Quasi 1D Flow

CodeForQuasi1DFlow

The minimum number of cycles for solution convergence is dependent on CFL number, mesh size and time stepping. Keeping mesh and time stepping constant and varying the CFL number. For a grid size of 31 points and CFL based time step, higher CFL <= 1.1 number leads to faster convergence.

For CFL of 0.5 convergence is achieved at 500 cycles for conservation form and 700 cycles for nonconservation form.

The image attached is for values at the nozzle throat.

The small difference in values between the two form is possible due to CFL based time stepping. Increasing the CFL number or making the grid finer, makes the solution match.

Attached below is the graph with CFL 0.75

It is to be noted that increasing the CFL number (for constant grid size) increases the error in field variables compared to the exact analytical solution.

Attached below are two images comparing error for density and Mach number for a grid size of 31 and CFL 0.25 and 0.75 respectively.

A particular distinction between nonconservation and conservation form of equations is, the nonconservation form solves the equation for the field variables whereas the conservation form solves the equation which is modified for the flux terms. The difference is significantly noticed while comparing the mass flow rates across the length of the nozzle.

Notice the wiggles in the nonconservation form. These wiggles do not die down for either refinement in grid size or that of CFL number. On the flip side the, the graph of conservation form is smooth with minimal undulations or wiggles. Also overall, the value of conservation form is closer to the actual analytical value of the mass flow rate. The conservation form also has faster convergence than nonconversation.

Overall the conservation form is fasters as well as provides better results.

Attached below are extra graphs comparing the two forms.