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Investigation of shallow mixing layers by BGK finitevolume modelMohamed S. Ghidaoui a & Jun Hong Liang ba Hong Kong University of Science and Technology, Kowloon, Hong Kong, P.R. Chinab University of California, Los Angeles, CA, USAVersion of record first published: 24 Jul 2008.
To cite this article: Mohamed S. Ghidaoui & Jun Hong Liang (2008): Investigation of shallow mixing layers by BGK finitevolume model, International Journal of Computational Fluid Dynamics, 22:7, 523-537
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Investigation of shallow mixing layers by BGK finite volume model
Mohamed S. Ghidaouia* and Jun Hong Liangb
aHong Kong University of Science and Technology, Kowloon, Hong Kong, P.R. China; bUniversity of California, Los Angeles,CA, USA
(Received 14 May 2008; final version received 29 May 2008 )
Turbulent shallow mixing layers and their associated vortical structures are ubiquitous in rivers, estuaries and coasts.Examples of these flows can be found in compound/composite channels, at the confluence of two rivers, at harbour entrancesand at groyne fields. A finite volume 2D model, based on the averaging of the 3D shallow water equations with respect todepth and in which the numerical fluxes are obtained from the Bhatnagar–Gross–Krook (BGK) Boltzmann equation, isapplied to shallow mixing layers for which experimental results are available. This model is hereafter referred to as the BGKmodel or BGK scheme. The BGK scheme is explicit, second order in time and space and conserves bothmass andmomentum.The BGK relaxation time is locally evaluated from the classical turbulence model of Smagorinsky. The BGK modelaccurately represents the mean flow field such as mean velocity profile, mean spread of the mixing layer, mean position of themixing layer centreline and mean surface water profile. In addition, the Kelvin–Helmholtz (KH) instability includinginception, vortex roll up, vortex growth by pairing and the eventual decay of the vortices by bed shear is well represented bythe model. On the other hand, the magnitude of the turbulence intensity is over-predicted by the shallow water model. Thisdiscrepancy is partly due to the fact that the turbulence forcing assumed may not represent the actual random perturbationsthat may exist in the laboratory experiments and partly due to the inability of the depth-averaged shallow water equations toallow for the redistribution of turbulent energy along the vertical direction, since these governing equations do not model the3D turbulence. Thus, the depth-averaged shallow water equations are well suited for investigating the KH stability and forpredicting mean flow field including velocity profiles and transversal mixing of mass momentum in shallow environments.Accurate prediction of turbulence statistics would require resolving the small 3D scales with respect to water depth.
Keywords: shallow mixing layers; stability of shallow shear flows; BGK modelling of shallow flows; coherent structures;quasi-2D turbulence
1. Introduction
Turbulent mixing layers are commonly observed in various
engineering applications such as combustion, propulsion
and environmental flows. Turbulent mixing layers are
susceptible to Kelvin–Helmholtz (KH) instabilities and
these instabilities are known to result in complex flow
phenomena such as vortex roll up and pairing. Turbulent
deep mixing layers, where the length scale perpendicular to
the plane of the flow is much larger than the length scale in
the plane of the flow, have been extensively investigated
through stability theory as well as laboratory and numerical
experiments. It is found that the growth of KH instabilities
lead toKHvortices. The subsequent pairing of these vortices
lead to organised nearly 2D vortical structures. These
organised structures are themselves susceptible to secondary
instabilities, which are responsible for the breakdown of the
2D organised structures into 3D turbulence. The breakdown
of organised structures into 3D turbulence is associated with
the stretching and tilting of vortices and follows the classical
energy cascade principle, where turbulent kinetic energy
flows from large to small scales.
Turbulent shallow mixing layers and their associated
vortical structures are ubiquitous in rivers, estuaries and
coasts. Shallow mixing layers in surface waters are
characterised by having a large horizontal length scale in
comparison to their vertical length scale. The spectacular
Naruto vortical structures (whirlpools) generated by tidal
exchange between the Pacific Ocean and the Seto Inland
sea in Japan are a typical example of shallow mixing
layers. The understanding of shallow mixing layers is
important for modelling water quantity and quality, and
for the analysis and prediction of the transversal exchange
and spread of mass, momentum and energy in shallow
water environments.
The behaviour of vortical structures in shallow mixing
layers is distinct from that in deep flows. Unlike in deep
flows, vortical structures in shallow flows are subjected to
bottom shear stresses and their vertical extent is limited by
the water depth. Indeed, stability analysis and experimental
investigations show that the dynamics of shallow mixing
layers is controlled by the bottom shear and by the
shallowness of the flow (e.g. Chu et al. 1991, Chen and Jirka
ISSN 1061-8562 print/ISSN 1029-0257 online
q 2008 Taylor & Francis
DOI: 10.1080/10618560802238283
http://www.informaworld.com
*Corresponding author. Email: [email protected]
International Journal of Computational Fluid Dynamics
Vol. 22, No. 7, August 2008, 523–537
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1995, 1997, Ghidaoui and Kolyshkin 1999, Uijttewaal and
Booij 2000, Kolyshkin and Ghidaoui 2002, van Prooijen and
Uijttewaal 2002a, 2002b).
The role of shear stresses and flow shallowness on
the behaviour of turbulent shallow mixing layers is as
follows. KH instabilities result in the formation of
vortical structures extending from the flow bed to the
water surface. The confinement of the flow by the water
depth suppresses the vortex stretching mechanism, forcing
the KH vortical structures to move in horizontal plane and
to exhibit a strongly 2D behaviour. The presence of the
vortex pairing mechanism and the absence of the vortex
stretching mechanism result in a reverse energy cascade,
which leads to a horizontal growth of the vortical
structures along the downstream direction. The shallow
flow experiments of Uijttewaal and Booij (2000) and
Uijttewaal and Jirka (2003) show that there is a range
of scales for which the slope of energy spectrum follows
the -3 law. The -3 law provides clear evidence for the
reverse energy cascade, also called enstrophy cascade and
explains the flow of energy from small to large scales
evidenced by the growth of the vortical structures in the
downstream direction. The bottom friction, on the other
hand, dissipates energy. This dissipation increases as the
scale of the vortices increases. As a result, the bottom
friction reduces the flow of energy from the small to the
large scales and limits the extent of the enstrophy cascade.
Indeed, the shallow flow experiments of Chu and
Babarutsi (1988) and Uijttewaal and Booij (2000) show
that the growth/spread rate of the mixing layer decreases
with downstream distance and eventually becomes
negligibly small.
