Janzen–Rayleigh expansion

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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

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

 

where  , the sound speed is expressed as a function of the velocity magnitude   For a polytropic gas, we can write

 

where   is the specific heat ratio,   is the stagnation sound speed (i.e., the sound speed in a gas at rest) and   is the stagnation enthalpy. Let   be the characteristic velocity scale and   is the characteristic value of the sound speed, then the function   is of the form

 

where   is the relevant Mach number.

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

 

Substituting this governing equation and collecting terms of different orders of   leads to a set of equations. These are

 

and so on. Note that   is independent of   with which the latter quantity appears in the problem for  .

Imai–Lamla method

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A simple method for finding the particular integral for   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   formally regarded as the function of   and its conjugate  ; here   is the stream function, defined such that

 

where   is some reference value for the density. The perturbation series of   is given by

 

where   is an analytic function since   and  , being solutions of the Laplace equation, are harmonic functions. The integral for the first-order problem leads to the Imai–Lamla formula[7][8]

 

where   is the homogeneous solution (an analytic function), that can be used to satisfy necessary boundary conditions. The series for the complex velocity potential   is given by

 

where   and[9]

 

References

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  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.