Stream Function - Potential Function coordinates
 

I'm trying to study sound propagation due to reflections from a cylinder. Since stream function follows the surface of the cylinder I thought it might be a good idea to use it as a coordinate system. Has anyone tried or can provide me with any references on how to go about it.

-Harish

Re: Stream Function - Potential Function coordinat
 

From what I understand, I recommend that you look at "the Method of Conformal Mapping", described in most basic books on Fluid Mechanics. A good reference is "Fluid Mechanics" by P. Kundu.

Re: Stream Function - Potential Function coordinat
 

Can you explain your problem a little more? Is this sound in a fluid flowing over a cylinder or in a cylinder? If so, what is the Mach number? Is this 2-dimensional or 3?

Re: Stream Function - Potential Function coordinat
 

It is a 2-d case .I'm solving the linearised euler equation in polar coordinates.The sound is generated in the fluid by an initial pulse and i want to study the directivity of sound due to reflection from the surface of the cylinder for different locations of the pulse.For the mean flow i use the potential flow solution for a non lifting cylinder.The mach number of the problem is 0.5.I use the characteristic BC in the far field and reflection BC on the surface of the cylinder.I'm using the fourth order compact scheme in space and 4-th order r-K for time marching. I'm facing the problem of dispersion and it blows up my solution.

-Harish

Re: Stream Function - Potential Function coordinat
 

A few questions out of curiosity -- what is the ratio of the radius of the problem domain to the cylinder radius? What is the ratio of the initial width of the pulse (pulse duration times the sound speed?) to the mesh spacing? Is the pulse source up- or down-stream? How many radii is the source from the cylinder? How bad are reflections from the outer boundary? How quickly does the pulse blow up from dispersion? Could you start the pulse as a (larger) ring, avoiding a localized initial pulse?

Your high order method and integration should reduce dispersion. That leaves the fineness of the mesh and time step size as possible contributors to excessive dispersion.

Re: Stream Function - Potential Function coordinat
 

I forgot... What is the amplitude of the pulse velocity? Small compared to the Mach number?

Re: Stream Function - Potential Function coordinat
 

what is the ratio of the radius of the problem domain to the cylinder radius? The domain is [ r , 5 r ] r being the radius of the cylinder. What is the ratio of the initial width of the pulse (pulse duration times the sound speed?) to the mesh spacing? The courant number is set at 0.1 and I varied it to 0.01 and did not improve my results. Is the pulse source up- or down-stream? How many radii is the source from the cylinder? The pulse is upstream of the cylinder.It is about 2r away from the cylinder. How bad are reflections from the outer boundary? The boundary does not give me any problems. The problems occur at the back centerline of the cylinder. How quickly does the pulse blow up from dispersion? The blowing up is delayed when i increase the size of the pulse but it eventually blows up.

The compact scheme should be working without any problem in this case since the initial condition is smooth. The reason for my problem might be the one sided finite difference scheme on the downstream part of the cylinder.

-Harish

Re: Stream Function - Potential Function coordinat
 

You said the (global) Mach number was 0.5 . The veolcity increases at the surface of the cylinder (for Euler flow). What is the local Mach number there?

You said "The problems occur at the back centerline of the cylinder." I am thinking of a Huygen wavelet construction. It would seem that the wavefront could be swept around the cylinder with the fronts on either side colliding at the back. Does this make sense?

Re: Stream Function - Potential Function coordinat
 

In fact the numerical scheme which I'm using captures that.But what happens is I use a polar grid and i need to use one sided Finite differences at theta=0 and 2 pi which causes errors that grow with time and finally spoils the solution. The fine meshing does not seem to help me overcome that problem.So i'm not sure what else to try.

-Harish