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Expansion solution for subsonic compressible flow

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Abstract

An expansion solution in the physical plane is developed for subsonic compressible fluid flow past an obstacle. Assuming that the stream is inviscid, isentropic, irrotational and steady, it is shown that the velocity potential may be expressed as a series of homogeneous Heun functions and radial distance terms.

The basis of this analysis is Ludford's formal discussion of corresponding singularities in Bergman's Linear Integral Operator Method. A modification of these results permits reduction of the governing nonlinear partial differential equation to an ordinary, nonhomogeneous, linear differential equation.

The expansion solution is compared with the Rayleigh-Janzen method and the Prandtl-Glauert theory. The comparison indicates that this expansion gives better results than other methods currently used. The simplicity and economy of this expansion solution facilitates direct practical application.

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Abbreviations

a i :

acceleration of fluid element, (ft/sec2)

a n (θ):

angle dependent function in expansion solution

A n :

integration constant

B n :

integration constant

b :

stream parameter

c :

local speed of sound, (ft/sec)

c 0 :

stagnation speed of sound, (ft/sec)

C :

closed curve of integration in (3)

C i :

integration constant

F :

Heun's homogeneous function

F i :

body force in the i-direction

F(x i):

equation of obstacle surface

g :

determinant of the fundamental tensors

g ij :

associated metric tensor

g ij :

fundamental metric tensor

H n (θ):

homogeneous Heun function determined by the index n

m :

source strength, (ft2/sec)

M 0 :

stagnation Mach number

M ∞ :

free stream Mach number

M :

local Mach number

n :

surface normal vector

N p :

number of index permutations

P(θ):

general term in ordinary differential equation (26)

p :

local pressure magnitude

q :

velocity magnitude (ft/sec)

Q(θ):

general term in ordinary differential equation (26)

R(θ):

remainder term defined by (30)

r :

radial polar coordinate (ft)

s :

entropy magnitude (ft2/sec2−0 R)

S :

surface area (ft2)

u 1 :

curvilinear surface coordinates (ft)

u 2 :

curvilinear surface coordinates (ft)

U ∞ i (x i):

free stream velocity function (ft/sec)

U ∞ :

free stream velocity magnitude (ft/sec)

v :

velocity magnitude (ft/sec)

v i :

velocity component in i-direction (ft/sec)

v(θ):

nonhomogeneous solution function used in (25)

\(\bar v_r \) :

radial velocity found by expansion solution (ft/sec)

\(\bar v_\theta \) :

angular velocity found by expansion solution (ft/sec)

w :

vorticity vector, (radians/sec)

x i :

curvilinear coordinates

y i :

alternate curvilinear coordinate

y n (θ):

homogeneous solution function used in (25)

Z :

transformed independent variable

i, j, k, l, n, p :

indices

α :

parameter in Heun's equation (21)

β :

Mach number parameter

γ :

ratio of specific heats

δ :

parameter in Heun's equation (21)

ρ :

local stream mass density

ρ 0 :

stream stagnation mass density

ε(r, θ):

disturbance velocity potential in (45)

η :

specific heat ratio parameter

θ :

angular polar coordinate (radians)

\(\bar \theta \) :

direction of the velocity vector (radians)

μ :

coefficient of stream viscosity

ξ :

complex variable

σ :

coefficient of bulk viscosity

φ :

velocity potential (ft2/sec)

φ 0 :

incompressible velocity potential (ft2/sec)

φ0:

velocity potential of expansion solution (ft2/sec)

φ*:

modified velocity potential

Γ :

circulation (ft2/sec)

Γ 0 :

constant circulation magnitude (ft2/sec)

ψ :

stream function (ft2/sec)

ψ*:

modified stream function

[ij, k]:

Christoffel three-index symbol of the first kind

\(\mathop {jk}\limits^i \) :

Christoffel three-index symobl of the second kind

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Authors and Affiliations

  1. Bell Telephone Labs, Denver, Colorado, USA

    P. J. Whalen

  2. New Mexico State University, Las Cruces, New Mexico, USA

    G. P. Mulholland

Authors
  1. P. J. Whalen
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  2. G. P. Mulholland
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Whalen, P.J., Mulholland, G.P. Expansion solution for subsonic compressible flow. Appl. Sci. Res. 25, 445–473 (1972). https://doi.org/10.1007/BF00382316

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