Howarth–Dorodnitsyn transformation explained
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
coordinate (usually taken as the coordinate normal to the predominant flow direction) 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,[6] [7] a transformation that extends the Howarth–Dorodnitsyn transformation to compressible flows. The transformation reads as[8]
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
\xi=
\right)(3\gamma-1)/(\gamma-1) dx,
where
is the
specific heat ratio.
Notes and References
- Dorodnitsyn, A. A. (1942). Boundary layer in a compressible gas. Prikl. Mat. Mekh, 6(6), 449-486.
- Howarth, L. (1948). Concerning the effect of compressibility on laminar boundary layers and their separation. Proc. R. Soc. Lond. A, 194(1036), 16-42.
- Stewartson, K. (1964). The theory of laminar boundary layers in compressible fluids. Oxford: Clarendon Press.
- Rosenhead, L. (Ed.). (1963). Laminar boundary layers. Clarendon Press.
- Lagerstrom, P. A. (1996). Laminar flow theory. Princeton University Press.
- 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.
- 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.
- N. Curle and HJ Davies: Modern Fluid Dynamics, Vol. 2, Compressible Flow