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FIGURE 6.14 Typical grid used for simulations (40 x 40).

FIGURE 6.14 Typical grid used for simulations (40 x 40).

The residuals (R) for each of the governing equations need to be reduced to an adequately low level. Definition of an 'adequate' degree of convergence may be problem dependent. In general, an examination of the history of residuals and variables at key locations, along with the integral balances may provide a good indication of the degree of convergence. The convergence behavior of a numerical method depends on several factors discussed in this chapter. For a given algorithm and discretization scheme, values of the under-relaxation factor control the rate of convergence.

For the problem considered in the previous sections, the influence of under-relaxation factors on the reduction in residuals is shown in Fig. 6.18. It can be seen that up to a certain limit, as the under-relaxation factor increases, the rate of residual reduction increases. However beyond a certain limit, further increase in under-relaxation parameters may lead to divergence (see, for example, results for an under-relaxation parameter of 0.85, where residues increase with further iterations). For an under-relaxation parameter of 0.6 (for velocities), residual and variable history are shown in Fig. 6.19. These profiles and an examination of integral balances indicate that adequate convergence occurs in about 150 iterations. Further simulations lead to further reduction in residuals, however, the predicted results are almost independent of the actual value of residuals beyond 150 iterations. This fact further confirms that convergence level obtained at 150 iterations (with an under-relaxation parameter of 0.6) was satisfactory. Comparison of the converged results obtained using different under-relaxation parameters confirms that the value of under-relaxation parameter has no influence on the converged results. After ensuring adequate convergence, it is important to examine possible errors in the simulated results.

of stream function: 10 uniform contours between 0 to 0.001 (Lowest level: A; Highest level: J).

Distance from the Bottom, m

FIGURE 6.16 Influence of grid size of predicted results (Velocity profile at x = 0.5 m).

Distance from the Bottom, m

FIGURE 6.16 Influence of grid size of predicted results (Velocity profile at x = 0.5 m).

Distance from the Bottom, m

FIGURE 6.17 Influence of solution domain of predicted results (Velocity profile at x = 0.5 m; Grid: 40 x 40).

Distance from the Bottom, m

FIGURE 6.17 Influence of solution domain of predicted results (Velocity profile at x = 0.5 m; Grid: 40 x 40).

There are several ways of evaluating possible errors in the simulated results. Values of residuals in the discretized equations are one of the indicators of errors. In addition, it is often useful to examine integrated fluxes of the quantity of interest (for example, mass flow rate) and compare them with the expected values. For example,

FIGURE 6.18 Influence of under-relaxation parameters on convergence (Pressure residual; Grid: 40 x 40).

in the case considered, net mass flow passing through any constant I plane should be equal to the inlet mass flow rate. Simulated results reveal that computed net mass flow rate is the same as the inlet flow rate, indicating adequate convergence. All the results discussed so far were obtained with a first-order upwind differencing scheme. In order to assess the quality of results, simulations were carried out for two higher order schemes: a second-order upwind method and QUICK (with SMART limiter to prevent non-physical oscillation). The predicted profiles at the mid-I plane of the base case are compared with the results of these two cases in Fig. 6.20. It can be seen that the difference in predicted results of the three discretization schemes is not significant. Thus, simulated results are not dependent on any of the numerical parameters: number of grids, discretization schemes, convergence criterion and so on. After ensuring this, further evaluation of simulated results can be done with the help of experimental data, if available. After such validation, the computational model may be used to understand the flow process under consideration. Here we illustrate the possible use of the computational model to understand the influence of the fluid viscosity on the fluid dynamics of the considered configuration.

6.5.3. Simulations with Low Viscosity Fluid

Starting with simulations of fluid with viscosity 100 Pa.s, further simulations were carried out for fluids with viscosity 10 Pa.s, 1 Pa.s and 0.001 Pa.s. Initially all simulations were carried out for 40 x 40 grids. Typical predicted results are shown in Figs 6.21 and 6.22. It can be seen that fluid viscosity has a pronounced influence on fluid dynamics. As the viscosity decreases, the penetration depth of the incoming jet increases, leading to circulatory flow within the domain. For lower viscosity fluids, much sharper profiles exist within the solution domain. It will be of interest

Iteration number

FIGURE 6.19 Residual and variable history (Under-relaxation factor for velocity = 0.6; Grid: 40 x 40). (a) Residuals; (b) Velocity (top: J = 36; middle: J = 22; bottom: J = 7).

Iteration number

FIGURE 6.19 Residual and variable history (Under-relaxation factor for velocity = 0.6; Grid: 40 x 40). (a) Residuals; (b) Velocity (top: J = 36; middle: J = 22; bottom: J = 7).

to examine whether the conclusion of grid adequacy is valid for the low viscosity fluids. Predicted results for the lowest viscosity fluids, obtained with different grids, are shown in Fig. 6.23. It can be seen that the predicted results are functions of grid size and are no longer grid independent even for the finest grid size used (640 x 640). This can be understood with reference to the phenomenon of turbulence discussed in Chapter 3. For a fluid with viscosity of 0.001 Pa.s, the set boundary conditions generate flow with sharp gradients, through which flow instability and turbulence sets in. In turbulent flow widely different scales co-exist and flow becomes inherently unsteady. It can be seen that even 400 000 computational cells turn out to be inadequate to capture all the small-scale features of turbulence (without using unsteady state equations). It may be necessary to use an appropriate turbulence model in such a

Distance from the bottom, m

FIGURE 6.20 Influence of discretization scheme on predicted results (Velocity profile at x = 0.5; Grid: 40 x 40).

Distance from the bottom, m

FIGURE 6.20 Influence of discretization scheme on predicted results (Velocity profile at x = 0.5; Grid: 40 x 40).

case. The solution of model equations describing turbulent and other complex flows is discussed in the next chapter.

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