Howarth–Dorodnitsyn transformation

In fluid dynamics, Howarth–Dorodnitsyn transformation (or Dorodnitsyn-Howarth transformation) is a density-weighted coordinate transformation, which reduces variable-density flow conservation equations to simpler form (in most cases, to incompressible form). The transformation was first used by Anatoly Dorodnitsyn in 1942 and later by Leslie Howarth in 1948.^{[1][2][3][4][5]} The transformation of ${\displaystyle y}$ coordinate (usually taken as the coordinate normal to the predominant flow direction) to ${\displaystyle \eta }$ is given by

${\displaystyle \eta =\int _{0}^{y}{\frac {\rho }{\rho _{\infty }}}\ dy,}$

where ${\displaystyle \rho }$ is the density and ${\displaystyle \rho _{\infty }}$ 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,^[6][7] a transformation that extends the Howarth–Dorodnitsyn transformation to compressible flows. The transformation reads as^[8]

${\displaystyle \xi =\int _{0}^{x}{\frac {c}{c_{\infty }}}{\frac {p}{p_{\infty }}}\ dx,}$

${\displaystyle \eta =\int _{0}^{y}{\frac {\rho }{\rho _{\infty }}}\ dy,}$

where ${\displaystyle x}$ is the streamwise coordinate, ${\displaystyle y}$ is the normal coordinate, ${\displaystyle c}$ denotes the sound speed and ${\displaystyle p}$ denotes the pressure. For ideal gas, the transformation is defined as

${\displaystyle \xi =\int _{0}^{x}\left({\frac {c}{c_{\infty }}}\right)^{(3\gamma -1)/(\gamma -1)}\ dx,}$

${\displaystyle \eta =\int _{0}^{y}{\frac {\rho }{\rho _{\infty }}}\ dy,}$

where ${\displaystyle \gamma }$ is the specific heat ratio.

References

1. Dorodnitsyn, A. A. (1942). Boundary layer in a compressible gas. Prikl. Mat. Mekh, 6(6), 449-486.
2. Howarth, L. (1948). Concerning the effect of compressibility on laminar boundary layers and their separation. Proc. R. Soc. Lond. A, 194(1036), 16-42.
3. Stewartson, K. (1964). The theory of laminar boundary layers in compressible fluids. Oxford: Clarendon Press.
4. Rosenhead, L. (Ed.). (1963). Laminar boundary layers. Clarendon Press.
5. Lagerstrom, P. A. (1996). Laminar flow theory. Princeton University Press.
6. Stewartson, K. (1949). Correlated incompressible and compressible boundary layers. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 200(1060), 84-100.
7. Illingworth, C. R. (1949). Steady flow in the laminar boundary layer of a gas. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 199(1059), 533-558.
8. N. Curle and HJ Davies: Modern Fluid Dynamics, Vol. 2, Compressible Flow