drag & lift main topics - ulyssisdreaden.ulyssis.org/fm.pdf · drag flow over a flat plate...

William D’haeseleer - 2007Fluidummechanica & Warmteoverdracht FM&WO 85

Main Topics�The Boundary-Layer Concept�Boundary-Layer Thicknesses� Laminar Flat-Plate Boundary Layer: Exact

Solution�Momentum Integral Equation�Use of the Momentum Equation for Flow

with Zero Pressure Gradient�Pressure Gradients in Boundary-Layer Flow�Drag� Lift


Forces from the surrounding fluid on a two-dimensional object:

(a) pressure force,

(b) viscous force,

(c) resultant force (lift and drag).

Drag & Lift

Drag force parallel to

Lift force perpendicular to

Ur

Ur


Drag & Lift

Pressure and shear forces on a small element of the surface of a body.


Drag & Lift

Pressure and shear forces on a small element of the surface of a body.

( )cos ( )sin

cos sin

x w

x w

dF p dA dA

Drag dF p dA dA

θ τ θ

θ τ θ

= +

= = +∫ ∫ ∫

( ) sin ( ) cos

sin cos

y w

y w

dF p dA dA

Lift dF p dA dA

θ τ θ

θ τ θ

= − +

= = − +∫ ∫ ∫


…………………….

Drag & Lift

Example

Lift = 0

Drag = 0.0992 lb (shear drag)


…………………….

Drag & Lift

Example:

Lift = 0

Drag = 55.6 lb (pressure drag)


Drag & Lift

When plate is tilted as shown; then both finite drag and lift


Drag & Lift

Often difficult to integrate drag and lift force equations; need detailed distribution of pressure and shear stress at surface body!

Customary to define dimensionless coefficients; to be determined by experiment or simplified analysis

2 21 12 2

LL

FLiftC

U A V Aρ ρ≡ ≡

2 21 12 2

DD

FDragC

U A V Aρ ρ≡ ≡


Drag

Recall from dimensional analysis: drag on a cylinder or sphere

1( , , , )DF f d V µ ρ=

After Buckingham Pi: 22 2DF Vd

fV d

ρρ µ

= 2 / 4A dπ=With 3 32

(Re)DF Vdf f

V A

ρρ µ

= =

With

� (Re)DC f=

Valid for incompressible flow over ANYbody !!

{William D’haeseleer - 2007Fluidummechanica & Warmteoverdracht FM&WO 94

Drag

�Pure Friction Drag: Flat Plate Parallel to the Flow

�Pure Pressure Drag: Flat Plate Perpendicular to the Flow

�Friction and Pressure Drag: Flow over a Sphere and Cylinder

�Streamlining


Drag




�Streamlining


Drag

Flow over a Flat Plate Parallel to the Flow :

Friction Drag (or shear drag)

Boundary Layer can be 100% laminar, partly laminar and partly turbulent, or essentially 100% turbulent; hence several different drag coefficients are available


Drag


For laminar flow, it was found:2

0.6641 Re2

wf

x

CU

τ

ρ= =

2

1...

12

DD f

A

FC C dA

AV Aρ≡ = ∫Therefore:

Result laminar BL:


Drag


For turbulent flow, it was found: 1/ 5

2

0.05941 (Re )2

wf

x

CU

τ

ρ= =

2

1...

12

DD f

A

FC C dA

AV Aρ≡ = ∫Again:

Result turbulent BL:


Drag

Flow over a Flat Plate Parallel to the Flow:


Drag


Drag




�Streamlining


Drag

Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag

Drag coefficients are usually obtained empirically


Drag

Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag (Continued)


Drag




�Streamlining


RECALL:

Inviscid flow past a circular cylinder: (a) streamlines for the flow if there were no viscous effects. (b) pressure distribution on the cylinder’s surface, (c) free-stream velocity on the cylinder’s surface.

Drag


Character of the steady, viscous flow past a circular cylinder:

(a) low Reynolds number flow,

(b) moderate Reynolds number flow,

(c) large Reynolds number flow.

Drag


Boundary layer characteristics on a circular cylinder: (a) boundary layer separation location. (b) typical boundary layer velocity profiles at various locations on the cylinder, (c) surface pressure distributions for inviscid flow and boundary layer flow.

Drag


Drag

Flow over a Sphere and Cylinder: Friction and Pressure Drag (Continued)


Drag

Flow over a Sphere and Cylinder: Friction and Pressure Drag


Drag

Sphere


Drag


Drag




�Streamlining


Drag

Streamlining:Used to Reduce Wake and hence Pressure Drag

Value of CD at min is about 0.06; being about 1/5 of a cylinder with the same thickness!


Drag



Drag



Main Topics�The Boundary-Layer Concept�Boundary-Layer Thicknesses� Laminar Flat-Plate Boundary Layer: Exact

Solution�Momentum Integral Equation�Use of the Momentum Equation for Flow

with Zero Pressure Gradient�Pressure Gradients in Boundary-Layer Flow�Drag� Lift


Lift

To be read; For details, see course aerodynamics


Symmetrical and nonsymmetrical airfoils.

Lift


Inviscid flow past an airfoil: (a) symmetrical flow past the symmetrical airfoil at a zero angle of attack: ( b) same airfoil at a nonzero angle of attack—no lift, flow near trailing edge not realistic: ( c) same conditions as for ( b) except circulation has been added to the flow—nonzero lift, realistic flow: (d) superposition of flows to produce the final flow past the airfoil.


Inviscid flow past a circular cylinder: ( a) uniform upstream flow without circulation. ( b) free vortex at the center of the cylinder, ( c) combination of free vortex and uniform flow past a circular cylinder giving nonsymmetric flow an d a lift.


Lift

�Examples: NACA 23015; NACA 66 2-215

CL


Lift

�Examples: NACA 23015; NACA 66 2-215

CD


Lift


Lift


Lift

�Induced Drag (Continued)

Reduction in Effective Angle of Attack:

Finite Wing Drag Coefficient:


Lift


Lift�Induced Drag (Continued)

drag & lift main topics - ulyssisdreaden.ulyssis.org/fm.pdf · drag flow over a flat plate...

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