0.10 Numerical simulation  (Page 6/8)

We can see that this equation is of the same form as before with only the coefficients $e$ and $f$ of the vorticity equation modified.

Since the variable in the linear system of equations is now $\Delta w$ rather than $w$ , the equations for the stream function also need to be modified.

These modifications to the coefficients should be made after the coefficients are updated for convection and boundary conditions.

The numerical calculations will start from some initial value of vorticity and proceed to steady state. The size of the time step is important if accuracy of the transient solution is of interest. The truncation error of the time finite difference expression is approximately the product $\Delta t\frac{{\partial }^{2}w}{\partial t^2}$ . The time step size can be chosen as to keep this value approximately constant. Thus the time step size will be chosen as to limit the maximum change in the magnitude of $w$ over a time step. Let $\Delta w_{max}$ be the maximum change in $w$ over the previous time step and $\Delta w_{spec}$ be the specified value of the desired maximum change in $w$ . The new time step can be estimated from the following expression.

The initial time step size needs to be specified, and the new time step may be averaged with the old or constrained to increase by not greater than some factor.

Now that we have a means of calculating the transients solution, this approach can be used to treat the nonlinear terms. Recall that when the equations were linear, the system of equations had constant coefficients and the conjugate-gradient linear solver solved the linear system of equations.

The convective derivatives have a product of velocity and vorticity, which can be expressed as a product of the derivative of stream function and vorticity. One approximation would be to use the value of the stream function from the old time step to calculate the coefficients for the new time step. An alternative is use a predictor-corrector approach, similar to the Runge-Kutta solution for quasilinear ordinary differential equations. The "predictor" step estimates the solution at the new time step using the coefficients calculated from the stream function of the previous time step. This gives an estimate of the stream function at the new time step. This estimated stream function is then used to evaluate the coefficients for the "corrector" step. The steps in the calculations are as follows.

The choice of the finite difference expression for the convective derivative is very important for the stability of the solution. We will illustrate here treatment of only the $x$ derivative. Suppose $i-1$ , $i$ , and $i+1$ are the grid points where the stream function and vorticity are evaluated and $i-1/2$ and $i+1/2$ are the mid-points where the velocity-vorticity products are evaluated.

The dependent variables are known only at the grid points and interpolation will be needed to evaluate in between location. If the product is evaluated at $i-1/2$ and $i+1/2$ by using the arithmetic average of the values at the grid points on either side, the numerical solution will tend to oscillate if the convective transport dominates the diffusive (viscous) transport. It is preferable to determine the upstream direction from the sign of $u_{i+1/2}$ or $u_{i-1/2}$ and use the vorticity from the upstream grid point. This is illustrated below.