Flow Over Immersed Bodies

Chapter 9Flow over Immersed

Bodies

We consider flows over bodies that are immersed in a fluid and the flows are termed external flows.We are interested in the fluid force (lift and drag) over the bodies. For example, correct design of cars, trucks, ships, and airplanes, etc. can greatly decrease the fuel consumption and improve the handling characteristics of the vehicle.

9.1 General External Flow Characteristics

A body immersed in a moving fluid experiences a resultant force due to the interaction between the body and the fluid surrounding it.Object and flow relation:stationary air with moving object or flowing air with stationary objectFlow classification:

Body shape:Streamlined bodies or blunt bodies

Figure 9.2Flow classification: (a) two-dimensional, (b) axisymmetric, (c) three-dimensional.

9.1.1 Lift and Drag ConceptsWhen any body moves through a fluid, an interaction between the body and the fluid occursThis can be described in terms of the stresses-wall shear stresses due to viscous effect and normal stresses due to the pressure P

Drag, Dthe resultant force in the direction of the upstream velocity Lift, Lthe resultant force normal to the upstream velocity

Resultant force--lift and drag

Resultant force ( ) cos ( )sinand

( )sin ( )cos

x w

y w

dF pdA dA

dF pdA dA

θ τ θ

θ τ θ

= +

= − +

The net and components of the force on the object are,

= cos sin

sin cos

x w

y w

x y

D dF p dA dA

L dF p dA dA

τ θ

θ τ

= +

= = − +

∫ ∫ ∫∫ ∫ ∫

Nondimensional lift coeff. CL and drag coeff. CD

AUDC

AULC DL 2

212

21

,ρρ

== A: charateristic area, frontal area orplanform area?

Note: When CD or CL=1, then D or L equals the dynamic pressure on A.

Velocity and pressure distribution around an air foil

(From S.R. Turns, Thermal-Fluid Sciences, Cambridge Univ. Press, 2006)

Drag and lift due to pressure and friction: example of an air foil

(From S.R. Turns, Thermal-Fluid Sciences, Cambridge Univ. Press, 2006)

9.1.2 Characteristics of Flow Past an Object

For typical external flows the most important of parameters are

the Reynolds number

the Mach number

the Froude number, for flow with a free surface

Re Ulρμ

=

UMa c=

UFrgl

=

Flows with Re>100 are dominated by inertia effects, whereas flows with Re<1 are dominated by viscous effects.

Flow past a flat plate (streamlined body)

Figure 9.5Character of the steady, viscous flow past a flat plate parallel to the upstream velocity: (a) low Reynolds number flow, (b) moderate Reynolds number flow, (c) large Reynolds number flow.

Steady flow past a circular cylinder (blunt body)

The velocity gradients within the boundary layer and wake regions are much larger than those in the remainder of the flow fluid.Therefore, viscous effects are confined to the boundary layer and wake regions.

V9.2 Streamlined and blunt bodies

9.2 Boundary Layer Characteristics9.2.1 Boundary Layer Structure and Thickness on a Flat Plate

Figure 9.7Distortion of a fluid particle as it flows within the boundary layer.

Boundary layer thickness, δδ =y where u=0.99U

5 6Re 2 10 ~ 3 10

depending on surface roughness and amount of turbulence in upstream flow

xcrUxρμ

= = × ×

Figure 9.9Typical characteristics of boundary layer thickness and wall shear stress for laminar and turbulent boundary layers.

Transition occurs at

Boundary Layer Structure and Thickness on a Flat Plate

V9.3 Laminar boundary layerV9.4 Laminar/turbulent transition

Boundary layer displacement thickness

( ) or

*

0

*

0

1

bU U u bdy

u dyU

δ

δ

δ

∞

∞

= −

⎛ ⎞= −⎜ ⎟⎝ ⎠

∫

∫

Figure 9.8 (p. 472)Boundary layer thickness: (a) standard boundary layer thickness, (b) boundary layer displacement thickness.

(9.3)

Boundary layer displacement thicknessRepresents the amount that the thickness of the body must be increased so that the fictitious uniform inviscid flow has the same mass flow rate properties as the actual viscous flow.It represents the outward displacement of the streamlines caused by the viscous effects on the plate.

