JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. C5, PAGES 9295-9310, MAY 15, 2001
Turbulent
flow over a wavy surface- Stratified
case
Ronald J. Calhoun, Robert L. Street, and Jeffrey R. Koseff
EnvironmentalFluid MechanicsLaboratory,Departmentof Civil and EnvironmentalEngineering
Stanford University, Stanford California
Abstract. Stably stratifiedturbulent flow over wavy terrain is investigatedusing
large-eddy simulation for laboratory-scale domains. Basic near-wall turbulence
structureand productionmechanismsappear similar to the neutral case,though
dampened.The state of turbulencein the stratifiedfluid is analyzedby graphing
the relationshipsbetweenkey nondimensional
parameters.A shiftingand smearing
of the curveof flux Richardson
numberRif versusturbulentFroudenumberFrt
causedby the wavy boundary(comparedwith a fiat boundary)demonstrates
the
higher level of turbulencenear the wavy wall and showsstrong mixing in the
detachedshear layer over the trough. The relationshipdescribingthe state of
turbulencein a stratifedfluid givenby Ivey and Imberger[1991]is approximately
satisfied. Conditionalsamplingis usedto analyze the nature of Reynoldsstress
events.
1.
Introduction
0,
Calhoun and Street [this issue](herein referred to
as CS1) examinedneutral flow over wavy boundaries.
Increasing Reynolds stressesassociatedwith flow over
increasing amplitude wavy boundaries serve to more
strongly couple the flow near the wall with the outer
flow. However, stable stratification has the opposite ef-
fect, i.e., to decouplethe inner and outer flows [Garg
1996; Garg et al., 1997; Herring and Metals, 1989;
Metals and Herring, 1989;and Fallon et al., 1997]. The
objective of this paper is to investigatethe combination of these two effects and to compare the results
with those of the neutral casewhich was presentedby
CS1. Our specificapproachis to numericallysimulate a
laboratory-scaleflow with stable stratification; indeed,
we have chosenthe same configuration given by CS1
and focusedon a waveamplitude(a) to wavelength(,•)
ratio, 2a/,• - 0.1. Again, the bulk Reynoldsnumber
is -7000, basedon maximum mean streamwisevelocity
and total channel depth 2h; y0 refers to the deviation
of the bottom
from its mean value.
The equations and numerical method are similar to
those for the neutral flow scenario,the differencesbe-
ing the buoyancyterm in the momentumequations(for
the cartesianvelocities•i), an equation relating density to the scalar (temperatureT), and a scalartransport equation. Following typical large-eddy formulations, spatially filtered variables are denoted with an
(1)
O(J-Z•i)
(2)
Ot
O(J-ZT)
(3)
+
ot
where
Rm- UmT-(I,i;
q-I,i;T)Gmno•-•, (6)
-
0
ST-- -C20q•m(J-l'•jPj).
(7)
In equations(1)-(7), a typical notation is usedfor velocities, pressure,accelerationof gravity, etc. For the
__
curvilinear coordinate system, Um is the resolved vol-
ume flux (contravariantvelocitymultiplied by the in-
verseof the Jacobian[j-z] or thevolumeof the computational cell) normal to the surfaceof constant•m, and
G"m, which is calledthe "meshskewness
tensor," measuresthe skewness
of a grid cell. In the scalartransport
equation, T is the scalar variable and Tr is a reference
scalar value. The values of C1 and C2 can take the value
overbar.
Copyright 2001 by the American GeophysicalUnion.
0 or I or may be determineddynamicallydependingon
the type of subgrid-scalemodel being used;VT and nT
are the "eddy viscosity"and the "eddy diffusivity," re-
Paper number 2001JC900002.
spectively.
L•], the "modified
Leonardterm," and Pj
0148-0227/01/2001JC900002$09.00
are defined as
9295
9296
CALHOUN
=
ET AL: STRATIFIED
-
FLOW
OVER
A WAVY
SURFACE
wall
(s)
stress
__
__
Numericaldetailsare givenby Zang [1993],Zang et
al. [1993,1994],and Zang and Street[1995]. In this
work, the dynamic two parameter model of $alvetti and
Banerjee[1995](seealsoSalvettiet al. [1997])wasused
to determinethe eddy viscosityand diffusivity(related
to the turbulent Prandtl number) and the valuesof C•
and C2 dynamically.
In the following sectionswe provide a discussionof
an important nondimensional parameter, a description
of the simulations, a local characterization of flow conditions, and instantaneous and statistical results.
2.
Richardson
al. [1997]foundthe friction Richardsonnumberto be
useful in characterizing stratified turbulent flows in a
The basic idea of the Richardson
numbers
is
to compare the relative strengths of buoyancy and inertial forces. The friction Richardson number is typically
defined
Lx
•
Figure 1. Balance
offorces,
wherePi andPorepresent
themeanpressure
at theinletandoutlet,respectively.
Pattern representssolid walls.
Numbers
There is a bewildering array of nondimensionalparameters available to describe stratified flows. Garg et
channel.
•
as
Ri• =
Ap gh
(10)
'
channelflowswhich are driven by the samepressuregradient, regardlessof bottom topography. All the flowsin
this paper may therefore be nondimensionalizedby the
same friction velocity. The second reason is that the
value of u,e is essentiallyequivalentto (within experimental error) the friction velocityestimatedfrom the
experimentof Itudsonet al. [1996].The third reasonis
that in the caseof flat topography, u,e reducesto the
standard definition used, for example, by Garg et al.
[1997].
To justify the third reason, consider a fully developed
channel flow where a balance of forces exists bewhere Ap is the differencein density between the uptween
the drag on the upper and lower physical walls
per and lower boundaries; other variables were defined
previously. Thus Ri• is seenas the ratio of the buoyant and the mean pressuregradient (Figure 1). Equating
unit force Ap gh and the dynamic unit force. In flow the difference in the pressure force between the inlet
over topography,however,rw is a function of positionon and outlet surfacesand the drag force on the upper and
the lower boundary. Friction velocity is therefore also a lower boundaries gives
function of position if it is defined in the usual manner,
PiLyLz - PoLyLz= 2rwLxLz,
i.e.,V/rw/pr.CS1used,forthepurposes
ofcomparisons
(12)
with experimental results, an effective friction velocity where Pi and Po represent the mean pressure at the
that was derived from experimental results by extrapo- inlet and outlet, respectively. Simplifying yields
lating Reynoldsstressfrom the outer flow to a "mean"
{P• - Po)
lower boundary. For more details on how this is done
Lx Ly- 2rw.
in an experimentalsetting,seeHudsonet al. [1996].
