278 lines
4.6 KiB
Plaintext
278 lines
4.6 KiB
Plaintext
#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\begin_header
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\textclass article
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\index Index
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\shortcut idx
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\end_index
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\end_header
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\begin_body
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\begin_layout Title
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1D Euler time domain
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{eqnarray*}
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\frac{\partial\rho}{\partial t}+\frac{\partial m}{\partial x} & = & 0\\
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\frac{\partial m}{\partial t}+\frac{\partial\left(\rho u^{2}+p_{0}+p\right)}{\partial x} & = & 0\\
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\frac{\partial E}{\partial t}+\frac{\partial\left[\left(E+p\right)u\right]}{\partial x} & = & 0
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\end{eqnarray*}
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\end_inset
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\end_layout
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\begin_layout Standard
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with
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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\begin_inset Formula $\rho e=\rho c_{v}T=\frac{c_{v}}{R}\rho R_{s}T=\frac{c_{v}}{R}\left(p_{0}+p\right)=\frac{1}{\gamma-1}\left(p_{0}+p\right)$
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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E=\rho e+\rho\tfrac{1}{2}u^{2}=\frac{p_{0}+p}{\left(\gamma-1\right)}+\rho\tfrac{1}{2}u^{2}
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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If we replace
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\begin_inset Formula $E$
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\end_inset
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with
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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E=\hat{E}+\frac{p_{0}}{\gamma-1}
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\end{equation}
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\end_inset
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then
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\begin_inset Formula
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\[
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\frac{\partial\hat{E}}{\partial t}+\frac{\partial\left(\hat{E}+p_{0}+p\right)u}{\partial x}=0
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\]
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\end_inset
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so
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\begin_inset Formula $E$
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\end_inset
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is internal energy per unit volume!
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\end_layout
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\begin_layout Section
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Scheme
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\end_layout
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\begin_layout Standard
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If we say:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\frac{\partial U}{\partial t}+\frac{\partial}{\partial x}\left(F(U)\right)=0
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\]
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\end_inset
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which is
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\begin_inset Formula
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\begin{equation}
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\frac{U_{i}^{n+1}-\left(\frac{1}{2}\left(U_{i+1}+U_{i-1}^{^{2}}\right)\right)}{\Delta t}+\frac{F(U_{i+1}^{n})-F(U_{i-1}^{n})}{2\Delta x}=0
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\end{equation}
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\end_inset
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in its discrete form
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\end_layout
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\begin_layout Standard
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then
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\begin_inset Formula
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\[
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U=\left\{ \begin{array}{c}
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\rho\\
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\rho u\\
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\rho E
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\end{array}\right\}
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\]
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\end_inset
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and Lax-Friedrichs says for a middle node
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\begin_inset Formula
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\[
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U_{i}^{n+1}=\frac{1}{2}\left(U_{i+1}^{n}+U_{i-1}^{n}\right)-\lambda\left(F\left(U_{i+1}^{n}\right)-F(U_{i-1}^{n})\right)
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\]
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\end_inset
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with
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\lambda=\frac{\Delta t}{2\Delta x}
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\]
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\end_inset
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Total domain length:
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\begin_inset Formula $L$
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\end_inset
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.
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If number of gridpoints is 3, one left, one right, than
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\begin_inset Formula $dx=L/(gp-1$
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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L
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\]
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\end_inset
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\end_layout
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\begin_layout Section
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Right wall bc
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\end_layout
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\begin_layout Standard
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At a wall:
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\begin_inset Formula
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\[
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m=0
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\]
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\end_inset
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and
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\begin_inset Formula
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\[
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\frac{\partial}{\partial x}\left(F(U)\right)\approx\frac{F\left(U_{i}^{n}\right)-F(U_{i-1}^{n})}{\Delta x}
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\]
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\end_inset
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but
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\end_layout
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\begin_layout Section
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Left pressure bc
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\frac{\partial}{\partial x}\left(F(U)\right)\approx\frac{F\left(U_{i+1}^{n}\right)-F(U_{i}^{n})}{\Delta x}
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\]
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\end_inset
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for continuity, energy for momentum:
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\begin_inset Formula
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\[
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\frac{\partial}{\partial x}\left(F(U)\right)\approx\frac{\left[\rho u^{2}+p\right]_{i+1}-(\rho u^{2})|_{i}-P_{pres}}{\Delta x}
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\]
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\end_inset
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\end_layout
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\end_body
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\end_document
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