Pressure and Friction Drag I

Hydromechanics VVR090

ppt by Magnus Larson; revised by Rolf L Jan 2014

SYNOPSIS

1. General description

2. Applications

3. Forces on immersed bodies

4. Dimensional analysis

5. Drag coefficients and lift coefficients

6. Examples/Problems

7. Flow separation. Introduction.

1. General Description

Fluid Flow About Immersed Objects

Flow about an object may induce:

• drag forces

• lift forces

• vortex motion

Asymmetric flow field generates a net force

Drag forces arise from pressure differences over the body (due to its

shape) and frictional forces along the surface (in the boundary layer)

D’Alemberts Paradox

Jean d’Alembert

(1717-1783)

Ideal fluid no viscosity (implies no friction, no separation)

No net force on an object submerged in a flowing fluid.

2 Applications

• Naval architecture and aerodynamics.

• Structural engineering (e.g., houses, bridges)

• Vehicle design (e.g., cars, trains).

Applications

3. Forces on an Immersed Body

sin cos

cos sin

o

o

dD pdA dA

dL pdA dA

Drag and lift on a small element:

Total drag and lift force on the body:

sin cos

cos sin

o

A A

o

A A

D dD p dA dA

L dL p dA dA

Effects of the shear stress on lift negligible:

cos A

L p dA

sin cos o

A A

D p dA dA

Drag force:

Pressure drag (Dp) Frictional drag (Df)

(form drag)

Dp = function of the body shape and flow separation

Df = function of the boundary layer properties (surface

roughness etc)

Examples of Flow around Bodies I

sin 0, cos 1

0,

p f o

A

D D dA

No pressure drag

sin 1, cos 0

, 0

p f

A

D pdA D

No frictional drag

Examples of Flow around Bodies II

Dp >> Df Dp Df Dp << Df

D1 >> D2 >> D3

1. 2. 3.

4. Dimensional Analysis of Drag and Lift

Assumed relationships:

1

2

, , , ,

, , , ,

o

o

D f A V E

L f A V E

Derived -terms:

1

2 22

2 2

3 2

3 2

Re

M

o

o o

o

o

AV

V V

E a

D

A V

L

A V

(drag)

(lift)

Results of Dimensional Analysis

2 2

3

3

1 1Re,M

2 2

Re,M

o D o

D

D f AV C AV

C f

Total drag force:

Total lift force:

2 2

4

4

1 1Re,M

2 2

Re,M

o L o

L

L f AV C AV

C f

5. Drag coefficients CD and lift coefficients CL

Bodies of same shape and alignment with the flow have the

same CD and CL, if Re and M are the same.

Re describes the ratio between intertia forces and viscous

forces

M describes the ratio between inertia forces and elastic

forces

M only important at high flow velocities

(e.g. supersonic flows)

Drag Coefficient for Various Bodies

2D3D

Re Re

CDCD

Drag Coefficient for a Sphere

Mach number effects

Ernst Mach

(1838-1916)

Drag Coefficient at Sound Barrier

CD as a function of M

Example I: Drag Force on an Antenna Stand

30 m

0.3 m

What is the total drag force on the stand and

moment at the base?

wind

35 m/s

Standard atmosphere

(101 kPa, 20 deg)

6. Examples / problems

Example II: Sphere Dropped from an Airplane

Plastic sphere falling in air – what is the

terminal speed?

FG

FD

50 mm

Relative density of

sphere S = 1.3

Standard atmosphere

(101 kPa, 20 deg)

7. Flow Separation

contraction expansion

airfoilcylinder

Flow around an airfoil

small angle of attack large angle of attack

(drag from frictional losses) (drag from pressure losses)