Module 5: Solution of Navier-Stokes Equations for Incompressible Flows Using SIMPLE and MAC Algorithms
   
 

 

Problems

  1. Develop the MAC and SIMPLE Algorithms described in this chapter for a steady one dimensional flow field driven by a constant (but as yet undetermined) pressure gradient. For definiteness, flow in a parallel plate channel can be considered.
  2. For a two-dimensional flow, show that the Mac type iterative correction of pressure and velocity field through the implicit continuity equation is equivalent to the solution of Poisson equation for pressure. How does the procedure avoid the need of directly applying pressure- boundary- conditions?
  3. Compare the solutions obtained by the stream function-vorticity method with those presented in this chapter for the following problems:

    (a) Developing flow in a parallel plates channel;

    (b) Flow past a square cylinder (Re < 200)

    (c) Flow past a periodic array of square cylinders.

  4. Apply MAC method to solve the share driven cavity flow problem shown in Figure 36.2. Take a grid size of 51 ×51 and solve the flow equations for Re=400

  1. The Navier-Stokes equations in three-dimensional spherical-polar coordinate system in direction may be written as

The continuity equation is given by

Discretize the weak conservative form of the - momentum equation on a grid (in spherical polar coordinate) using a weighted average scheme.

Figure 36.2; Shear Driven Capacity

  1. Obtain the exact solution of the equation given by

Where, is the convective flux at the pointhas been defined. is the diffusion coefficient. Both and are constant over the control volume. The boundary conditions are: at and at

Figure 36.3: Flow over a Backward Facing Step

  1. Solve three-dimensional Navier-Stokes equations for the flow over a backward facing step as illustrated in Figure 36.3. Find out the reattachment length for the Reynolds numbers of 200, 300, and 400. The Reynolds number is given by
  2. Consider the channel flow and backward-facing-step configurations once again. Combine the flow and thermal energy equations and determine the local and global heat transfer rates when one of the solid surfaces is heated, the incoming fluid is cold and all other solid surfaces are thermally insulated.