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Wave Amplitudes for NonReflective Boundaries 

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April 18, 2006, 14:23 
Wave Amplitudes for NonReflective Boundaries

#1 
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Hello All:
I'm currently implementing a set of nonreflective boundary conditions following the work of Kevin Thompson ("Time Dependent Boundary Conditions for Hyperbolic Systems")and Poinsot and Lele ("Boundary Conditions for Direct Simulations of Compressible Viscous Flows"). I need to derive the Characteristic Wave Amplitudes, called (cursive) L in both papers, for the generic equation that I'm currently working with. (It's different than the standard generic equation.) The problem is, I can't seem to determine exactly how Thompson derives them. He seems to pull them out of the air. Poinsot and Lele also neglect their derivation in their paper. Have any of you generated these yourself? Can you explain the process you used? This is absolutely critical work for me at the moment, so I'll appreciate any help you can give me. Thanks, Rich 

April 18, 2006, 14:47 
Re: Wave Amplitudes for NonReflective Boundaries

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The BC of Thompson is more like deriving the compatibility equations for any system.Check Tanehill or Laney for more details on how to obtain the compatibility equations.Once we have this equation depending on the eigen values we denote the wave as L or R going and apply the BC.
H 

April 19, 2006, 04:12 
Re: Wave Amplitudes for NonReflective Boundaries

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Hello,everyone,
I am very intersted in this topic.But I cann't find the papers you mentiond. Could you sent me the papers? Kevin Thompson: Timedependent boundary conditions for hyperbolic systems and Poinsot TJ, Lele S K. , Boundary Conditions for Direct Simulations of Compressible Viscous Flows. My email is vivan_lee@163.com. Thanks a lot! Vivian 

April 19, 2006, 12:53 
Re: Wave Amplitudes for NonReflective Boundaries

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Thanks for your reply.
I understand the theory that drives the process and results in using characteristic analysis. In fact, I've taken the process to its one end result and generated the split, upwind and downwind fluxes for my set of governing equations. The problem I'm having is that Thompson doesn't explain clearly how he generates the wave amplitudes that he uses in his paper. He might as well say, "...and then a miracle happened..." because he's left very little trail to follow. Poinsot and Lele simply say that the wave amplitude equations that they use came from Thompson. No other explanation. And unfortunately, those wave amplitudes are the crux of their paper! I've generated the Jacobian matrix for my governing equations, as well as the eigenvalues and left and right eigenvector matrices. I just need to know how to apply them to derive the same wave amplitude equations that Thompson cites. Any ideas? If you prefer not to elaborate, can you give me complete citations for "Tanehill and Laney"? I'm not familiar with them. Thanks again. 

April 19, 2006, 13:57 
Re: Wave Amplitudes for NonReflective Boundaries

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Here are the citations for the papers:
 Thompson, Kevin W., "Time Dependent Boundary Conditions for Hyperbolic Systems", Journal of Computational Physics, Volume 68, Number 1, January 1987, pages 124  Poinsot, T.J. and Lele, S.K., "Boundary Conditions for Direct Simulations of Compressible Viscous Flows", Journal of Computational Physics, Volume 101, Number 1, 1992, pages 104129 

April 19, 2006, 14:19 
Re: Wave Amplitudes for NonReflective Boundaries

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Consider that you have derived your compatibility equations and the eigen values are 1 0 and 1 and the characteristic variables are u+p v and up .
The BC is applied as follows.v is interpolated from the interior.Consier the right boundary .The wave carrying u+p moves out of the domain and the wave carrying up is entering the domain which is impossible due to radiation BC.Hence on the left boundary you calculate only u+p. if we define w1=u+p and w2= up u=w1+w2/2 p=w1w2/2 and since w2 is noty there for this case u=w1/2 p=w1/2 what you do is you solve the compatibility equation for w1 numerically and calculate u and p at the boundaries. H Computational Gas dynamics  Laney Computational Fluid mechanics and heat transfer tanehill 

April 20, 2006, 12:24 
Re: Wave Amplitudes for NonReflective Boundaries

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What Thomson does is fairly straightforward, although demonstrating the math on this forum is not practical. The gist of the approach is to first write the set of equations in terms of derivatives tangent to the boundary and normal to the boundary. Let x be tangent to the boundary and y be normal to the boundary. Assuming no sources, the equation can be written in the form
dq/dt = Adq/dx  Bdq/dy where each of the terms on the RHS can be interpreted as the contribution to the time variation of q at the boundary due to signals in the appropriate directions. Along the tangential direction standard differencing techniques can be used; however normal to the boundary onesided differences can introduce errors if the direction of signal propagation is not considered. To do this a local transformation to characteristic variables is made using the auxiliary equation dq/dty = Bdq/dy = XLX(1)*dq/dy where the ()y denotes the time variation due to normal waves, X is the matrix of eigenvectors of B, L is the diagonal matrix of eigenvalues of B, and X(1) is the inverse of X. Making the assumption that the dq in the auxiliary equation can be related to a set of characteristic variables in the standard fashion results in a set of characteristic equations that are approximately valid in the normal direction. This is where Thomson's L functions come from. The idea then is that for a nonreflective boundary incoming waves contribute nothing to dq/dty. Hope this helps, it's been almost 10 years since I used that approach in my doctoral work. As an alternative to the Poinsot and Lele formulation, you might try simply adding the tangential viscous terms directly and dropping the norma terms at the boundary, once you get Thomson's approach down. 

April 20, 2006, 14:00 
Re: Wave Amplitudes for NonReflective Boundaries

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Yes, that's helpful. Thanks for your reply. It tells me that I do indeed understand what I'm reading. Let me get very specific about my point of confusion:
In one step, Thompson takes the quasilinear generic equation, dU/dt + [A] dU/dx = 0 where [A] is the Jacobian matrix of the system, and multiplies the equation by the left eigenvector matrix, [L] dU/dt + [L][A] dU/dx = 0 This is my sole source of confusion. He offers no rationale for doing this. What's the point here? This equation is the same as: [L] dU/dt + [Lambda][L] dU/dx = 0 where [Lambda] is the eigenvalue matrix. Is this significant? It's also unclear whether the final operation on the second term should be a matrix multiplication, taking the product of [[L][A]] and multiplying by the vector dU/dx or whether each row of dU/dx is simply being multiplied to each row of the product [[L][A]]. Likewise, there's the first term, [L] dU/dt If [L] can't be moved into the differential, then it seems that this is a matrix operation, and EACH of the equations that results will have the term a dU(1)/dt + b dU(2)/dt + c dU(3)/dt, etc. on its lefthand side, making the system nearly unreducible. Or, does this simply indicate that I'm supposed to multiply each row of [L] by its respective dU(i)/dt? Any ideas? 

April 20, 2006, 16:35 
Re: Wave Amplitudes for NonReflective Boundaries

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The rationale is to construct a form that can be used to get a characteristic form in the normal direction. All he's doing is making use of the similarity transformation for the flux Jacobian
[A] = [Linv][D][L], so [L][A] = [D][L]. Premultiplying the portion of the equation related to the normal direction by [L] allows the replacement of the term multiplying the spatial derivative  [L][A]du/dx becomes [D][L]du/dx. The kicker is that the characteristic variables W are defined by the relation dWi/dUj = [L], so that [L]dU/dx = dW/dx and similarly for the time derivative by virtue of the chain rule. You need to make sure you use the proper eigenvalues (right vs. left, etc) in doing these transforms, but in the end you get a diagonalized system. The trick is in making the definition of the characteristic variables Wi. 

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