Blasius - Formulación PDF

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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:

∂u ∂v + =0 ∂x ∂y

(1)

⎛ ∂u ∂ 2u ∂u ⎞ + v ρ ⎜u ⎟=µ 2 ∂y ⎠ ∂y ⎝ ∂x

(2)

Conservation of Mass:

Momentum Balance (x-direction):

u( y = 0) = v ( y = 0 ) = 0 and u( y = δ) = U

Boundary Conditions: Variable Substitution:

η=

U y y = Rex , νx x

u df ψ , and = f ʹ(η) = U dη ν xU

f (η) =

f ʹʹʹ + 12 f f ʹʹ = 0

Reduced Equation:

f ( y = 0) = f ʹ( y = 0 ) = 0 and

Boundary Conditions:

(3)

f ʹ( y → ∞) = 1

Solution:

δ 5.0 = Rex x

Disturbance Thickness:

δ* ⎛ δ * ⎞ ⎛ δ ⎞ 1.72 = ⎜ ⎟⎜ ⎟ = Rex x ⎝ δ ⎠⎝x⎠

Displacement Thickness:

θ ⎛ θ ⎞ ⎛ δ ⎞ 0.665 =⎜ ⎟⎜ ⎟= x ⎝δ ⎠ ⎝ x ⎠ Rex

Momentum Thickness:

Wall Shear Stress:

from η = 5.0 where u/U = 0.99

τ w =µ

⎛ ∂f ʹ ∂η⎞ ∂u U 0.332 ρ U 2 =µU⎜ = = 0 η µ f U = ) ν ʹ( ⎟ ∂y y =0 x Rex ⎝ ∂η ∂y ⎠ y =0

cf =

Friction Coefficient:

Drag Coefficient for Friction:

CD, f =

1 A

∫c

f

1 2

τw 0.664 2 = ρU Rex

dA =

1 ℓ

∫

ℓ x=0

1.328 0.664 dx = Reℓ Rex

Blasius solution for laminar flow over a flat plate.

η=y

U νx

f (η)

f ʹ (η ) =

u U

f ʹʹ(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 2.0

0.3701 0.6500

0.4868 0.6298

0.3026 0.2668

2.5 3.0

0.9964 1.3969

0.7513 0.8461

0.2174 0.1614

3.5

1.8378

0.9131

0.1078

4.0

2.3059

0.9555

0.0642

4.5 5.0

2.7903 3.2834

0.9795 0.9916

0.0340 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

8 7 6

η

5 4 3 2 1 0 0

0.1

0.2

0.3

0.4

0.5 u/ U

0.6

0.7

0.8

0.9

1

Dimensionless velocity profile from Blasius solution for laminar flow over a flat plate....