Shock Waves

, Volume 6, Issue 4, pp 241–248

# Computational analysis of dense gas shock tube flow

• B. M. Argrow
Article

## Abstract

Nonclassical phenomena associated with the classical dynamics of real gases in a conventional shock tube are studied. A TVD predictor-corrector (TVD-MacCormack) scheme with reflective endwall boundary conditions is used for the one-dimensional Euler equations to simulate the evolution of the wave field of a van der Waals gas. Depending upon the initial conditions of the gas, wave fields are produced that contain nonclassical phenomena such as expansion shocks, composite waves, splitting shocks, etc. In addition, the interactions of waves reflected from the endwalls produce both classical and nonclassical phenomena. Wave field evolution is depicted using plots of the flow variables at specific times and withx-t diagrams.

### Key words

Riemann problem Shock tube Dense gas TVD scheme

### Nomenclature

a

nondimensional speed of sound,$$a = {{\bar a} \mathord{\left/ {\vphantom {{\bar a} {\left( {\bar R\bar T_c } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\bar R\bar T_c } \right)}}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}$$

$$\bar b$$

van der Waals covolume parameter,$$\bar b = {{\bar R\bar T_c } \mathord{\left/ {\vphantom {{\bar R\bar T_c } {8\bar p}}} \right. \kern-\nulldelimiterspace} {8\bar p}}_c :\left[ {m^3 \cdot kg^{ - 1} } \right]$$

$$\bar c_\upsilon$$

specific heat at constant volume [J·kg−1·K−1]

e

nondimensional specific internal energy,$$e = {{\left( {\bar e - \bar e_c } \right)} \mathord{\left/ {\vphantom {{\left( {\bar e - \bar e_c } \right)} {\left( {\bar R\bar T_c } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\bar R\bar T_c } \right)}}$$

et

nondimensional specific total energy,e t=e+u2/2

F

flux vector

$$\bar L$$

shock tube length [m]

M

Mach number

n

number of time steps

p

nondimensional pressure,$$p = {{\bar p} \mathord{\left/ {\vphantom {{\bar p} {\bar p_c }}} \right. \kern-\nulldelimiterspace} {\bar p_c }}$$

Q

vector of conserved variables

$$\bar R$$

gas constant [J·kg−1·K −1]

s

nondimensional specific entropy,$$s = {{\left( {\bar s - \bar s_c } \right)} \mathord{\left/ {\vphantom {{\left( {\bar s - \bar s_c } \right)} {\bar R}}} \right. \kern-\nulldelimiterspace} {\bar R}}$$

t

nondimensional time coordinate,$$t = {{\bar L\bar t} \mathord{\left/ {\vphantom {{\bar L\bar t} {\left( {\bar R\bar T_c } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\bar R\bar T_c } \right)}}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}$$

T

nondimensional temperature,$$T = {{\bar T} \mathord{\left/ {\vphantom {{\bar T} {\bar T}}} \right. \kern-\nulldelimiterspace} {\bar T}}_c$$

u

nondimensional velocity,$$u = {{\bar u} \mathord{\left/ {\vphantom {{\bar u} {\left( {\bar R\bar T_c } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\bar R\bar T_c } \right)}}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}$$

x

nondimensional space coordinate,$$x = {{\bar x} \mathord{\left/ {\vphantom {{\bar x} {\bar L}}} \right. \kern-\nulldelimiterspace} {\bar L}}$$

Z

compressibility factor,$$Z_c = \left( {{{\bar p} \mathord{\left/ {\vphantom {{\bar p} {\bar \rho \bar R\bar T}}} \right. \kern-\nulldelimiterspace} {\bar \rho \bar R\bar T}}} \right)_c = {3 \mathord{\left/ {\vphantom {3 8}} \right. \kern-\nulldelimiterspace} 8}$$

### Greek characters

$$\bar \alpha$$

van der Waals force parameter,$$\bar \alpha = {{27\bar R^2 \bar T_c^2 } \mathord{\left/ {\vphantom {{27\bar R^2 \bar T_c^2 } {\left( {64\bar p_c } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {64\bar p_c } \right)}}$$

δ

$${{\bar R} \mathord{\left/ {\vphantom {{\bar R} {\bar c_\nu }}} \right. \kern-\nulldelimiterspace} {\bar c_\nu }}$$

Δt

time step

Δx

space step

Γ

fundamental derivative of gas dynamics

Φ

flux limiter function

λl

eigenvalue of the Euler equations flux-vector Jacobian matrix,l=1,2,3

ν

nondimensional specific volume ν=1/ρ

ρ

nondimensional density,$${{\bar \rho } \mathord{\left/ {\vphantom {{\bar \rho } {\bar \rho _c }}} \right. \kern-\nulldelimiterspace} {\bar \rho _c }}$$

σ

Courant number, σ=vΔtx

v

spectral radius of the Euler equations flux-vector Jacobian matrix

n

temporal index

### subscripts

c

critical point value

i

spatial index

r

reduced variable

s

shock quantity

0

reference value

### operators

dimensional quantity

predictor value

T

transpose

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