Most current channel models are still based on 1D or
2D shallow water equations due to their relatively low
computational cost. However, their ability in modelling
complex flow structures governed by 3D dynamics, which
features channels with complex topographical conditions,
is still questionable. For example, Ghidaoui et al. (2006)
found that the shallow water model underpredicts the
bubble size in shallow recirculation zone in the wake of a
circular cylinder by about 30% when compared to
laboratory experiments. It is conjectured by Ghidaoui
et al. (2006), that the difference is partly due to the failure
of modelling unsteady flow conditions by friction
coefficients derived from steady flows.
The current paper investigates the ability of the
shallow water equations in modelling shallow mixing
layers using a finite volume numerical solution on the
basis of the Boltzmann equation. The classical Smagor-
insky turbulence model is used to estimate the collision
term. The model is applied to the experimental set-up of
Uijttewaal and Booij (2000). The study investigates the
ability of the shallow water model to reproduce the mean
flow field, the turbulence statistics and the general flow
features such as vortex roll up and pairing.
2. Governing equations
Governing equations for shallow turbulent flow can be
obtained by integrating the Navier–Stokes equations
(Batchelor 1967) over the vertical dimension and averaging
over resolved scales. The depth-averaged shallow water
equations, obtained by integrating the incompressible 3D
Navier–Stokes equations, invoking the kinematic condition
at the free surface, the no slip condition at the bed and the
hydrostatic pressure assumption gives
›h
›tþ ›huk
›xk¼ 0; ð1Þ
›hua›t
þ›ðhuaukþghh=2Þ›xk
¼ ghSa2 tarþ y ›
›xkh›ua›xk
� �;
ð2Þ
where t is time; x is spatial coordinates;k ¼ 1, 2 anda ¼ 1, 2with 1 indicating the streamwise direction and 2 indicating
the cross-stream direction;g is the gravitational acceleration;
h is water depth; u is the depth-averaged velocity; S is the bed
slope; r is the density of water; n is the kinematic viscosity ofwater; and t is the bottom shear stress (shear between thefluid and the bed).
Similar to compressible flows, it is convenient to
introduce the mass-weighted (Favre) filtering as follows
(e.g. Xu et al. 2005):
~f ¼ hf�h; ð3Þ
where ~f is the Favre filtering of f. Applying the Favre
filtering to systems (1)–(2) gives:
›�h
›tþ ›
�h~uk
›xk¼ 0; ð4Þ
›�h~ua›t
þ ››xk
�h~ua ~ukþdak 12g�h2
� �
¼ g�hSa2 �tarþ y ›
›xk�h›~ua›xk
� �þ ››xk
ð�h~ua ~uk2 �h~ua ~ukÞ:
ð5Þ
The last term on the right hand side of (5) requires a
turbulence closure model. The effects of unresolved
motions on resolved motions are conventionally related to
resolved motions by an expression similar to that for
viscous stress except that the kinematic viscosity is
replaced by the eddy viscosity. Various turbulence models
are devised to relate the eddy viscosity with the resolved
flow field. In the current study, the Smagorinsky model
(Pope 2002) is used,
y e ¼ CSlm2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2SkaSka
p; Ska ¼ 1
2
›~ua›xk
þ ›~uk›xa
� �; ð6Þ
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where CS is the Smagorinsky constant; ye is eddyviscosity; lm is the mixing length, which is related to the
grid size by lm ¼ffiffiffiffiffiffiffiffiffiffiffiDxDy
p. Although there is no theoretical
foundation for why the Smagrorinsky model can be
applied to 2D flows, the adoption of this model outside its
domain of applicability has generally led to encouraging
results. For example, the model adopted in this study has
been successfully applied by Zhou (2004) to 2D channel
flows and by Zhou (2004) as well as Ghidaoui et al. (2006)
to shallow wake flows. In addition, the Princeton ocean
model (POM) uses the Smagorinsky relation to estimate
the horizontal component of the turbulent diffusion
(Mellor 2004). POM has been successfully applied to a
wide range of problems in ocean circulations (e.g. Oey
et al. 2005). It is worthwhile noting that the 2D version of
the Smagorinsky model belongs to the general class of
mixing-length models (Pope 2002), where the mixing-
length in the Smagorinsky model is related to the
computational grid-size. The linkage between the
Boltzmann equation and the filtered shallow water
equation requires that the collision time be given by
t ¼ ðy þ y eÞ=g�h. The code evaluates the collision time atevery computational node in order to account for the
variability of eddy viscosity with time and space.
For free-surface flows, shear stresses are commonly
modelled by the following quadratic friction law
(Schlichting and Gersten 2000, Pope 2002),
�tar¼ cf ~ua
ffiffiffiffiffiffi~u 2k
p2
; ð7Þ
where cf is the friction coefficient. For flows over a smooth
bottom, the following semi-empirical law is often
implemented for the friction coefficient (Schlichting and
Gersten 2000, Pope 2002),
1ffiffiffifficf
p ¼ 24 log 1:254Re
ffiffiffifficf
p� �
ð8Þ
3. Numerical model
Systems (4)–(8) are solved using a conservative finite
volume method on irregular grids, where the algorithm for
the mass and momentum fluxes at the control surface of
the finite volume is obtained from the solution of the
Bhatnagar–Gross–Krook (BGK) Boltzmann equation.
Such approach is selected because of its conservation
properties, its ability to handle complex geometry, and to
solve waves and turbulence without the need for operator
splitting (Ghidaoui et al. 2001, Liang et al. 2007). Such
properties are critical for the authors’ current and future
research on shallow flows, which entails the study of 2D
and 3D instabilities in shallow flows in geometries with
varying degrees of complexities and the interaction
between turbulence and waves.
The discretised form of (4) and (5) is (details are
available in Refs. Ghidaoui et al. (2001), Liang et al.
(2007)),
Unþ1i; j ¼Uni; jþSni; jDt
21
Vi; j
XSi; js¼1
Ls g1þg3 ››t
� �ðAsiþAsoÞ
�
þ Dt2g1þg6 ››t
� �Esþðg2þg3Þ
£ ½ns·7ðBsiþBsoÞþts·7ðDsiþDsoÞ�
þl5ðns·7Gsþts·7IsÞ�: ð9Þ
Where U¼ b�h; ~ua �hc ; S ¼ [›b/›xa]; (i, j) denotes spatialposition; n indicates time step; V is the area of a grid;Ls is the length of the side s of a grid; ns is the unit vectornormal to the side s; ts is the unit vector normal to the side s;
g1¼t 2te2Dt=t; g2¼2t2þðt2þtDtÞe2Dt=t; g3 ¼ tg1;g5¼2t Dtþ2t2ð12e2Dt=tÞ2Dtte2Dt=t; g6¼tDt2t2ð12e2 Dt=tÞ, with t¼ ðy þyeÞ=gh. The matrices in (9) aregiven in the Appendix. It is important to note that the collision
time and node (n,i,j) is given by:
t ni; j¼yþðyeÞni; jghni; j
: ð10Þ
In what follows, the streamwise direction x1 is denoted by x
and the cross-stream direction x2 by y.
