Module 5: Solution of Navier-Stokes Equations for Incompressible Flows Using SIMPLE and MAC Algorithms
  Lecture 31:
 

 

Solution of the Unsteady Navier-Strokes Equations

The MAC method of Harlow and Welch is one of the earliest and most useful methods for solving the Navier-Stokes equations. This method necessarily deals with a Poisson equation for the pressure and momentum equations for the computation of velocity. It was basically developed to solve problems with free surface, but can be applied to any incompressible fluid flow problem. A modified version of the original MAC methods due to Hirt and Cook (1972) has been used by researchers to solve a variety of flow problems.

The text discusses the modified MAC method and highlights the salient features of the solution algorithm so that the reader will be able to write a computer program with some confidence. The important ideas on which the MAC algorithm is based are:

  1. Unsteady Navier-Strokes equations for incompressible flows in weak conservative form and the continuity equation are the governing equations.

  2. Description of the problem is elliptic in space and parabolic in time.  Solution will be marched in the time direction. At each time step, a converged solution in space is obtained but this converged solution  at any time step may not be the solution of the physical problem.

  3. If the problem is steady, in its physical sense, then after a finite number of steps in time direction, two consecutive time steps will show identical solutions. However, in a machine computation this is not possible hence a very small upper bound, say “STAT” is predefined. Typically, STAT may be chosen between 10-3 and 10-5.  If the maximum discrepancy of any of the velocity components for two consecutive time steps for any location over the entire space does not exceed STAT, then it can be said that the steady solution has been evolved.

  4. If the physical problem is basically unsteady in nature, the aforesaid maximum discrepancy of any dependant variable for two consecutive time steps will never be less than STAT. However, for such a situation, a specified velocity component can be stored over a long duration of time and plot of the velocity component against time (often called as signal) depicts the character of the flow. Such a flow may be labeled simply as “unsteady”.

  5. With the help of the momentum equations, we compute explicitly a provisional value of the velocity components for the next time step.