NUMERICAL PREDICTION ON CHARACTERISTICS FOR VERTICAL PLANE NEGATIVELY BUOYANT JETS

Wenxin Huai, Wei Li and Zhonghua Yang

Dept. of River Engineering, Wuhan University, Wuhan, Hubei 430072, China

Tel: 027-67802205,Fax: 027-87884496, E-mail: wxhuai@wuhee.edu.cn

Abstract: The k-e turbulent model is used to obtain a mathematical model of a vertical plane negatively jet in a static homogeneous ambient. The hybrid finite analytic method, with a non-uniform staggered grid, is used to calculate the whole field of this flow pattern. The variation of centerline velocity and density along the axial line for a given exit densimetric Froude number take a good regularity under the unified scaling law derived by Chen and Rodi. It is found that the flow can be divided into three regions. The profiles of mean velocity and mean density difference is given. The equation of the maximum height of rise for vertical plane negatively buoyant jets is given also. This equation could be used to estimate the region of pollution.

Keywords: buoyant jets, buoyancy effects, plane Jets, numerical simulation

1  INTRODUCTION

One of the flow configurations important in the environmental processes, with significant buoyancy effects, is a vertical negatively buoyant jet discharged into a stagnant environment. An example is an industrial discharge such as brine, which is released into the ocean through multiport diffusers. In the case of a jet issuing vertically upwards into fluid of which the density lighter than one of jet, the fluid of the jet cannot penetrate in a vertical direction beyond a certain ceiling level (See Fig. 1). In the flow pattern of a jet issuing vertically upwards into lighter ambient fluid distinction must be made between the actual jet with an upflow and the surrounding downflow region. For a homogeneous ambient fluid the downflow region extends downwards to the nozzle. Near the ceiling level, (i.e. the highest level reached by the fluid of the jet) jet fluid leaves the upflow region to enter into the downflow region. Thus near the ceiling level the vertical flux of jet fluid and the vertical flux of a tracer carried by the jet decrease with height.

In the past, Goldman and Jaluria(1986),Campbell and Turner (1989) ,Baines et al (1990) and Zhang and Baddour(1997) analyzed and tested the problem of a jet of heavy fluid issuing vertically upward into homogeneous lighter ambient fluid using the integral methods. In the integral methods, the entertainment rate and the profiles of both the velocity and the density must be assumed. Recently, the differential method was presented, which is not based on the entertainment and profile assumptions. Instead the differential method employs a turbulence model to predict the turbulence behavior at each point in the negatively buoyant jets. However, the accuracy of the differential method depends on the turbulence method used.

For positive buoyant jets, Madni and Pletcher (1976), Chen and Rodi (1975), Chen and Chen (1979), Chen and Nikitopoulos (1979), Li and Chen (1985), Li and Huai (1995) have been used the turbulent model, (such as a mixing length model, two equation k-e model) to calculated the behavior of buoyant jets in uniform and non-uniform environments. Few numerical simulations have been reported on plane negative buoyant jet in a homogeneous ambient. Recently, Huai and Li (1999) presented the behavior of vertical round negatively buoyant jet by k-e model.

In this paper the k-e turbulence model is adopted to close the governing equations of buoyant jets. For this kind flow, the unified equation of the closed equations is convective-diffusion type equation. The Hybrid Finite Analytic (HFA) Method and a staggered grid are used to solve the convective-diffusion type equation in this study (Li and Yang, 1990). The hybrid finite analytic method was applied to solve many fluid dynamics and heat transfer problems and shown to be stable and accurate (Huai and Li, 1994 a, b, 1995,Li and Huai, 1993,1995).

A unified scaling law (Zhang and Baddour, 1997) is employed to correlate the predicted results. It is found that the agreement between the calculated results and experimental data for the height of buoyant jet rise is quite satisfactory. The predicted velocity and density are given also.

2  MATHEMATICAL MODEL

Fig.1 shows a plane negatively buoyant jet in static ambient. If the Boussinesq approximation is made and the k-e turbulence model is used, and it is assumed that the flow is steady. The mathematical model of this flow pattern can be presented as

Continuity equation:

                                 (1)

x-momentum equation:

+            (2)

y-momentum equation:

            (3)

Density equation:

               (4)

k-equation:

       (5)

e-equation:

                               (6)

 = Eddy viscosity coefficient.                  