Ex 9.3 Determine the velocity U=U(x) of the air within the duct but outside of the boundary layer with * 1 20.004( )xδ =

Boundary layer momentum thickness

Boundary layer momentum thickness θ

( ) 2

0

0

( )

1

u U u dA b u U u dy bU

u u dyU U

ρ ρ ρ θ

θ

∞

∞

− = − =

⎛ ⎞= −⎜ ⎟⎝ ⎠

∫ ∫

∫ (9.4)

9.2.2 Prandtl/Blasius Boundary Layer Solution

2-D Navier-Stokes Equations

2 2

2 2

2 2

2 2

1

1

0

u u p u uu vx y x x y

v v p v vu vx y y x y

u vx y

νρ

νρ

⎛ ⎞∂ ∂ ∂ ∂ ∂+ = − + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

⎛ ⎞∂ ∂ ∂ ∂ ∂+ = − + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

∂ ∂+ =

∂ ∂

Elliptic equation

The Navier-Stokes equations are too complicated that no analytical solution is available. However, for large Re, simplified boundary layer equations can be derived.

(9.5)

(9.6)

Boundary layer assumption

so that and x v ux y

δ ∂ ∂<< << <<

∂ ∂

2

2

Thus the equations become,

0parabolic equation

1

u vx y

u u p uu vx y x y

νρ

∂ ∂+ =

∂ ∂

∂ ∂ ∂ ∂+ = − +

∂ ∂ ∂ ∂

Physically, the flow is parallel to the plate and any fluid is convected downstream much more quickly than it is diffused across the streamlines.For boundary layer flow over a flat plate the pressure is constant. The flow represents a balance between viscous and inertial effects, with pressure playing no role.

The boundary layer assumption is based on the fact that the boundary layer is thin.

(9.8)

(9.9)Detailed derivation is given in Cengel & Cimbala, Fluid Mechanics, 2006, pp. 516-519.

Boundary conditions

2

2 2

BCs: 0 , 0 ,

It can be argued that the velocity profile should be similar =

In addition, order of magnitude analysis leads to:

, ~

u v yu U y

u ygU

u u u uu uu vx y y x

δ

ννδ

= = =→ →∞

⎛ ⎞→ ⎜ ⎟⎝ ⎠

∂ ∂ ∂+ =

∂ ∂ ∂1

22

,

~ , ~x xU Uν νδ δ ⎛ ⎞

⎜ ⎟⎝ ⎠

Similarity variables

( ) ( ) ( )

12

12

Define the dimensionless similarity variable

~ ,

and stream function

where is an unknown function

U yyx

xU f f

ην δ

ψ ν η η

⎛ ⎞= ⎜ ⎟⎝ ⎠

=

( ) ( ) ( )

( ) ( )

121

2

12

Thus

where 4

Uu xU f Ufy x

U dv f f dx x

ψ ν η ην

ψ ν η η

⎛ ∂ ⎛ ⎞⎜ ′ ′= = =⎜ ⎟∂⎜ ⎝ ⎠⎜

∂ ⎛ ⎞⎜ ′′= − = − =⎜ ⎟⎜ ∂ ⎝ ⎠⎝Then the parabolic equation (9.9) becomes 2 0with the boundary conditions (0) (0) 0 at 0, ( ) 1 as

f ff

f f fη η

′′′ ′′+ =

′ ′= = = ∞ → →∞

(9.14a)

(9.14b)

Blasius solutions

( )

12

from the numerical solution 0.99 5.0 (Table 9.1)

5 , 5

5or ReRe x

x

u fU

uU

U xx U

Uxx

η

η

νη δ δν

δν

′=

≈ =

⎛ ⎞= = =⎜ ⎟⎝ ⎠

= =

HW: Derive (9.14&9.18) and solve using Matlab to get Table 9.1

Blasius solution

Figure 9.10Blasius boundary layer profile: (a) boundary layer profile in dimensionless form using the similarity variable ŋ. (b) similar boundary layer profiles at different locations along the flat plate.

Blasius solution

*

32

0

1.721displacement thickness: from (9.3) Re

0.664momentum thickness: from (9.4) Re

boundary layer Shear stress:

0.332

x

x

wy

x

x

u Uy x

δ

θ

ρμτ μ=

→ =

→ =

∂= =

∂

(9.16)

(9.17)

(9.18)

: For fully developed pipe flow shear stress, (why?)w Uτ ∝Note

9.2.3 Momentum-Integral Boundary Layer Equation for a Flat Plate

One of the important aspects of boundary layer theory is the determination of the drag caused by shear forces on a body.Consider a uniform flow past a flat plate

Figure 9.11Control volume used in the derivation of the momentum integral equation for boundary layer flow.

