Boundary Layer Flow: Blasius solution for laminar …kshollen/ME347/Handouts/Blasius.pdfBoundary...
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Boundary Layer Flow: Blasius solution for laminar flow over a flat plate Assume: Steady, constant property, 2-D flow of a Newtonian fluid with negligible body forces Governing Equations: Conservation of Mass: ∂u ∂x + ∂v ∂y = 0 (1) Momentum Balance (x-direction): ρ u ∂u ∂x + v ∂u ∂y ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = μ ∂ 2 u ∂y 2 (2) Boundary Conditions: uy = 0 ( ) = vy = 0 ( ) = 0 and uy = δ ( ) = U Variable Substitution: η = U ν x y = Re x y x , f η () = ψ ν xU , and u U = df dη = ʹ f η () Reduced Equation: ʹ ʹ ʹ f + 1 2 f ʹ ʹ f = 0 (3) Boundary Conditions: fy = 0 ( ) = ʹ f y = 0 ( ) = 0 and ʹ f y →∞ ( ) = 1 Solution: Disturbance Thickness: δ x = 5.0 Re x from η = 5.0 where u/U = 0.99 Displacement Thickness: δ * x = δ * δ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ δ x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = 1.72 Re x Momentum Thickness: θ x = θ δ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ δ x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = 0.665 Re x Wall Shear Stress: τ w = μ ∂u ∂y y = 0 = μ U ∂ ʹ f ∂η ∂η ∂y ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ y = 0 = μ U ʹ ʹ f η = 0 ( ) U ν x = 0.332 ρ U 2 Re x Friction Coefficient: c f = τ w 1 2 ρ U 2 = 0.664 Re x Drag Coefficient for Friction: C D, f = 1 A c f dA ∫ = 1 ℓ 0.664 Re x dx x=0 ℓ ∫ = 1.328 Re ℓ
Transcript of Boundary Layer Flow: Blasius solution for laminar …kshollen/ME347/Handouts/Blasius.pdfBoundary...
Boundary Layer Flow: Blasius solution for laminar flow over a flat plate Assume: Steady, constant property, 2-D flow of a Newtonian fluid with negligible body forces Governing Equations:
Conservation of Mass:
€
∂u∂x
+∂v∂y
= 0 (1)
Momentum Balance (x-direction):
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ρ u∂u∂x
+ v∂u∂y
⎛
⎝ ⎜
⎞
⎠ ⎟ = µ
∂ 2u∂y 2
(2)
Boundary Conditions:
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u y = 0( ) = v y = 0( ) = 0 and
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u y = δ( ) =U
Variable Substitution: η =Uν x
y = Rexyx
,
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f η( ) =ψν x U
, and
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uU
=dfdη
= ʹ f η( )
Reduced Equation:
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ʹ ́ ́ f + 12 f ʹ ́ f = 0 (3)
Boundary Conditions:
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f y = 0( ) = ʹ f y = 0( ) = 0 and
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ʹ f y →∞( ) =1 Solution:
Disturbance Thickness:
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δx
=5.0Rex
from
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η = 5.0 where u/U = 0.99
Displacement Thickness:
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δ*
x=δ*
δ
⎛
⎝ ⎜
⎞
⎠ ⎟ δx⎛
⎝ ⎜ ⎞
⎠ ⎟ =
1.72Rex
Momentum Thickness:
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θx
=θδ
⎛
⎝ ⎜ ⎞
⎠ ⎟ δx⎛
⎝ ⎜ ⎞
⎠ ⎟ =0.665Rex
Wall Shear Stress:
€
τw = µ∂u∂y y =0
= µ U∂ ʹ f ∂η
∂η∂y
⎛
⎝ ⎜
⎞
⎠ ⎟
y =0
= µ U ʹ ́ f η = 0( ) Uν x
=0.332 ρU 2
Rex
Friction Coefficient:
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cf =τw
12 ρU 2 =
0.664Rex
Drag Coefficient for Friction: CD, f =1A
cf dA∫ =1ℓ
0.664Rex
dxx=0
ℓ
∫ =1.328Reℓ
Blasius solution for laminar flow over a flat plate.
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η = y Uν x
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f η( )
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ʹ f η( ) =uU
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ʹ ́ f η( )
0.0 0.0000 0.0000 0.3321 0.5 0.0415 0.1659 0.3309 1.0 0.1656 0.3298 0.3230 1.5 0.3701 0.4868 0.3026 2.0 0.6500 0.6298 0.2668 2.5 0.9964 0.7513 0.2174 3.0 1.3969 0.8461 0.1614 3.5 1.8378 0.9131 0.1078 4.0 2.3059 0.9555 0.0642 4.5 2.7903 0.9795 0.0340 5.0 3.2834 0.9916 0.0159 5.5 3.7807 0.9969 0.0066 6.0 4.2798 0.9990 0.0024 6.5 4.7795 0.9997 0.0008 7.0 5.2794 0.9999 0.0002 7.5 5.7794 1.0000 0.0001 8.0 6.2794 1.0000 0.0000
Dimensionless velocity profile from Blasius solution for laminar flow over a flat plate.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
8
u/ U
η
