Flow Over Immersed Bodies

Chapter 9

Flow over Immersed

Bodies

z We consider flows over

bodies that are immersed in a fluid and the flows are

termed external flows.

z 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

9.1 General External Flow Characteristics

z A body immersed in a moving fluid experiences a

resultant force due to the interaction between the body and the fluid surrounding it.

z Object and flow relation:

stationary air with moving object or flowing air with stationary object

z Flow classification:

z Body shape:

Streamlined bodies or blunt bodies

Figure 9.2

9.1.1 Lift and Drag Concepts

z When any body moves through a fluid, an interaction

between the body and the fluid occurs

z This can be described in terms of the stresses-wall

shear stresses due to viscous effect and normal stresses due to the pressure P

z Drag, D

the resultant force in the direction of the upstream velocity

z Lift, L

the resultant force normal to the upstream velocity

Resultant force--lift and drag

z Resultant force ( ) cos ( ) sin and ( ) 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 = + = = − +

∫

∫

∫

∫

∫

∫

z Nondimensional lift coeff. C_L and drag coeff. C_D

A U D C A U L C_{L D 2} 2 1 2 2 1 ρ , = ρ

= A: charateristic area, frontal area or

planform area?

Velocity and pressure distribution around an air foil

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

9.1.2 Characteristics of Flow Past an Object

z

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

ρ

Ul

μ

= U Ma c = U Fr gl =

z Flows with Re>100 are dominated by inertia effects, whereas

Flow past a flat plate (streamlined body)

Figure 9.5

Character 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)

z The velocity gradients

within the boundary

layer and wake regions are much larger than those in the remainder of the flow fluid.

z Therefore, viscous

effects are confined to the boundary layer and wake regions.

9.2 Boundary Layer Characteristics

9.2.1 Boundary Layer Structure and Thickness on a Flat Plate

Figure 9.7

Distortion of a fluid particle as it flows within the boundary layer. z Boundary layer thickness,

δ

5 6

Re 2 10 ~ 3 10

depending on surface roughness and amount of turbulence in upstream flow

xcr Ux ρ μ = = × × Figure 9.9

Typical 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 layer

Boundary layer displacement thickness

(

)

or * 0 * 0 1 bU U u bdy u dy U δ

δ

δ

∞ ∞ = − ⎛ ⎞ = _⎜ − _⎟ ⎝ ⎠

∫

∫

Figure 9.8 (p. 472)

Boundary layer thickness: (a) standard boundary layer thickness, (b) boundary layer displacement thickness.

Boundary layer displacement thickness

z Represents 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.

z 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 2

0.004( )x

Boundary layer momentum thickness

z

Boundary layer momentum thickness θ

(

)

2 0 0 ( ) 1 u U u dA b u U u dy bU u u dy U U ρ ρ ρ θ θ ∞ ∞ − = − = ⎛ ⎞ = _⎜ − _⎟ ⎝ ⎠

∫

∫

∫

(9.4)

9.2.2 Prandtl/Blasius Boundary Layer Solution

z

2-D Navier-Stokes Equations

2 2 2 2 2 2 2 2 1 1 0 u u p u u u v x y x x y v v p v v u v x y y x y u v x y ν ρ ν ρ ⎛ ⎞ ∂ ∂ ∂ ∂ ∂ + = − + _⎜ + _⎟ ∂ ∂ ∂ _⎝ ∂ ∂ _⎠ ⎛ ⎞ ∂ ∂ ∂ ∂ ∂ + = − + _⎜ + _⎟ ∂ ∂ ∂ _⎝ ∂ ∂ _⎠ ∂ ∂ + = ∂ ∂ Elliptic equation

z 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)

Boundary layer assumption

so that and x v u x y δ << << ∂ << ∂ ∂ ∂ 2 2

Thus the equations become,

0 parabolic equation 1 u v x y u u p u u v x y ρ x ν y ∂ ₊ ∂ ₌ ∂ ∂ ∂ ₊ ∂ _{= −} ∂ ₊ ∂ ∂ ∂ ∂ ∂

z Physically, the flow is parallel to the plate and any fluid is

convected downstream much more quickly than it is diffused across the streamlines.

z 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.

z 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

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 y u U y u y g U u u u uu u u v x y y x δ ν ν δ = = = → → ∞ ⎛ ⎞ → _{⎜ ⎟} ⎝ ⎠ ∂ ₊ ∂ ₌ ∂ ∂ ∂ ∂ 1 2 2 , ~ x , ~ x U U ν ν δ δ ⎛_⎜ ⎞_⎟ ⎝ ⎠

Similarity variables

(

)

( )

( )

1 2 1 2

Define the dimensionless similarity variable

~ ,

and stream function

where is an unknown function

U y y x xU f f η ν δ ψ ν η η ⎛ ⎞ = ⎜ ⎟_{⎝ ⎠} =

(

)

( )

( )

(

)

( )

1 2 1 2 1 2 Thus where 4 U u xU f Uf y x U _d v f f d x x ψ _{ν η η} ν ψ ν _η η ⎛ _{∂ ⎛ ⎞} ⎜ = = ′ _{⎜ ⎟} = ′ ∂ ⎜ ⎝ ⎠ ⎜ ∂ _{⎛ ⎞} ⎜ _{= − = ′− ′ =} ⎜ ⎟ ⎜ _{∂ ⎝ ⎠} ⎝

