lecture6 boundary layers

Boundary Layers

Hydromechanics VVR090

Boundar Layer on a Flat Plate

Boundary layer: the zone in which the velocity profile is governed by frictional action

1

Boundary layer characterized by a Reynolds number:

Rδ =

Vo δ ν

Rδ < 3900

laminar flow

Rδ > 3900

turbulent flow

Viscous sublayer exist close to the surface.

Mathematical implications of boundary layer:

∂u ∂u ∂v ∂v  , , ∂y ∂x ∂x ∂y uv →

∂p ∂p dp ≈ 0, ≈ ∂y ∂x dx

von Karman momentum integral equation

2

Von Karman momentum integral equation

Apply momentum equation for control volume ABCD. Height of control volume extends beyond the edge of the boundary layer (to the outer flow). At edge of the boundary layer: po(x), Vo(x) (known). Small curvature along body.

Conservation of mass: h ⎞ d ⎛ ⎜ ∫ ρudy ⎟ + [ρv ] y =h = 0 dx ⎝ 0 ⎠

Conservation of momentum:

−

h ⎞ dpo d ⎛ h − τo + [ τ] y =h = ⎜ ∫ ρu 2 dy ⎟ + [ρuv ] y =h dx dx ⎝ 0 ⎠

3

Develop conservation of mass equation:

ρo vo = −

h ⎞ d ⎛ ⎜ ∫ ρudy ⎟ dx ⎝ 0 ⎠

Develop conservation of momentum equation:

h ⎞ dpo d ⎛ h − τo = ⎜ ∫ ρu 2 dy ⎟ + ρoVo vo − dx dx ⎝ 0 ⎠

Combine conservation of mass and momentum:

−

h h ⎞ ⎞ dpo d ⎛ d ⎛ h − τo = ⎜ ∫ ρu 2 dy ⎟ − Vo ⎜ ∫ ρudy ⎟ dx dx ⎝ 0 dx ⎝ 0 ⎠ ⎠

Euler eqution for the outer flow:

dpo + Vo dVo = 0 ρo dpo dV = −ρoVo o dx dx

4

Combining:

−τo =

h h ⎞ ⎞ dVo d ⎛ d ⎛ 2 u dy V h ρ − ⎜∫ ⎟ o ⎜ ∫ ρudy ⎟ − ρoVo dx ⎝ 0 dx dx ⎠ ⎝0 ⎠

Define displacement thickness: h ⎛ ρu δ1 = ∫ ⎜1 − ρoVo 0⎝

h

ρoVo δ1 = ∫ ( ρoVo − ρu ) dy 0

⎞ ⎟ dy ⎠

h

∫ ρudy

< ρoVo δ

0

Flow rate with boundary layer

and

without

In the same manner, define momentum thickness:

u ⎞ ρu ⎛ ⎜1 − ⎟ dy Vo ⎠ ρV 0 o o ⎝ h

δ2 = ∫

Substitute in d1 and d2 in momentum equation and express τo in terms of a local friction coefficient cf:

cf 2

=

d δ2 dx

(constant density, no pressure gradients)

5

Laminar Boundary Layer along a Flat Plate Assume parabolic velocity profile:

u=

Vo ( 2δy − y 2 ) δ2

2 δ 15 cf τo 2 dδ = = 2 ρVo 2 15 dx

δ2 =

τo =

2V 2 dδ ρVo2 =μ o 15 dx δ

Integrate the equation:

2V 2 dδ ρVo2 =μ o 15 dx δ δ 2 15μx = 2 ρVo δ 30 = x Rx Substitute in dd/dx in shear stress expression:

τo =

ρVo2 8 2 15 Rx

6

Local friction coefficient:

cf ≈

8 15 Rx

Mean friction coefficient along the plate:

x

1 32 C f = ∫ c f dx = x0 15 Rx

Relationship between Rx and Rd:

Rx =

Rδ2 30

Rd = 3900

Rx = 500,000

(transition between laminar and turbulent conditions in the boundary layer)

7

Drag Coefficient for Smooth, Flat Plates

Df = C f

1 ρVo2 A 2

A: surface area of plate

8