Solutions of the Laminar Boundary Layer Equations · Solutions of the Laminar Boundary Layer...
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Solutions of the Laminar Boundary Layer Equations The boundary layer equations for incompressible steady flow, i.e., 2 2 0 y u dx dp y u v x u u y v x u e ∂ ∂ + − = ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ µ ρ ρ 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, constant ← = ∞ p p e 0 = ⇒ dx dp e Blasius was able to show that the boundary later equations could be rewritten to only depend on a parameter, vx V y 2 ∞ ≡ η and its derivatives The resulting solution has been tabulated and compared to experiments on the following page. Note: ∞ ∞ = ∞ ′ ′ = ∂ ∂ = = ′ ′ = V vx f V y u d df f f V y x u y w / 2 ) 0 ( where ) ( ) , ( 0 µ µ τ η η These values from the solution of ) ( η f can be used to find: ∞ = − ≡ V y x u y 99 . 0 ) , ( % 99 which at location δ Note: since ) ( set we , 0 x p p y p e = = ∂ ∂ , i.e. the boundary layer edge pressure.
Transcript of Solutions of the Laminar Boundary Layer Equations · Solutions of the Laminar Boundary Layer...
Solutions of the Laminar Boundary Layer Equations
The boundary layer equations for incompressible steady flow, i.e.,
2
2
0
yu
dxdp
yuv
xuu
yv
xu
e
∂∂
+−=∂∂
+∂∂
=∂∂
+∂∂
µρρ
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, constant ←= ∞ppe
0=⇒dxdpe
Blasius was able to show that the boundary later equations could be rewritten to only depend on a parameter,
vx
Vy
2∞≡η
and its derivatives The resulting solution has been tabulated and compared to experiments on the following page. Note:
∞
∞
=
∞
′′=
∂∂
=
=′′=
VvxfV
yu
ddfffVyxu
yw /2
)0(
where )(),(
0
µµτ
ηη
These values from the solution of )(ηf can be used to find: ∞=−≡ Vyxuy 99.0),(%99 whichat locationδ
Note: since )(set we,0 xppyp
e==∂∂ ,
i.e. the boundary layer edge pressure.
Solutions of the Laminar Boundary Layer Equations
16.100 2002 2
From the table, 5.3at 99.0)( ≈=′ ηηf :
xVvxvx
Vvx
Vy
25.3
25.3
2
%99
%99
as grows layerboundary ←=⇒
=
=
∞
∞
∞
δ
δ
η
Typically, this result is written “non dimensionally” as:
x
vxV
x xx
on based number sReynold'
where ∞≡= ReRe
0.5%99δ
We can also find:
x
wf
x
x
VC
x
x
Re664.0
21
Re664.0Re7208.1
2
*
==
=
=
∞∞ρ
τ
θ
δ
Comment:
∗ At leading edge of a flat plate ∞→→ fCx gives this and 0 ! ∗ In reality, the leading edge of an infinitely thin plate would have very large,
but not infinite fC . ∗ 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
0→x ?
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