A mean pressuregradient drives all the flows in this
paper. The imposed pressure gradient is the parameter which
determines
the level of stresses that
will
be
present in the flow. Therefore a form of the friction
velocity based on the pressuregradient rather than on
wall stress would be useful. Using dimensional argu-
(13)
Recognizingthe fraction on the left-hand side as the
mean pressuregradient, introducing a rearrangementof
2
the standard definition of friction velocity, rw -u,pr,
and substituting2h for Ly yields:
•)•dp
ments [Pan and Banerjee,1995], an effectivefriction
velocity based on the pressuregradient may be defined
(14)
Equation(14) showsthat for channelflow overa fiat
as follows:
•g,e
--
• hdp
_
boundary the friction velocity based on the mean pressure gradient is equivalent to the standard friction velocity. We now may definea friction Richardsonnumber
(11)
where dp/dx is the mean pressuregradientdriving the
basedon effectivefriction velocity (basedon the mean
pressuregradient)as follows:
flow.
This
definition
is useful for several
reasons.
The first
is that one value may be used to nondimensionalizeall
Ri• =
Apgh
(15)
CALHOUN
ET AL' STRATIFIED
FLOW
0.98
Another form of this equation uses a rearrangement
of the definitionof the coefficientof expansion,(Ap =
C•prAT), to obtain
Ri•=ctATgh
(U,e)2
,
OVER
A WAVY
SURFACE
9297
--
0.96 -
(16)
0.94-
0.92
where AT is the differencein temperature between the
lower and upper boundaries. Becausethis definition
makes sensein channel flows over topography and re-
ducesto Garg's[1996]definitionfor the fiat channel
case, it will be used to characterizethe flows in this
paper, and the subscript"e" will be dropped.
3. Description of Simulations
0.90
0.0-
0
I
5
I
10
I
15
I
20
time
Figure 2. Flow rate versustime (nondimensionalized
by u,A). Case Ri• -62.
The initial conditionfor the stably stratified runs was
created by coolingthe bottom boundary and heating flow experiencesa readjustmentperiod during which
the upper boundaryof a neutrally stratified flow. Tem- both the gravitational potential energyand the volumeperature differenceswere allowedto be passivelytrans- averagedturbulent kinetic energy(TKE) decrease.
An exampleof the readjustmentperiod for the Riz =
ported in the flow by turning of[ the buoyancy term
31
case was verified in a specificsimulation sequence.
in the momentum equations. The flow evolved in time
Over
the durationof the sequence
(about6 flowthrough
until the mean temperature profile was approximately
fixed. This became the initial condition for all of the
times, where 1 flow through time is how long it takesfor
stably stratified runs. This technique was used by Garg a particle of fluid moving at the averagefluid velocity
et al. [1997]to establishinitial conditionsfor stably to traversethe length of the channel),heavierfluid fell
stratified flowsovera flat bed. At the beginningof each toward the wavy boundary,while lighter fluid rose. By
stably stratified run, the buoyancyterm in the momen- the end of the simulation, larger ejections of heavier
tum was turned on and temperature became an active fluid into the core of the flow were suppressed.
The time developmentof the flow can be roughly
scalar. Different Richardson numbers were obtained by
described
with three stages: (1) a "fall," or period of
adjusting the gravitational constant.
decreasing
TKE and potential energies,(2) "readjustGarget al. [1997]ran finelarge-eddysimulation(LES)
ment,"
when
the flow is adusting to a new balance of
and direct numericalsimulation(DNS) resolutionstudies to determine that for their flows with Prandtl
numforces,and (3) "quasi-equilibrium,"when an approxiber equalto 0.71 (thermallystratifiedair) the grid res- mate balance has been achieved and the mass flow rate
olution used for neutral flows was adequate for stably is roughlystabilized.Garget al. [1997]hasdocumented
stratified flowsin the rangesthey studied (Ri• = 0 -• a similar developmentin stratified flows over a flat bed.
60). Similarly, we use a Prandtl number of 1.0 so that They alsofound that flowswith higher Richardsonnumthe Oboukhov scale (the scale at which the dissipa- bers took longer to stabilize.
In our case, for the Ri• - 31, extensive data sets
tion of temperaturefluctuationsoccurs) and the Kolwere
gathered during stage I and stage 3, but statistimogorovscale(the scaleat whichdissipationoccursfor
cal
data
presentedwere calculated from stage 3. Stage
the energyassociatedwith turbulent motions)are com3
data
were
gathered(for about 6 flow throughtimes)
parable. We thereforeusegrid resolutionssimilar to the
neutral case.
beginning 12 simulated flow throughs after the activaWe present two stratified flows with friction Richard- tion of the gravity force. Time averaging(in additionto
spanwisedirection) was
son numbers of 31 and 62, a buoyancy-controlledand a averagingin the homogeneous
buoyancy-dominatedflow, respectively. In both of the performed on the data, since the flow rate is virtually
flows,when the gravity term is switchedon, cold fluid in constant(within 5%) overthe averagingperiod.
In the Ri• - 62 casethe flowrate (Figure 2) increases
the upper part of the channel slowly settles and warmer
slowly.As wasreportedby Garget al. [1997]for their
fluid makes its way upward.
The activation of the gravity force causesan increase simulations, here the more highly stratified casesdid
in mass flux. When buoyancy effects are turned on, not achieve a quasi-equilibrium state within the amount
We ran
turbulence is suppressedand the fluid has a tendency to of time we could afford to run the simulation.
move more in horizontal planes. Stronger stratification the simulation gathering data on the center plane and
requires more kinetic energy to move a parcel of fluid a transverseplane through the trough region for about
the same distanceupward against the density gradient. 9 flow through times and then gathered full field data
Sincethe dissipatingturbulent motionsare suppressed, over 3 flow through times. Averaging was performed
a faster fluid velocity is required to balance the same in the spanwisedirectionand over the shorter time (3
pressureforce. Therefore bulk velocity increases. The flow throughs). The figuresthat follow are represen-
9298
CALHOUN
ET AL: STRATIFIED
FLOW OVER A WAVY SURFACE
0.5 I
0.4
-
•_____---
0.3
-
•
•
•5
0.1
- --
•4 •__•
10-
••
9•--
•••
-6•
•
•
0.0-
-0.1 -
0.3
0.4
0.5
0.6
0.7
x/(2;•)
Figure 3. Contoursof meanstreamwise
velocity(nondimensionalized
by u,). The 2a/& - 0.1.