4. Hydraulic and numerical parameters
The numerical test rigs correspond to the experiments
conducted by Uijttewaal and Booij (2000) and van
Prooijen and Uijttewaal (2002a). Shallow mixing layers
develop from parallel streams of different velocities
separated by a splitter at the inflow boundary. Hydraulic
parameters for the tests are summarised in Table 1.
A variety of Smagorinsky coefficients were tested. It is
found that Cs ¼ 0.2 gives the best correlation with theexperimental data. The sensitivity of the results to the
Smagorinsky coefficient is given in Figure 17.
Table 1. Hydraulic conditions for shallow mixing layer tests.
Case U1 (m/s) U2 (m/s) H (m) cf ( £ 1023)1 0.23 0.11 0.042 6.42 0.32 0.14 0.067 5.4
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At the inflow (i.e. x ¼ 0), zero lateral velocity isimposed and a mean flow profile is prescribed for
streamwise velocity. In order to minimise the reflection
from the outflow boundary, a radiation boundary condition
(Durran 1999) is imposed at x ¼ Lx. At the lateralboundary (i.e. y ¼ ^ (1/2)W), a free-slip condition isprescribed. It is shown by Socolofsky and Jirka (2004) that
the stability of co-flowing shallow mixing layers are of
convective nature. This means that the flow behaves like a
‘noise-amplifier’ (Huerre and Monkewitz 1990), and
large-scale vortical structures inside a shallow mixing
layer are sustained by perturbation prescribed at the inflow.
In natural and laboratory flows, the flow is perturbed by the
presence of developed turbulence, as well as environmen-
tal (natural) factors. In numerical simulations, however,
artificial forcing is introduced at the inflow boundary to
mimic the perturbations in the laboratory experiments. In
the current study, white noise is prescribed at the inflow.
Sensitivity of flow evolution to inflow perturbations will be
explored in a later section.
4.1 Mean inflow profile
In all laboratory set-ups (see Refs. Chu and Babarutsi
1988, Uijttewaal and Booij 2000), a splitter plate is located
at the inflow to separate streams of different velocities.
Due to the boundary layers attached to the splitter plate,
wakes develop downstream of the splitter plate. The effect
of splitter wake is reflected by a velocity deficit at the low
speed side of the mean velocity profile at the inflow and it
is shown to be obvious in laboratory measurements (see
Figures 4 and 5 in Ref. Uijttewaal and Booij 2000). In the
current study, the effect of splitter wake is modelled in the
mean inflow profile by adding a secant hyperbolic term
which is typical to wakes (Monkewitz 1988) to the
hyperbolic tangent profile typical to mixing layers (van
Prooijen and Uijttewaal 2002a) as follows,
UðyÞ ¼ Uc þ DU0 tanh 2yd
� �2 0:5 1 þ sinh2 y
d
� �h i� �:
ð11ÞFigure 1 compares the measured inflow profile, the
mean inflow profile with splitter wake effects given by
Equation (11) (continuous curve in Figure 1) and the
hyperbolic tangent profile (dashed curve in Figure 1). It
shows that the measured profiles just after the edge of
the splitter plate collapse nicely into Equation (11), while
the conventional hyperbolic profile hardly matches the
measured data.
The development of mixing layers from an inflow
profilewith andwithout the effect of splitterwakes is shown
in Figure 2. It can be seen that the development of shallow
mixing layers is generally the same for both mean inflow
profiles. The mixing layer width for the mean profile with
the effect of splitter wake is larger near the inflow
(x , 2 m). This can be ascribed to the fact that the meaninflow profile with the effect of splitter wake actually
broadens the mixing layer (refer to Figure 3). At the region
where x . 2 m, the mean flow recovers to the traditional
–0.4
0
0.4
0.8
1.2
–2 –1.5 –1 –0.5 0 0.5 1 1.5 2
tanh profiletanh with splitter wakedata at 0.05m for 42mm flowsdata at 0.05m for 67mm flows
y/δ
(u – u1) / (u2 – u1)
Figure 1. Comparison of analytical and measure velocityprofile at the inlet of a mixing layer (d tanh denotes mixing layerwidth based on tangent hyperbolic profile; d wake denotesmixing layer width based on mean flow profile with wake effect).
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 2 4 6 8 10 12 14 16
x (m)
With wakeWithout wake
δ (m)
Figure 2. Development of mixing layer width from inflowprofiles with and without the effects of splitter wakes (case 1).
0
0.2
0.4
0.6
0.8
0 10 15
x (m)
0.1u0.2u0.3u0.4u
δ (m)
5
Figure 3. Development of shallow mixing layers under inflowforcing of different amplitudes (case 1).
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tangent hyperbolic profile in a manner similar to the
laboratory tests (see Figures 4 and 5 in Ref. Uijttewaal and
Booij 2000). In addition, the computed mixing layer widths
from two inflow profiles are comparable in this region.
4.2 Inflow perturbations
Co-flowing mixing layers are highly susceptible to inflow
perturbations due to their convective instability nature.
Laboratory experiments in deep mixing layers also showed
that the development of mixing layers is highly sensitive to
inflow perturbations, in particular, the forcing at the most
unstable frequency (harmonic) and its subharmonics (for a
review on experimental findings, see Ref. Ho and Huerre
1984). In numerical studies, several attempts have been
made to simulate a natural perturbation both in deepmixing
layers (e.g. Stanley and Sarkar 1997) and in shallowmixing
layers (e.g. van Prooijen and Uijttewaal 2002b). A note of
caution, all this sophisticated forcing can never duplicate
the real turbulent field since a lot of detailed information,
like amplitudes and phases of different modes, is uncertain.
The imposed forcing will undergo an adjustment stage to
evolve into the real turbulence.
In the current study, white noise is prescribed at the
inflow. Stanley and Sarkar (1997) and Yang et al. (2004)
in their studies of deep mixing layers found that a naturally
developed mixing layer forms under this inflow forcing. To
investigate the sensitivity of shallow mixing layers to the
perturbation, perturbations of amplitudes ranging from
10% of the inflow velocity difference (DU0) to 40% of theinflow velocity difference are imposed in both tests.
