Non-reflecting BC's
 

Hello, I have a question regarding "non-reflecting boundary conditions". Has anybody ever tried implementing Gile's non-reflecting BC's? -Girish

Re: Non-reflecting BC's
 

I have not heard about it. Will you please give me some references on Gile's BCs? Are you dealing with incompressible flow?

GS

Re: Non-reflecting BC's
 

Non reflecting BC is usually used for solving unsteady compressible flows in turbomachinery cascades. Gile's BC is based upon the characterastic BC for perturbations in Unsteady flows. I was looking for the right way of implementation of these BCs. GB

Re: Non-reflecting BC's
 

Does the Gile method consist of linearizing the equations and making a first order perturbation:

e.g. f=f0+f1

where f0 is the steady solution (or an approximation to it) and f1 is the perturbation.

By solving the NS equations like that at the boundaries, a matrix equations (for the velocities, density and pressure) is written and then diagonalized to find the eigenvectors (characteristics) and eigenvalues (propagation velocities of the characteristics) of the system.

From the eigenvalues one can find which characteristic is incoming and/or outgoing at each boundary. Incoming characteristics are given value from outside (the actual boundary conditions; these can be set to zero sometimes) while outgoing characteristics take values extrapolated from inside the computational domain through the boundary. Then the primitive variables (velocties, pressure and density) are obtained from the values of the characteristics.

Is that the method you are refering to? If it is the case I can help you and give you some references and/or answer some of your specific questions.

Cheers, Patrick

Re: Non-reflecting BC's
 

Thanks Parick..

This is exactly I am refering to. I know the theory behind it and you are absolutely right in putting it in the words. But my question is regarding the implimentation. Lets take a case of Inflow BC.

1.Do we need to take in to account the ghost cell,since we have to solve the NS equation on inflow grid points also?

2.If yes, then what should be the ghost cell values?

3.If no, then how will you keep the incoming perturbations fixed at inflow?

Same questions are applicable for outflow BC. I have the references by Giles and by Hall K.C. Do you have any other references to add?

Thanks.

Girish

Re: Non-reflecting BC's
 

If you use a second order finite difference (or element) scheme, then at each points in the computational domain, in order to solve the NS eq, you need to know the neighbouring points. This holds up to the boundary points. At the boundary points you use the method of charactersitics to solve for the variables. So inside the domain you need to know only up to the boundary points, while at the boundary you dont need to know the ghost points, since the method you are using is different (though you might need to assume a value for the ghost points, this is usualy pretty straightforward).

The best is to take a concrete example.

Assume that at the boundary you have (for simplicity only) 3 characteristics, say CI, CII, and CIII, (all the C are function of rho, P and v) propagating at velocity v+c, v-c and v respectively (where v is the flow speed and c is the sound speed). Assume also that the boundary at which you are is such that if v is posititve you have an outflow and if v is negative you have an inflow. Assume that the speed in the flow is subsonic, such that v is always less than c (all the following treatment can be done for more general cases, etc..).

Then at this boundary you have

CI is always outgoing CII is alway incoming CIII is outgoing if v is positive and incoming if v is negative.

Then you assume that outside the boundary the velocities, pressure and density are given (say close to some kind of steady state). Inside the boundary these variable are computed and their values can be (wisely) extrapolated to the boundary itself.

Then to CI you assign the extrapolated values (calculated from the extrapolated values of rho, P and v), to CII you assign the values given outside (calculated from the outside values given to rho, P and v; these are the physical boundary conditions that you wish to impose) and CIII will take extrapolated values when v (at the boundary) is positive and given values from outside when v (at the boundary) is negative.

CI, CII, CIII are functions of rho, P and v and rho, P and v (al at the boundary) are then obtained through the values of the Cs.

That's how you get the values of the (primitive) variables (rho, P, v) at the boundary.

Patrick

Re: Non-reflecting BC's
 

Thanks Patrick... I got your well defind explaination. But still, can you put a little focus on a problem such as an Incoming Vortical gust? I mean to say that, we have a sinosoidal variation in incoming U velocity with certain magnitude. There is no perturbation in density or V velocity. That is, rho_pert=0, v_pert=0, u_pert=u_mag*cos(theta), where 'theta' varies along the y direction. Now how should I go about putting these conditions in Non Reflecting BC? Thanks for your views.

GB

Re: Non-reflecting BC's
 

The sinusoidal U and the non-perturbed V and rho are actually what I called the physical boundary conditions. THese are the exact values given outside the boundary and have to be imposed on the incoming characteristic(s) at the boundary. WHile the outgoing characteristic(s) will take value extrapolated from the computational domain. Then U, V and rho solutions that you are looking for are obtained by explicitly writing U,V and rho as a function of the incoming and outgoing characteristics.

When you check to see which characteristic is incoming and which is outgoing (by looking at v+c, v-c and v) you need to consider v and c computed, obtained from the previous time step at the boundary by the above characteristic method.

Have a look at the paper

Godon and Shaviv, 1993, Computer Methods in Applied Mechanics and Engineering, volume 110, pages 171-194 (and the references therein). The equations are first linearized and then written for the perturbation (no matter if outside you chose zero perturbation) and solved. Then you can replace the perturbation by the difference between the solution and the state you linearized about (say an approximation to the steady state: v=v0+deltav; then deltav=v-v0).

If this does not help, then have a look at the Revie paper of Givoli (you can find there different way of implementing non-reflective and transmitting boundary conditions)

Givoli, J.Comput.PHys, 1991,vol.94, page 1.

It would be very difficult to try to explain more just by writting messages here. So I hope this helps.

Patrick

Re: Non-reflecting BC's
 

Thanks a lot Patrick.

GB