Case Ri• - 31.
tative
of the flow field at nondimensional
times
15 to
The key point is that the basic near-wall turbulence
20. The assumption here is that over relatively short structure and production mechanismsappear the same
times the flow rate is roughly constant, though it will for this flow, though apparently slightly dampened by
actually continue to increase,and turbulence may even- the stratification. Analysis of the turbulence statistics
tually be extinguished[cf. Garg et al., 1997]. Trends allows us to quantify this impression.
with increasingstratification can be deduced,but exact
values of the maximums
and minimums
of the contours
will change as the flow field continuesto develop. The
results presented for RiT = 62 represent a snapshot
5. Mean Quantities
As done by CS1, statistical quantities are averaged
(averagedover a shorttime) of the flow field as it is de- both in time and in the spanwisedirection. The means
caying and are not representative of a final stationary
are nondimensionalizedby a friction velocity obtained
from the experiment describedby CS1 (cf. section
2). Comparingthe mean streamwisevelocityfor the
Ri•= 31 (Figure 3) and the neutral case(CS1, Fig4. Instantaneous
Field
ure 10) showsthat the recirculationzonehas decreased
The qualitative conclusionfrom comparing the in- in size though not dramatically in magnitude. Higher
stantaneousfield of a stably stratified flow (Ri•- = 31) velocity fluid from the outer flow has moved lower towith the neutral case(cf. CS1) is that the flow vari- ward the wavy surface,as can be seen,for example, by
ability and near-wall turbulent structure are quite sim- the loweringof contour lines 9 and 10. However,the 8
ilar. We summarize the most important points. As contourhas movedslightly upward, indicating that the
in the neutral case, the flow may be characterized by coupling between the inner and outer flows has been
state.
regions, i.e., outer, recirculation, boundary layer, shear slightly reduced.
The contoursfor the Ri•- = 62 case(Figure 4) furlayer, separation, and reattachment regions. Transverse
planes appear to show ejectionsof slowermoving fluid. ther support the reduced coupling idea. Notice that
0.5 I
0.4 I
0.3 I
0.2 m
0.1 w
I
0.3
I
0.4
I
0.5
I
0.6
I
0.7
x/(2•)
Figure 4. Contoursof meanstreamwise
velocity.CaseRi•
-
62.
CALHOUN
ET AL: STRATIFIED
FLOW
OVER
A WAVY
SURFACE
9299
0.5--
0.4--
0.3--
0.2-
7
0.1
0.0
-0.1
I
I
I
i
I
0.3
0.4
0.5
0.6
0.7
x/(2X)
Figure 5. Contoursof meantemperature(degrees
C). CaseRi• - 31.
comparedwith the Ri•. - 31 case,the 8 contour moves
clearly upward and the 10 contour movesonly slightly
upward. The trend to noticeis that the vertical distance
between the 8 and 10 contours decreases with
increas-
6. Reynolds Stresses
Contours of turbulence intensities for the Ri•- case all have the same basic form
as the neutral
31
case
[Calhoun,1998]. We presenthereonlythe spanwise(as
givenby CS1) turbulenceintensitycontours(Figure 6).
ing stratification. We speculatethat this is causedby
the tendency for stable stratification to reduce vertical
turbulent exchangeallowingthe outer flow to increase
in velocity until the additional shear associatedwith
the highervelocitygradients(in the vertical direction)
establishesenoughvertical momentum exchange. The
recirculation region in the Ri•- - 62 caseis suppressed.
The negativevaluesare belowthe -1 contourand so are
not pictured in Figure 4.
With increasingstable stratification, vertical velocities maintain basically the same shape, though weakenedin magnitude[Calhoun,1998].
Mean temperature contours are shown in Figure 5.
The bulge in the contours in the recirculation region
demonstratesthat colder fluid is higher above the surface at this location than at any other point. It is reasonableto expect slow moving and recirculating fluid
the magnitudes by an additional 10 to 15% that occurs
for the Ri•. - 62 case. Vertical turbulence intensity
is reducedrelatively more than the other turbulence in-
to be colder because it has more time to be cooled by
the lower boundary.
the hill height is lowered.
For each turbulence intensity field, however,the magnitudes of the intensity decrease-10% overall compared
with the neutral flow. In the neutral flow over wavy
terrain with lower amplitude hills, the different turbulent intensities were reduced nonuniformly; that is, the
neutral flows became increasinglyanisotropicwith decreasinghill height.
Stablestratification,however,appearsto suppressall
of the intensitiesmoreuniformlywhile maintainingapproximately the same shape of the contoursin the near
field. The sameis approximatelytrue whencomparing
the two stratified cases,where there is a suppressionof
tensities, but the effect is not as noticeable as that when
0.4
0.2
--
1.2•
0.1
--
•
0.0-I
I
0.3
0.4
1.6
••
•
I
I
I
0.5
0.6
0.7
x/(2X)
Figure 6. Contoursof Wrms/U,. CaseRi•. - 31.
9300
CALHOUN
ET AL- STRATIFIED
FLOW OVER A WAVY SURFACE
0.5
0.4
0.3
0.8
0.2
0.1
1.4
1.2
0.0
t
-0.1
I
I
I
I
I
0.3
0.4
0.5
0.6
0.7
x/(2x)
Figure 7. Contoursof-•'-ff/u,.2 Case
-31.
The peak Reynoldsshearstress(Figure 7) is reduced [1997]in stratifiedfiat-channelflows. The decoupling
by roughly 20% with the first level of stratification increaseswith increasingRichardsonnumber.
(Ri,- = 31) and by another 20% in the strongercase
(Ri• = 62) (cf. CS1, Figure 14).
8. TKE Budget
The mathematical
7. Vorticity
Mean spanwisevorticity contours for the Ri• - 31
caselook similar to thoseof the neutral case(CS1, Figure 8). Only minor differencesare noticeabledespite
the stratification.
The shear layer still exists in the
mean and has the same streamwise
extent.
There
definitions for the terms of the tur-
bulent kinetic equationsin this sectionare givenby CS1.
The TKE budget was calculated only for the lower
level of stratification Ri• = 31, where a larger time
averaging was possible. The TKE budget may be analyzed with Figures 8 through 13. Compared with the
neutral case,mean TKE showsa reductionof 13% in its
peak magnitude with only small changesin the shape
of the contours. The peak mean production, located
againat (0.43, 0.1), decreases
by almost25%. Both the
positiveand negativemean velocity transport contours
are reducedas well (the negativeby -20% and the positive decreased
lesssignificantly).This is expectedsince
there is less TKE to be transported by the mean field.
The turbulent transport contours have the familiar
three-layer form that was displayed in neutral flow,
is a
small decreasein the strength of the maximum vorticity over the upslope boundary. Since the averageflow
rgte has increasedafter stratification, one might expect
the strain rates and vorticity to be stronger near the
surface. This is not the case, indicating that the increase in flow rate comes through an increase in the
mean streamwise velocity of the interior flow. There is
an increaseddecouplingof the inner and outer flows.
