Chapter 11 Boundary layer theory

4 CHAPTER 11. BOUNDARY LAYER THEORY
l = Re −1/2 L in equation 11.14, to get the scaled momentum equation in
the streamwise direction,
u ∗ ∂u ∗ x
x
∂x ∗ + u∗ y
∂u ∗ y
∂y ∗ = −∂p∗ ∂x ∗ + ∂2 u ∗ x
∂y ∗2 (11.15)
Next, we analyse the momentum equation in the cross-stream direction,
11.11. This equation is expressed in terms of the scaled spatial co-ordinates,
velocities and pressure, to obtain,
ρU∞ 2 l (
u ∗ u ∗ y
L 2 x
x + ∂u ∗ )
⎛
∗ u∗ y
y = − ρU2 ∞ ∂p ∗
∂y ∗ l ∂y + µU ∞ ⎝ ∂2 u ∗ ( ) ⎞ 2
y l ∗ lL ∂y + ∂ 2 u ∗ y ⎠
∗2 L ∂x ∗2 (11.16)
By examining all terms in the above equation, it is easy to see that the
largest terms is the pressure gradient in the cross-stream direction. We divide
throughout by the pre-factor of this term, and substitute (l/L) = Re −1/2 , to
obtain,
Re
(u −1 ∗ u ∗ y
x
x + ∂u ∗ )
∗ u∗ y
y = − ∂p (
∗ ∂ 2
∂y ∗ ∂y + u ∗
∗ Re−1 y
∂y + u ∗ )
∗2 Re−1∂2 y
(11.17)
∂x ∗2
In the limit Re ≫ 1, the above momentum conservation equation reduces to,
∂p ∗
∂y ∗ = 0 (11.18)
Thus, the pressure gradient in the cross-stream direction is zero in the leading
approximation, and the pressure at any streamwise location in the boundary
layer is the same as that in the free-stream at that same stream-wise location.
This is a salient feature of the flow in a boundary layers. Thus, the
above scaling analysis has provided us with the simplified ‘boundary layer
equations’ 11.12, 11.15 and 11.18, in which we neglect all terms that are o(1)
in an expansion in the parameter Re −1/2 . Expressed in dimensional form,
these mass conservation equation is 11.9, while the approximate momentum
conservation equations are,
(
)
∂u x
ρ u x
∂x + u ∂u x
y = − ∂p
∂y ∂x + u x
µ∂2 (11.19)
∂y 2
∂p
∂y = 0 (11.20)