Figures 3 and 4 show the development of the mixing
layer width for both cases. As is also found by van Prooijen
and Uijttewaal (2002b), the growths of the mixing layer
width forced by disturbances of different amplitudes differ
largely (see Figures 3 and 4). Near the inflow (x , 1.5 m),growths of mixing layer width are almost the same for
inflow forcing of different amplitudes. Regardless of the
amplitude of forcing imposed, no vortices are present in this
region. Thus, mixing between two streams and sub-
sequently growths of mixing layers is totally due to
turbulent diffusion. From the position at around x ¼ 2 monward, mixing layers forced by perturbation of larger
amplitudes grow faster. For mixing layers forced by 0.3DU0perturbations, vortices roll up between the position x ¼ 2 mand the position x ¼ 3 m, while vortices roll up between theposition x ¼ 3 m and the position x ¼ 4 m for mixing layersforced by 0.1DU0 perturbations. Furthermore, the inten-sities of coherent structures are weaker for the case forced
by 0.1DU0. This illustrates the importance of vorticalstructures in exchange and mixing between two streams. In
both tests, perturbation of amplitude 0.3DU0 is found toprovide the closest results to experimental data from
Uijttewaal and Booij (2000). This forcing amplitude will be
adopted for later tests.
5. General flow features
Tests under both hydraulic conditions shown in Table 1 are
carried out. All results displayed in this section are forced by
white noisewith 30%of the velocity difference at the inflow.
Figures 5–8 display the vorticity field, as well as the passive
scalar field and illustrate the spatial development of the KH
instability. The roll up mechanism, which results in the
formationofKHvortices, occurs in the regionbetween x ¼ 2and 3 m. This indicates that the instability begins at small
values of x and growth of disturbances reaches the saturation
state near x ¼ 2–3 m, where vortices roll up. The vorticesexperience growth until around x ¼ 7 m for case 1 andaround x ¼ 13 m for case 2. Vortex pairing is obviousthroughout the growth of the mixing layers. The general
features of vortex roll-up and pairing are consistent with
those in Plates 7 and 8 (Lesieur 1997). Comparing the
contour of the passive scalar (Figures 6 and 8) and the
contour of vorticity (Figures 5 and 7), it is clear that mixing
and the subsequent growth of the shallow mixing layer is
much more prominent in the region after the roll-up of
vortices, indicating that the vortices are important for mixing
between two streams.
The effects of flow confinement and bed friction can be
understood by invoking the bed friction parameter S (Chen
and Jirka 1997, Chu and Babarutsi 1988, Ghidaoui and
Kolyshkin 1999),
Sh ¼ cfðU1 þ U2Þ=4hðU1 2 U2Þ=d ; ð12Þ
where the subscript h emphasises the dependence of S on the
water height. The local value of Sh is a measure of the local
ratio of energy output from the KH vortices by the bed
friction to the energy input into the KH vortices by the
Reynolds stresses (Chu et al. 1991, Uijttewaal and Booij
2000). The local value of S for the case h ¼ 42 mm (S42) is
0
0.2
0.4
0.6
0.8
1
0 10 15
x (m)
0.1u0.2u0.3u0.4u
δ (m)
5
Figure 4. Development of shallow mixing layers under inflowforcing of different amplitudes (case 2).
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larger than that for the case h ¼ 67 mm (S67), for all x. Thefact that S42 . S67 explains why the vortical structures aremore evident and experience larger spatial growth for
h ¼ 67 mm than for h ¼ 42 mm (refer to Figures 5 and 7).Although, in both cases, one finds that the KH vortices roll up
and merge and that the confinement of the flow by the water
depth suppresses the vortex stretching mechanism leading to
the enstrophy cascade (Lesieur 1997), the extent of the
enstrophy cascade is much smaller for the shallower case
than for the deeper case. This is a result of the fact that
S42 . S67 (i.e. the ratio of the energy supply to the largerscales by the pairing of KH vortices to the energy loss from
the larger scales by the bottom friction is greater for
h ¼ 67 mm than for h ¼ 42 mm). This conclusion isconsistent with the autocorrelation function and energy
spectrum found in Uijttewaal and Booij (2000). They show
that (i) the range of scales for which the slope of energy
spectrum follows the -3 law is much more evident for the case
with h ¼ 67 mm and (ii) the spatial growth of the vorticalstructures is much more apparent for the case h ¼ 42 mm.
Large scale vortical structures in shallow flows are
often called coherent structures (Jirka 2001, Jirka and
Uijttewaal 2004). The coherence of large-scale vortical
structures in the shallow mixing layers is recognised by
x (m)
Y (
m)
0 2 4 6 8 10 12 14
–1
0
1
Figure 5. Vorticity contour for mixing layer corresponding to the test with h ¼ 42 mm inflow forcing amplitude ¼ 0.3DU0.
Figure 6. Scalar contour for mixing layer corresponding to the test with h ¼ 42 mm inflow forcing amplitude ¼ 0.3DU0. Available incolour online.
x (m)
Y (
m)
0 2 4 6 8 10 12 14
–1
0
1
Figure 7. Vorticity contour for mixing layer corresponding to the test with h ¼ 67 mm inflow forcing amplitude ¼ 0.3DU0.
Figure 8. Scalar contour for mixing layer corresponding to the test with h ¼ 67 mm inflow forcing amplitude ¼ 0.3DU0. Available incolour online.
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invoking a spatial temporal correlation coefficient as
follows (Roshko 1976),
Corrð~uðx0; t0Þ; ~uðx0 þ x 0; t0 þ tÞÞ
¼ Covð~uðx0; t0Þ; ~uðx0 þ x0; t0 þ tÞÞ
SDð~uðx0; t0ÞÞSDð~uðx0 þ x 0; t0 þ tÞÞ ; ð13Þ
where Corr stands for correlation, Cov denotes covariance,
SD indicates standard deviation, x0 and x0 þ x 0 are twospatial sampling, and t can be related to x 0 by theconvective velocity of perturbations (Uconvective) as
follows, t ¼ x 0/Uconvective.The correlation coefficient defined by Equation (9)
measures how much change the flow structures experience
when they travel from x0 to x0 þ x0 at the speed ofUconvective. When the correlation coefficient equals unity,
the flow structures are unchanged between x0 to x0 þ x0. In amixing layer, growth and decay of vortices and disturbances
contributes to the decrease of this correlation coefficient.
Figures 9–11 compare correlation coefficients under
different convective velocities. It is shown that the
downstream velocity fluctuations are positively correlated
when the mean velocity of two streams is chosen to be the
convective velocity (see Figure 9). When other velocities
are chosen as convective velocities, negative correlation
coefficients are prominent (see Figures 10 and 11).
A negative correlation coefficient between points (x0, t0)
and (x0 þ x 0, t0 þ t) means that flow features at positionx0 þ x 0 and time t0 þ t are no longer at the same phase asthe flow features at position x0 and time t0.
