Boundary Layer Flows

Flat Plate (No Pressure Gradient) Boundary layer properties for laminar and turbulent flows are summarized in this section, Uo

l

Laminar Boundary Layer 1 − δ = 4.96 Re x 2 , x

Transition Re crit

−

1 2

Cf = 1.328 Re l ,

Re l =

U 0l ν

 3.2 × 10 5 ~ 10 6 ≈ 5 6 5 × 10 ~ 3 × 10

Turbulent Boundary Layer

Smooth Plate − δ = 0.37 Re x 5 , x 1

Cf =

0.074 Re

Rcrit A

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1 5 l

−

−

1

C ′f = 0.0577 Re x 5

A Re l

5 × 10 5 < Re x < 10 7

5 × 10 5 < Re l < 10 7

Values of parameter A for smooth plates 3× 10 5 5 × 10 5 10 6 1050 1700 3300

1

3× 10 6 8700

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Prandtl-Schlichting (Drag Coefficient for Smooth Plates)

Cf =

0.455

(log Re l )

2.58

−

A Re l

Re l > 10 7

Schlichting Formula (Drag Coefficient) Rough:

l  Cf = 1.89 + 1.58 log  e 

Smooth:

Cf =

0.031 Re

1 7 l

−

A Re l

−2.5

Rough Zone

Transition Zone

Admissible Roughness

e odm = l

100 Re l

Boundary Layer Thickness − δ = 0.16 Re x 7 x 1

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Turbulent Boundary Layer Boundary Layer Transition R δ | critical ≈ 1220 ,

R θ |critical ≈ 420

(

R L | critical = 5 × 10 5 ~ 3 × 10 6

) Transition

Laminar Turbulent

Boundary Layer Equations 1 dP0 ∂2U ∂ ∂U ∂U u ′v ′ U =− +ν 2 − +V p dx ∂y ∂y ∂y ∂x P0 = P + ρ v′ 2 ∂U ∂V + =0 ∂x ∂y Integral Equation (von Karman Momentum Integral) d 2θ + δ * dU 0 τ θ+ = 02 dx U0 dx ρU 0 ∞  U  dy = Displacement Thickness δ* = ∫ 1 − U 0  0

θ=

∞

0

H=

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U  U  1 −   U dy = Momentum Thickness 0  0 

∫U

δ* = shape factor θ

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Power Law Velocity Approximation 1

1 U  y n =   = ηn U0  δ 

U av n = U0 n +1

where U av = average velocity

1 U0 n +1 c R = 1 δ u* 2 − cf −2 n +1 c R = 1 ν 2

δ* 1 = , δ n +1 H=

θ n = δ (n + 1)(n + 2 )

δ* n + 2 = n θ n +1

U 0 θ  U 0 x  n +3 =  c2  ν ν   Note

1 corresponds to the Blasius resistance law. 7 −

1

 U δ 4 cf ~  0  .  ν  1

 u *y  7 U  = 8.3 u* ν  

U0 − U y  = 0.61 −  * u  δ

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2

y > 0.15 δ

for

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Friction Coefficient Ludwieg - Tillmann Expression C f = 0.246 × 10

Here H is a function

− 0.678H

 U0 θ     ν 

− 0.268

10 3 < Rθ < 10 4 ,

Rθ =

U 0θ ν

u* or C f . U0

In this range, H ≈ 1.36 ,

Cf ≈ 0.03R −θ0.268

which corresponds to a power-law velocity with n = 6 .45 .

R θ = 0.045R 0x.79 ,

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Rx =

U0 x ν

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Experimental Data for Turbulent Boundary Layer (Hinze, Turbulence, McGraw-Hill, 1975)

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