Solutions of the Laminar Boundary Layer Equations
The boundary layer equations for incompressible steady flow, i.e.,
∂u ∂v
+
=0
∂x ∂y
ρu
∂p
= 0, we set p = p e ( x) ,
∂y
i.e. the boundary layer edge pressure.
Note: since
dp
∂u
∂u
∂ 2u
+ ρv
=− e +µ 2
∂x
∂y
dx
∂y
have been solved for a handful of important cases. We will look at the results for
a flat plate and a family of solutions called Falkner-Skan Solutions.
Flat Plate (Laminar): Blasius Solution
For a flat plate, p e = p ∞ ← constant
⇒
dp e
=0
dx
Blasius was able to show that the boundary later equations could be rewritten to
only depend on a parameter,
η≡y
V∞
2vx
and its derivatives
The resulting solution has been tabulated and compared to experiments on the
following page. Note:
u ( x, y ) = V∞ f ′(η )
τw = µ
∂u
∂y
=
y =0
where f ′ =
df
dη
µV∞ f ′′(0)
2vx / V∞
These values from the solution of f (η ) can be used to find:
δ 99% ≡ y − location at which u ( x, y ) = 0.99V∞
Solutions of the Laminar Boundary Layer Equations
From the table, f ′(η ) = 0.99 at η ≈ 3.5 :
V∞
2vx
η=y
3.5 = δ 99%
V∞
2vx
⇒ δ 99% = 3.5
2vx
V∞
← boundary layer grows as x
Typically, this result is written “non dimensionally” as:
δ 99%
=
x
5.0
Re x
where Re x ≡
V∞ x
v
Reynold' s number based on x
We can also find:
δ*
x
θ
x
=
=
1.7208
Re x
0.664
Cf =
Re x
τw
1
ρ ∞V∞2
2
=
0.664
Re x
Comment:
∗ At leading edge of a flat plate x → 0 and this gives C f → ∞ !
∗ In reality, the leading edge of an infinitely thin plate would have very large,
but not infinite C f .
∗ The problem is that near the leading edge of a thin plate, the boundary
layer equations are not correct and the Navier-Stokes equations are
needed. Question: Why did the boundary layer approximation fail at
x → 0?
16.100 2002
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