simulating flow over circular spillways by using different turbulence models-libre

Engineering Applications of Computational Fluid Mechanics Vol. 6, No.1, pp. 100–109 (2012)

Received: 14 May. 2011; Revised: 18 Aug. 2011; Accepted: 6 Oct. 2011

100

SIMULATING FLOW OVER CIRCULAR SPILLWAYS BY USING DIFFERENT TURBULENCE MODELS

H. Rahimzadeh *, R. Maghsoodi **, H. Sarkardeh # and S. Tavakkol †

* Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran E-Mail: [email protected] (Corresponding Author)

** Department of Civil Engineering, Shahrood Branch, Islamic Azad University, Shahrood, Iran # Hydraulic Structures Division, Water Research Institute (WRI), Tehran, Iran

† Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran

ABSTRACT: Fluent software has been used to simulate flow over a circular spillway and results were compared

with experimental data. As the flow over a circular spillway is turbulent and has a free surface, its characteristics are

complex and often difficult to be predicted. This study assesses the performance of some turbulence models to

predict the hydraulic condition of flow over circular spillways. The Volume of Fluid (VOF) method is applied to

obtain the free surface in each case. Such cases include highly swirling flows, stress-driven secondary flows and

flows over circular spillways. Finally it is concluded that the results of RSM, RNG k-, Realizable k-, SST k-ω

turbulence models agree well with experimental data.

Keywords: circular spillway, numerical simulation, fluent, turbulence models, VOF

1. INTRODUCTION

Circular spillways are typically used for

measuring discharges and as a control device in

water systems. A circular spillway is an overflow

structure with a circular crest. In fact, they

provide a unique relationship between the

upstream head and the discharge. The

characteristics of the flow over circular spillways

have been a subject of interest to many

researchers. Vo (1992) experimentally found the

discharge coefficient of circular spillways as a

function of the dimensionless total head of the

approaching flow. Ramamurthy and Vo (1993a)

applied these equations to predict the velocity

distribution over a cylindrical spillway.

Ramamurthy and Vo (1993b) compared results of

their experiments with others. Chanson and

Montes (1997) described experiments of circular

spillways, with eight cylinder sizes, for several

spillway heights and for five types of inflow

conditions including partially-developed inflow,

fully developed inflow, upstream ramp and

upstream hydraulic jump. Heidarpour and

Chamani (2006) developed a method to predict

the velocity distribution based on the potential

flow past a cylindrical spillway. Hargreaves et al.

(2007) described the validation of CFD for

modelling free surface flows past a broad-crested

weir. Castro-Orgazet et al. (2008) presented a

generalized one-dimensional model with the

assumption of critical flow in a curvilinear

domain. Castro-Orgazand and Chanson (2009)

developed the Bernoulli theorem along a

streamline to flow in open channels. Tadayon

(2009) analysed mean characteristics of

curvilinear flows by using Computational Fluid

Dynamics (CFD). Pettersson and Rizzi (2009)

used Fluent Software to compare two different

turbulence models accuracy in computing local

boundary layer properties with wind tunnel

measurements. Bagheri and Heidarpour (2010)

simulated flow over a circular-crested spillway

with an irrotational vortex to determine the

spillway discharge coefficient and velocity values

over the crest. Yazdi et al. (2010) simulated flow

around a spur dike with free-surface flow by

using fully three-dimensional, Reynolds-averaged

Navier–Stokes equation. They also applied the

volume of fluid method with geometric

reconstruction scheme to model the free-surface

flow. Rahimzadeh et al. (2010) also simulated

flow over stepped spillways by using Fluent

software. Unal and Goren (2011) presented a

comparative study based on the 3D computational

simulations of the flow around a circular cylinder.

They used three different two-equation turbulence

models in their simulations.

2. EXPERIMENTAL DATA

Experimental tests used in this numerical study

were conducted in a smooth channel by Vo

(1992). Six fixed-bed tests were conducted in a

flume with La = 1.800 m, R = 0.152 m and w =

1.164 m (Fig. 1). Other parameters are given in

Table 1.

Engineering Applications of Computational Fluid Mechanics Vol.6, No. 1 (2012)

101

Fig. 1 Schematic view of circular spillway.

Table 1 Selected experimental parameters.

