Chapter 4
Numerical Treatment of Plane Shocks
4.1 Introduction
This chapter describes the numerical procedure that is used to solve a number of fluid
flow problems involving plane shock waves. An alternative form of the differential
equations of fluid flow in Lagrangian form is derived. A dissipative mechanism in
the form of an artificial viscosity is introduced into the equations in order to deal with
shocks. A set of difference equations corresponding to the differential equations is
presented and the numerical solution of these difference equations is obtained. The
numerical results are then compared with the theoretical predictions.
4.2 The Need for Numerical Techniques
An analytic solution of the partial differential equations describing the flow of a
compressible fluid is, in general, a difficult task due to the nonlinearity of the
equations and this difficulty demonstrates the need for more powerful techniques
to solve the equations. Accordingly, it is necessary to turn to numerical methods in
which the differential equations are replaced by finite difference equations that
describe the properties of the fluid at discrete positions and at discrete times. Once
the initial and boundary conditions have been specified, one can follow the changes
that evolve in the properties of the fluid by solving the set of difference equations at
successive time steps (generally known as the time-marching procedure). However,
a solution is severely hampered by the presence of shocks. In the case of non-viscous
flow the shock front is a propagating discontinuity and across which the hydrodynamic functions of pressure, density, temperature and velocity exhibit abrupt
changes. The boundary conditions that connect the values on either side of the
shock are supplied by the Rankine-Hugoniot equations but their application is
complicated as the shock is in motion. In addition, the motion of the shock surface
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
S. Prunty, Introduction to Simple Shock Waves in Air, Shock Wave and High
Pressure Phenomena, https://doi.org/10.1007/978-3-030-63606-7_4
131
Numerical Treatment of Plane Shocks
4.1 Introduction
This chapter describes the numerical procedure that is used to solve a number of fluid
flow problems involving plane shock waves. An alternative form of the differential
equations of fluid flow in Lagrangian form is derived. A dissipative mechanism in
the form of an artificial viscosity is introduced into the equations in order to deal with
shocks. A set of difference equations corresponding to the differential equations is
presented and the numerical solution of these difference equations is obtained. The
numerical results are then compared with the theoretical predictions.
4.2 The Need for Numerical Techniques
An analytic solution of the partial differential equations describing the flow of a
compressible fluid is, in general, a difficult task due to the nonlinearity of the
equations and this difficulty demonstrates the need for more powerful techniques
to solve the equations. Accordingly, it is necessary to turn to numerical methods in
which the differential equations are replaced by finite difference equations that
describe the properties of the fluid at discrete positions and at discrete times. Once
the initial and boundary conditions have been specified, one can follow the changes
that evolve in the properties of the fluid by solving the set of difference equations at
successive time steps (generally known as the time-marching procedure). However,
a solution is severely hampered by the presence of shocks. In the case of non-viscous
flow the shock front is a propagating discontinuity and across which the hydrodynamic functions of pressure, density, temperature and velocity exhibit abrupt
changes. The boundary conditions that connect the values on either side of the
shock are supplied by the Rankine-Hugoniot equations but their application is
complicated as the shock is in motion. In addition, the motion of the shock surface
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
S. Prunty, Introduction to Simple Shock Waves in Air, Shock Wave and High
Pressure Phenomena, https://doi.org/10.1007/978-3-030-63606-7_4
131