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Decomposing gradients into wave strengths

Consider the 2D hydro linearization of the wave equation in each dimension

\partial_t W_i = A_{ij} \partial_x W_j

where A_{ij} = \frac{\partial F_i}{\partial W_j}

The matrix A has eigenvalues u-a, u, u, u+a corresponding to left propagating sound wave, contact, shear discontinuity, and right moving sound wave. And we can construct the matrix of left eigenvalues L (where each row is a left eigenvector)

0	-\frac{\rho}{2a}	0	\frac{1}{2a^2}
1	0	0	-\frac{1}{a^2}
0	0	1	0
0	\frac{\rho}{2a}	0	\frac{1}{2a^2}

and the matrix of right eigenvectors R (where each column is a right eigenvector)

1	1	0	1
-\frac{a}{\rho}	0	0	\frac{a}{\rho}
0	0	1	0
a^2	0	0	a^2

and the diagonal matrix of eigenvalues \Lambda where each diagonal term is an eigenvalue

u-a
	u
		u
			u+a

The eigenvectors are orthogonal

\mathbf{LR} = \mathbf{I}

and we also have

\mathbf{L{^-1}R^{-1}} = \mathbf{I}

\mathbf{RL} = \mathbf{I}

\mathbf{AR=R \Lambda}

\mathbf{LA = \Lambda L}

which combining gives

\mathbf{L A R = \Lambda}

\mathbf{A = L^{-1} \Lambda R^{-1}}

u	\rho	0	0
0	u	0	\frac{1}{\rho}
0	0	u	0
0	\rho a^2	0	u

Now since the eigenvectors are orthonormal, we can multiply any left eigenvector by an arbitrary normalization constant, provided we divide the corresponding right eigenvector by the same amount.

Rewriting the equation

\partial_t \mathbf{W}=\mathbf{A}\partial_x\mathbf{W}

\partial_t \mathbf{W}=\mathbf{RLA}\partial_x\mathbf{W}

\partial_t \mathbf{W}=\mathbf{R\Lambda L}\partial_x\mathbf{W}

\partial_t \mathbf{W}=\sum \lambda_i \mathbf{r_i} \mathbf{l_i^T} \partial_x \mathbf{W}

Now if we renormalize the left eigenvectors

0	-\frac{1}{2a}	0	\frac{1}{2 \rho a^2}
\frac{1}{\rho}	0	0	-\frac{1}{\rho a^2}
0	0	1/a	0
0	\frac{1}{2a}	0	\frac{1}{2\rho a^2}

and the right eigenvectors

\rho	\rho	0	\rho
-{a}	0	0	a
0	0	a	0
\rho a^2	0	0	\rho a^2

then

\partial_t \mathbf{W}=\sum \alpha_i \lambda_i \mathbf{r_i}

where

\alpha_i=\mathbf{l_i^T} \partial_x \mathbf{W}

has units of \frac{1}{dx}

This gives us the relative strength of each wave at each cell. This looks at gradients in the fluid variables and how they correlate with the various types of waves.

For example, if we look at the Sod Shock Tube test problem whose solution entails a left rarefaction, contact, and right moving shock, we see that the 3 different waves are correlated with the amplitude of the projections of the gradients onto the left eigenvectors associated with each wave.

And if we look at the Brio Wu shock tube which consists of a fast rarefaction, a slow compound wave, a contact, a slow shock and a fast rarefaction, we see the following wave strengths

Posted: 9 years ago (Updated: 9 years ago)
Author: Jonathan
Categories: (none)

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Screen Shot 2015-11-04 at 4.08.01 PM.png (177.7 KB) - added by Jonathan 9 years ago.
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waves0010.png (3.3 KB) - added by Jonathan 9 years ago.

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Decomposing gradients into wave strengths

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