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ME433 (2020SS) Project I - 1D Unsteady Compressible

Navier-Stokes

Due 5pm on April 27, 2020

1 Shock Tube or Riemann Problem

The shock tube problem constitutes a particularly useful test case, as it simulates a shock wave, a contact

discontinuity and an expansion wave region. Initially, a diaphragm in a long 1D tube separates two initial gas

states (both air) at different pressures and densities, but both stationary (u = 0). At t = 0, the diaphragm breaks

down suddenly, creating three types of waves. (1) A shock wave moving to the right with speed W , across

which all variables jump in value, (2) an expansion wave region moving to the left, across which all variables

change in value continuously, and (3) a contact discontinuity (moving with the flow to the right), across which

u and P are continuous, but other variables experience a jump. These waves lead to 4 regions in the tube.

Region 1: Undisturbed portion of driven section, with original values of u1(= 0), P1, ρ1, T1

Region 2: Processed by shock wave propagating through it, P2 > P1, u2 > 0, ρ2 > ρ1.

Region 3:Processed by expansion waves (isentropic waves), P3 = P2, u3 = u2, s3 6= s2; thus T3 6= T2,

ρ3 6= ρ2.

Region 4: Undisturbed protion of driver section, with original P4, ρ4, T4, u4(= 0).

For further descriptions and derivation of analytical solution for inviscid flow, see Charles, Hirsch,1990,

Numerical computation of internal and external flows, Chapter 16.

Tasks:

1. Compare inviscid (analytical) results (µ = 0, K = 0) with the viscous (numerical) results.

2. Effects of grid resolution: Nx = 21, 101, 1001.

3. Compare 2nd-order (in space) with 4th-order scheme.

5 Report requirement

1. Calculate P , ρ, u, T and plot them vs. x at different times.

2. Plot/Compare numerical (viscous) and analytical (inviscid) results in the same figure.

3. Attach the main part of your program to the report.

4. Submit an electronic copy of your report before/on due date.

5. Report must have: Introduction, Formulation/Numerical Procedure, Results/Discussions, Conclusions,

Appendix (code, ...)

Note: Undergraduate students can work in pairs. Sign both names.

3

6 Reference: Analytical solution for inviscid flow

Regions 1 and 4: values of P , T , ρ, u etc. are the same as the initial conditions.

Region 2: PR = P2/P1 can be calculated (given here as a known value). Obtained the values of other

variables based on PR:

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