A similar effect has been documented by Garg et al.
I
I
I
I
I
0
1
2
3
4
.................
,.....................
:..............
•.i•,i•..Z
............
::::':ZZZ'
2.8....................
•:•
.......................................
0.2-
'•'•"
r....... •........
5.7;
........
:::::::::::::::::::::::::::::::::::::::::::::::::::::::
32 3.4•:.;.•.
3.6.................
.......
•",•, •
0.1
x E ::.
......,........ ......................
........
•.
. ..
5
....... ......,'
.....
:•:;•:::•E;.;;:•4;.•.:•.;•:::.•[::,,,•,•i...,,,,,..
..........
,.,........
• ( /'
C• .......
-• / ........
.........
./,. (......... • •
0.0
I
0.3
0.2
I
0.4
'
I
0.5
I
0.6
.
I
0.7
x/(2X)
2 CaseRi•
Figure 8. Contoursof mean TKE nondimensionalizedby u,.
-31.
CALHOUN
ET AL' STRATIFIED
I
0.0
I
0.5
FLOW OVER A WAVY SURFACE
I
1.0
I
1.5
I
2.0
9301
I
2.5
0.4--
0.3--
0.2--
0. l --
••..•`.•.••.•:•...`
..`• .•i•...i..:..:g..:•::.•.•:.:...:•:•..:..::.:•.::::•.:....:.•.•.:::::::::•:::..
2.2::::'"i
0.0--
I
I
I
I
I
0.3
0.4
0.5
0.6
0.7
x/(2•)
Figure
9. Contours
of meanproductionof TKE (•P) nondimensionalized
by 100u,•/A. Case
Ri• - 31.
but the strongestpart of the negativelayer has shifted the neutral and Ri•- = 31 cases(in dissipatingturbulent
downstream. The peak in the upper positive region motions).
has actually increasedslightly and shifted downstream
Profilesof the TKE budgetnear the upperwall [Calto (0.54, 0.2) nondimensionalunits. The lower layer houn, 1998] showthat the wavy boundaryhas only a
weakens and shifts upstream -0.2 units.
small effect on the generation and transport of turbuAs might be expected, the less energetic turbulent lent motions near the upper wall.
motionsin the Ri•- = 31 caserequirelessdissipation
see CS1 section6.3 for the mathematical definition). 9. Local Characterization
of Flow
Most of this dissipation is viscousand occurs on the Conditions
upslope boundary. The decreasealong this boundary
So far, we have described the stratified flows in this
is •11%. The subgrid-scale(SGS) dissipationis also
of lowermagnitude,
but the ratio of the SGSto vis- paper using a bulk friction Richardson number similar
cous dissipation remains the same. Therefore the SGS to that usedby Garget al. [1997].However,it is useful
model carries approximately the same burden in both to characterizeinhomogeneousstratified flowslocally as
-0.8
-0.4
0.0
0.4
0.8
0.4--
0.3 w
0.1
0.0
I
I
I
I
I
0.3
0.4
0.5
0.6
0.7
x/(2•)
Figure 10. Contoursof meanvelocitytransport(Tin) nondimensionalized
by 100u,3/X.Case
Ri• - 31.
9302
CALHOUN
ET AL' STRATIFIED
FLOW OVER A WAVY SURFACE
I
I
I
I
I
I
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.5--
0.4--
0.3--
0.1
•:-•:...•.:•.•
..............
:::::
..........................
':::
.....
ß .•..........
:::::::::::::::::::::::::::::::::
:::::::
-•'"'•••::• ß • ......
;•;•':;
.....•-0.2:•:•-0.3
0.0
I
I
I
I
I
0.3
0.4
0.5
0.6
0.7
x/(2•)
Figure
11. Contours
of meantransportof TKE (Tt) nondimensionaliked
by 100u•,/A.Case
Ri•- - 31.
well as globally. Local flow charaterizationsmay bet- the solid boundarieson the mean velocity profile in the
ter portray how the relationshipbetweenthe buoyancy channelflow. In contrast,Garg et al. [1997]foundthe
and inertial effectschangesin variousregionsof the flow flux Richardsonnumber to be useful in characterizing
field. Garget al. [1997]plottedboth the gradientand stratified channel flows. In the following we present
flux Richardson numbers in stratified channel flows over length scalesand nondimensional
parametersbasedon
and spatiallyaveraged(in the hoa fiat bed. They found that the gradient Richardson a short-time-averaged
spanwisedirection)portion of the evolution
number varied from 3 to 5 orders of magnitude over mogeneous
the flow field and did not predict decay rates of turbu- of the more stronglystratifiedcase(Ri• - 62) which
was presented earlier.
There are severallength scalesthat are important in
lence intensities in their stratified channel flows. They
suggestedthat it would be misleadingto attempt to
correlate turbulent statistics with the gradient Richard- characterizinglocal flow conditionsin a stratified flow.
son number. They speculatedthat this might be due For an in-depth description of these length scalesand
to the fact that the gradient Richardsonnumber does nondimensionalnumbers discussedbelow, see Ivey and
Imberger[1991],and Itsweireet al. [1993].We summa-
not account for the kinematic constraints imposed by
I
I
I
I
-0.4
-0.3
-0.2
-0.1
0.4--
0.3--
,E
,.•
•
0.2•
_------ -0.2
-0.1
I
I
I
I
I
0.3
0.4
0.5
0.6
0.7
x/(2•)
Figure 12. Contours
of meantotaldissipation
31.
OOu,/A. CaseRi• =
CALHOUN
ET AL- STRATIFIED
FLOW
OVER A WAVY
i
i
i
-60
-40
-20
SURFACE
9303
xlO-3
•"b
-0.02::.:
...........................................
.............................................................
:......
0.04.......
6"69
'"""
:•'
...............
.........................
•...........................
?2....:...:.:'.
............................................
ß ...............
..........................
..................'
'..... (
..........
:.............
..............
0.06 ...................................................................
'.....
'x
•:•:
>•
•
'.................
;;:::' '.........................
i................
",..,,:, (...........................
•-0.07
•---,, ...........
.:, '•:::• ":i
'.......................
.......................................
...........................
.11.....-.":Ci}?:::::':'::::
.........................
0.0-
-0.1 -
I
I
I
I
I
0.3
0.4
0.5
0.6
0.7
x/(2•.)