Figures 9 and 12 display the correlation coefficients of
transverse velocities originating from different downstream
positions in shallowmixing layers for both case 1 and case 2.
–0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15
x' (m)
Corrx0 = 0.0m
x0 = 0.05m
x0 = 0.25m
x0 = 0.5m
x0 = 1.0m
x0 = 2.0m
x0 = 3.0m
x0 = 5.0m
x0 = 7.0m
x0 = 9.0m
x0 = 11.0m
Figure 9. Correlation coefficients of transverse velocities in a shallow mixing layer with 42mm water depth, convective velocity is Uc.
–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.8
1
0 5 10x' (m)
Corrx0 = 0.0m
x0 = 0.05m
x0 = 0.25m
x0 = 0.5m
x0 = 1.0m
x0 = 2.0m
x0 = 3.0m
x0 = 5.0m
x0 = 7.0m
x0 = 9.0m
x0 = 11.0m
15
Figure 10. Correlation coefficients of transverse velocities in a shallow mixing layer with 42mm water depth, convective velocity is0.9Uc.
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Velocity fluctuations at a certain spatial position are all, to
some extent, correlated to downstream velocity fluctuation.
Correlation coefficients all decrease in the downstream
direction. In shallow mixing layers, vortices undergo
different processes including roll-up, growth due to
entrainment, amalgamation and decay. These vortex
evolutions all contribute to the decrease in correlation
coefficients. Comparison of Figures 9 and 12 shows that the
decrease in correlation for case 2 is more significant than for
case 1 for x 0 , 11 m. This is ascribed to the fact that thevortices in case 2 continue to change within the range
x 0 , 11 m by the process of pairing, while the vortices incase 1 exhibit little change in structure beyond x 0 . 7 m, dueto the suppression of pairing by the bottom friction.
6. Mean flow quantities
This section investigates how well the depth-averaged
shallow water equations model mean flow quantities of
shallow mixing layers. Simplified analytical solutions,
semi-empirical expressions as well as experimental data
(Uijttewaal and Booij 2000, van Prooijen and Uijttewaal
2002a) are adopted for comparison.
6.1 Free surface water profile
To achieve steady state in free surface flows, the drag force
by bottom friction must be balanced by either a pressure
gradient or gravity force by tilting the channel bed or both.
In the laboratory experiments, the channel bed is horizontal
and the flow is driven by the a pressure gradient which is
given as follows,
dh
dx¼ 2cf 1
gh=U2c�
2 1: ð14Þ
Figure 13 shows that the computed results are in good
agreement with both experimental measurements and
Equation (14).
–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15x' (m)
Corrx0 = 0.0m
x0 = 0.05m
x0 = 0.25m
x0 = 0.5m
x0 = 1.0m
x0 = 2.0m
x0 = 3.0m
x0 = 5.0m
x0 = 7.0m
x0 = 9.0m
x0 = 11.0m
Figure 11. Correlation coefficients of transverse velocities in a shallowmixing layer with 42mmwater depth, convective velocity is 1.1Uc.
–0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15
x' (m)
Corr
x0 = 0.0m
x0 = 0.05m
x0 = 0.25m
x0 = 0.5m
x0 = 1.0m
x0 = 2.0m
x0 = 3.0m
x0 = 5.0m
x0 = 7.0m
x0 = 9.0m
x0 = 11.0m
Figure 12. Correlation coefficients of transverse velocities in a shallow mixing layer with 67mm water depth, convective velocity is Uc.
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6.2 Velocity difference between two ambient streamsof the mixing layer
In deepmixing layers, the velocity differences of the ambient
streams are the same in the flow direction (Pope 2002). This
fact does not hold for shallow mixing layers. In a shallow
mixing layer, friction is bigger for the faster stream and
smaller for the slower stream. In addition, since transverse
water depth gradient is negligible (van Prooijen and
Uijttewaal 2002a), i.e. the streamwise pressure gradient
(›p/›x) is the same for both the fast and the slow streams.Thus, the fast stream slows down and the slow stream speeds
up in a shallow mixing layer. The velocity difference along
the stream is derived by van Prooijen and Uijttewaal (2002a)
based on a quasi-1D shallow water model. Laboratory
observations (van Prooijen and Uijttewaal 2002a) suggest
that the variation of mean velocity in the streamwise
direction (Uc) is within 3% and thus, can be assumed to be
constant. Velocity difference is,
DUðxÞ ¼ U1 2 U2 ¼ DU0 exp 2 cfhx
� �; ð15Þ
where the subscript 0 denotes variable at the inflow.
Figures 14 presents the variation of velocity difference
across a shallow mixing layer for both 42 and 67mmwater
depth, respectively. The computed results show good
agreement with expression (15) and laboratory data.
6.3 Velocity distribution in lateral direction
In a series of laboratory tests, van Prooijen and Uijttewaal
(2002a) noted that the transverse distributions of
streamwise velocity away from the inflow collapse
approximately into a curve of tangent hyperbolic profile, i.e.
Uðx; yÞ ¼ Uc þ DUðxÞ2
tanhy2 ycðxÞ
12
� dðxÞ
!; ð16Þ
where d is the width of a mixing layer and the subscript cindicates that the variable is evaluated at the centreline of a
mixing layer.
Figures 15 and 16 plot the computed and theoretical
lateral profiles of the mean streamwise velocity at different
streamwise locations for the mixing layer with water
heights of 42 and 67mm, respectively. It is shown that the
velocity profiles at different spatial locations away from
the inflow boundary (x . 2.0 m) essentially collapse ontoa single curve (Equation (16)). Near the inflow boundary
condition (x , 2 m), the splitter wake is obvious. More-over, computed results at the high speed side of the mixing
layer with 42 mm water depth depart from the empirical
relation (16). The combined effects of streamwise pressure
gradient and bottom friction result in acceleration of the
low-speed flows and deceleration of fast-speed flows. This,
in turn, results in the widening of a shallow mixing layer at
the fast speed side. All these characteristics are also
observed in laboratory tests (see Figures 4 and 5 in Ref.
Uijttewaal and Booij 2000).
6.4 Development of shallow mixing layer widths
The growth of a shallow mixing layer implies that the scale
of the vortices inside the mixing layer is growing in the
0.035
0.045
0.055
0.065
0.075
0.085
0 2 4 6 8 10 12 14 16
x(m)
h(m)Computed results (67mm)Measurements (67mm)Integral model (67mm)Computed results (42mm)Measurements (42mm)Integral model (42mm)
Figure 13. Comparison of water depth.