(deg) (deg) H1 (m) Q (L/s/m)

Test 1 90 90 0.1237 85.39

Test 2 90 90 0.1762 152.17

Test 3 90 90 0.2093 203.43

Test 4 60 45 0.0796 41.34

Test 5 60 45 0.1185 80.04

Test 6 60 45 0.1482 115.71

3. TURBULENCE MODELS

In this section different turbulence models which

are used in the present research are briefly

described. In Reynolds averaging:

i i iu u u

(1)

where iu and iu are the mean and fluctuating

velocity components, respectively. Substituting

expressions of this form for the flow variables

into the instantaneous continuity and momentum

equations and simplifying (and dropping the

overbar on the mean velocity, u ), we have

0ii

ut x

(2)

i i jj

u u ut x

2

3

ji lij

i j j i l

i jj

uu up

x x x x x

u ux

(3)

where equations 2 and 3 are called Reynolds-

averaged Navier-Stokes (RANS) equation,

i ju u is called Reynolds stresses, and should

be modelled. Hinze (1975) related the Reynolds

stresses to the mean velocity gradients:

2

3

ji ii j t t ij

j i i

uu uu u k

x x x (4)

The Boussinesq hypothesis is used in the k-, and

k- models.

3.1 Standard k- model (Launder and Spalding, 1972)

The effective viscosity is modelled as follows:

2

t

kC (5)

where C is a constant (Table 2), k is the

turbulence kinetic energy, and is the turbulence

dissipation rate. The following equations are the

transport equations for the standard k- model.

ii

k kut x

t

j k j

k b M

k

x x

G G Y

(6)

ii

ut x

2

1 3 2

t

j j

k b

x x

C G C G Ck k

(7)

Standard constants of the k- model are listed in

Table 2, and were used in the model.

3.2 Renormalization-group (RNG) k- model (Yakhot and Orszag, 1986)

Effective viscosity and transport equations for the

RNG k- model are as follows:

2

31.72

1 100

effkd d

(8)

effˆ

ii

k kut x

k eff k b Mj j

kG G Y

x x

(9)

Engineering Applications of Computational Fluid Mechanics Vol. 6, No. 1, (2012)

102

Table 2 Standard turbulence models constants.

C1 C2 C1 C2 C k

Standard k- 1.44 1.92 - - 0.09 1.0 1.3

RNG k- 1.42 1.68 - - - - -

Realizable k- 1.44 - - 1.9 - 1.0 1.2

Standard k- - - 0.555 0.83 0.09 2 2

ii

ut x

1 3

32 20

2 3

1

1

eff k bj j

C G C Gx x k

CC

k k

(10)

where k and are computed using the

following formula:

0.6321 0.3679

0 0

1.3929 2.3929

1.3929 2.3929

mol

eff

(11)

where 0 1.0

and

0, 4.38, 0.012Sk

Standard constants of RNG k- model are listed

in Table 2, and were used in the model.

3.3 Realizable k- model (Shih et al., 1995)

The effective viscosity is modelled as follows:

2

t

kC (12)

The difference between the realizable k- model

and the standard and RNG k- models is that C

is no longer constant. It is computed from

*

*

0

1,

2 ,

ij ij ij ij

s

ij ij ijk k ij ij ijk k

C U S SkU

A A

where ij is the mean rate-of-rotation tensor

viewed in a rotating reference frame with the

angular velocity k and

1

0

14.04, 6 , cos 6 ,

3sA A W

, ,ij jk ki

ij ij

S S SW S S S

S

1

2

j iij

i j

u uS

x x

Transport equations of the realizable k- model

are as follows:

ji

k kut x

t

i k j

k b M

k

x x

G G Y

(13)

jj

ut x

1

2

2 1 3

t

j j

b

C Sx x

C C C Gkk

(14)

where

1 max 0.43, ,5

kC S

Standard constants of realizable k- model are

listed in Table 2, and were used in the model.

3.4 The standard k- models (Wilcox, 1998)

The effective viscosity and transport equations of

this model are as follows:

*

t

k

(15)


103

where * damps the turbulent viscosity causing a

low-Reynolds-number correction.

ii

k kut x

tk k

j k j

kG Y

x x

(16)

ii

ut x

t

j j

G Yx x

(17)

Standard constants of k- model are listed in

Table 2, and were used in the model.

3.5 Shear-stress transport (SST) k- model (Menter, 1994)

The effective viscosity and transport equations are

modelled in SST k- model, by equations 18, 19

and 20.

2

*

1

1

21max ,

t

ij ij

k

F

(18)

ii

k kut x

tk k

j k j

kG Y

x x

(19)

ii

ut x

t

j j

G Y Dx x

(20)

where

1 ,1 1 ,2

1,

1k

k kF F

1 ,1 1 ,2

1

1F F

,1 ,1 ,2 ,21.176, 2.0, 1.0, 1.168k w k w

ij is the mean rate-of-rotation tensor,

* is

damps the turbulent viscosity causing a low-

Reynolds-number correction and 1F ,

2F are the

blending functions.

In k- and k-models, kG , bG , G , D and

MY (Sarkar and Balakrishnan, 1990) are the

generation of turbulence kinetic energy due to the

mean velocity gradients, the generation of

turbulence kinetic energy due to buoyancy, the

generation of , the cross-diffusion term, and the

contribution of the fluctuating dilatation in

compressible turbulence to the overall dissipation

rate, respectively.