Figure 13. Contoursof mean subgrid-scale
(SGS) dissipation(esgs)nondimensionalized
by
100u,3/A.CaseRi•- - 31.
rize three important length scaleshere: the Ellison scale The Ellison scale is the overturning scale. The Ozmi(Le), the Ozmidovscale(Lo), andthe Kolmogorovscale dov scalerepresentsthe scaleof motion at which buoyancy and inertial forcesare in balance. The Kolmogorov
scale is the scale where inertial
v/<
>
and viscous forces are
in balance.
Le=O<•]•y'
(17)
One may form nondimensionalparametersfrom these
length scales. For example, the turbulent Froude and
Lo- (•)«,
(18)Reynoldsnumbermaydefinedas [cf. Ivey andIrnberger,
Lk--(•-)¬,
(19)
1991].
where • is kinematic viscosity and N representsthe
buoyancyfrequencydefined as
N_•gO<p>
Pr
Oy
(20)
Frt - (LolLs)•,
(21)
Ret-(L•/L•)•.
(22)
In addition, a flux Richardson number based on local
flow conditionsmay be defined as
Rif = Bk/(B• + e),
(23)
The buoyancyfrequencyis a measureof the frequency where B• is the buoyancyflux or buoyancysink term in
of oscillationif a fluid particle is vertically displaced. the TKE equation.Ri/is a measureof how important
-0.001
0.8-0.002
-0.003
.........................................................................................................
0.4 - •
-o.oo4
....................................................................................................
/
....
'::::::::::.::::::::::'::'i:'"":i::::::::::>
.....................................................................................
o.oo•
...........................................................
I
o.a-1.....................
:::"'":
•;•
.....
i'...............................
> ...............
::::::::111;;•111;:i;ii.11;:::;•:•iii'.•:'-'•iiii:::'•:•!•::::::::::::O
'øø•
...............
::.::- . .........................
:::"':::::":'2-:
....................
I
I
I
I
I
I
I
1 .•
1.3
1.4
1.5
1.6
1.7
1.8
x/(2X)
Figure
14.
100u,3/A).
Contoursof buoyancysink term of the TKE equation (nondimensionalized
by
9304
CALHOUN ET AL: STRATIFIED
1.5
FLOW OVER A WAVY SURFACE
I
1.0 m
0.5
I
_.•
O.O•
,,; •--••"•
..........................
• ...................................
.; .......... -0.02....
I
I
0.3
0.4
I
I
0.5
0.6
• ;;
I
0.7
Figure 15, Contoursof flux Richards•.number.
the buoyancy term is in the TKE equation relative to
the sum of the dissipationand the buoyancyflux.
Ivey andImberger[1991]haveshownthat the essence
of the state of stratified
turbulence
can be character-
ized with the nondimensionalparameters Ret, Frt, and
the flux Richardsonnumber is less usefully interpreted
as a measureof efficiencyof stirring.
Becausethe buoyancysinkterm (B•) is an important
term in the calculation of the flux Richardson number,
it is presentedin Figure 14. As expected in a stably
Rif. The buoyancyfrequencyIN], the Ellisonscale stratified flow, the buoyancyterm is negative, indicat[L•], and the Ozmidovscale[Lo] representonly com- ing a net drain from the TKE of the turbulent field.
ponents of the nondimensional parameters on which we
focusour attention here. Ivey and Imberger arguedthat
the magnitude of the flux Richardsonnumber is determined by the local values of the turbulent Froude and
Reynolds numbers. We begin with a discussionof the
flux Richardson
number.
The flux Richardson number is sometimes interpreted
The region with the strongest negative buoyancy sink
is shownby the contoursaboveand to the left of the center of the trough. It is not surprisingthat the maximum
is near the strongesttemperature fluctuations and not
far from the peak vertical turbulence intensities. Notice
that the buoyancy term decreasesin magnitude vertically but not as significantly as the dissipationterm.
as the mixing efficiencyor, more appropriately, as a Weak fluctuations in the recirculation zone result in
measureof stirring due to the turbulence. (Rehmann
negligible buoyancy sink values as can be seen by the
[1995]pointsout that mixingis an irreversibledestruc- zero contour in Figure 14.
tion of temperaturegradientsand takesplaceultimately
by moleculardiffusionprocesses
at the Oboukhov-Corrsin The flux Richardsonnumber is shown in Figure 15.
scales.) The conversionof turbulent kinetic energyto Near both the wavy boundary and the upper solidwall
available potential energy is more efficient if relatively the flux Richardsonnumber is low. One might have exless energy is used overcomingviscousdissipationpro- pecteda larger phasedependencenear the wavy boundcesses.Thereforea higherRf representsmoreefficient ary than is shownhere. In order to better understand
stirring. However, Garg [1996] commentsthat in the why this is the case, considerthat the flux Richardpresenceof counter gradient fluxes and internal waves son number will be small in two cases: (1) when the
.........
1.5
w
1.0
I
2
1.5
0.5•
::?'::::::
....................
:::-':::
• 5 •:•:""':•'•?"'?•:':?""•'"¾:::::::;:::::::
................
:'• • 4.5 :'•i!?i;;i;i:•;';i;;:iii.
':•...
......
-'.'... i:ii!i!!!!i!!
3.5 '-'-j•
6 o •':::"•-"•'•i•i:-:•!.'-'-:;;:::::::::::::';;.
.....................
;:'"'"'"'z;"';
5.5
O.O--
.......
"_:'"'""'.',":•
'-" /.• '"•'•:.:.-::•
:;,¾..'
.................................
'-' •"•"•'"'..
.........
•;';2'"'""•"•':•
'
' '"..:'•'2_
.,•
5.5
6 ";..'-'.::•
•.b"'"•
"•,
" '"'"':•:,•',•,,•"•
•.5_ ,,.,"•
•.•
i
I
i
0.5
1.0
1.5
x/(2X)
Figure 16, Contoursof the turbulent Froude number Frt.
CALHOUN
ET AL: STRATIFIED
FLOW
OVER
A WAVY
SURFACE
9305
0.20
0.15 ................
0.10-
0.05 ...............................
0.00 .................................
-0.05
0
2
4
6
8
10
Figure 17. Plot of spanwiseaveragedFrt versusRif at all pointsin the verticalplane of the
flow.
