TURBULENCE SIMULATION BY LAGRANGIAN BLOCK METHOD

Altai W. and Chu, V. H.

Department of Civil Engineering and Applied Mechanics,

McGill University

817 Sherbrooke W., Montreal, Canada, H3A 2K6

Tel: 514-398-6863, Fax: 514-398-7361, E-mail: chu@civil.lan.mcgill.ca

Abstract: highly non-diffusive numerical scheme, known as the Lagrangian Block Method (LBM), is employed to simulate the chaotic advective process in a turbulent plane jet.  Simulations were conducted using a diffusion coefficient equal to sub-grid scale of the turbulent jet. The results are in close agreement with the experimental observation. The simulation is also conducted in a pure advection experiment.  The result of the pure advection experiment shows that the numerical diffusion associated with the LBM is extremely small.  This is in contrast with the conventional computational scheme in which the numerical diffusion is of the same order of magnitude as the sub-grid scale viscosity.  Because the numerical diffusion associated with the LBM is very small, the method is most suitable for the simulation of the transport processes in the coastal waters of lakes and oceans where the diffusion coefficient is often quite small.

Keywords: recirculating flow, turbulent mixing, coherent structure, bed-friction effect, open-channel Flow

1  INTRODUCTION

The transport of pollutants and sediments in coastal waters of lakes and oceans are advective dominant process.  The diffusion process is often determined the depth of the flow, but the size of computational grid can be much greater the depth.  Numerical diffusion can be overwhelming compared with the diffusion associated with the physical processes.  The challenge in the simulation of the process has been the problem of numerical diffusion and unphysical oscillations. The source of the numerical diffusion can be traced to the problem of cellular averaging.  Using the Lagrangian Block Method (LBM), masses are contained in blocks that are moved in the direction of the velocity.  New set of blocks are re-generated at each increment in time. The masses are re-distributed to the new blocks without ever passing on to the cells.  In this ways, the problem of cellular averaging is eliminated.  The original implementation of the LBM by Egan and Mahony (1972) was based on a hybrid procedure and was not generally a stable procedure.  Chu and Altai (1998, 2000) have improved the method to eliminate the instability problem and to include the diffusion as part of the simulation. The new LBM is mass conserving, positive definite and unconditional stable and therefore is most suitable for the simulation of the advection dominant process.  In this paper, this new LBM is employed to conduct Large Eddy Simulation (LES) of a turbulent plane jet.

2  LAGRANGIAN BLOCK METHOD

Figure 1 shows the LBM by Chu and Altai (1998, 2000).  The method is distinguished with the conventional computation method by the use of the blocks as the computational element.  Masses are contained in the blocks that move across the fixed computational grid lines.  The block as shown in the figure moves from one position at time t to a new position at a later time t + Dt.  Due to diffusion the size of the block increases with time.  The rate is related to the diffusion coefficient D by the following equations:

                                        (1)

In the diffusion problem of the random walk (Einstein, 1905; Taylor, 1921), The root-mean-square displacements, s_x and s_y, are related to the widths of blocks, w_x and w_y, by

                                              (2)

The mass is contained in a block as shown in the shaded region.   The block moves with the flow velocity components u(x_c, y_c, t) and v(x_c, y_c, t).  Over the period of one time step, from time t = n Dt to t = (n+1) Dt, the mass center of the block moves from (x_cⁿ, y_cⁿ) to (x_cⁿ⁺¹, y_cⁿ⁺¹).  The displacements in the x- and y-directions over the period are given by the integrals

             (3)

The positions of the blocks are determined from these integrals by iteration method. The widths of block are increased due to diffusion as follows:

             (4)

At the end of the time increment, the mass in the block is divided into ‘portions' and re-distributed onto a new set of blocks. Aspects of the implementation details of the LBM have been reported in Chu and Altai (1998, 2000). An application of the method for turbulent flow simulation using a K-e model was given by Altai, Zhang and Chu (1999). The method is unconditionally stable, non-negative and mass conserving. More computer memory is required for the implementation of the method. Compared with the conventional method, the LBM is more efficient because accurate results can be obtained with the LBM using a value of the Courant number very close to unity. The performance of the LBM is shown in Figure 2 where a block of red dye is advected by a circular velocity field associated with the solid-body rotation. The profile at the lower-right-hand corner shows the concentration distribution of a 16 by 16 blocks. There is very little numerical diffusion associated with method. After 16 rotations, the dye tracer is not dispersed more than one row of cells.