(1) (2)

2 2 2

(1) (2) 0

-component of momentum equation:

or

( )

x

x w wplate plate

x

F uV ndA uV ndA

F D dA b dx

D U U dA u dA D U bh b u dyδ

ρ ρ

τ τ

ρ ρ ρ ρ

= +

= − = − = −

− = − + → = −

∑ ∫ ∫

∑ ∫ ∫

∫ ∫ ∫

uv v uv v

2

0 0 0

2 2 2

0 0 0 0

2 2 2

0

Since ,

( )

(1 ) , and

Uh udy U bh bU udy b Uudy

D U bh b u du b Uudy b u dy b u U u dy

u u dD dD bU dy bU bUU U dx dx

δ δ δ

δ δ δ δ

δ

ρ ρ ρ

ρ ρ ρ ρ ρ

θρ ρ θ ρ

= = =

= − = − = −

= − = =

∫ ∫ ∫

∫ ∫ ∫ ∫

∫

The increase in drag per length of the plate occurs at the expense of an increase of the momentum boundary layer thickness, which represents a decrease in the momentum of the fluid.

2 momentum integral equation by von Karmanw w wdD ddD bdx b Udx dx

θτ τ τ ρ= → = ∴ = →Q

Momentum integral equation

If we know the velocity distributions, we can obtain drag or shear stress.The accuracy of these results depends on how closely the shape of the assumed velocity profile approximates the actual profile.

2 momentum integral equationwdUdxθτ ρ= →

Example 9.4

2

0

w

w

Uyu y u U y

dUdx

U

δ δδ

θτ ρ

τ μδ

= ≤ ≤ = >

=

=

( )

0 011 2 3

2

0 0 02

22

1 1

12 3 6

6 , 6

6 12 2

u u u udy dyU U U U

y y y ydy y y dy

U U d d dxdx U

x xU U

δ

δ

θ

δδ δδ δ

μ ρ δ μδ δδ ρδ μ μδ

ρ ρ

∞ ⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎛ ⎞= − = − = − =⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

= =

= ⇒ =

∫ ∫

∫ ∫

3 32 2 2

or 3.46 compare with Blasius solution 5

3.46

0.5766

0.289 0.332w w

xU x Ux

x Ux

xU

dU U Udx x x

μ δ μδρ ρ

δ μρ

δ μθρ

θ ρμ ρμτ ρ τ

= ⇔ =

=

= =

= = ⇔ =

Figure 9.12 Typical approximate boundary layer profiles used in the momentum integral equation.

Momentum integral equation for general profile of u/U

1 21 2

2

1 20

2 2

2 0.664 2 (Blasius solution: )1 Re Re2

8 1.328 , where Re (Blasius solution: )1 1 Re Re2 2

wf f

x x

wf

Df l Df

C Cc C C c

UxU

b dxD C C UC CU b U b

τ μρρ

τρμρ ρ

= = = =

= = = = =∫l

l l

l

l l

12 2 2

10 0

200

2 1

Let ( ) for 0 1, 1 for 1 , where

B.C.: (0) 0 , (1) 1

(1 ) ( )[1 ( )]

we can obtain that

2 / (

Re

wYy

x

u u yg Y Y Y YU U

g g

u uD bU dy bU g Y g Y dY bU CU U

u U dg U Cy dY

C Cx

δ

δ

ρ ρ δ ρ δ

μ μτ μδ δ

δ

==

= ≤ ≤ = > =

= =

= − = − =

∂= = =

∂

=

∫ ∫

31 2 22

5Blasius solution: )Re

2

x

w

x

C CU C Ux

δ

μ ρμτδ

=

= =

1

10

20

( )[1 ( )]

Y

C g Y g Y dY

dgCdY =

= −

=

∫

9.2.4 Transition from Laminar to Turbulent Flow

The analytical results are restricted to laminar boundary layer along a flat plate with zero pressure gradient.Transition to turbulent boundary layer occurs at

5 6,crRe 2 10 3 10x = × ×

Figure 9.9 Typical characteristics of boundary layer thickness and wall shear stress for laminar and turbulent boundary layers.

