Jacobian Coefficient matrices for the viscous flux
 

Hi there,

does anyone know of a CFD book where the Jacobian coefficient matrices for the viscous fluxes are derived explicitly? Hirsch & Anderson only deal with the inviscid fluxes.

Regards

Andy

Re: Jacobian Coefficient matrices for the viscous
 

What's the problem - take a pencil into Your hand and a scratch of paper and go ahead - I made that myself many times and checked with Maple V - very useful thing in such affairs

Re: Jacobian Coefficient matrices for the viscous
 

Obviously, I have tried to do it by hand myself but have run into some difficulties, specifically, with the elements of the coefficient matrices that come from the energy equation.

The viscous fluxes can be written in the form:

df/dx+dg/dy+dh/dz=0 (eqn 1)

where the d's here are partial derivatives.

The idea is then to introduce the primitive variables v via the chain rule such that

(df/dv).(dv/dx)+(dg/dv).(dv/dy)+(dh/dv).(dv/dz)=0

where the jacobian matrices are

A=df/dv, B=dg/dv, C=dh/dv.

For the continuity and momentum equations it is fairly easy to write them in the form of (eqn 1) above (in order to obtain expressions for f, g and h) and then to differentiate f, g and h with respect to the primitive variables to obtain the corresponding elements of the jacobian coefficient matrices.

But, for the energy equation, it is not so straightforward and it is here that I am having problems because the energy equation contains terms such as udv/dx, vdw/dz etc. which cannot to my knowledge be written in the form of (eqn 1) above i.e. udv/dx cannot be written in the form d/dx[]. Therefore, I do not know what the energy elements of f, g and h are and therefore I cannot differentiate them with respect to the primitive variables and so cannot obtain the energy elements of the jacobian coefficient matrices.

Any useful comments would be very much appreciated.

Kind Regards

Andy

Re: Jacobian Coefficient matrices for the viscous
 

Which numerical method are you using? Finite element, finite difference, finite volume, or other method? When I use finite element method, I write the equations in weak form first, and then do the derivatives of the weak form with respect to unknowns. The derivatives are complicated, it might take you some time. We have a whole 1" binder to store all the math derivatives.

Re: Jacobian Coefficient matrices for the viscous
 

I am using the finite volume method.

Regards

Andy

Re: Jacobian Coefficient matrices for the viscous
 

Re: Jacobian Coefficient matrices for the viscous
 

When deriving visous Jacobian matrix, remeber two things: (1) an entry of the resulting Jacobian matrix may be a derivative operator, not purely a number; (2) assume the transport coefficent such as thermal conductivity coef. as constant. Let (p,q) be the primitive variable, for f= u * dv/dx, rewrite it as u(p,q)* d v(p,q)/dx. Now, d f / d(p,q) *(Dp,Dq) -->: (du/dp *dv/dx) Dp+ (du/dq *dv/dx )/dp *Dq + (u d [dv/dp] /dx )Dp +(u d [dv/dq] /dx ) Dq. The terms in small bracket give the element of viscous Jacobian. Dp, Dq is increment of p, q.

Re: Jacobian Coefficient matrices for the viscous
 

You can find the viscous jacobian in

''Computational fluid Dynamics for Engineers'' from K.A. Hoffmann

Re: Jacobian Coefficient matrices for the viscous
 

For perfect gas, Beam and Warming has derived a detailed formula in AIAA Journal vol.16, No.4 1978; For chemical reaction gas, you can also derive the counterpart by yourself under their principles. Good luck!