Dear sir,I want to slove the compressible low speed flow in microchannel.We assume(1)The flow is unsteady(2)The flow is assumed to 2-D flow(3)The pressure in the channel is a function of the channel axis only, p=p(x).I am wonder,u,v,p,density are variables,but just have three equations(continuity,x-momentum,y-momentum equation) and equation of state.How to solve the problem of density of continuity equation,and how to solve these problem,what is the procedure?Please suggest me? Thank you very much.

Best regards.

Nagoo sander

Incompressible but variable density? Unusual.

Perhaps you also need to account for energy conservation to calculate the temperature? With that addition, people will often use the Boussinesq approximation (density is a linear function of temperature) to calculate a variable density. But this is not incompressible.

I'm sorry, probably you make a mistake.What I mean is compressible flow in microchannel.What is the procedure?Please suggest me? Thank you very much.

Best regards.

sander Nagoo

Sorry myself! You clearly say 'compressible'!

The equation of state provides a relationship between pressure and density. But the EOS for a compressible fluid normally contains a dependence on the fluid temperature. In the general case, the energy equation is coupled with the momentum, continuity and EOS; they must be solved simultaneously.

Sometimes this can be avoided by assuming isothermal conditions or some simple algebraic relationship between temperature and density. In that case, either the pressure or density can be removed from the problem.

Unless the microchannel allows such a simplification, you'll need to use a 'general' compressible CFD method.

Thank you for your help.May I have one more question?If I have four variables u,v,p,density, four equations(x-momentum,y-momentum,continuity equation and equation of state).We will solve for primitive variables p,u,v,rho.If I want to solve pressure,which equation I should use?(continuity equation? or not)what is the procedure?Please suggest me? Thank you very much.

Best regards.

sander Nagoo

Hi sander Nagoo,

As suggested by Jim Park, you actually have another equation - the energy transport - which, in compressible flow, is coupled with the other transport equations (i.e., mass and two momentum components) through the EOS (equation of state).

The most naive solution would then use density, velocity components and enthalpy (or energy or temperature, etc) as the solved for variables. For low Mach numbers, however, this approach often leads to numerical difficulties. A common remedy would be to use pressure rather than density as an independent variable, and apply one of the SIMPLE scheme variants. You may find details in

Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere.

I hope this helps,

Rami

Thank you for your help.Thank you.

I would be using the continuity equation myself.

The method developed at Los Alamos in the mid 60's (ICE - Implicit Compressible Eulerian) is a time-marching technique. The momentum equations are tentatively solved with an explicit step. The continuity equation is converted into a Poisson equation for the pressure correction. After the Poisson equation is solved, the velocities are updated. Those are used in the energy conservation law to update the temperatures. This method is a bit fussy about where the equation of state is introduced, so do check the published work. Look in the JCP in the mid-sixties. Look for Harlow, Amsden. A modern version was published at the end of the 70's, also at Los Alamos. Look in JCP for Cloutman, Hirt.

But the suggestion below about using Patakar's SIMPLE method or variants has merit; that is a direct steady state technique. Once you figure out how to fiddle with all of the relaxation factors, you have a steady state solution without messing with the intermediate time history.

Thank you for your help.May I have one more question.I use CFD software FEMLAB(finite element method) to solve this problem.We assume the flow is steady and isothermal(dont't need solve energy equation? or need).So we will solve for the primitive variables p,u,v.First I substituting for density by using the EOS(rho=p/RT). governing eqn become:

(px*u+p*ux+py*v+p*vy)/(RT)=0 (continuity)

(px*u^2+p*2u*ux+py*u*v+p*uy*v+p*u*vy)/(RT)= 1/Re*(uxx+uyy)-px+... (x-mom)

(px*u*v+p*ux*v+p*u*vx+py*v^2+p*2v*vy)/(RT)= 1/Re*(vxx+vyy)-py+... (y-mom)

We use x and y-mom to solve u and v,use continuity equation to solve pressure,but appear the following messages:"Inf or NaN repeatedly found in solution".Please suggest me? Thank you very much.

Best regards.

sander Nagoo

I'm sorry, but I don't use the FE method.

However, in the FD world, one must be careful that the pressure (Poisson) equation is not singular. That is, the boundary conditions need to be applied according to BC's on the velocities.

One of the tipoffs that the pressure equation is singular is the appearance of underflows, overflows, and NaN's, which are generated by singular coefficient matrices.

Thank you for your help.Thank you very much.