Example 9.5 Boundary layer transition

Figure 9.14 (p. 484)Typical boundary layer profiles on a flat plate for laminar, transitional, and turbulent flow (Ref. 1).

Figure 9.13 (p. 483)Turbulent spots and the transition from laminar to turbulent boundary layer flow on a flat plate. Flow from left to right. (Photograph courtesy of B. Cantwell, Stanford University.

V9.5 Transition on flat plate

9.2.5 Turbulent Boundary Layer Flow

The structure of turbulent boundary layer flow is very complex, random, and irregular, with significant cross-stream mixing.Empirical power-law velocity profile is a reasonable approximation.

2

To begin with the momentum integral equaion:

wdUdxθτ ρ=

Example 9.6

17 1

7

14

2

, 1Assume

, 1and the experimentally determined formula:

0.0225w

yY Yu y YU u U Y

UU

δδ

ντ ρδ

⎧ = <⎪⎛ ⎞= = ⎨⎜ ⎟⎝ ⎠ ⎪ = >⎩

⎛ ⎞= ⎜ ⎟⎝ ⎠ (1)

( )

2

1 11 1

7 7

0 0 0

14

2 2

7 1 1 172

where is still unknown. To determine , we combine Eq. 1 with Eq. 2 to get the ODE:

7 0.0225 72

wdUdx

u u u udy dY Y Y dYU U U U

dU UU dx

θτ ρ

θ δ δ δ

δδ

ν δρ ρδ

∞

=

⎛ ⎞ ⎛ ⎞= − = − = − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ =⎜ ⎟⎝ ⎠

∫ ∫ ∫

141

4

15 4

51

5

or 0.231

with BC: 0 0, we obtain

0.370 0.370 , or Rex

d dxU

at x

xU x

νδ δ

δ

ν δδ

⎛ ⎞= ⎜ ⎟⎝ ⎠

= =

⎛ ⎞= =⎜ ⎟⎝ ⎠

(2)

( )11 1 51 4* 7 5

0 0 01

5 4 *5

1 1 1 0.04638

0.036 , 72

u udy dY Y dY xU U U

xU

δ νδ δ δ

δ νθ θ δ δ

∞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − = − = − = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞= = < <⎜ ⎟⎝ ⎠

∫ ∫ ∫

( )

14

22

1 4 15 5 5

15

2 21

50 0

152

0.0288 0.02250.37 / Re

0.0288 0.036 , Re

0.072 1 Re2

: The above results are valid for smooth flat p

w

x

f w

fDf

UUU U x

AD b dx b U dx U A bUx

DC

U A

ν ρτ ρν

ντ ρ ρ

ρ

⎡ ⎤⎢ ⎥= =⎢ ⎥⎣ ⎦

⎛ ⎞= = = =⎜ ⎟⎝ ⎠

= =

∫ ∫

Note

l l

l

l

l

5 7x

4 15 5

1 12 2

lates with 5 10 <Re < 10

Turbulent B.L.: ,

Laminar B.L.: ,

w

w

x x

x x

δ τ

δ τ

−

−

×

Friction drag coefficient for a flat plate

Although Fig. 9.15 is similar with Moody diagram (pipe flow) of Fig. 8.23, the mechanisms are quite different.

For fully developed pipe flow, fluid inertia remains constant and the flow is balanced between pressure forces and viscous forces.

For flat plate boundary layer flow, pressure remains constant and the flow is balanced between inertia effects and viscous forces.

Figure 9.15 Friction drag coefficient for a flat plate parallel to the upstream flow.

Empirical equations for CDf

Example 9.7

9.2.6 Effects of Pressure GradientInviscid flow over a cylinder

Viscous flow over a cylinder

Viscous flow

Viscous flow over cylinder: velocity-pressure

Separated flow

No matter how small the viscosity, provided it is not zero, there will be a boundary layer that separates from the surface, giving a drag that is, for the most part, independent of the value of μ.

