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
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