Basics of Fluid Mechanics, 2014a

10.2. POTENTIAL FLOW FUNCTION 337
To absorb the density, dimensionless density is inserted into the definition of the stream
function as
∂ψ
dy = ρU x
(10.56)
ρ 0
and
∂ψ
dx = −ρU y
(10.57)
ρ 0
Where ρ 0 is the density at a location or a reference density. Note that the new stream
function is not identical to the previous definition and they cannot be combined.
The stream function, as it was shown earlier, describes (constant) stream lines.
Using the same argument in which equation (10.46) and equation (10.48) were developed
leads to equation (10.49) and there is no difference between compressible flow
and incompressible flow case. Substituting equations (10.56) and (10.57) into equation
(10.49) yields
( ) ∂ψ ∂ψ
dy +
∂y ∂x dx ρ0
ρ = ρ 0
dψ (10.58)
ρ
Equation suggests that the stream function should be redefined so that similar expressions
to incompressible flow can be developed for the compressible flow as
dψ = ρ 0
ρ U · ŝdl (10.59)
With the new definition, the flow crossing the line 1 to 2, utilizing the new definition
of (10.59) is
ṁ =
∫ 2
1
10.2.2.1 Stream Function in a Three Dimensions
∫ 2
ρU · ŝd ′ l = ρ 0 dψ = ρ 0 (ψ 2 − ψ 1 ) (10.60)
Pure three dimensional stream functions exist physically but at present there is no known
way to represent then mathematically. One of the ways that was suggested by Yih in
1957 34 suggested using two stream functions to represent the three dimensional flow.
The only exception is a stream function for three dimensional flow exists but only for
axisymmetric flow i.e the flow properties remains constant in one of the direction (say
z axis).
Advance material can be skipped
The three dimensional representation is based on the fact the continuity equation
must be satisfied. In this case it will be discussed only for incompressible flow. The
3 C.S. Yih “Stream Functions in Three–Dimensional Flows,” La houille blanche, Vol 12. 3 1957
4 Giese, J.H. 1951. “Stream Functions for Three–Dimensional Flows”, J. Math. Phys., Vol.30, pp.
31-35.
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