We assume a steady, two-dimensional (σ = 0) or axisymmetric (σ = 1) flow of a perfect gas that contains a shock wave. In addition, no sweep or swirl and a uniform upstream flow are assumed. A Cartesian coordinate system initially is utilized, as sketched in Figure 3.1, where x1 is aligned with the uniform freestream velocity. It is convenient to introduce a transverse radial position vector:

R x x|ˆ |ˆ2 2 3 3

= + σ (3.1a)

where

R x x R x

x R

R x

x R

, ,2 2

3( )= + σ ∂∂ = ∂∂ = σ (3.1b) and its normalized form is

R R

x R

x R

ˆ |ˆ |ˆR 2

3ε = = + σ (3.1c)

The constant freestream velocity is given by Equation (2.29). For the derivative analysis, a known shock shape is presumed. Of course, from

a computational fluid dynamics (CFD) point of view, the shock’s location is generally not known but must be found. For our purposes, however, the assumption is warranted, and the resulting jump and derivative relations hold, whether or not the shock’s location is actually known.