investigation of shallow mixing layers by bgk finite...

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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

M. Ghidaoui and J.H. Liang524

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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

International Journal of Computational Fluid Dynamics 525

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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.

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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


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investigation of shallow mixing layers by bgk finite...

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