Chapter 1
Brief Outline of the Equations of Fluid Flow
1.1 Introduction
Those who are familiar with compressible fluid flow are aware that the equations of
motion are nonlinear and, as such, it is very difficult to obtain analytical solutions.
As a consequence, numerical methods are generally employed and the differential
equations are approximated by finite difference equations and these in turn are
solved in a stepwise manner. In many examples involving compressible fluid flow
shocks appear and their presence is a complicating factor since they are characterized
by very steep gradients in the variables describing the flow, such as, in the velocity,
density, pressure and temperature. In fact, the gradients become infinitely steep when
the effects of viscosity and thermal conduction are neglected: this introduces discontinuities in the solutions and, as a result, it requires the application of boundary
conditions connecting the values across the shock front but the implementation of
this technique can be quite complex. However, the need for any boundary conditions
can be avoided by using a method proposed in 1950 by Von Neumann and
Richtmyer [1] where an artificially large viscosity is introduced into the numerical
calculations: the present text utilizes this technique. Instead of obtaining a discontinuous solution at the shock front, the shock acquires a thickness comparable to the
spacing of the grid points used in the numerical procedure so that the shock appears
as a near-discontinuity and across which velocity, pressure etc. vary rapidly but
continuously.
A brief review of the fundamental equations of fluid dynamics [2–6] is provided
in this chapter so that the reader can have to-hand the appropriate governing
equations. The one-dimensional form of the equations is presented as they apply
to non-viscous
1
flow, so that any physical effects involving friction and thermal
conduction are neglected. A treatment involving the derivation of the full
1 Although the term “non-viscous” implies the absence of viscosity or friction, it also implies “nonconducting” as well.
© 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_1
1
Brief Outline of the Equations of Fluid Flow
1.1 Introduction
Those who are familiar with compressible fluid flow are aware that the equations of
motion are nonlinear and, as such, it is very difficult to obtain analytical solutions.
As a consequence, numerical methods are generally employed and the differential
equations are approximated by finite difference equations and these in turn are
solved in a stepwise manner. In many examples involving compressible fluid flow
shocks appear and their presence is a complicating factor since they are characterized
by very steep gradients in the variables describing the flow, such as, in the velocity,
density, pressure and temperature. In fact, the gradients become infinitely steep when
the effects of viscosity and thermal conduction are neglected: this introduces discontinuities in the solutions and, as a result, it requires the application of boundary
conditions connecting the values across the shock front but the implementation of
this technique can be quite complex. However, the need for any boundary conditions
can be avoided by using a method proposed in 1950 by Von Neumann and
Richtmyer [1] where an artificially large viscosity is introduced into the numerical
calculations: the present text utilizes this technique. Instead of obtaining a discontinuous solution at the shock front, the shock acquires a thickness comparable to the
spacing of the grid points used in the numerical procedure so that the shock appears
as a near-discontinuity and across which velocity, pressure etc. vary rapidly but
continuously.
A brief review of the fundamental equations of fluid dynamics [2–6] is provided
in this chapter so that the reader can have to-hand the appropriate governing
equations. The one-dimensional form of the equations is presented as they apply
to non-viscous
1
flow, so that any physical effects involving friction and thermal
conduction are neglected. A treatment involving the derivation of the full
1 Although the term “non-viscous” implies the absence of viscosity or friction, it also implies “nonconducting” as well.
© 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_1
1