Rieff . The second seriesof simulations covers only a limited range of Richardsonnumbers that result in growth rates similar to those observedin the first series of simulations.

An additional series of simulations performed to studythe influence of the shear inclination angle on the turbulenceevolution at a large gradient Richardson number is now dis-cussed. The objective of these simulations is to confirm thatthe sensitivity to the inclination angleu, as seen in the resultspresented in Sec. V with Ri50.2, persists in a strongly strati-fied medium with Ri52.0. All simulations were started fromthe same initial conditions with Ri52.0, Pr50.72, initialRel533.54, and initialSK/e52.0. Figure 17 shows the evo-lution of the turbulent kinetic energyK as a function of thenondimensional timeSt for different anglesu. For all casesthe turbulent kinetic energyK decays due to the strong stablestratification. However, the growth ratesg vary for differentanglesu. Figure 18 shows the dependence of the growth rateg as a function of the angleu for Ri52.0 ~open diamonds!and Ri50.2 ~filled diamonds!. Both series show a qualita-tively similar increase of the growth rateg with increasingangleu. In agreement with the Ri50.2 case, the asymptotic

FIG. 16. Dependence of the effective Richardson number Rieff on the incli-nation angleu. The diamonds represent the direct numerical results, thedashed line represents Eq.~33! and the solid line represents Eq.~34!.

FIG. 17. Evolution of the turbulent kinetic energyK for different shearinclination anglesu in the case of strong stratification with Ri52.0.

1165Phys. Fluids, Vol. 10, No. 5, May 1998 F. G. Jacobitz and S. Sarkar

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value of the flux Richardson number Rif50.3– 0.4 wasfound to be relatively insensitive to the angle variation in thestrongly stratified Ri52.0 case. The large value of the turbu-lent Prandtl number Prt55 – 7 agrees with the results ofSchumann and Gerz.18

Now consider only the case of purely horizontal shear(u5p/2). The Ri50.2 case shows strong growth ofK asdiscussed in Sec. V. The Ri52.0 case, on the other hand,shows a slight decay ofK. Therefore strong stable stratifica-tion is able to suppress the growth ofK in the case of hori-zontal shear despite the absence of the direct mechanism thatis responsible for the suppression of growth in the case ofvertical shear. However, the value of the critical Richardsonnumber Ricr at which growth is suppressed is about an orderof magnitude larger in the case of horizontal shear comparedto the case of vertical shear.

The shear numberSK/e has an important influence onthe turbulence evolution in vertically sheared and stratifiedflow as discussed in Jacobitzet al.11 Therefore additionalsimulations with high initial shear numbers were performedin purely vertical shear (u50) and in purely horizontal shear

(u5p/2) to ascertain if the shear number effect persists inthe case of purely horizontal shear. Figure 19 shows the evo-lution of the turbulent kinetic energyK for different initialshear numbers in the case of vertical shear. The simulationswere started from the same initial conditions with Ri50.1,Pr50.72, and Rel533.54. The turbulent kinetic energyKgrows for the cases with initialSK/e52.0 andSK/e56.0but decays for the case with initialSK/e514.0. Figure 20shows the evolution ofK in the case of horizontal shear. Thesimulations were started with the same initial conditions withRi50.2, Pr50.72, and Rel533.54. The turbulent kinetic en-ergy grows strongly forSK/e52.0 but grows only mildly forSK/e514.0. Figure 21 shows the corresponding growthratesg. The cases with high initial shear numbers finallyresult in an evolution with a smaller exponential growth ratethan the corresponding cases with low initial shear numbers.Therefore the stabilizing effect of high shear number flowthat was observed by Jacobitzet al.11 in the case of verticalshear persists in the case of horizontal shear.

FIG. 18. Dependence of the growth rateg on the shear inclination angleu inthe case Ri50.2 ~filled diamonds! and Ri52.0 ~open diamonds!.

FIG. 19. Evolution of the turbulent kinetic energyK for different shearnumbersSK/e in the case of purely vertical shear (u50.0).

FIG. 20. Evolution of the turbulent kinetic energyK for different shearnumbersSK/e in the case of purely horizontal shear (u5p/2).

