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February 1, 2001, 17:18 
eigenvalues in propagation problems

#1 
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Dear sir,
I read from one book that the eigenvalues of NavierStokes equations are equal to "u+a", "u", and "ua", where u is the flow velocity, and a is the sound speed. What is the physical meaning of eigenvalues? Refer to the topic about speed of propagation of information along the characteristic curves, I think that the mass, momentum and energy are convected with the flow at speed equals "u". Then what information is propagated at speed equal "u+a" and "ua"? Thank you very much sir. Best regards, Atit Koonsrisuk 

February 2, 2001, 03:06 
Re: eigenvalues in propagation problems

#2 
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Dear Atit Koonsrisuk,
u+a and ua are the reciprocal of the slopes of left and right running characteristics. They CARRY information about PRESSURE AND VELOCITY. Hence, if you are using backword marching method, you can get pressure and velocity at next time step by solving the left and right running characteristics simultaneously. u is the reciprocal of slope of pathline and it carries information about density. Density can be obtained by using the calculated pressure and velocity valus and pathline. vijay pargaonkar 

February 2, 2001, 13:18 
Re: eigenvalues in propagation problems

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Hi Atit,
In the flow there are quantities that are (as you correctly said) advected at a velocity u (the velocity of the flow). However, there are other means of propagation in addition to advection. Advection just means that the flow is taking them with it as it moves. For example pressure changes propagates at the sound speed, denoted by 'a' in your message. Now the sound waves themselves can propagate in the same direction as the flow or in the opposite direction. For an observator in the inertial frame of reference (in which the computational domain is fixed and in which the equation are written) the velocity of propagation can therefore be u+a or ua, and for quantities that do not propagate at the sound speed it is simply a. The quantities that propagate at these velocities (eigenvalue) are the eigenvectors of the flow equations. In one dimension these are called the Riemann invariants, in more than one dimension, there are actually no Riemann invariants. The eigenvectors are call the Characteristics of the flow. THe flow equations are usually written in the 'primitive' form, so that the variables (density, velocity, energy) are actually what we call the primitive variables. However, the system of equations can be transformed, so that new variables are obtained, and these new variables are then in their 'natural' form, these are the characteristics of the flow. They are the actual physical quantities that really propagate in the flow. While the primitive variables are variables that we can measure in the Lab (temperature, pressure, density, velocities, etc...). They are an 'artifact' of our ability to quantify the flow. A perturbation in the pressure will propagate in the flow at the sound speed and will be advected in the flow at the speed flow. So all in all a pressure change can propagate at speeds u+a ('blueshift') or ua ('redshift'). This is a quantity (information) that is moving at the characteristic speed. Cheers, Patrick Godon 

February 2, 2001, 17:44 
Re: eigenvalues in propagation problems

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Dear sir, I'm sorry, but I need more explaination as follows: 1.As Patrick said "the velocity of propagation can therefore be u+a or ua, and for quantities that do not propagate at the sound speed it is simply a", I wonder why does the last eigenvalue is "u", not "a"? 2.What are the physical meaning of "eigenvectors" of the flow equations? 3.As Patrick said about the new variables, do you mean "(rho)(u)", "(rho)(v)", "(rho)(w)", and "(rho)(e)"? 4.As Patrick said, what are "blueshift" and "redshift"? 5.Actually I ask the question about eigenvalues because I wonder about the stiff system. I read from the book that in the low speed flow, that a>>u, the system is stiff. I know what the stiff system is, but I don't understand why does it occur in the very low speed flow?
Thank you very much sir. Atit Koonsrisuk 

February 5, 2001, 14:35 
Re: eigenvalues in propagation problems

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IF the sound speed is extremely high, then your time step is reduced due to the Courant (CFL) condition and you can advance only very slowly as you intergrate explicitly. The flow will advance at a speed u much smaller than the sound speed a. This is stiff.
The new variables are not rho*v etc.. (that's momentum) The new variables can be for example a combination of the velocity, the pressure and the density. See for example the the Rieman invariants of a one dimensional (polytropic) flow in the book of Landau and Lifshitz, Hydrodynamics (Pergamon Press, reprinted and translated from the Russion Gidrodinamika). For treatment of charateristics in 2D you will need to visit the library and work a little more, 'cause it is not from 'chating' here that you can get all the formalism. So check for example the charateristic equation in these papers: Abarbanel et al., 1991, Journal of Fluid Mechanics, volume 225, page 557 Givoli, 1991, Journal of Computational Physics, volume 94, page 1 (see the references in this review paper). In these papers, the characteristic equations are used to treat boundary conditions. See also the paper on the matrix equations and its linearization to obtain the equivalent of the Riemann invariants in 2D: Abarbanel and Gottlieb, 1981, Journal of Computational PHysics, vol. 41, p. 1. ("Optimal time splitting for 2 and 3 dimensional Navier Stokes Equations with Mixed derivatives"). Forget about blueshift and redshift, this is just by analogy of what would happen if you drive fast towards an ambulance, then the tone of the siren is higher, while if you drive away from it, it is lower (due to the same effect of addition or substration of the sound speed and the velocity of the flow relatively to you). Cheers, Patrick Godon 

February 21, 2001, 13:34 
Re: eigenvalues in propagation problems

#6 
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What you talked about is applicable to compressible flow, not to incompressible flow, right?
Is this discussion of eigenvalues also applicable to geophysical flow? Thank you! 

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