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flow direction, due to both vortex pairing (Roshko 1976)
and vorticity induction (Dimotakis 1986). Laboratory
observations indicate that for deep mixing layers, the
spatial growth rate of a mixing layer is approximated by
(see Ref. Balaras et al. 2001),
dd
dx¼ aDUðxÞ
Uc; ð17Þ
where a is entrainment coefficient and has an empiricalvalue of 0.85 (van Prooijen and Uijttewaal 2002a).
Integration over the streamwise direction (x) results in
the expression for width of shallow mixing layers (van
Prooijen and Uijttewaal 2002a),
dðxÞ ¼ aDU0Uc
h
cf1 2 exp 2
cf
hx
� �h iþ d0; ð18Þ
where d0 is the mixing layer width at the inflow and isapproximately equal to the water depth.
Figures 17 and 18 depict the computed mixing layer
thickness, the empirical solutions given by Equation (18)
and data measured by van Prooijen and Uijttewaal
(2002a). The computed mixing layer width shown in
Figure 17 is performed for Cs ¼ 0.1, 0.2 and 0.3. The valueof 0.2 provides the closest fit with the data. Indeed, this
value is used throughout the paper.
0.01
0.1
10 2 4 6 8 10 12 14 16
x (m)
Computed results (67mm)Measurements (67mm)Integral model (67mm)Computed results (42mm)Measurements (42mm)Integral model (42mm)
u1 - u2 (m/s)
Figure 14. Comparison of velocity difference.
–0.2
0
0.2
0.4
0.6
0.8
1
1.2
–2 –1.5 –1 –0.5 0 0.5 1 1.5 2
0.05m
0.25m
0.5m
1.0m
1.5m
2.0m
2.5m
4.0m
5.8m
7.5m
Theory
y/δ
(u-u1)/(u2-u1)
Figure 15. Comparison of mean streamwise velocities (case 1).
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The computed shallow mixing layer experiences slow
growth just downstream of the inflow boundary, causing it
to deviate from the laboratory data (see Figures 17 and 18).
This can be attributed to the different flow conditions
between the model and the experiment. In the simulation,
the flow undergoes linear KH instability and mixing in this
region is mostly due to turbulent diffusion, while in
laboratory tests, vortical structures in splitter wakes play an
important role in mixing between two streams. The slow
growth just downstream of the inflow boundary is also
reported by Stanley and Sarkar (1997), Li and Fu (2003),
and Yang et al. (2004) for 2D deep mixing layer
simulations forced by stochastic disturbances. Expression
(18) seems able to model the mixing layer growth near the
inflow, which is not surprising if one notices that expression
(18) is valid when vortices in mixing layers play an
important role in the growth of mixing layers and it seems
that mixing due to vortices in splitter wakes can also be
modelled by this expression. Downstream of the position at
3 m, where vortices begin to roll up in the simulation, the
model predicts a slightly larger growth of the mixing layers
compared to the data and the semi-empirical relation (18).
Considering that 2D direct numerical simulation can
reproduce correctly the growth rate of mixing layers
(Stanley and Sarkar 1997), this slightly larger prediction is
ascribed to the use of the Smagorinsky parameterisation
schemes for turbulent diffusion.
6.5 Centre of shallow mixing layer
An interesting phenomenon in mixing layer flows is that
the centre of the mixing layer will be displaced in the
lateral direction to the low velocity side. In deep mixing
layers, Dimotakis (1986) proposed that the differences
between the entrainment fluxes from high-speed stream
–0.2
0
0.2
0.4
0.6
0.8
1
1.2
–2 –1 0 1 2
0.05m
0.25m
0.5m
1.0m
2.0m
5.8 m
11m
Theory
y/δ
(u-u1)/(u2-u1)
Figure 16. Comparison of mean streamwise velocities (case 2).
0
0.1
0.2
0.3
0.4
0.5
0.6
0 2 4 6 8 10 12
Computed results (42mm Cs = 0.1)Computed results (42mm Cs = 0.2)Computed results (42mm Cs = 0.3)Measurements (42mm)Integral model (42mm)
δ (m)
x(m)
Figure 17. Comparison of shallow mixing layer width (case 1).
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 2 4 6 8 10 12
x(m)
Computed results (67mm)Measurements (67mm)Integral model (67mm)
δ(m)
Figure 18. Comparison of shallow mixing layer width (case 2).
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and those from the low-speed streams due to the increasing
spacing between vortices contribute to this shift of
centreline. In shallow mixing layers, van Prooijen and
Uijttewaal (2002a) noted that the deceleration of the high
velocity side and the acceleration of the low velocity side
is the major factor for the shift of mixing layer centre.
Assuming that the centreline of the mixing layer is a
streamline, the lateral shift of the centreline can be
estimated from the conservation of mass as follows (van
Prooijen and Uijttewaal 2002a),
ðycðxÞ2W=2
HUðx; yÞdy ¼ HW2U2ðx ¼ 0Þ; ð19Þ
where W is the width of the channel.
Figures 19 and 20 show acceptable agreements between
the computed and analytical positions of the centre of the
mixing layer (yc), where the solution for yc is obtained from
the mass conservation principle (Equation (19)).
7. Turbulent intensities
In this section, the ability of shallow water equations in
modelling turbulence intensities of shallow mixing layers
is examined. Figures 21–24 show the spatial development
of streamwise turbulence intensities and spanwise
turbulence intensities for both shallow mixing layers of
42 and 67mm water depth, respectively. In all figures,
turbulence intensities decay near the inflow (x , 1 m).Disturbances undergo a selective amplification process in
this region, where only a small portion of modes are
unstable and grow, while others are stable and decay
rapidly. In the region from x ¼ 1 to 5.8 m, turbulenceintensities experience substantial growth, indicating that
energy extracted from the horizontal velocity shear is
larger than energy consumed by bottom friction and
horizontal diffusion in this region. Further downstream,
turbulence intensities decay implying the dominance of
damping due to bottom friction and horizontal diffusion
over instability due to horizontal velocity shear. These
decays and growths of turbulence intensities are also
observed in laboratory measurements (see Figures 10–13
in Ref. Uijttewaal and Booij 2000).
–0.5
–0.4
–0.3
–0.2
–0.1
00 2 4 6 8 10 12 14 16
x(m)
Computed results (42mm)Measurement (42mm)Integral Model (42mm)
yc(m)
Figure 19. Comparison of mixing layer center (case 1).
–0.5
–0.4
–0.3
–0.2
–0.1
00 2 4 6 8 10 12 14 16
x(m)
Computed results (67mm)
Measurements (67mm)Integral model (67mm)
yc(m)
Figure 20. Comparison of mixing layer centre (case 2).
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
–2 –1 0 1 2
0.05m0.25m0.5m1.0m1.5m2.0m2.5m4.0m5.8m7.5m
y/δ
u' (m/s)
Figure 21. Profile of streamwise turbulence intensities atvarious downstream positions of a shallow mixing layer (case 1).