3.6 Reynolds stress model (RSM) (Launder et al., 1975; Gibson and Launder, 1978; Launder, 1989a and b)

The turbulence stress components are,

ij

i j k i jk

Local Time Derivative C Convection

u u u u ut x

,

,

T ij

L ij

i j k kj i ik jk

D Turbulent Diffusion

i jk k

D Molecular Diffusion

u u u p u ux

u ux x

Pr

Prij

ij

j ii k j k i j j i

k kG Buoyancy oduction

P Stress oduction

u uu u u u g u g u

x x

Pr

Pr

2

2

ijij

ij

j ji i

j i k k

Dissipationessure Strain

k j m ikm i m jkm

F oduction by System Rotation

u uu up

x x x x

u u u u

(21)

Of the various terms in the above transport

equation, ijC , ,L ijD , ijP , and ijF do not require

any modeling. However, ,T ijD (Lien and

Leschziner, 1994), ijG , ij (Gibson and Launder,

1978;, Fu et al., 1987;, Launder, 1989a and b),

and ij need to be modeled to close the equations.

To simulate the pressure strain (ij), Linear

pressure-strain is used.

4. FREE SURFACE MODELLING

The volume of fluid (VOF) method appears to be

a powerful computational tool for the analysis of

free-surface flows (Hirt and Nichols, 1981). The

tracking of the interface(s) between the phases is


104

accomplished by the solution of a continuity

equation for the volume fraction of one (or more)

of the phases. For the qth

phase, this equation has

the following form:

. . 0q

qt

(22)

where q is the volume fraction of qth phase. In

each control volume, the volume fractions of all

phases sum up to unity. The following three

conditions are possible for each cell:

0q : the cell is empty (of the qth phase).

1q : the cell is full (of the qth phase).

0 1q : the cell contains the interface

between the qth phase and one or more other

phases.

It can be assumed the free surface is on the

volume fraction of 0.5 (Fluent Manual 2005,

Dargahi 2006). In each cell the average properties

are computed according to the volume fraction of

each phase. For example, density and dynamic

viscosity in each cell of two phases are:

1 1 1 21 , 1 1 1 21 ,

respectively. The phases are represented by the

subscripts 1 and 2.

In this research, the geometric reconstruction

method of Young (1982) is employed. This

method represents the interface between fluids

using a piecewise-linear approach. It assumes that

the interface between two fluids has a linear slope

within each cell, and uses this linear shape for

calculation of the advection of fluid through the

cell faces.

5. BOUNDARY CONDITIONS AND MESH GENERATION

Both structured and unstructured meshes have

been used. Denser mesh was used around the

spillway, close to free surface and in the boundary

layer in order to provide a higher accuracy and

considering the viscous flow. The first grid

surface of the solid boundaries was at

0.0005y , which ensures that the first grid

surface off the wall is located almost everywhere

at *1.0y y u y and that at least two

grid surfaces are located within the laminar sub-

layer (y+< 5.0), where y is the distance of the

first grid from the solid wall, *u is wall shear

stress, and is kinetic viscosity (Fig. 2).

(a)

(b)

(c)

Fig. 2 Computational grid in the vicinity of spillway:

(a) 3D view, (b) horizontal view, (c) near the

crest.

Fig. 3 Solution domain and boundaries for modelled

circular spillway.

Boundary conditions which were employed in this

investigation are (Fig. 3): Two different inlets

were needed to define the water flow (Inlet I) and

air flow (Inlet II). These inlets were defined as

stream-wise velocity inlets that require the values

of velocity. To estimate the effect of walls on the

flow, empirical wall functions known as standard

wall functions (Launder and Spalding, 1974) were

used. The upper boundary above the air phase

was specified as a symmetry condition, which

enforces a zero normal velocity and a zero shear

stress.


105

To complete the description of the CFD

simulation, the PRESTO pressure discretization

scheme was applied as it showed the best

convergence in this simulation. The first order

upwind momentum and turbulent kinetic energy

discretization scheme was applied (Karki and

Patankar, 1989). The PISO pressure-velocity

coupling algorithm (Issa, 1986) was used purely

because it is designed specifically for transient

simulations. The unsteady, free-surface

calculations required fine grid spacing and small

initial time steps. The grid spacing used was

adequate for solution convergence and has shown

good agreement with the experimental results. A

time step equal to 0.001 was selected. During the

3D simulation runs, solution convergence and the

water-surface profiles were monitored.

Convergence was reached when the normalized

residual of each variable was in the order of 0.001.

