1 Short description
This tutorial solves the Taylor Green Vortex Case.
The case is used to validate the temporal accuracy of the Euler and Crank-Nicolson schemes and spatial accuracy of the central difference scheme.
This presents an accuracy study on the OpenFOAM framework by using the Taylor Green Vortex case which has an analytical solution. The numerical solution of the Navier-Stokes is subject to errors due to the time and spatial discretizations. The analytical equation for the velocity may be expressed as a function of the numerical solution
were is velocity, is the time step, is the grid size in the x direction and is the grid size in the y direction, and a and b are the order of the discretization errors. The error of the numerical solution may be calculated by subtracting the numerical solution from the analytical solution.
Defining a constant value of and making the equation can be rearranged to
Taking the logarithm on both sides of the equation and defining
The same can be done in order to obtain the error as a function of time-step
3 Analytical Solution
The analytical solution gives velocity in the x and y components and the pressure.
The computational period was in the and directions. The first case simulations were run by letting and fixing . The second case simulations were run by fixing This allows the time and spatial discretization errors to be studied independently. Different time-steps were chosen with a final time of 0.34 seconds which is close to the half life of the decaying vortex. The error was calculated by extracting the numerical solution from the analytical solution.