FIG. 21. Dependence of the growth rateg on the initial value of the shearnumberSK/e in the case of purely horizontal shear~filled diamonds! andpurely vertical shear~open diamonds!.

1166 Phys. Fluids, Vol. 10, No. 5, May 1998 F. G. Jacobitz and S. Sarkar

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VIII. CONCLUSIONS

The effect of both vertical and horizontal shear compo-nents in a vertically stably stratified fluid has been investi-gated using direct numerical simulations. To the best of ourknowledge, no such study, either experimental or numerical,has been performed previously. It was found that not onlythe magnitude of the shear but also its orientation relative tothe vertical direction of gravity and stratification has a sig-nificant effect on the flow dynamics.

Two series of direct numerical simulations were initiallyperformed. In the first series, the shear inclination angleubetween the direction of stratification and the gradient of themean streamwise velocity was varied fromu50, corre-sponding to purely vertical shear, tou5p/2, correspondingto purely horizontal shear. The gradient Richardson numberRi50.2 based on the magnitude of the shear rateS wasfixed. In the second series, the gradient Richardson numberRi was varied from Ri50, corresponding to unstratifiedshear flow, to Ri50.2. The angleu50 was fixed in thissimulation. All simulations were started from the same initialconditions taken from a simulation of decaying isotropic tur-bulence with no density fluctuations. The initial value of theTaylor microscale Reynolds number Rel533.54, the initialvalue of the shear numberSK/e52.0, and the Prandtl num-ber Pr50.72 were fixed for all simulations.

The turbulent kinetic energy was found to evolve expo-nentially after an initial period of decay for all simulations.For the first series of simulations, the exponential growthrateg was found to increase with the inclination angleu asshown in Fig. 4. This increase is due to a strong increase ofthe horizontal turbulence productionP2 caused by the hori-zontal component of shear asu is increased. Although thevertical turbulence productionP3 decreases whenu is in-creased, the net turbulence productionP5P21P3 increases.An examination of the transport equations for the second-order moments shows that there is a direct influence of thebuoyancy on the vertical turbulence production but no directinfluence on the horizontal turbulence production. Thereforethe turbulence production in the case of purely horizontalshear is stronger than the turbulence production in the caseof purely vertical shear, but due to the three-dimensionalcharacter of turbulence, the turbulence production in the caseof purely horizontal shear is still smaller than the turbulenceproduction in unstratified shear flow. Therefore the growthrateg is smaller in the case of horizontal shear with respectto that in the unstratified case.

For the second series of simulations, the exponentialgrowth rateg was found to decrease approximately linearlywith increasing gradient Richardson number Ri over therange 0

tions of the turbulence evolution in a uniformly sheared and stably strati-fied flow,’’ J. Fluid Mech.342, 231 ~1997!.

12H.-J. Kaltenbach, T. Gerz, and U. Schumann, ‘‘Large-eddy simulation ofhomogeneous turbulence and diffusion in stably stratified shear flow,’’ J.Fluid Mech.280, 1 ~1994!.

13W. Blumen, ‘‘Stability of non-planar shear flow of a stratified fluid,’’ J.Fluid Mech.68, 177 ~1975!.

14J. M. Chomaz, P. Bonneton, and E. J. Hopfinger, ‘‘The structure of thenear wake of a sphere moving horizontally in a stratified fluid,’’ J. FluidMech.254, 1 ~1993!.

15G. R. Spedding, F. K. Browand, and A. M. Fincham, ‘‘Turbulence, simi-larity scaling and vortex geometry in the wake of a towed sphere in astably stratified fluid,’’ J. Fluid Mech.314, 53 ~1996!.

16S. I. Voropayev, Xiuzhang Zhang, D. L. Boyer, and H. J. S. Fernando,‘‘Horizontal jets in a rotating stratified fluid,’’ Phys. Fluids9, 115~1997!.

17R. S. Rogallo, ‘‘Numerical experiments in homogeneous turbulence,’’NASA Tech. Memo. , 81315~1981!.

18U. Schumann and T. Gerz, ‘‘Turbulent mixing in stably stratified shearflows,’’ J. Appl. Meteorol.34, 33 ~1995!.

1168 Phys. Fluids, Vol. 10, No. 5, May 1998 F. G. Jacobitz and S. Sarkar

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