Howarth–Dorodnitsyn transformation

In fluid dynamics, Howarth–Dorodnitsyn transformation is a density-weighted coordinate transformation, which reduces variable-density flow conservation equations to simpler form. The transformation was first used by Anatoly Dorodnitsyn in 1942 and later by Leslie Howarth in 1948. The transformation of coordinate to is given by
where is the density and is the density at infinity. The transformation is extensively used in boundary layer theory and other gas dynamics problems.

Stewartson–Illingworth transformation

Keith Stewartson and C. R. Illingworth, independently introduced in 1949, a transformation that extends the Howarth–Dorodnitsyn transformation to compressible flows. The transformation reads as
where is the streamwise coordinate, is the normal coordinate, denotes the sound speed and denotes the pressure. For ideal gas, the transformation is defined as
where is the specific heat ratio.