Where x and y denote the axial and horizontal directions of the plane negatively buoyant jet; u and v denote the corresponding velocity components; r and r_a denote the density and local ambient density; k and e are the turbulent kinetic energy and its dissipation; g is the gravity constant. The modeling constants are adopted as follows (W.Rodi, 1980): C_m =0.09, C_e1=1.44, C_e2=1.92, s_k =1.0, s_e=1.3, S_c=0.614,where S_c is taken as 0.614 by computer test. The following boundary conditions are adopted

x³0,y=0; 

    x³0,y=L; 

y³0,x=H >x_m ,  u=v=k=e=0, r=r_a

in which H is the depth of water, x_m is the maximum height of rise for buoyant jet. When the solution starts x_m is roughly given by using the empirical formula (Eq. 21). Most of the literature on experimental measurements does not provide the profiles for the initial turbulent kinetic energy,k₀ and its dissipation rate,e₀. In present study, Gaussian profiles were assumed for them just as Chen and Nikitopoulos did. These profiles are

x=0, 0£y£B/2; u=u₀, v=0, r=r₀

k=k_0mexp(-1.7y²) , k_0m=0.06u₀²

e=e_0mexp(-1.7y²) , e_0m=0.06u₀³/B

In which the centerline values of k and e at the exit are denoted by k_0m and e_0m respectively. It is shown by Chen and Nikitopoulos (1979) that the predicted results of the centerline velocity in the zone of development for buoyant jet agrees best with the experimental data when the above conditions at the exit are adopted.

At solid surfaces (y>B/2,x=0), wall functions are used to relate the values at the first grid points outside the viscous sublayers to the boundary conditions. Assuming a universal log-law for the flow near the solid surface, then

In which v_* is the resultant friction velocity, k is von Karman’s constant (k=0.4); E is a roughness parameter, and x_p is normal distance between the first grid point and the solid surface, The boundary conditions on k and e are

In process of calculations, the horizontal component of velocity at the first grid points,v_p, is obtained by iteration.In addition, the velocity component and the gradient of density normal to the surface equals zero.

3  NUMERICAL METHOD

The partial differential equations in mathematical model can be written as the following generalized form:

                              (7)

The edge of the calculation domain in the horizontal direction is far away from the axis of jet to guarantee the full development of jets; in the vertical direction, the calculation is made from exit to the water surface. The jet discharge width at exit is small. In order to reduce the numbers of grids, the non-uniform scheme is used in this study. Hybrid finite analytic method (Huai and Li, 1994b) is used to calculate this flow. The staggered grid technique is adopted to obtain the correct pressure and velocity at each grid point. The calculated domain is taken as (x,y)Î(0,H)´(0,3H) (H=1.2x_m) and the number of grid points with non-uniform space is taken to be 61 and 161 in x and y directions respectively.

4  MODELING VERIFY

Zhang and Baddour (1997) conducted an experimental study on a plane negatively buoyant jet in a homogeneous ambient for the plane case. They injected a heavy salt solution upward into a tank of fresh water and measured the ceiling height of the jet. They used dimensional arguments to relate the results to the volume, momentum, and buoyancy fluxes at the jet source:

             x_m=C_mM₀B₀^-2/3                              (8)

In which, the proportionality constant evaluated from the data is 1.59 for large Froude number. For a plane jet, these source parameters are M₀=Bu₀² and B₀=Bu₀g(r_a-r_a)/r_a and u₀=source exit velocity; B= slot width; g= gravitational acceleration;r₀=source density;r_a= uniform ambient density. Substituting for M₀ and B₀, we have C_m=x_m/(BF₀^2/3), where C_m= a constant that must be determined from experiment; and F₀=sourse densimetric Froude number defined as F₀=u₀²/[(r₀-r_a)gB/r_a]. The maximum jet penetration reported in previous studies is summarized in table 1, and the data in table 1 is given based on F₀.

