Liu C.R.  Ma W.J.   HuHe A.D. 

AbstractThe CFD code of FLUENT with the method of Volume of Fluid (VOF) is employed to simulate the flows over a weir. The effects of Reynolds number, Froude number, and the ratio of the weir height to the water depth, D/H, on the flow patterns are discussed. The vertical profile of velocity and the distribution of shear stress on the bed at downstream of the weir are also calculated. The numerical results agree well with the experiment.

  Weir is the construction widely used in hydraulic engineering. It is very important to investigate the behaviors of the flows over a weir and the local scour caused by such flows.

  In general when water streams over a weir it behaves various flow patterns according to different conditions that govern the flows.  Different flow patterns result in different inference on the vertical profile of velocity and the distribution of shear stress on the bed at downstream of the weir.  Four types of patterns of the flows over a plate obstacle can be identified: surface jet; surface wave; plunging condition and limited jump (Iwao^[1] 1997). 

  There are a number of experimental and numerical studies on water flows over obstacles. Most of the numerical works are based on the inviscid model. (Lamb^[2](1945). Forbes and Schwartz^[3](1982) Dias and Broeck^[4](1989)). For simulation of water jump, Fennema and Chaudhry^[5] (1990) used MacCormack scheme to simulate transcritical flow. Navarro et al. ^[6] (1992) used a Total Variation Diminishing (TVD) version of the MacCormack scheme which can simulate water jump more accurately. Meselhe^[7] (1997) developed a two-step, predictor-corrector, implicit numerical scheme which has reasonable shock-capturing capabilities, and can easily simulate hydraulic structures . All of the researches mentioned above did not consider the viscous effect.

  Dani and Peter^[8](1997) considered the viscous effect. The incompressible, laminar, free surface fluid flow over a submerged obstruction has been investigated by the Surface Marker and Micro Cell technique (SMMC) in Dani's calculation. It shows that the flow over an obstacle for the condition of small deformational free surface can be simulated by inviscid model. However the work of Dani and Peter did not deal with the situation in which the free surface undergoes large deformation and the turbulence exists.  For the conditions of plunging and limited jump, the free surface undergoes large deformation and the water has mixed with the air in some region. So far few numerical research can be found concerning the plunging condition with viscous and turbulent effects.

  In this paper the viscous and turbulence effects are considered. The method of Volume of Fluid (VOF) is adopted to track the free surface. And the standard model is used as turbulence closure.  Four types of flow patterns are presented in this simulation. The effects of Reynolds number, Re, Froude number, Fr, and the ratio of height to depth, D/H, on the flow patterns are discussed. The vertical profile of the velocity and the shear stress on the bed at downstream of the weir are also obtained in the simulation. The numerical results agree fairly well with the experimental results.

In this study a two-dimensional semicircular weir is considered. The computing domain is a rectangle with 50D long and 2.2D high (fig.1). Where D is the weir height and H is the water depth, which changes from 1.17D to 1.5D. Above the water is an area of air.

The governing equations are the unsteady incompressible two-dimensional Continuity Equation and Reynolds-averged Navier-Stokes equation.

 

 

 
Where  are the velocity components; the density, and  are the volume fraction of air and water respectively,  and  are the density of air and water respectively;  is the pressure;  is the gravity acceleration. The viscosity ,  and  are the viscosity of air and water respectively;  is the eddy viscosity and can be expressed as

  These equations are coupled with the standard  turbulent model: 

 Where  is the turbulent kinetic energy,  the turbulent dissipation rate. The empirical coefficients in the turbulent model are recommended:   (Rodi ^[9]1980)

  To track the location of the free surface with large deformation the Volume of Fluid method (VOF) is used. The volume fraction equation is

The region where  is occupied by the water, and the area where  is filled with the air.  According to the value of , the location of the free surface can be determined.

Four types of boundaryconditions are introduced: velocity inlet, outflow, wall and pressure inlet.

