Introduction_to_Generalized_Functions_with_Applications_in_Aerodynamics_and_Aeroacoustics-Farasat.pdf

NASA Technical Paper 3428

Introduction to Generalized Functions With

Applications in Aerodynamics and

Aeroacoustics

F. Farassat

Langley Research Center Hampton, Virginia

Corrected Copy (April 1996)

National Aeronautics and Space Administration

Langley Research Center Hampton, Virginia 23681-0001

May 1994

ADDENDUM

( ƒ ) in a Multidimensional Space,

Journal of Sound and Vibration , Volume 230, No. 2, February 17, 2000,

p. 460-462

d¢

ftp://techreports.larc.nasa.gov/pub/techreports/larc/2000/jp/NASA-2000-jsv-ff.ps.Z

http://techreports.larc.nasa.gov/ltrs/PDF/2000/jp/NASA-2000-jsv-ff.pdf

F. Farassat: The Integration of

Contents

Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. What Are Generalized Functions? . . . . . . . . . . . . . . . . . . . . . . 2

2.1. Schwartz Functional Approach . . . . . . . . . . . . . . . . . . . . . . 2

2.2. How Can Generalized Functions Be Introduced in Mathematics? . . . . . . . 6

3. Some Denitions and Results . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2. Generalized Derivative . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3. Multidimensional Delta Functions . . . . . . . . . . . . . . . . . . . . 19

3.4. Finite Part of Divergent Integrals . . . . . . . . . . . . . . . . . . . . 24

4. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2. Aerodynamic Applications . . . . . . . . . . . . . . . . . . . . . . . 30

4.3. Aeroacoustic Applications . . . . . . . . . . . . . . . . . . . . . . . 34

5. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

iii

Symbols

A(x) coecient of second order term of linear ordinary dierential equation

A() lower limit of integral in Leibni z rule depending on parameter

a constant

BC, BC 1 , BC 2 boundary conditions

B(x)

coecient of rst order term in second order linear ordinary dierential equation

B()

upper limit of integral in Leibniz rule depending on parameter

constant

C; C 1 ; C 2

constants

C(x)

coecient of zero order term (the unknown function) in second order linear

ordinary dierential equation

constant, also speed of sound

space of innitely dierentiable functions with bounded support (test functions)

D 0

space of generalized functions based on D

E 1 ; E 2

expressions in integrands of Kirchho formula for moving surfaces

E()

function dened by equation (3.70)

E h

shift operator E h f(x) = f(x+ h)

E ij

viscous stress tensor

in F [] , denes linear functional on test function space; generalized function

F (y; x; t)

= [f(y; )] ret = f(y; t c )

F (y; x; t)

= [f(y; )] ret = f(y; t c )

f(x); f(x)

arbitrary ordinary functions

f 1 (x)

arbitrary function

f i ()

components of moving compact force, i = 1 to 3

f(x; t)

equation of moving surface dened as f(x; t) = 0, f > 0 outside surface

f(x; t)

moving surface dened by f(x; t) = 0 intersection of which with f(x; t) = 0

denes edge of open surface f = 0, f > 0

g(x; y); g(x; y) Green’s function

g = t + c

g 1 (x; y); g 2 (x; y) dene Green’s function for x< y and x > y, respectively

g (2)

determinant of coecients of rst fundamental form of surface

g(x); g(x)

arbitrary functions

in H[], linear functional R 0

(x) dx based on Heaviside function

H f

local mean curvature of surface f = 0

H(x; )

function dened by equation (3.71)

constant

h(x)

Heaviside function

h " (x)

function of x indexed by continuou s parameter "

interval on real line, expression given by integral; expression

= p 1; index

index

in K[], denes l inear functional on test function space; generalized function

nonnegative integer

k(x); k(x; t)

equation of shock or wake surface given by k = 0

in dL, length parameter of edge of surface given by F = F = 0

‘

in ‘u, second order linear ordinary dierential equation

Mach number vector

M n

= M n; local normal Mach number

M r

= M r

= M

index of summation of Fourier series

unit normal to F = 0

nonnegative integer

local unit outward normal to surface

n 0

local unit inward normal to surface

n 1

vector (n 1 , 0, 0) based on n = (n 1 ; n 2 ; n 3 )

in o("), smal l order of "

principal value

P ij

compressive stress tensor

blade surface pressure

p 0

acoustic pressure

Q(x; t); Q(x; t) source strength of inhomogeneous term of wave equation

= jx yj

r i

components of vector r = x y, i = 1 to 3

r i

components of unit radiation vector

r , i = 1 to 3

in dS, surface area of given surface; space of rapidly decreasing test functions

S 0

space of generalized functions based on S

S k

portion of surface k = 0 inside surface @

s(t)

position vector of compact force in motion

T i j

Lighthill stress tensor

variable; time variable

t 1

unit vector in direction of projection of r onto local tangent plane to f(x; t) = 0

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