I'm trying to develop numerical method in reacting flow using PDF method. I am confused with the SIMPLE algorithm. From the mixture fraction PDF I will obtain density, temperature, and reacting scalars. Do I still need to execute SIMPLE algorithm to obtain velocity field or just solve the momentum equation , since I can express the pressure field through state equation?

Remember compressible versus incompressible. In most turbulent combustion applications you use low Mach number assumption, which allows to un-coupled the momentum from the energy equation and therefore use SIMPLE type algorithms (see the Wiki for tons of examples). The pressure you solve with SIMPLE in the momentum equation is NOT the thermodynamic pressure.

If you are dealing with a pure compressible flow (Ma>0.7) then you have to change the numerical method to something like Riemann solvers or so. In this case methods based on scalar PDFs are questionable (they may assume constant pressure internally) and you must check thet the density obtained satisfies continuity.

Hello Salvador,

Could you explain further what you mean by: 'The pressure you solve with SIMPLE in the momentum equation is NOT the thermodynamic pressure'? What is the difference between these two 'pressures' you seem to be referring to?

Furthermore, the say that the density obtained using a PDF transport equation in high Mach No. flows does not necessarily satisify continuity: Does this mean that the PDF equation is not conservative? How can we make the density field continuity satisfying?

I am primarily interested in an answer to the first of my questions.

'reduced' pressure, which contains a 1/3 div(v) inside. This term appears from the stress modelling and dissapears if no variations of density are used. The pressure solved in the momentum equation is the pressure created/needed by moving/to move the fluid. The absolute value of pressure is of no significance in incompressible flow only differences. Pressure is only used to satisfy continuity. In compressible flow you need the continuity equation and and an energy equation as well.

Thre are many PDF type methods, if you are using a flamelet solution Y(Z). This flamlet profiels are obtained a priori assuming certain Pressure (thermodynamic) and thereore giving certain profiles rho(Z). If the pressure used a priori and the actual are not the same the density field will not satisfy continuity.

In many open flames pressure is assumed constant so low Mach number assumptions are valid.

So, is it valid when I calculate the pressure field from the state equation using mixture fraction transport equation (Z transport equation)? Then I calculate the velocity field from the momentum equation using the pressure obtained before. Thank you for the response.

Well the velocit field obtained will not satisfy continuity. Best (low Mach) ---------

1)Compute u,v,w 2) Then P (or pressure correction depends method) 3) Correct velocities u.v,w (now they wil satisfy continuity) 4) Advance Z (and Z") using new u,v,w 5) Compute T(Z),rho(Z) using flamelets

Go to (1) and repeat until steady state or until you want to advance the solution one time step, in the latter yopur new rho will enter in equation 2) torugh the term drho/dt.

If your code is unsteady and have large denity changes ypu may have isntability problems

thanks for the details