Where  represents

  For the turbulence field near the solid boundary the wall function is applied on the surface of the bed bottom and the weir

Pressure inletcondition

Where  is constant.

Initial condition 

Time dependent simulation is conducted with the initial conditions of U_x=constant, U_y=0 in the entire computing domain,

3. Numerical Methods

A control-volume-based technique is used to convert the governing equations to a set of algebraic equations that can be solved iteratively^[10]. The method of Volume of Fluid (VOF) with geometric reconstruction approach is used to track the location of the free surface more accurately^[11].The VOF model can model two or more immiscible fluids by solving a single set of momentum equations and determining the volume fraction of each of the fluids throughout the domain, thus implement the transient tracking of the liquid-gas interface. 

  As the free surface is almost flat in the areas far away from the weir, the structured mesh is adopted where. In the region near the weir the unstructured mesh is employed (Fig.2). The general mesh size is 0.2D. To ensure the stability of the calculation, the mesh near the free surface must be refined, and the mesh where free surface undergoes large deformation must be further refined. Near the wall the boundary layer mesh is used. The time step is chosen to satisfy

  In this paper the flow fields and the bed shear stress were calculated numerically for different Re, Fr_u and D/H, where is the mean flow velocity at upstream of the weir. The differences of results treated from the rigid-lid assumption and VOF method are discussed.

The parameter h/H is important in identifying the flowpatterns. Where h is the water depth of downstream. Let Fr_u and D/H remains constant, the flows for different Re are simulated. The relationship of h/H and flow patterns with the Re is shown in Fig.3. The results of simulation show that h/H and flow patterns have little altered with the change of Re in the range  of  our  investigation (2000<Re<4.5×10⁶).  It seems that the flow pattern over a weir is predominated by Fr and D/H. 

The flows for different Fr_u and D/H are then simulated in this work. The results show that with the increase of Fr_u and D/H, h/H decreases, and the flow patterns change from the types of surface jet and surface wave to those of plunging condition and limited jump.

When Fr_u and D/H are small, h/H is about 0.9-1.0, the flow patterns over the weir exhibit the surface jet or surface wave. For these conditions, the higher velocity flow at downstream of the weir is along the free surface. A clockwise rotating circulation is formed behind the weir (Fig.4). When Fr_u and D/H increase, h/H is smaller than 0.9, the flow patterns become plunging conditions, and the higher velocity flow streams from the surface of the weir to the bottom of the bed, which induces the higher velocity near the bed bottom. Two vortices are formed behind the weir. One is a counter-clockwise rotating vortex at the cornerof the weir and the bed. The other is a clockwise rotating vortex near the free surface (Fig.5). The vertical profiles of the velocity at near and far downstream are given in fig.6. When Fr_u and D/H increase to make Fr_d=>1.0, water jump will appear at downstream of the weir, where U_d is the mean velocity of the flow at downstream of the weir. Fig.7 shows the relationship of the flow patterns with Fr and D/H obtained by this simulation. The experimental results^[12]  are also plotted in the figure The computed vertical profiles of velocity agree well with the experimental results. 

  The distribution of shear stress on the bed bottom at downstream of the weir is also achieved by the simulation. It is found that in the range of the full development flow, the shear stress on the bed has little change. So the shear stress on the bed in the range of full development flow can be assumed a constant . The relationship of the dimensionless shear stress on the bed with L/D is shown in Fig.8.  It is seen that the flow patterns have a great effect on the distribution of shear stress. For the conditions of surface jet and surface wave, there are two apices of shear stress on the bed at downstream due to the effect of circulation (Fig8-a). One apex is in the range of bed reverse flow; the other is in the range of transitional flow. Both apices are not very high (less than 1.5). For the conditions of plunging condition and limited jump, the shear stress on the bed is very high (Fig8-b). Its maxim value can reach 30. In the range of transitional flow, the shear stress on the bed decreases greatly. In the range of full development channel flow, the shear stress on the bed approaches a constant . These results agree with our experiments^[12] quite well (see Fig8).    

