Shock Waves

, Volume 28, Issue 2, pp 379–399 | Cite as

Development of a particle method of characteristics (PMOC) for one-dimensional shock waves

Original Article

Abstract

In the present study, a particle method of characteristics is put forward to simulate the evolution of one-dimensional shock waves in barotropic gaseous, closed-conduit, open-channel, and two-phase flows. All these flow phenomena can be described with the same set of governing equations. The proposed scheme is established based on the characteristic equations and formulated by assigning the computational particles to move along the characteristic curves. Both the right- and left-running characteristics are traced and represented by their associated computational particles. It inherits the computational merits from the conventional method of characteristics (MOC) and moving particle method, but without their individual deficiencies. In addition, special particles with dual states deduced to the enforcement of the Rankine–Hugoniot relation are deliberately imposed to emulate the shock structure. Numerical tests are carried out by solving some benchmark problems, and the computational results are compared with available analytical solutions. From the derivation procedure and obtained computational results, it is concluded that the proposed PMOC will be a useful tool to replicate one-dimensional shock waves.

Keywords

Particle method of characteristics (PMOC) One-dimensional shock waves Rankine–Hugoniot relations Shock tube problems 

List of symbols

A

Coefficient matrix

a

Channel area in open-channel flow

b

Flow width in open-channel flow

c

Convection velocity or flow sound speed

Cr

Courant number, \(c{\Delta } x/{\Delta } t\)

F

Flux function in conservation form

f

Generalized fluid density function

h

Flow piezometric head

Np

Total particle number

\(n_\mathrm{C}\)

Initial particle number

p

Pressure

R

Right-side eigenvector matrix

t

Time

U

Conserved solution vector

u

Particle conveying velocity

V

Primitive solution vector

v

Fluid velocity

W

Characteristic variable vector or Riemann invariant vector

w

Flow characteristic

x

Horizontal coordinate

Greek symbols

\(\beta \)

Barotropic coefficient

\({\Delta } t\)

Computational time step

\(\delta \)

Initial particle spacing

\(\delta t\)

Incremental time step to derive Rankine–Hugoniot relation

\(\delta x\)

Incremental spatial step to derive Rankine–Hugoniot relation

\(\varepsilon \)

Solution error

\(\eta \)

Flow depth in open-channel flow

\(\varvec{{\varLambda }}\)

Diagonal matrix

\(\lambda \)

Characteristic transmitting velocity

\(\rho \)

Fluid density

\(\sigma \)

Shock speed

\(\phi \)

Flow driving function

\(\varphi \)

Flow characteristic function

\(\psi \)

Reduced flow characteristic function

Subscripts

B

Boundary

ex

Exact solution

H

Open-channel flow

i

Grid index

in

Inserted particle

p

Particle

L

Left neighboring particle

m

Dividing location for shock tube problems

min

Criterion for particle insertion spacing

R

Right neighboring particle

ref

Reference state in two-phase flow

s

Shock

sL

Left side of a shock

sR

Right side of a shock

0

Reference state for open-channel flow

Superscripts

n

Previous time step

\(*\)

Fictitious or intermediate states

\(+\)

Right-running

Left-running

Notes

Acknowledgements

The author would like to express his gratitude to two anonymous referees for their constructive criticisms and suggestions which were very beneficial in the revision of this article.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Marine EngineeringNational Kaohsiung Marine UniversityKaohsiungTaiwan

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