The free surface was defined by a value of VOF =

0.5, which is a common practice for volume

fraction results (Fluent Manual, 2005 and Dargahi,

2006). After the convergence of the numerical

solution, in order to obtain more accurate results,

the mesh was refined according to gradients of

two phases and velocities, and the model was run

subsequently. The final number of mesh in

various conditions changed in the range of

187026–202881 cells. A sensitivity analysis was

used and the number of mesh increased to over

two times, which has shown the results of the

model were valid.

6. VERIFICATION

It was necessary to ensure the accuracy of the

model before employing the numerical model.

For this purpose, experimental cases which were

mentioned in previous sections were employed.

To evaluate the free surface, the first case was

selected regarding the available flume data.

Existing experimental results used for validation

include water surface profiles, pressure

distributions and streamwise velocity distributions

for flow over the circular spillways. Fig. 4

compares experimental results of Tests 1, 2 and 3

(considering Table 1 parameters) with computed

water surface profiles over the crest.

Fig. 4 shows that different turbulence models

have no meaningful effect on predicting the free

surface in this simulation. Therefore all numerical

results are on one line. Results of velocity

simulation were compared with all experimental

tests (Table 1) and have been shown in Fig.5.

This figure shows good agreement between

measured and computed velocities by different

turbulence models except the standard k-, the

standard k-, and the k- RNG models.

Fig. 6 compares the pressure profile at crest,

obtained from simulation and experimental results

in different tests.

Fig. 5 and 6 show that the results from standard

models do not have a good agreement with

experimental data. It could be concluded that in

standard turbulence models (k- and k-), both

models under curvature conditions do not have

good performance in predicting hydraulic

characteristics.

In Fig. 7 and Fig. 8, numerical velocity vector

field and the corresponding flow patterns

represented by the streamlines for the flow

upstream of the spillway are given for two

extreme available cases.

7. CONCLUSIONS

In the present numerical study, flow over circular

spillways was simulated by using a three

dimensional code (Fluent Software). Different

turbulence models with the VOF method for free

surface modelling were employed to simulate

fully 3D flow. The numerical simulation solved

the Navier–Stokes equations within the flow

domain upstream and downstream from a

spillway. In the present research, focus was on

performance of different turbulence models to

simulate the flow over the circular spillways. By

comparing the 3D simulation results with the

flume data obtained by other researchers, the

simulation was found to produce flow over a

circular spillway with sufficient accuracy by all

turbulence models except the standard k- and the

standard k-. The RSM turbulence model had the

best agreement among all turbulence models with

experimental data.

Fig. 4 Water surface profiles for flow over circular

spillway.

Engineering Applications of Computational Fluid Mechanics Vol. 6, No. 1 (2012)

106

(a) Test 1 (b) Test 2

(c) Test 3 (d) Test 4

(e) Test 5 (f) Test 6

Fig.5 Comparison between computed and measured horizontal velocity profiles at the crest of circular spillway by

employing different turbulence models.

NOMENCLATURE

ui velocity component

iu mean velocity component

iu fluctuating velocity component

*u wall shear stress

jiuu Reynolds stresses

density of fluid

p pressure

µ viscosity of fluid

µt turbulent viscosity

µeff effective viscosity

ij Kronecker delta

k turbulence kinetic energy

turbulence dissipation rate

Gk generation of turbulence kinetic energy

due to the mean velocity gradients

Gb generation of turbulence kinetic energy

due to buoyancy

G generation of turbulence kinetic energy

due to

YM contribution of the fluctuating dilatation

in compressible turbulence to the overall

dissipation rate

Yk dissipation of k due to turbulence

Engineering Applications of Computational Fluid Mechanics Vol. 6, No. 1 (2012)

107

(a) Test 1 (b) Test 2

(c) Test 3 (d) Test 4

(e) Test 5 (f) Test 6

Fig.6 Comparison between computed and measured pressure head distributions at crest of spillway by employing

different turbulence models.

Y dissipation of due to turbulence k turbulent Prandtl number for k

turbulent Prandtl number for turbulent Prandtl number for

ij the mean rate-of-rotation tensor viewed

in a rotating reference frame with the

angular velocity k

S modulus of the mean rate-of-strain

tensor specific dissipation rate

ij mean rate of rotation tensor

F1& F2 blending functions

α* Parameter used to damp the turbulent

viscosity causing a low-Reynolds-

number correction term

Cij convection term

D cross diffusion term

DT,ij turbulent diffusive transport

DL,ij molecular diffusion

Pij stress production

Gij effects of buoyancy on turbulence ij pressure strain

ij dissipation rate

Fij production by system rotation

αq volume fraction of qth phase

y distance of first grid from solid wall


108

Case 1

Case 6

Fig.7 Numerical field velocity vectors.

Case 1

Case 6

Fig.8 Numerical field velocity streamlines.

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simulating flow over circular spillways by using different turbulence models-libre

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