Bk term is near zero (assumingdissipationis not near
zero),and (2) whendissipation
is very largeandtherefore dominatesthe Bk term. Even though there is a
strongphasedependenceof the buoyancyterm near the
wavy boundary, this phase dependenceis much weaker
for the flux Richardsonnumber. Most of the trough
betweenthe crestof the hills displaysrelativelystrong
cal coordinatevalueof 1.0 corresponds
to the midheight
level of the flow domain. Dissipation attenuates more
rapidly toward the center of the trough than the buoyancy sink term. Therefore the buoyancy sink term is
relatively more important in the TKE equation near
the center of the trough. The small magnitude of the
buoyancyterm near the wavy boundary relative to the
other terms in the TKE equation helps to explain why
dissipationcomparedwith the outer flow. Where this is
not the case,as in the recirculationzone, the low fluctu- the flow appearsslowto extinguish(with increasinglevations alsoresult in very low buoyancysink values. The els of stable stratification) the characteristicturbulent
straight contoursof dissipationover the valleys(Fig- structures near the boundary.
ure 12) showthat the strongerturbulencepresentin
Turbulent Froude number [see Ivey and Iraberger,
the trough has the effectof "filling in" the downward 1991]is shownin Figure 16. At the midheightlevel
undulation of the topography.
the turbulent Froudenumberis small, indicatingthat
The flux Richardsonnumber becomeslarger at the stratification plays a strongerrole. Toward both the
midheight level as was documented in the simulations upper and lower wavy boundary the Froude number
of Garget al. [1997].(Notethat the buoyancysinkterm becomeslarge. By this measureone shouldexpectthat
is low here,but the relativeimportanceof buoyancyver- the effectsof stratificationare relatively strongerat the
susdissipationdeterminesthis parameter.) The verti- midheight level.
O.2O
-
0.15
-
0.10
i
0.05
0.00
I
I
2
4
I
6
Frt
Figure 18. Frt versusRif, crestprofile.
lO
9306
CALHOUN
ET AL' STRATIFIED
FLOW
OVER
A WAVY
SURFACE
0.20
0.15
•:• O.lO-
O.O5 flat
0.00
....
-
I
I
I
I
2
4
6
8
lO
Frt
Figure 19. Frt versusRi/, trough profile.
ivey and Iraberger[1991]analyzeddata from several dicates the upper fiat boundary. Near the solid boundexperiments in stratified turbulence and discusseda re-
aries the turbulent Froude number is very large, and
lationshipbetweenRif and Frt. Figure 17 showsthis this is the reasonfor the asymptotic tail to the right. As
relationshipbetweenRif and Frt for stratifedflowover the vertical profile proceedstoward the midheight level,
a wavy bottom boundary. The valuesfor the nondimen- the effectsof stratification becomemore significant,dissional parameters were averagedin the spanwisedirec- sipation decreases,and mixing efficiencyincreases.The
tion to obtain a vertical streamwise plane of data. All resultis a peak in the Frt versusRif plot. As the verof these points are plotted in Figure 17. It is interesting that all of the points in the domain fall on the
approximate shape describedby Ivey and Imberger.
Since this type of plot contains no information on
where the individual points are located in the physical
tical profile increasesfarther toward the fiat boundary,
the turbulent Froude number again increasesand the
(throughthe physicaldomain)of data beginningfrom
ure 20 are because of the recirculation
the lower wavy boundary and proceedingupward to the
fiat boundary. In the Figures 18 and 19, the arrows indicate increasingvertical distancefrom the lower wavy
boundary, and the upper curve indicatesthe points being plotted are near the wavy boundarywhile lower in-
part of the curve near the wavy boundary representsthe
part of the profile that passesdirectly through the recirculation bubble. The buoyancyterm is approximately
zero in the bubble becauseof the very low levelsof fluctuations. In fact, small positive valuesof the buoyancy
Frt versusRif pointsmovedownwardand toward the
asymptotic tail. Notice that in Figure 20 the largest deviation from the fiat curve occursfor the vertical profile
domain (rather than the Rif versusFrt plane), Fig- which passesthrough the recirculation zone in the lee
ures 18, 19, and 20 are presented. These figuresdisplay of the hill.
the Rif versusFrt valuesfor individualverticalprofiles The largest variations in the curves shown in Fig-
0.20
0.15
O.10 -
upslope
crest
•k•.•••.......•....•
........................
'"'
0.00- lee/recirc.
zone
0
I
I
2
4
I
I
6
8
Frt
Figure 20. Frt versusRi/, four profilesalongthe wavy surface.
10
zone. The lower
CALHOUN
ET AL: STRATIFIED
.........
FLOW
OVER
A WAVY
SURFACE
9307
25
............................
,30"
.,
........................
35...............
:•":":.:;:';•.......:;...:
..................
:::::::::::::::::::::::::::::::::::
............
::::":.;.;¾;:';;;;:.::.:.•;::•;'?::':;:';::•;;'•;;;;:;;;'40:'::'
"'"'•.•.....::.;:.::....
.........
.:......:.::.:;:::..:::.:.'.;;;2..;.;:
.......
..:.::::;.;;(:•'"'z..:•;E.
E55•.:;;;'::;
.............
/;::::'z;?,
,......
;Z.:';:=:::';::';:
................
:.::
:::::;;:X1:•2•;•'½.•
':2!½;•
"=
.................
'.....................
I
I
0.5
I
1.0
1.5
Figure 21. Contours of turbulent Reynolds number Ret.
term exist near the wall in the lee/recirculationzone.
The wavy surface causesan upward shifting and a
Becausebuoyancyterm switchessign in this region, the smearing of the expected shape given by Iraberger and
flux Richardsonnumber actually has small negative val- Ivey [1991]. However,the major point of Ivey and Imues. This somewhatsurprisingbehavior only occursin bergeris that the flux Richardsonnumber is determined
the recirculation
zone near the wall. Note that the turby the turbulent Froude and Reynolds numbers. Therebulent Reynolds number is quite small in this region. fore it is important to examine the local Reynolds numNa [1991]comments
that insidethe recirculationof his ber in the vicinity of the wavy and flat boundaries. Noflow with an adversepressuregradient over a fiat wall tice in Figure 21 that there are significantdifferencesin
there is a laminar-like
behavior.
the behavior of the turbulent Reynolds number between
Moving higher on the same vertical line, the profile the upper half of the flow domain near the flat surface
exits the recirculation zone and enters the shear or mixand lower half of the domain near the wavy surface.
ing layer. The buoyancy sink term is relatively strong In fact, the Reynolds number is in general higher near
in this layer comparedto the dissipation, and therefore the wavy boundary. The differencesin the flow condithe mixing efficiencyas measuredby the flux Richard- tions near the lowerwavy surface(crestversustrough)
son number is high. The curve for the wavy part of the cause an increasedscattering of the data points. The
boundary indicates this behavior by the upward devia- increased buoyancy flux term toward the wavy-surface
tion in the curve at about Frt = 4. Higher vertically side of the channel is not counterbalanced by the disalongthe profile,the Frt versusRif curvereturnsto the sipation, which decays rapidly from the surface. The
expected curve. The "trough" profile partially passes result is an increasedmixing efficiencybecauseof the
through the downstreamend of the recirculation zone wavy surface.
and then through the mixing layer, which is also quite
strong over the trough. Hence the "trough" profile dis10. Anisotropy
plays an upward deviation from the other profiles simOne would intuitively expect that as a flow becomes
ilar to the curveprovidingFrt and Rif near the wavy
boundary.
increasingly stratified and fluid motion in the vertical
0.6 I
0.5 I
0.4 m
0.2
-•-
•
0.55
•
0.1
0.0
-0.1
I
I
I
I
I
0.3
0.4
0.5
0.6
0.7
x/(2X)
Figure 22. AnisotropyparameterVrms/Urms.CaseRir - 31.