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
–2 –1 0 1 2
0.05m0.25m0.5m1.0m1.5m2.0m2.5m4.0m5.8m7.5m
y/δ
v' (m/s)
Figure 22. Profile of spanwise turbulence intensities at variousdownstream positions of a shallow mixing layer (case 1).
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Table 2 compares the maximum measured turbulence
intensities by Uijttewaal and Booij (2000) and the
maximum computed ones in the current study. The
computed streamwise turbulence intensities are compar-
able to the measured one, while the computed spanwise
turbulence intensities are significantly larger than the
measured ones. The computed distribution of turbulence
intensities also spread more across the mixing layers than
the measured ones. These phenomena are also noted in the
study of the deep mixing layer by Stanley and Sarkar
(1997), where they compare their 2D numerical results
with 3D direct numerical simulation results and laboratory
data. Indeed, the 3D energy cascade processes are totally
absent in 2D models and energy can only be dissipated by
bottom friction after they are transferred to large scale.
8. Conclusions
Turbulent shallow mixing layers and their associated
vortical structures are ubiquitous in rivers, estuaries and
coasts. Examples of these flows can be found in compound
and/our composite channels, at the confluence of two
rivers, at harbour entrances, and at groyne fields. A 2D
BGK-based finite volume model for shallow water is
applied to shallow mixing layers for which experimental
results are available. The BGK scheme is explicit, second
order in time and space and conserves both mass and
momentum. The BGK relaxation time is locally evaluated
from the classical turbulence model of Smagorinsky. The
KH instability and the mean flow quantities are well
represented by the shallow water model. The following
remarks in relation to the applicability of the shallow water
model to shallow mixing layers can be made.
(1) The model provides the spatial development of the KH
instability. The instability develops in the region very
close to the splitter plate. Downstream of this region,
the disturbances roll up, resulting in the formation of
KHvortices.This is then followedbya regionwhere the
vortices undergo pairing and result in the formation of
coherent vortical structures. While vortex pairing acts
to increase the length scale of the vortical structures, the
bed friction tends to suppress these structures. The
degree of suppression becomes more significant as
the structures become larger. The scale of the final size
of these structures at large distances from the splitter
plate is governed by the bed friction parameter.
(2) Shallow mixing layers are highly sensitive to
perturbations at the inflow boundary. These pertur-
bations are selectively amplified in their linear region
and cease to grow when they reach saturation. Further
downstream where bottom friction is dominant, the
perturbations are dissipated.
(3) Coherence of large-scale vortical structures in shallow
mixing layers decreases due to growth, amalgamation
and decay of these vortices. It is also found that these
large-scale vortices are advected at the mean speed
between two streams in the shallow mixing layer, a
result which is the same as that for deep mixing layers.
(4) The Smagorinsky scheme for 2D shallowwater flows is
a bit diffusive since it predicts a slightly larger growth
rate than observed. However, other mean flow
quantities, like the velocity difference, centreline and
water surface, are well predicted by the current model.
(5) Shallow water model is incapable of accurately
predicting turbulent intensities and their distribution.
A 3Dmodel would be required to accurately predict the
distribution of turbulence intensities.
Table 2. Comparison of measured and computed turbulenceintensities.
u 0max (m/s) v 0max (m/s)
Measured (42mm) 0.095 0.007Computed (42mm) 0.0085 0.12Measured (67mm) 0.028 0.018Computed (42mm) 0.24 0.03
0
0.005
0.01
0.015
0.02
0.025
0.03
–2 –1 0 1 2
0.05m0.25 m0.5m1.0m2.0m5.8m11.0m
y/δ
u' (m/s)
Figure 23. Profile of streamwise turbulence intensities atvarious downstream positions of a shallow mixing layer (case 2).
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
–2 –1 0 1 2
0.05m0.25m0.5m1.0m2.0m5.8 m11.0m
y/δ
v' (m/s)
Figure 24. Profile of spanwise turbulence intensities at variousdownstream positions of a shallow mixing layer (case 2).
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(6) The close agreement between the model and exper-
iments suggests that the processes of flow instability,
and its associated transversal mixing of mass
momentum and flow entrainment are well represented
by the shallow water model.
Although, the computed and measured turbulent
intensities do not agree, flow instabilities and mean flow
quantities are well represented by the shallow water model.
It is plausible to conclude that the shallowwater models can
be used to conduct stability of shallow flows and to
determine mean flow features of shallow flows. Shallow
water model is not be suitable for investigating small scale
features such as flow intermittency, instantaneous structure
of the flow and possible breakdown of vortical motion into
3D turbulence. Such breakdown is expected to occur in
shallow flows, where the water depth is of the order of the
bottomboundary layer. Such very shallowflowcan occur in
laboratory studies. Indeed, experiments show that the
vortical motion eventually disintegrates for the case with
water depth equal to 42mm.
Acknowledgements
The authors wish to thank the Research Grants Council of HongKong for financial support under Project HKUST6227/04E andProject 613306.
References
Balaras, E., Piomelli, U., and Wallace, J., 2001. Self-similarstates in turbulent mixing layers. Journal of FluidMechanics, 446, 1–24.
Batchelor, G.K., 1967. An introduction to fluid dynamics.Cambridge: Cambridge University Press.
Chen, D. and Jirka, G.H., 1995. Experimental study of planeturbulent wakes in shallow water layers. Fluid DynamicsResearch, 16, 11–41.
Chen, D. and Jirka, G.H., 1997. Absolute and convectiveinstabilities of plane turbulent wakes in a shallow waterlayer. Journal of Fluid Mechanics, 338, 157–172.
Chu, V.H., Wu, J.H. and Khayat, R.E., 1991. Stability oftransverse shear flows in shallow open channels. Journal ofHydraulic Engineering, ASCE, 117, 1370–1388.
Chu, V.H. and Babarutsi, S., 1998. Confinement and bed-frictioneffects in shallow turbulent mixing layers. Journal ofHydraulic Engineering, ASCE, 114, 1257–1274.
Dimotakis, P.E., 1986. Two-dimensional shear-layer entrain-ment. AIAA Journal, 24, 1791–1796.
Durran, D.R., 1999. Numerical methods for wave equations ingeophysical fluid dynamics. New York: Springer-Verlag.
Ghidaoui, M.S. and Kolyshkin, A.A., 1999. Linear stabilityanalysis of lateral motions in compound open channel.Journal of Hydraulic Engineering, 125, 871–880.
Ghidaoui, M.S., Deng, J.Q., Gray, W.G. and Xu, K., 2001.A Boltzmann based model for open channel flows.International Journal for Numerical Methods in Fluids, 35(4), 449–494.