3  CHAOTIC ADVECTION IN A TURBULENT JET

Figure 3 shows the Large Eddy Simulation (LES) of a turbulent plane jet obtained using the LBM. The sub-grid scale viscosity in the vorticity equation is determined using the Smorgorinsky model.  The velocity field is then determined from solution of the Poisson equation. Advection is dominant over the diffusion in this simulation. The sub-grid scale viscosity is n_SG = C_s² D² |S_ij| where D² = (Dx)²+(D_y)² is the filter length and S_ij is the strain-rate tensor.  Being equal to the product of D² and a small coefficient C_s²  (= 0.014), the sub-grid viscosity is small.  In some computational method, the sub-grid viscosity of the Smagorinsky model can be smaller than the numerical diffusion of the method.

To demonstrate the capability of the LBM, the distribution of a tracer (red dye) is calculated for the same turbulent velocity field but with two diffusion coefficients.  Figure 3b shows the dye concentration distribution in the turbulent jet obtained using a diffusion coefficient equal to the sub-grid viscosity. Figure 3d is the results of a pure advection experiment obtained by setting the dye diffusion coefficient to zero. Although the sub-grid diffusivity is very small, its contribution to the distribution of the dye in the turbulent flow is huge. Figure 3a is a histogram showing the number of blocks in which the dye concentration is in the range from c to c + Dc. In this histogram, the interval Dc is equal to 0.05 c_o, where c_o is the source concentration; the number of blocks in each interval is plot versus concentration.  The red line in the figure marks the source concentration. Diffusion causes the dye to mix with the surrounding fluid and causes the dye to reduce its concentration. In this case the majority of the blocks have a dye concentration equal to 40% of the source concentration (see the location of the peak in the histogram). Figure 3d shows the result of the pure advection experiment by setting the dye diffusion coefficient to zero. The LBM method now produces a distribution of dye in close agreement with the exact solution. As shown in the histogram in Figure 3c, most of the blocks has been able to maintain the same dye concentration as the source. Only a small number of blocks are affected by the false numerical diffusion.

4  CONCLUSION

The computation for the chaotic advection of the dye in a turbulent jet has shown that the numerical diffusion associated Lagrangian Block Method is negligible.  In the coastal waters of lakes and oceans, because the length scale in the horizontal direction of the flow field is large compared with the depth of the flow, the size of the computational grid would have to be large compared with the depth.  Under this condition, the numerical diffusion associated with most of the conventional methods would be significant and could seriously affect the accuracy and stability of the simulation.  Therefore, the LBM with almost no numerical diffusion, is most suitable for the simulation of the processes in the coastal waters.

References

[1]  Chu, V. H. and Altai, W. (1998) “Generalized second-moment method for advection and diffusion processes,” Proc. 2^nd International Symposium on Environmental Hydraulics, Hong Kong, Balkema Publisher, pp. 357-2362.

[2]  Chu, V. H. and Altai, W. (2000) “Lagrangian block method,” Proc. 1^st International Conf. Computational Fluid Dynamics, Kyoto, Lecture Notes in Physics, Springer-Verlag (in press).

[3]  Egan, B. A. and Mahoney, J. R. (1972) “Numerical modeling of advection and diffusion of urban source pollution,” J. Appl. Meteorology, Vol. 11, pp. 312-322.

[4]  Einstein, A. (1905): Ann. d. Physik.  This and other articles of Einstein is collected in a book entitled Investigations on the theory of the Brownian Movement, Edited by R. Füth, translated by A. D. Cowper (Dutton, New York).

[5]  Taylor, G. I. (1921) “Diffusion by continuous movement,” Proc. Lond. Math. Soc., Vol. 20, No. 2, pp. 196-212.

Fig. 1  Lagrangian block Method. (a) the block at time t = n Dt. (b) The block at time t = (n+1) Dt, after the displacement and the enlargement of the block by advection and diffusion, respectively. at the end of the time step, the block is divided by the grid lines into

Fig. 2  Rotation in a circular velocity field with diffusion  D = 0 and the Péclet number Pe equal to infinity.

Fig. 3  LES of turbulent jet by LBM.  (b) Dye diffusion coefficient equal to sub-grid scale viscosity. (d) Dye diffusion coefficient equal to zero.  (a) and (c) are histograms for the number of blocks versus the dye concentration.

Fig. 4  (a) Mean velocity and concentration profiles obtained using the LBM with C_s = 0.12 and CDS with C_s = 0.075. (b) profiles by LBM and CDS compared with experimental data. (c) profiles by LBM and CDS compared with experimental data.