Janzen–Rayleigh expansion explained

In fluid dynamics, Janzen–Rayleigh expansion represents a regular perturbation expansion using the relevant mach number as the small parameter of expansion for the velocity field that possess slight compressibility effects. The expansion was first studied by O. Janzen in 1913^[1] and Lord Rayleigh in 1916.^[2]

Steady potential flow

Consider a steady potential flow that is characterized by the velocity potential

\varphi( x). Then \varphi satisfies(c ^{2 )\varphi} _{xx} + (c ^{2 )\varphi} _{yy} + (c ^{2 )\varphi} _{zz} -2 (\varphi _{x\varphi} _{y\varphi} _{xy} + \varphi _{y\varphi} _{z\varphi} _{yz} + \varphi _{z\varphi} _{x\phi} _{zx} ) =0

where

c = c(v ^{2)} , the sound speed is expressed as a function of the velocity magnitude v ^{2=(\nabla} \varphi) ^{2.} For a polytropic gas, we can writec ^{2} = c ^{2} _{0} - (\gamma -1)/(2) v ^{2}

where

\gamma is the specific heat ratio, c ^{2} _{0} = h _{0(\gamma-1)/2} is the stagnation sound speed (i.e., the sound speed in a gas at rest) and h _{0} is the stagnation enthalpy. Let U be the characteristic velocity scale and c _{0} is the characteristic value of the sound speed, then the function c(v ^{2)} is of the form(c ^{2})/(U ^{2}) = (1)/(M ^{2}) - (\gamma -1)/(2) (v ^{2})/(U ^{2}) .

where

M = U/c _{0} is the relevant Mach number.

For small Mach numbers, we can introduce the series^[3]

\varphi = U (\varphi _{0} + M ^{2} \varphi _{1} + M ^{4} \varphi _{2} + … )

Substituting this governing equation and collecting terms of different orders of

Ma leads to a set of equations. These are\begin{align} \nabla ^{2 \varphi} _{0} & = 0 ,\\ \nabla ^{2 \varphi} _{1} & = \varphi ^{2 \varphi} _{0 ,xx} + \varphi ^{2 \varphi} _{0 ,yy} + \varphi ^{2 \varphi} _{0 ,zz} +2 (\varphi _{0,x} \varphi _{0,y} \varphi _{0,xy} + \varphi _{0,y} \varphi _{0,z} \varphi _{0,yz} + \varphi _{0,z} \varphi _{0,x} \phi _{0,zx} ), \end{align}

and so on. Note that

\varphi _{1} is independent of \gamma with which the latter quantity appears in the problem for \varphi _{2} .

Imai–Lamla method

A simple method for finding the particular integral for

\varphi _{1} in two dimensions was devised by Isao Imai and Ernst Lamla.^{[4] [5] [6]} In two dimensions, the problem can be handled using complex analysis by introducing the complex potential f(z,\overline z) = \varphi + i\psi formally regarded as the function of z = x + iy and its conjugate \overline z = x - iy ; here \psi is the stream function, defined such thatu = (\rho _{infty})/(\rho) (\partial\psi)/(= \partial y) (\partial\varphi)/(\partial x) , v =- (\rho _{infty})/(\rho) (\partial\psi)/(= \partial x) (\partial\varphi)/(\partial y)

where

\rho _{infty} is some reference value for the density. The perturbation series of f is given byf(z,\overline z) = U[f _{0(z)} + M ^{2} f _{1(z,\overline} z) + … ]

where

f _{0=f} _{0(z)} is an analytic function since \varphi _{0} and \psi _{0} , being solutions of the Laplace equation, are harmonic functions. The integral for the first-order problem leads to the Imai–Lamla formula^{[7] [8]} f _{1(z,\overline} z) = (1)/(4) (df _{0})/(\overline{\int\left( dz) (df _{0})/(dz) \right) ^{2dz}} + F(z)

where

F(z) is the homogeneous solution (an analytic function), that can be used to satisfy necessary boundary conditions. The series for the complex velocity potential g = u - iv is given byg(z,\overline z) = U[g _{0(z)} + M ^{2 g} _{1 (z,\overline} z) + … ]

where

g _{0=df} _{0/dz} and^[9] g _{1(z,\overline} z) = (1)/(4) (d ^{2 f} _{0})/(\overline{\int\left( dz ^{2}) (df _{0})/(dz) \right) ^{2dz}} + (1)/(4) \overline{ (df _{0})/(dz) }\left(\frac\right)^2 + \frac.

Notes and References

1. O. Janzen, Beitrag zu eincr Theorie der stationaren Stromung kompressibler Flussigkeiten. Phys. Zeits., 14 (1913)
2. Rayleigh, L. (1916). I. On the flow of compressible fluid past an obstacle. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 32(187), 1-6.
3. Von Karman, Th. "Compressibility effects in aerodynamics." Journal of spacecraft and rockets 40, no. 6 (1941): 992-1011.
4. IMAI, Isao. "A new method of successive approximations for dealing with the two-dimensional subsonic flow of a compressible fluid." Proceedings of the Physico-Mathematical Society of Japan. 3rd Series 24 (1942): 120-129.
5. Lamla, E. (1942). On the symmetrical potential flow of compressible fluid past a circular cylinder in the tunnel in the subcritical zone (No. NACA-TM-1018).
6. Imai, Isao, and Takasi Aihara. On the subsonic flow of a compressible fluid past an elliptic cylinder. Aeronautical Research Institute, Tokyo Imperial University, 1940.
7. JACOB, C. 1959 Introduction Mathématique a la Mécanique des Fluides. Gauthier-Villars.
8. Barsony-Nagy, A. "Extension of the Blasius force theorem to subsonic speeds." AIAA journal 23, no. 11 (1985): 1811-1812.
9. Carabineanu, Adrian. "A boundary integral equations approach for the study of the subsonic compressible flow past a cusped airfoil." Nonlinear Analysis: Theory, Methods & Applications 30, no. 6 (1997): 3449-3454.