XI. Thwaites' Integral method for Laminar Incompressible Boundary Layers
This is an empirical method based on the observation that most laminar boundary layers obey
the following relationship (Ref: Thawites, B., Incompressible Aerodynamics, Clarendon Press, Oxford,
1960).:
ue
ν
d
dx
θ
2
( )=A−B
θ
2
ν
due
dx
(1)
Thwaites recommends A = 0.45 and B = 6 as the best empirical fit.
The above equation may be analytically integrated yielding
θ
2
=0.45
ν
u
e
6
u
e
5
x=0
x
∫
dx +
θ
2
(x=0) u
e
6
(x=0)
u
e
6
(x)
(2)
For blunt bodies such as airfoils, the edge velocity ue is zero at x=0, the stagnation point. For
sharp nosed geometries such as a flat plate, the momentum thickness θ is zero at the leading edge.
Thus, the term in the square bracket always vanishes.
The integral may be evaluated, at least numerically when ue is known.
After θ is found, the following relations are used to compute the shape factor H and the shear
stress at the wall τw.
For 0 ≤
λ
≤0.1
H=2.61 −3.75
λ
+5.24
λ
2
For −0.1≤
λ
≤0
H=2.472 +0.0147
0.107 +
λ
where ,
λ
=
θ
2
ν
due
dx
(3)
and,
τ
w
=
µ
u
e
θλ
+0.09( )
0.62
Despite the empiricism involved in the above formulas, Thwaites' integral method is considered to
be the best of a variety of integral boundary layer methods.
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