matlab code for

This highly compressible transonic unsteady flow is produced in the long one$-$dimensional tube by sudden breakdown of the diaphragm, separating two initial gas states at different pressures and densities.

The viscous effects are neglected, and it is assumed that there are no shock wave reflections at the tube ends $($long tube assumption$).$

After the bursting of the diaphragm at t$=0,$ the pressure discontinuity $($shock wave$)$ propagates to the right in the low$-$pressure gas and simultaneously an expansion fan propagates to the left in the high$-$pressure gas $($see Figure$).$ The $1-$D Euler equations describe the flow.

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The $1-$D Euler equations describe the flow

The initial conditions at t$=0$ are given by

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The computational domain is

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The computational domain is

The boundary conditions at x$=0$ and x$=2$xo may be set equal to the initial conditions in undisturbed gas $($Left or Right state, respectively$).$

Consider two cases in this assignment:

Case $1$: PR$=1\mathrm{.}2$ x $104$ Pa$,$ R$=0\mathrm{.}125$ kg$/$m$3,$ tf$=0\mathrm{.}005$ s

Assume PL$=105$ Pa $($atmospheric pressure$),$ L$=1\mathrm{.}0$ kg$/$m$3,$ velocities UL$=$UR$=0,$ and x$0=5$m

$1.$Model the $1-$D flow from t$=0$ to t$=$tf $$using the MacCormack two$-$step predictor$-$corrector scheme $($see Textbook, $3$rd edition, p $119$; $4$th edition pp$.157)$ for $201$ grid nodes per computational domain $($that$$ is$,200$ intervals$).$

Note: To choose the $$right$$ time step, take care about stability condition!

$2.$ Obtain numerical solution for N$=401$ and N$=801$ grid nodes per computational domain. Present computed pressure distribution p$=$p$($x$)$ for both numerical methods and all three numerical grids.

$3.$ Present velocity distribution u$($x$)$ and temperature distribution T$($x$)$ for the finest grid.

$4.$ Report about possible oscillations in the solution for Case $2$:

$$Case $2$: PR$=1\mathrm{.}2$ x $103$ Pa$,$ R$=0\mathrm{.}01$ kg$/$m$3,$ tf$=0\mathrm{.}003$ s

For both cases $1$ and $2$ PL$=105$ Pa $($atmospheric pressure$),$ L$=1\mathrm{.}0$ kg$/$m$3,$

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Notes:

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The stability requirement that the Courant number needs to be less than one? yes. It is the same as Lax$-$Wendroff, see Textbook $3$rd edition, p$.118$ or $4$th edition$-$p$.157$ and lecture notes.

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Take maximum Courant equal to $0\mathrm{.}8$ for safety.

The speed of sound is c$=$sqrt $($kRT$),$ where k$=1\mathrm{.}4$ R gas constant for air, T static temperature. See my lecture notes Lecture$19$CFD Methods for Compressible Flow and Shock Wave$$ for details.