ME 116 Fluid Mechanics Boundary Layer Flows

Fall 2014

Deify Law

Pioneer of Boundary Layer Flows ~

Ludwig Prandtl

Large Reynolds number flow fields consist of viscous region in the boundary layer and

inviscid region elsewhere

No-slip condition at the wall or solid boundary: the fluid sticks to the surface

Boundary Layer Over a Flat Plate

Reynolds number (Rex) where Laminar Boundary Layer becomes Turbulent

Boundary Layer is about 2 x 105 to 3 x 106.

Boundary Layer Thickness

Definitions

Boundary Layer Thickness,

y 0.99u UAt

Boundary Layer Displacement

Thickness, *

*0

bU U u bdy

b is the depth

*

01

udy

U

Based on

Incompressible Continuity

Applicable

For

Incompressible

(Laminar or Turbulent),

constant or

variable pressure,

steady flow

Boundary Layer Momentum Thickness,

Based on momentum flux

01

u udy

U U

Boundary Layer Displacement and

Momentum Thicknesses

Used for the Momentum Integral Equation

Used for calculating local wall shear stress and drag force.

Momentum Integral Equation with

Nonzero Pressure Gradient for Flows

Past a Flat Plate (Von Karman)

2 *wd dU

U Udx dx

Boundary Layer Equations for a

Laminar Flow past a Flat Plate

Order of Magnitude Analysis (Scale Analysis) with assumptions reduce Navier-Stokes equations to

these boundary layer equations

v ux y

0u v

x y

2

2

u u uu v

x y y

2D, Laminar, Incompressible

Pressure is constant

so pressure gradient

is negligible, Steady Flows

Continuity:

X-Momentum:

Scale:

Prandtl/Blasius Dimensionless

Variable

u yg

U

1/2

~x

U

1/2U

yx

1/2

vxU f

Substitute u and v into the previous boundary layer equations

and take the other derivatives with

chain rule involving

1/2

vxU f

'

1/2

'

4

u Ufy

Uv f f

x x

Blasius Equation: Conversion from

PDE to ODE

Boundary conditions: y=0; u=0, y=0; v=0

y=infinity; u=U

''' ''2 0f ff

'0, 0f f 0

' 1f

Blasius Solution Laminar Flow Past a Flat Plate without Pressure Gradient

u/U = 0.99 when = 5.0

Displacement Thickness

Momentum Thickness

5x

yU

* 1.721 1.721

Rexx Ux

0.664

Rexx

Nondimensional Height vs.

Nondimensional Streamwise Velocity

Determination of Friction Drag Force

over a Flat Plate with Momentum

Integral Equation

2(1) (2)

Drag U U dA u dA

2Drag bU

Local Wall Shear Stress over a

Flat Plate

21w

dDrag dU

b dx dx

3

''

0 0

0.332wy

u U UUf

y x x

For Laminar Flow Past a Flat Plate:

Local Skin Friction or Local

Friction Drag Coefficient (Cf)

For Laminar Flow Past a Flat Plate:

2

0.664

1 Re

2

wf

x

c

U

21

2

wfc

U

Wall Shear Stress and

Friction Drag Coefficient

0

1 Lw wdx

L

0

1 LDf fC c dx

L

Blasius Solution

For Laminar Flat Plate:

1.328

ReDf

L

C

Momentum Integral Boundary

Layer Equation

Using assumed velocity profiles to predict boundary layer information.

For example, consider the laminar flow of an incompressible fluid past a flat plate at y=0.

The boundary layer velocity profile is

approximated as:

Determine the shear stress using momentum integral equation. Compare results with the exact

Blasius results.

u y

U

Comparison of Approximated Velocity

Profiles used in Momentum Integral

Equation with Exact Blasius Results

Transition from Laminar to

Turbulent Flows over a Flat Plate

Transitional Flow when: 5,Re 5 10x cr

Laminar and Turbulent Boundary

Layer Properties (Flat Plate)

Laminar (from

Blasius Exact)

Turbulent (from

Power Law)

Boundary Layer

Thickness

Wall Shear Stress

Friction Drag

Coefficient

5.0

Rexx

1/5

0.370

Rexx

3

0.332wU

x

2

1/50.0288

Rew

x

U

1.328

ReDf

L

C 1/5

0.0720

ReDf

L

C

Friction Drag Coefficient for a Flat

Plate Parallel to the Free Stream Flow

Boundary Layer Flows on Curved

Surface

Pressure gradient is not negligible.

Fluid velocity at the edge of boundary layer is not constant.

Effects of Pressure Gradient

The variation in the free-stream velocity, U, the fluid velocity at the edge of the boundary layer, is the cause of the existence of pressure gradient.

dp dUU

dx dx

Inviscid Flow Past a Circular

Cylinder

Viscous Flow Past a Circular

Cylinder Favorable Pressure Gradient

Diminishes Boundary Layer

Thickness

Adverse Pressure Gradient

Increases Boundary Layer

Thickness