boundary layer flows
DESCRIPTION
boundaryTRANSCRIPT
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ME 116 Fluid Mechanics Boundary Layer Flows
Fall 2014
Deify Law
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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
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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.
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Boundary Layer Thickness
Definitions
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Boundary Layer Thickness,
y 0.99u UAt
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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
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Boundary Layer Momentum Thickness,
Based on momentum flux
01
u udy
U U
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Boundary Layer Displacement and
Momentum Thicknesses
Used for the Momentum Integral Equation
Used for calculating local wall shear stress and drag force.
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Momentum Integral Equation with
Nonzero Pressure Gradient for Flows
Past a Flat Plate (Von Karman)
2 *wd dU
U Udx dx
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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:
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Prandtl/Blasius Dimensionless
Variable
u yg
U
1/2
~x
U
1/2U
yx
1/2
vxU f
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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
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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
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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
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Nondimensional Height vs.
Nondimensional Streamwise Velocity
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Determination of Friction Drag Force
over a Flat Plate with Momentum
Integral Equation
2(1) (2)
Drag U U dA u dA
2Drag bU
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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:
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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
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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
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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
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Comparison of Approximated Velocity
Profiles used in Momentum Integral
Equation with Exact Blasius Results
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Transition from Laminar to
Turbulent Flows over a Flat Plate
Transitional Flow when: 5,Re 5 10x cr
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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
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Friction Drag Coefficient for a Flat
Plate Parallel to the Free Stream Flow
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Boundary Layer Flows on Curved
Surface
Pressure gradient is not negligible.
Fluid velocity at the edge of boundary layer is not constant.
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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
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Inviscid Flow Past a Circular
Cylinder
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Viscous Flow Past a Circular
Cylinder Favorable Pressure Gradient
Diminishes Boundary Layer
Thickness
Adverse Pressure Gradient
Increases Boundary Layer
Thickness