Physics: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 [math]\displaystyle{ y }[/math] coordinate (usually taken as the coordinate normal to the predominant flow direction) to [math]\displaystyle{ \eta }[/math] is given by
- [math]\displaystyle{ \eta = \int_0^y \frac{\rho}{\rho_\infty} \ dy, }[/math]
where [math]\displaystyle{ \rho }[/math] is the density and [math]\displaystyle{ \rho_\infty }[/math] 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]
- [math]\displaystyle{ \xi = \int_0^x \frac{c}{c_\infty}\frac{p}{p_\infty} \ dx, }[/math]
- [math]\displaystyle{ \eta = \int_0^y \frac{\rho}{\rho_\infty} \ dy, }[/math]
where [math]\displaystyle{ x }[/math] is the streamwise coordinate, [math]\displaystyle{ y }[/math] is the normal coordinate, [math]\displaystyle{ c }[/math] denotes the sound speed and [math]\displaystyle{ p }[/math] denotes the pressure. For ideal gas, the transformation is defined as
- [math]\displaystyle{ \xi = \int_0^x \left(\frac{c}{c_\infty}\right)^{(3\gamma-1)/(\gamma-1)} \ dx, }[/math]
- [math]\displaystyle{ \eta = \int_0^y \frac{\rho}{\rho_\infty} \ dy, }[/math]
where [math]\displaystyle{ \gamma }[/math] is the specific heat ratio.
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
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