9.2.7 Momentum-Integral Boundary Layer Equation with Nonzero Pressure Gradient

Outside the boundary layer

Momentum integral boundary layer equation

2

constant2

fs

fsfs

Up

dUdp Udx dx

ρ

ρ

+ =

= −

( )2 * 2(for , )fsw fs fs w

dUd dU U U C Udx dx dx

θτ ρ θ ρδ τ ρ= − + ⇔ = =

This equation represents a balance between viscous forces ( ), pressure forces( ), and the fluid momentum( ).Eq. 9.35 can be used to provide information about the boundary layer thickness, wall shear stress, etc,

wτθ

fsfs

dUdp Udx dx

ρ= −

(9.34)

(9.35)

Derivation of equation (9.35)

( )

( ) ( ) ( )

( ) ( ) ( )

2

0 0 0 0

0 (a)

1 (b)

(b)- (a)1

1

u vx y

u u Uu v Ux y x y

u UUuU u U u vU vu

y x x y

Udy u U u dy U u dy vU vu dyy x x y

δ δ δ δ

τρ

τρ

τρ

∂ ∂+ =

∂ ∂∂ ∂ ∂ ∂

+ = +∂ ∂ ∂ ∂

−

∂ ∂ ∂ ∂− = − + − + −

∂ ∂ ∂ ∂

∂ ∂ ∂ ∂− = − + − + −

∂ ∂ ∂ ∂∫ ∫ ∫ ∫

( ) ( )

( ) ( )

( )

0 0 0

0 0

2

0 0

2 *

1

1 1

Udy u U u dy U u dyy x x

Uu U u dy U u dyx x

u u U uU dy U dyx U U x U

UU Ux x

δ δ δ

δ δ

δ δ

τρ

τρ

θ δ

∂ ∂ ∂− = − + −

∂ ∂ ∂

∂ ∂= − + −∂ ∂

⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞= − + −⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦∂ ∂

= +∂ ∂

∫ ∫ ∫

∫ ∫

∫ ∫

(9.35)

v=0 at y=0,δ

9.3 DragAny object moving through a fluid will experience a drag D -- a net force in the direction of flow due to pressure (pressure drag) and shear stress (friction drag) on the surface of the object.

( )212

Drag coefficient:

shape, Re, Ma, Fr, /DDCU A

φ ερ

= = l

These are determined experimentally, and very few can be obtained analytically.

9.3.1 Friction Drag

Drag due to the shear stress , on the object.wτ

For large Re, the friction drag is generally smaller than pressure drag.However, for highly streamlined bodies or for low Reynolds number flow, most of the drag may be due to friction drag.Drag on a plate of width b and length l

21 2

where is the friction drag coefficient.

f Df

Df

D U b C

C

ρ= l

L

b

For laminar flow, is independent of

For turbulent flow, is function of

along the surface of a curved body is quite difficult to obtain

Df

Df

Df

CD

CD

C

ε

ε

Friction Drag

Example 9.8 Friction drag coeff. for a cylinder

9.3.2 Pressure Drag

Drag due to pressure on the object.Pressure drag – also called form drag because of its strong dependency on the shape or form of the object.

0

2

02 2

cos : pressure coefficient12

cos cos : reference pressure1 1

2 2

p p

ppDp

p pD p dA CU

p dA C dADC p

AU A U A

θρ

θ θ

ρ ρ

−= =

= = =

∫

∫ ∫

Pressure drag--Large Reynolds number

( )

( )20

02

2 2

For large Reynolds number, inertial effect viscous effect1 dynamic pressure2

cos : pressure coefficient/ 2

cos cos 1 1

2 2Therefore, i

p p

ppDp

p

p p U

p pD p dA CU

p dA C dADC

AU A U A

C

ρ

θρ

θ θ

ρ ρ

− ∝

−= =

= = =

∫

∫ ∫

( )s independent of Reynolds number,

is also independent of Reynolds number Re 1DpC

( )

( )

2 2

For very small Reynolds number inertial effect viscous effect

viscous stress1

1 1 Re2 2

Comparisons:1For laminar pipe flow

ReFor large Reynolds number constant

pDp

w

U Up Dl lCU UlU A Ul

f

f

μ μ μρρ ρτ μ

⎧ ∝⎪⎪ =⎨⎪ ∝⎪⎩

( ) pipe flow

Pressure drag--low Reynolds number

Example 9.9 Pressure drag coeff. for a cylinder

For 0, the pressure drag on any shaped object (symmetricalor not) in a steady flow would be zero.For 0, the net pressure drag may be non-zero because of boundary layer separation.

μ

μ

=

≠

Viscosity DependenceInviscid, Viscous flows over object

9.3.3 Drag Coefficient Data and Examples

shape, Re, Ma, Fr, DC εφ ⎛ ⎞= ⎜ ⎟⎝ ⎠l

Figure 9.20Two objects of considerably different size that have the same drag force: (a) circular cylinder CD = 1.2; (b) streamlined strut CD = 0.12.

