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]
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 .
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]
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]
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 .
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]