Ghidaoui, M.S., et al., 2006. Linear and nonlinear analysis ofshallow wakes. Journal of Fluid Mechanics, 548, 309–340.
Ho, C.M. and Huerre, P., 1984. Perturbed free shear layers.Annual Review of Fluid Mechanics, 16, 365–424.
Huerre, P. and Monkewitz, P.A., 1990. Local and globalinstabilities in spatially developing flows. Annual Review ofFluid Mechanics, 22, 473–537.
Jirka, G.H., 2001. Large scale flow structures and mixingprocesses in shallow flows. Journal of Hydraulic Research,39, 567–573.
Jirka, G.H. and Uijttewaal, W.S.J., 2004. Shallow flows: adefinition. In: G.H. Jirka and W.S.J. Uijttewaal, eds. Shallowflows. Leiden: A.A. Balkema Publishers.
Kolyshkin, A.A. and Ghidaoui, M.S., 2002. Linear stabilityanalysis of lateral motions in compound open channel.Journal of Hydraulic Engineering, 128, 1076–1086.
Lesieur, M., 1997. Turbulence in fluids. Boston: KluwerAcademic Publishers.
Li, Q. and Fu, S., 2003. Numerical simulation of high-speedplanar mixing layer. Computers & Fluids, 32, 1357–1377.
Liang, J.H., Ghidaoui, M.S., Deng, J.Q. and Gray, W.G., 2007.A Boltzmann-based finite volume algorithm for surfacewater flows on cells of arbitrary shapes. Journal of HydraulicResearch, 45 (2), 147–164.
Mellor, G.L., 2004. User guide for a 3D, primitive equation,numerical ocean model. http://www.aos.princeton.edu/WWWPUBLIC/htdocs.pom/
Monkewitz, P.A., 1988. The absolute and convective nature ofinstability in two-dimensional wakes at Low Reynoldsnumbers. Physics of Fluids, 31, 999–1006.
Oey, L.-Y., Ezer, T., and Lee, H.C., 2005. Loop current, rings andrelated circulation in the Gulf of Mexico: a review ofnumerical models and future challenges. GeophysicalResearch Letters, 32, L1266, doi: 10.1029/2005GL023253.
Pope, S.A., 2002. Turbulent flows. Cambridge: CambridgeUniversity Press.
Roshko, A., 1976. Structure of turbulent shear flows: a new look.AIAA Journal, 14, 1349–1357.
Schlichting, H. and Gersten, K., 2000. Boundary layer theory.Berlin, Heidelberg: Springer-Verlag.
Socolofsky, S.A. and Jirka, G.H., 2004. Large-scale flowstructures and stability in shallow flows. Journal ofEnvironmental Engineering and Science, 3, 451–462.
Stanley, S. and Sarkar, S., 1997. Simulations of spatiallydeveloping 2D shear layers and jets. Theoretical andComputational Fluid Dynamics, 9, 121–147.
Uijttewaal, W.S.J. and Booij, R., 2000. Effects of shallowness onthe development of free-surface mixing layers. Physics ofFluids, 12 (2), 4392–4402.
Uijttewaal,W.S.J. and Jirka,G.H., 2003.Grid turbulence in shallowflows. Journal of Fluid Mechanics, 489, 325–344.
van Prooijen, B.C. and Uijttewaal, W.S.J., 2002a. A linearapproach for the evolution of coherent structures in shallowmixing layers. Physics of Fluids, 14 (12), 4105–4114.
van Prooijen, B.C. and Uijttewaal, W.S.J., 2002b. On theinitiation of large scale turbulence structures in the numericalsimulation of shallow mixing layers. In: Bousamr and Zech,eds. River flow 2002. Lisse: Swets & Zeitlinger.
Xu, X., Lee, J.S., and Pletcher, R.H., 2005. A compressible finitevolume formulation for large eddy simulation of turbulentpipe flows at low Mach number in Cartesian coordinates.Journal of Computational Physics, 203 (1), 22–48.
Yang, W.B., Zhang, H.Q., Chan, C.K., and Lin, W.Y., 2004.Large eddy simulation of mixing layer. Journal ofComputational and Applied Mathematics, 163, 311–318.
Zhou, J.G., 2004. Lattice Boltzmann methods for shallow waterflows. Berlin: Springer-Verlag.
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Appendix: Matrices in Equation (5)
Asi ¼ hsiffiffiffiffiffiffiffiffighsi
p2
Vnerfcð2VnÞ þ 1ffiffiffipp e2V 2nVnvþ n
ffiffiffiffigh
p2
� �erfcð2VnÞ þ v 1ffiffiffipp e2V2n
2664
3775si
Aso ¼ hsoffiffiffiffiffiffiffiffiffighso
p2
VnerfcðVnÞ2 1ffiffiffipp e2V2nVnvþ n
ffiffiffiffigh
p2
� �erfcðVnÞ2 v 1ffiffiffipp e2V2n
2664
3775so
Bsi ¼ghesi
2
V2n þ 12�
erfcð2VnÞ þ 1ffiffiffipp Vne2V2nV2nvþ v2 þ vnn�
erfcð2VnÞ þ 1ffiffiffipp Vnvþ ffiffiffiffiffighp n� e2V2n264
375si
Bso ¼gh2so
2
V2n þ 12�
erfcðVnÞ2 1ffiffiffipp Vne2V2nV2nvþ v2 þ vnn�
erfcðVnÞ2 1ffiffiffipp Vnvþ ffiffiffiffiffighp n� e2V2n264
375so
Dsi ¼ hsiffiffiffiffiffiffiffiffighsi
p2
vt Vnerfcð2VnÞ þ 1ffiffiffipp e2V2nn ovtffiffiffiffiffigh
pV2n þ 12�
erfcð2VnÞ þ 3ffiffiffipp Vne2V2nn ov2t þ gh2�
Vnerfcð2VnÞ þ 1ffiffiffipp e2V2nn o
26666664
37777775si
Dso ¼ hsoffiffiffiffiffiffiffiffiffighso
p2
vt VnerfcðVnÞ2 1ffiffiffipp e2V2nn ovtffiffiffiffiffigh
pV2n þ 12�
erfcðVnÞ2 3ffiffiffipp Vne2V2nn ov2t þ gh2� �
VnerfcðVnÞ2 1ffiffiffipp e2V2nn o
26666664
37777775so
Es ¼ hsvn
vnvþ gh2 n
24
35s
Gs ¼ hsv2n þ gh2v2nvþ 3gh2 vnn
24
35s
Is ¼ hsvnvt
vnvtvþ gh2 ðvtnþ vntÞ
24
35s
International Journal of Computational Fluid Dynamics 537
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