To check the present model, the calculations were made for the parameters, r₀/r_a =1.2, in which the ratio just as the one of density of salt water and fresh water, B=2cm. The velocity at exit are taken 5 different value, the corresponding exit Froude number are F₀=25.48, 57.34, 80.74, 123.34 and 159.28.

Fig.2 shows the calculated streamline for F₀=25.48. The jet moves upward near the axial line, when it reach the maximum height, it turns downward and meet with the solid surface, then spreads sideways to form a spreading layer. There is a recirclelation region between axial line and outline of buoyant jet. Beyond the outline of buoyant jet, there is a large recircleation region based on the fluid continuity. The point at which the line of zero-streamline intersects axial line is x_m, the maximum height of negatively buoyant jet. The values of C_m= x_m/BF₀^2/3 for five different Froude number are given in table 1. We can find that the agreement between the predicted results of maximum height with the ones of experiment evidently is good.

5  PREDICTION ON THE BEHAVIOR

The present model and numerical method were tested against laboratory data. They show good agreement. In this section, the model and numerical method are used to predict the behavior of vertical plane negatively buoyant jets.

5.1  Velocity and density distribution in section

The predicted x-direction relative velocity (u/u_c) distributions, with  F₀=25.48 at the different location x/d, are given in Fig.3, for this case, the maximum height is x_m/B=12.0. The results stated that the velocity u in cross section appear positive and negatively value, this means the jet advanced upward near the region of axial line. With the increase of distance from the axial line, the velocity u becomes zero at a certain distance, and then becomes negative. The width of actual jet increase as the value of x/B increases. From Fig.3, we know these relative downward velocities increase as the increase of x/B, it does not mean that the downward velocity u becomes large, because the axial velocity decreases at this time.

The profiles of mean density for plane buoyant jet with F₀=25.48 is shown in Fig. 4. From the results in Fig. 4, the density near the source decrease slowly, as there is a edgy downward fluid body near exit separate mixing of between jet and the ambient fluids.

5.2  Variation of centerline velocity and density

Fig.5 and 6 show the variation of the centerline velocity and density. We observe that all predicted results normalized by the unified scaling law proposed by Chen and Rodi (1978) take a good regularity. In these figures,

,

,

.

It is found that the flow can be divided into three regions (x₁£x_11, x₁₁<x₁£x₁₂ and x₁>x₁₂), i.e., x₁£x₁₁, in which x₁₁ (=0.5 for centerline velocity, =0.25 for centerline density), the centerline velocity u₁ and density C₁ keep constant; x₁₁<x₁£x₁₂, in which x₁₂(=1.0 for centerline velocity,=0.8 for centerline density), u₁ and C₁ decreases along x direction; x₁>x₁₂, u₁ and C₁ decreases rapidly.

6  CONCLUSIONS

This paper presents a mathematical model is for simulating the mean velocity and density (or concentration) field in flow situations where buoyancy plays an important role. The article focused attention on the simulation of the influence of buoyancy on these characteristics by turbulent model and effectively of hybrid finite analytic method with non-uniform and staggered grids. The centerline velocity and density, the maximum height of rise, velocity and density difference in cross section were calculated for F₀ from 25.48 to 159.28.The predicted maximum height of rise is in good agreement with measurement. The other predicted results take a good regularity under the unified scaling law derived by Chen and Rodi.

Meanwhile it is found that HFA method is one kind of effective numerical method for the calculation of the convective-diffusion type equation.

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Table 1  Comparison of experimental condition and values of coefficient C

 ** ^a Equivalent slot half-width.  ^b Large Froude number data.(F₀>50.)

Fig.1  Plane negatively buoyant jets in a homogeneous ambient

Fig.2  Streamline of plane negatively buoyant jets with F₀=25.48

Fig.3  Profiles of mean velocity for plane negatively buoyant jets with F₀=25.48

Fig.4  Profiles of mean density for plane negatively buoyant jets with F₀=25.48

Fig.5  Decay of centerline velocity for plane negatively buoyant jets

Fig.6  Decay of centerline density for plane negatively buoyant jets