9308
CALHOUN
ET AL: STRATIFIED
FLOW OVER A WAVY SURFACE
0.6 I
0.5 m
0.4 I
0.3 I
0.2 m
0.1
0.0
-0.1
I
I
I
I
I
0.3
0.4
0.5
0.6
0.7
x/(2X)
Figure 23. Anisotropyparameter•rms/Urms-CaseRi• -62.
direction is depressed,anisotropy would increase. This [Kim et al., 1987]. The analysisdividesthe Reynolds
has beenreportedby Garg et al. [1997]and Itsweireet shear stressinto four categoriesaccordingto the signs
al. [1993].As can be seenin contoursof the anisotropy of u" and v". The first quadrant, u" > 0 and v" > 0,
parameter (Figures22 and 23), this is also the casein contains outward movement of high-speed fluid. The
the flows presentedhere, where the parameter remains
consistently less than unity. The anisotropy parame-
ter is definedas •rms/Urms [seeBriggset al., 1998]. A
decreasein the anisotropy parameter correspondswith
a greater disparity between the scales of the vertical
and horizontal turbulent intensities, and therefore correspondswith greater anisotropy.
Anisotropy is the greatest near to the crests. Over
the troughs the turbulence is less anisotropic. The
secondquadrant, u" < 0 and v" > 0, containsthe motion associatedwith ejections of low-speed fluid away
from the wall. The third, u" < 0 and v" < 0, contains
inward motion of low-speedfluid. The fourth quadrant,
u" > 0 and v" > 0, containshigh-speedfluid moving
toward the wall. The fourth quadrant is often referred
to as a "sweep"event. Kim et al. [1987]showedthat
"sweep"eventsdominatenear the wall (y+ < 12) and
"ejections"
are strongerabovethis region(y+ = u,y/•).
trendof increasing
anisotropy
in morestratified
flows Garg et al. [1997] applied the techniqueto stably
and widely varying levelsof anisotropy in complex flows stratified channel flows and found a disruption of the
providesstrong evidenceof the importance of assuring "burst-sweep"processby stable stratification, which in
that turbulence models have the ability to adapt to lo- turn suppressesTKE production.
We have applied quadrant analysis to both casesof
cal anisotropy.
stably stratified turbulent channel flows over a wavy
bottom boundary. We present examples in Figures 24
11. Conditional
Sampling
and 25 of the full set of figures[Calhoun1998]. The
Quadrant analysis of the Reynolds shear stressprovides information on the contribution of various types
of turbulent events to the total turbulence production
Reynoldsshearstressesfor eventsthat were larger than
a specifiedthresholdare averagedover the spanwisedirection and over time (seeCS1) for four types of quad-
0.6 I
0.5•
0.4 I
• 0.3
- -0•.>
-0.5
•
• 0.2
-0.6
_'-•.7.
_
•
0.1
0.0
-0.1
I
I
I
I
I
0.3
0.4
0.5
0.6
0.7
x/(2x)
Figure 24. Averagevalueof Reynoldsshearstressin Q2 "ejection"eventsfor a thresholdof
0.13. Case Ri•- - 31.
CALHOUN
ET AL: STRATIFIED
0.6
t
FLOW
OVER
A WAVY
SURFACE
9309
v-0.15
0.5
0.4[ •
0.3
-
.
-0.15
-0.25
-0.35
0.2
0.1
0.0
-0.1
I
0.3
I
0.4
I
0.5
I
0.6
I
0.7
x/(2•)
Figure 25. Average value of Reynolds shear stressin Q2 "ejection" events for a threshold of
2.1. Case Ri•- - 31.
rant events, yielding therefore a vertical plane of stress 12.
Summary
contours(1) Q2 or "ejection"events,(2) Q4 or "sweep"
Stably stratified flows over a wavy surface display
events,(3) StableQ2 events(Op/Oy< 0), and (4) Stamany of the same features of the neutral flow, though
ble Q4 events(O•/Oy < 0).
The thresholds
were defined
as a fraction
of the max-
imum average Reynolds shear stress. Contour plots
showing the number of times each type of event oc-
curedin the data set are presentedby Calhoun[1998].
Therefore approximate location, size, and number of occurrencesof the major Reynolds shear stressevents can
be studied.
As was seen earlier, the maximum Reynolds stressis
located over the trough at (0.53, 0.1) nondimensional
with somekey differences. In both caseswe find streamwise vortices, a shear layer in the lee of the crest, and
recirculation. The detached shear layer is associated
with very strong mixing and the recirculation has relatively weak mixing.
Many of the characteristics of other stratified flows,
presented,for example,by Garg et al. [1997], Coleman
et al. [1992],and Holt et al. [1992],are presentin stratifed flow over a wavy boundary. For example, the evidence suggeststhat inner and outer regionsof the flow
becomemore decoupledwith increasingstratification.
Terms of the TKE budget near the wavy wall show
a structure similar to the neutral flow, but production,
transport, and dissipation levels are suppressed. The
wavy boundary has only a small effecton the generation
and transport of turbulent motions on the upper wall.
units. Analysis of the Ri• - 31 figuresin the area near
to the maximum showsthat both ejections and sweeps
are strong but ejections dominate. However, the number of occurrencesof each event is roughly the same,
indicating that the average magnitude of the ejection
events is larger. This is true also for the smaller events.