For the conditions of surface jet, the free surface is nearly flat. The problem whether the free surface can be modeled by a rigid-lid for the conditions of surface jet will be discussed in this paper. 

It is found that the velocity magnitude near the bed bottom calculated with VOF method are larger than those with the rigid-lid assumption within the circulation region behind the weir. At far downstream (L>10), the vertical profiles of velocity calculated with either VOF method or the rigid-lid assumption are nearly the same (fig.9). The bed stress calculated with VOF methodis higher than that for the rigid-lid assumption in the region of circulation. Out of the circulation region the bed stresses for both VOF method and the rigid-lid assumption are almost the same (fig.10). 

In the circulation region the difference of the numerical results calculated with VOF method and the rigid-lid assumption may be caused by the turbulence produced at the interface of water and air, where the turbulent kinetic energy will be entrained by the vortex to the region near the bed bottom for the treatment with VOF method. The turbulence is, however, depressed   for the treatment with the rigid-lid assumption. So the turbulent kinetic energy near the bed bottom for the VOF method will be higher than that for the rigid-lid assumption (fig.11). The increase of the turbulent kinetic energy will result in the increase of the bed stress.

At far downstream of circulation region, less turbulent kinetic energy produced at the interface of water and air will be entrain to the region near the bed bottom. Thus turbulent kinetic energy calculated for both treatments is almost same near the bed bottom at far downstream (fig.11).  

 

L/D=2                        L/D=4                    L/D=10

Fig. 9  The vertical profile of velocity

 

Fig.10  The bed stress distribution

 

         

 

           


L/D=2                        L/D=4                     L/D=10

.Fig.11  The turbulent kinetic energy profile

5. Conclusion

  The CFD software of FLUENT with the method of volume of fluid (VOF) is adopted to model the flows over a weir. Flow patterns of surface jet, surface wave and plunging condition are simulated. The vertical profile of velocity and the distribution of shear stress on the bed bottom are achieved by the simulation. The results agree well with our experiments. The phenomenon of air entrainment to the water for the situation of plunging condition is observed in the simulation. However the method of VOF is difficult to simulate this phenomenon accurately. In order to simulate the phenomenon of air entrainment better, new method of simulation should be developed.      

Reference

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2.  Lamb,H. , Hydrodynamics, Dover, 1945

3.  Forbes,L.K. and Schwartz,L.W., Free-Surface Flow Over a Semicircular Obstruction, Journal of Fluid Mechanics, Vol.114, 1982, P299

4.  Dias,F. and Vanden Broeck, J. M.,  Open Channel Flow with Submerged Obstacles, Journal of Fluid Mechanics, Vol.206, 1989, P155

5.  Fennema,R.J. and Chaudhry, M. H. , Explicit Methods for 2-D Transient Free-Surface Flows, Journal of Hydraulic Engineering, Vol.116, No.8,1990, P1013

6.  Navarro,G.P. et al., 1-D Open-Channel Flow Simulation Using TVD-MacCormack Scheme,  Journal of Hydraulic Engineering, Vol.118, No.8, 1992, P1359

7.  Meselhe E.A. et al. Numerical Simulation of Transcritical Flow in Open Channel, Journal of Hydraulic Engineering, Vol.123, No.9, 1997, P774

8.  Dani F. and Peter E.R. Open Channel Flow Over Submerged Obstructions: An Experimental and Numerical Study, Journal of Fluids Engineering, Vol.119,  1997,  P906

9.  Rodi,W. Turbulence Models and Their Application in Hydraulics A State-of-the-Art Review, IAHR Publication, 1980. 

10.    Patankar S.V. Numerical Heat Transfer and Fluid Flow. Hemisphere, Washington, D.C., 1980.

11.  Hirt C.W.and Nichols.B.D. Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries. J. Comput. Phys., 39:201-225, 1981. 

12.    Huhe A.D. and Liu C.R. Experimental Investigation of Flows over a Weir, 2000, (to be submitted )