Shape Dependence

Shape Dependence

Figure 9.19Drag coefficient for an ellipse with the characteristic area either the frontal area, A = bD, or the planform area, A = bl (Ref. 5).

Reynolds Number Dependence

2

2 22 2

viscous force pressure force ( , , )From dimensional analysis [ ( / ) ]

2 2 , Re1 Re2

D

D f U l

D C lU C U l lD C lU C UlC

U lU l

μ

μ μμ ρ

ρ μρ

≈=

= = ⋅

= = = =

Note: For Re<1, streamlining increases the drag due to an increase in the area on which shear force act.

Low Reynolds number, Re<1

Ex 9.10: particle in the water dragged up by upward motion

( )

2 2 2

2 2 2

2 2

3 3

2 2 2 2

0.1 , 2.3, to determine

= , , 6 6

24 Re 1Re

1 1 24 32 4 2 4 /

B

sand H O B H O H O

D

H O D H O H OH O H O

D mm SG UW D F

W SG D F D

C For

D U D C U D UDUD

π πγ γ γ γ

π πρ ρ πμρ μ

= == +

∀ = = ∀ =

= <

⎛ ⎞= = =⎜ ⎟⎜ ⎟

⎝ ⎠

( )

( )

2 2 2

2 2

2 2

3 3

23

33 2

36 6

6.32 1018

For 15.6 , 999 , 1.12 10

Re 0.564 Re 1

H O H O H O

H O H O

H O H O

SG D UD D

g

SG gD mU s

kg N SC m mUD

π πγ πμ γ

γ ρ

ρ ρ

μ

ρ μ

ρμ

−

−

∴ = +

=

−= = ×

° = = ×

= = <

V9.7 Skydiving practice

Moderate Reynolds NumberCD of cylinder and sphere

3 1/ 2

3 5

Re 10 Re [Re , (not )]

10 Re 10 constantD D

D

C C D

C

−< ↑ ↓

< <

Flow over cylinder at various Reynolds number

Re 1 , no separation Re 10, steady separation bubble< ≈

Re 100, vortex shedding starts at Re 47≈ ≈

V9.8 Karman vortex street

Flow over cylinder at various Reynolds number

4Re 5 10≈ × 5Re 4 10≈ × from M. Van Dyke, An Album of Fluid Motion

Laminar and Turbulent Boundary Layer Separation

from M. Van Dyke, An Album of Fluid Motion

Drag of streamlined and blunt bodiesDrag coefficientsThe different relations of CD with Re for objects with various streamlining are illustrated in Fig. 9.22.

Note: In a portion of the range 105<Re<106, the actual drag (not just CD) decreases with increasing speed.

For streamlined bodies, CD when the boundary layer becomes turbulent.For blunt objects, CD when the boundary layer becomes turbulent.

V9.9 Oscillating signV9.10 Flow past a flat plateV9.11 Flow past an ellipse

Example 9.11 Terminal velocity of a falling object

2 2

2 2

12

, 12 4

12 4

43

B

ice B air

air D B

ice air B

air D ice

ice

air D

W D FW V F V

U D C W F

W F

U D C W V

gDUC

γ γπρ

γ γπρ γ

ρρ

= += =

= −

∴

= =

⎛ ⎞= ⎜ ⎟⎝ ⎠

Compressibility EffectsWhen the compressibility effects become important (Re, Ma)DC φ=

Notes: 1. Compressibility effect is negligible for Ma<0.5.

2. CD increases dramatically near Ma=1, due to the existence of shock wave.

Surface Roughness Effects

In general,

Streamlines bodies:

Blunt bodies: constant (pressure drag)

until transition to turbulence occurs and the wake region becomes considerably narrower so that the pressure dr

D

D

CD

CD

ε

ε

↑→ ↑

↑→

ag drops. (Fig. 9.25)

Surface roughness influences drag when the boundary layer is turbulent, because it protrudes through the laminar sublayer and alters the wall shear stress.In addition, surface roughness can alter the transitional critical Re and change the net drag.

