Convection Velocity in SIMPLE Algorithm
 

Hello,

I am working on implementing an unsteady, compressible 1D Euler solver in finite volume framework using the SIMPLE algorithm initially, possibly changing to PISO in the future if worthwhile. Implicit formulation.

I am fairly happy with the overall approach but one of the things I'm not sure about is which velocity should be used in forming the convective flux 'F' coefficients for the 'U' momentum equation (rhoUU)_east - (rhoUU)_west

I are trying to solve for the value of U_star using a TDMA approach such that I can apply the correction at the next stage. But which velocity should I use to calculate the convective flux values F_east = (rho*U_east) since the obvious problem is that we don't have a result for U_east since it's what we eventually need to solver for. I am planning to use the "best guess" for U in the convective flux, so the velocity that was corrected in the previous inner iteration. The difference between the uncorrected and correct velocity should get smaller as the solution is converging, to a point where the fact that I'm using the previous corrected velocity for the flux term does not make a significant difference.

Hope that I've been able to explain my thinking adequately.

Thank you.

I did not understnadthe question, but I can suggest you to read the book from Patanakar, who is the author of the SIMPLE scheme.
In his book, he explained the details nicely.

The unsteady, compressible 1D Euler flow model has a fully hyperbolic character. You cannot think to use the SIMPLE method as it is

Is that because in hyperbolic flows, the pressure field is not elliptic anymore ? Thus yielding unreaslitic solution ?
Thanks for that important point !!

Thanks for your reply FMDenaro.

Are you saying it's not possible/typical to use the SIMPLE approach for the 1D unsteady, compressible Euler equation?

If so, which approach would you recommend or is typical for this type of situation?

Thanks.

I can recommend this book https://www.cambridge.org/core/books...D52EAD6909E2B9

I am familiar with that book from previously working on Godunov methods. I feel that the book is fairly focused towards high Mach number flows. My application is variable density due to combustion, the Mach numbers are fairly low.

Most of the commercial codes in this field seem to make use of the PISO algorithm of Issa.

But for 1D, unsteady, subsonic, inviscid flows, you could get singularity in the solution. Usually, the PISO for low-Mach flows is for viscous flows.

Compressible NS
 

SIMPLE type methods won't work in the compressible regime due to the mathematical and physical nature of pressure. In compressible flows, you can calculate pressure thermodynamically (use ideal gas law.) Therefore you do not need to solve a pressure poisson type equation as in the incompressible regime. If you want to solve compresssible flows, Euler or full NS, read a book as recommended by Dr. Denaro or the book on Riemann solvers by Toro.

Thank you for your post moshe.

Can you suggest suitable methods for variable density, low Mach number flows? Where density varies significantly due to temperature changes rather than pressure.

From previously working with Riemann solvers, I don't believe that they're suitable for low Mach number flows.

Thank you.

You can take a look at preconditioning methods.

Thanks for your suggestions.

I've considered the situation and I don't believe that I need to solve a momentum equation or any other for pressure in my case.

I am intending on solving a scalar transport equation for the progress variable (separates burnt & unburnt regions). The local density is a function of the progress variable only which can be calculated directly at each time step.

I should then be able to use the unsteady, compressible mass conservation equation to calculate the new velocity fluxes (assuming constant flux of unburnt gas at the inlet boundary).

I appreciate that the direction of this topic has changed from the original title. But my eventual aim is to write a 1D premixed flame solver that can deal with unsteady conditions and allows variable density to capture acceleration of gases after the "flame".

I appreciate if you have any further input.

If density depends only on the progress variable then it's just like an incompressible problem (i.e. like when density is temperature dependent. forget that your scalar transport is progress variable, it's just like temperature).


Are you sure? The continuity equation is not enough (it's a scalar equation, not a vector equation).

Thank you for your response LuckyTran.

Do you not think that the continuity equation is enough to solve for the new velocity in a 1D case? We are prescribing the inlet velocity and density, from integrating the rate of change of density between the inlet boundary and current location, I think it's possible to calculate what the flux density at the west face of each control volume needs to be.

Perhaps I've overlooked something..

Thank you.