Seventy-two percent of the magnitude of the Reynolds
The state of turbulence in the stratified fluid is anashear stresswhich comesfrom the ejection events is crelyzed
by graphing the relationshipsbetweenkey nondiated by the larger events. These larger events represent
mensional
parameters. The relationship describingthe
less than a third of the number of events which exceed
state of turbulencein a stratified fluid givenin Ivey and
the lowest threshold. Larger ejections occur slightly
Iraberger[1991]is approximatelysatisfied.
downstream of smaller ejections and do not occur near
Anisotropyvariessignificantlywith positionand stratthe boundary as can be seen in the gap between the
ification. Specifically,anisotropyis greatestnear to the
contoursand the surfacein Figure 25.
crests and is weaker in the trough and increaseswith
The contours of the sweepsalso reach a maximum stronger stratification. This suggeststhe importance
overthe trough,thoughslightlyupstreamof wherethe of the ability of turbulence models to adapt to local
ejectionspeak. The sweepsalso approach closerto the anisotropy.
bottom surface.In contrastto ejections,larger sweeps Using conditional sampling, we have analyzed the
occur slightly upstream of smaller sweeps.
major Reynolds stressevents. The strongestReynolds
A largerfraction of the eventsare stablewith stronger stressesoccur suspendedover the trough at approxistratification. Roughly 55 to 60% of ejectionsare stable mately the height of the crests. In this area, ejections
for the Ri• = 31. Over 70% are stable in the more dominate sweeps. However, the number of occurrences
stratified case. A larger percentageof Q4 events are of each event is roughly the same, indicating that the
stable,80 to 85% at Ri•: 31 and 87 percent at Ri• = averagemagnitude of ejection eventsis larger.
Features of the stratified flow over a wavy wall are
62. Garget al. [1997]and Holt et al. [1992]alsofound
that statically stable events increasedwith increasing similar to the neutral flow. The major differencesare
Richardson
number.
dampeningof many of the TKE terms, increasedgradi-
9310
CALHOUN
ET AL: STRATIFIED
FLOW OVER A WAVY SURFACE
ents of velocity (in the vertical directionwhich caused Kim, J., P. Moin, and R. Moser, Turbulence statistics in
fully developedchannel flow at low Reynolds number, J.
increasedvelocitiesin the interior of the flow domain),
Fluid Mech., 177, 133-166, 1987.
and greater anisotropy of the turbulence.
Metais, O., and J. Herring, Numerical experimentsin forced
stably stratified turbulence, J. Fluid Mech., 202, 117-48,
Acknowledgments.
The authors are deeply appreciative for the support of this work over the years by ONR
grantsN00014-92-J-1223(scientificofficersR. Abbeyand R.
Ferek) and N00014-94-0190(scientificofficerE. Rood) and
by NSF grant ATM-9526246(programdirectorR. Rogers).
References
1989.
Na, Y., Direct numerical simulation of turbulent boundary layers with adversepressuregradient and separation,
Ph.D. thesis,Dep. of Mech. Eng., StanfordUniv., Stanford, Calif., 1996.
Pan, Y., and S. Banerjee, A numerical study of free-surface
turbulence in channel flow, Phys. Fluids, 7, 1649-1664,
1995.
Rehmann, C., Effects of stratification and molecular diffuBriggs, D., J. Koseff, and J. Ferziger, Turbulent mixing in
sivity on the mixing efficiencyof decayinggrid turbulence,
a shear-free stably two layer fluid, J. Fluid Mech., 3•5,
Ph.D. thesis,Dep. of Civ. Eng., Stanford Univ., Stanford,
175-208, 1998.
Calif., 1995.
Calhoun, R., Numerical investigationsof turbulent flow over
Salvetti,
M., and S. Banerjee, A priori tests of a new dycomplex terrain, Ph.D. thesis, Dep. of Civ. Eng., Stannamic subgrid-scalemodel for finite-differencelarge-eddy
ford Univ., Stanford, Calif., 1998.
simulations, Phys. Fluids, 7, 2831-2847, 1995.
Calhoun, R., and R. Street, Turbulent flow over a wavy
Salvetti, M., Y. Zang, R. Street, and S. Banerjee, Largesurface: Neutral case, J. Geophys. Res., this issue.
eddy simulation of free-surface decaying turbulence with
Coleman, G., J. Ferziger, and P. Spalart, Direct simulation
dynamic subgrid-scale models, Phys. Fluids, 9, 2405of the stably stratified turbulent Ekman layer, J. Fluid
2419, 1997.
Mech., 2•, 677-712, 1992.
Fallon, B., M. Lesieur, F. Delcayre, and D. Grand, Large- Zang, Y., On the development of tools for the simulation
of geophysicalflows, Ph.D. thesis, Dep. of Mech. Eng.,
eddy simulations of stable-stratification effects upon a
Stanford Univ., Stanford, Calif., 1993.
backstepflow, Eur. J. Mech. B, 16, 625-643, 1997.
Garg, R., Physicsand modelingof stratifiedturbulent chan- Zang, Y., and R. Street, A compositemultigrid method for
calculating unsteady incompresibleflows in geometrically
nel flows, Ph.D. thesis, Dep. of Mech. Eng., Stanford
complex domains, Int. J. Nurner. Methods Fluids, 20,
Univ., Stanford, Calif., 1996.
341-361, 1995.
Garg, R., J. Ferziger, and S. Monismith, Hybrid spectral
Zang, Y., R. Street, and J. Koseff,A dynamicmixed subgridfinite difference simulations of stratified turbulent flows
scale model and its application to turbulent recirculating
on distributed memory architectures, Int. J. Fluids, 2•,
1129-58, 1997.
flow,Phys.Fluids,5, 3186-3196,
1993.
Herring, J., and O. Metais, Numericalexperimentsin forced Zang, Y., R. Street, and J. Koseff, A non-staggeredgrid,
fractional step method for the time-dependent incomstably stratified turbulence, J. Fluid Mech., 202, 97-115,
1989.
Holt, S., J. Koseff, and J. Ferziger, A numericalstudy of the
evolution and structure of homogeneousstably stratified
sheared turbulence, J. Fluid Mech., 237, 499-539, 1992.
Hudson, J., L. Dykhno, and T. Hanratty, Turbulent production in flow over a wavy wall, Exp. Fluids, 20, 257-265,
1996.
pressible Navier-Stokes equation in curvilinear coordinates, J. Cornput. Phys., 11•, 18-33, 1994.
R. J. Calhoun, J. R. Koseff, and R. L. Street, Department of Civil and Environmental Engineering,Stanford University, Stanford, CA 94305-4020 ([email protected];
[email protected];
[email protected])
Itsweire, E., J. Koseff, J. Briggs, and J. Ferziger, Turbulence
in stratified shear flows: Implications for interpreting
shear-inducedmixing in the ocean, J. Phys. Oceanogr.,
21, 1508-1522, 1993.
Ivey, G., and J. Iraberger, On the nature of turbulence in a
stratified fluid. Part 1: the energeticsof mixing, J. Phys.
Oceanogr., 21, 650-658, 1991.
(ReceivedNovember13, 1998; revisedSeptember20, 2000;
acceptedJanuary 2, 2001.)