Surface Roughness5

4

4

A well-hit golf ball has Re of O(10 ), and the dimpled golf ball has a critical Reynolds number 4 10 dimples reduce

Table tennis Reynolds number is less than 4 10 no need of dimplesDC× →

× →

,rough

,smooth

0.25 0.50.5

D

D

CC

= =

dimpled

Example 9.12 Effect of Surface Roughness for golf ball and table tennis ball

Froude Number Effects (flows with free surface)When the free surface is present, the wave-making effects becomes important, so that (Re, Fr)DC φ=

Figure 9.26 (p. 507)Typical drag coefficient data as a function of Froude number and hull characteristics for that portion of the drag due to the generation of waves.

Composite body dragApproximate drag calculations for a complex body by treating it as composite collection of its various parts.e.g., drag on an airplane or an automobile.The contributions of the drag due to various portions of car (i.e., front end, windshield, roof, rear end, etc.) have been determined. As a result it is possible to predict the aerodynamic drag on cars of a wide variety ouf body styles (Fig. 9.27).

Example 9.13 Drag on a composite body

V9.14 Automobile streamlining

9.4 Lift (L)

Lift -- a force that is normal to the free stream.

2, shape, Re, Ma, Fr, 1

2

L LLC C

lU A

εφρ

⎛ ⎞= = ⎜ ⎟⎝ ⎠

For aircraft, For car, for better traction and cornering ability

LL

↑

↓

9.4.1 Surface Pressure Distribution

Froude no. – free surface presentε is relatively unimportantMa is important for Ma > 0.8Re effect is not greatThe most important parameter that affects the lift coefficient

is the shape of the object.

Figure 9.31Pressure distribution on the surface of an automobile.

Surface Pressure DistributionCommon lift-generating devices (airfoils, fans,…) operate at large Re. Most of the lift comes from the surface pressure distribution.For Re<1, viscous effect and pressure are equally important to the lift (for minute insects and the swimming of microscopic organisms).

Example 9.14 Lift from pressure and shear distributions

Airfoil

For airfoils, the characteristic area A is the planformarea in the definition of both CLand CD. For a rectangular planform wing, A=bc, where c is the chord length, b is the length of the airfoil.Typical CL ~O(1), i.e., L~(ρU2/2)A,and the wing loading L/A~(ρU2)/2:

Aspect ratio A = b2/A (=b/c if c is constant)

2

2

Wright Flyer: 1.5 /Boeing 747: 150 /

lb ftlb ft

In general, , L DC C↑⇒ ↑ ↓A

Large A : long wing, soaring airplane, albatross, etc.Small A : short wing, highly maneuverable fighter, falcon

Lift and drag coefficient data as a function of angle of attack and aspect ratio

Figure 9.33 Typical lift and drag coefficient data as a function of angle of attack and the aspect ratio of the airfoil: (a) lift coefficient, (b) drag coefficient.

Ratio of lift to drag/Lift-drag polar

Although viscous effects contributes little to the direct generation of lift, viscosity-induced boundary layer separation can occur when α is too large to lead to stall.

V9.15 Stalled airfoil

Lift of airfoil with flapLift can be increased by adding flap

Figure 9.35 Typical lift and drag alterations possible with the use of various types of flap designs (Ref. 21).

From Cengel and Cimbala, Fluid Mechanics, McGraw Hill, 2006

V9.17 Trailing edge flap

V9.18 Leading edge flap

Ex. 9.15

9.4.2 Circulation2-D symmetric air foil

--by inviscid theory

--adding circulation

Note: The circulation needed is a function of airfoil size and shape and can be calculated from potential flow theory.(Section 6.6.3)L= -ρUΓ (Kutta–Joukowski Law)

Kutta condition—The flow over both the topside and the underside join up at the trailing edge and leave the airfoil travelling parallel to one another.

From Cengel and Cimbala, Fluid Mechanics, McGraw Hill, 2006

CirculationFlow past finite length wing

Figure 9.37 (p. 546)Flow past a finite length wing: (a) the horseshoe vortex system produced by the bound vortex and the trailing vortices: (b) the leakage of air around the wing tips produces the trailing vortices.

Vee-formation of bird -- 25 birds fly 70% farther than 1 bird

V9.19 Wing tip vortex

V4.6 Flow past a wing

CirculationFlow past a circular cylinder

A rotating cylinder in a stationary real fluid can produce circulation and generate a lift—Magnus effect

EXAMPLE 9.16 Lift on a rotating table tennis ball

Figure 9.38Inviscid 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 and a lift.