Then the parabolic equation (9.9) becomes

2 0

with the boundary conditions

(0) (0) 0 at 0, ( ) 1 as f ff f f η f η ′′′+ ′′ = ′ ′ = = = ∞ → → ∞ (9.14a) (9.14b)

Blasius solutions

( )

1 2

from the numerical solution 0.99 5.0 (Table 9.1)

5 , 5 5 or Re Re_x x u f U u U U x x U Ux x η η ν η δ δ ν δ ν ′ = ≈ = ⎛ ⎞ = = _{⎜ ⎟} = ⎝ ⎠ = =

HW: Derive (9.14&9.18) and

solve using Matlab to get Table 9.1

Blasius solution

Figure 9.10

Blasius 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

* 3 2 0 1.721 displacement thickness: from (9.3)

Re 0.664 momentum thickness: from (9.4)

Re boundary layer Shear stress:

0.332 x x w y x x u U y x δ θ ρμ τ μ = → = → = ∂ = = ∂ (9.16) (9.17) (9.18)

: For fully developed pipe flow shear stress, τ_w ∝U (why?)

9.2.3 Momentum-Integral Boundary Layer

Equation for a Flat Plate

z One of the important aspects of boundary layer theory

is the determination of the drag caused by shear forces on a body.

z Consider a uniform flow past a flat plate

Figure 9.11

Control 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 w plate plate x F uV ndA uV ndA F D dA b dx D U U dA u dA D U bh b u dy δ 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 d D bU dy bU bU U U dx dx δ δ δ δ δ δ δ δ ρ ρ ρ ρ ρ ρ ρ ρ θ ρ 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 Karman

w w w dD d dD bdx b U dx dx θ τ τ τ ρ = → = ∴ = → Q

Momentum integral equation

z If we know the velocity distributions, we can obtain

drag or shear stress.

z The accuracy of these results depends on how closely

the shape of the assumed velocity profile approximates the actual profile.

2

momentum integral equation

w d U dx θ τ = ρ → Example 9.4 2 0 w w Uy u y u U y d U dx U δ δ δ θ τ ρ τ μ δ = ≤ ≤ = > = =

(

)

0 0 1 1 2 3 2 0 0 ₀ 2 2 2 1 1 1 2 3 6 6 , 6 6 12 2 u u u u dy dy U U U U y y y y dy y y dy U U d d dx dx U x x U U δ δ θ δ δ δ δ δ μ ρ δ _{δ δ} μ δ ρ δ μ _δ μ ρ ρ ∞ ⎛ ⎞ ⎛ ⎞ = _⎜ − _⎟ = _⎜ − _⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ = _⎜ − _⎟ = − = _⎜ − _⎟ = ⎝ ⎠ _{⎝ ⎠} = = = ⇒ =

∫

∫

∫

∫

3 3 2 _{2 2}

or 3.46 compare with Blasius solution 5

3.46 0.576 6 _w 0.289 _w 0.332 x U x Ux x Ux x U d U U U dx x x μ δ μ δ ρ ρ δ μ ρ δ μ θ ρ θ ρμ ρμ τ ρ τ = ⇔ = = = = = = ⇔ =

Figure 9.12

Typical approximate boundary layer profiles used in the momentum

Momentum integral equation for general profile of u/U 1 2 1 2 2 1 2 0 2 2 2 _0.664 2 (Blasius solution: ) 1 _{Re Re} 2 8 _1.328

, where Re (Blasius solution: )

1 1 _{Re Re} 2 2 w f f x x w f Df l Df C C c C C c Ux U b dx D C C _U C C U b U b τ μ ρ ρ τ ρ μ ρ ρ = = = = = =

∫

= = = l l l l l l 1 2 2 2 1 0 0 2 0 0 2 1

Let ( ) for 0 1, 1 for 1 , where B.C.: (0) 0 , (1) 1

(1 ) ( )[1 ( )]

we can obtain that 2 / ( Re w Y y x u u y g Y Y Y Y U U g g u u D bU dy bU g Y g Y dY bU C U U u U dg U C y dY C C x δ δ ρ ρ δ ρ δ μ μ τ μ δ δ δ = = = ≤ ≤ = > = = = = − = − = ∂ = = = ∂ =

∫

∫

3 1 2 2 2 5 Blasius solution: ) Re 2 x w x C C U C U x δ μ ρμ τ δ = = = 1 1 0 2 0 ( )[1 ( )] Y C g Y g Y dY dg C dY ₌ = − =

∫

9.2.4 Transition from Laminar to Turbulent Flow

z The analytical results are restricted to laminar

boundary layer along a flat plate with zero pressure gradient.

z Transition to turbulent boundary layer occurs at

5 6

,cr

Re_x = ×2 10 3 10×

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.

9.2.5 Turbulent Boundary Layer Flow

z The structure of turbulent boundary layer

flow is very complex, random, and irregular, with significant cross-stream mixing.

z Empirical power-law velocity profile is a

reasonable approximation.

2

To begin with the momentum integral equaion: _w U d dx θ τ = ρ Example 9.6 1 7 ₁ 7 1 4 2 , 1 Assume , 1 and the experimentally determined formula:

_w 0.0225 y Y Y u y Y U u U Y U U δ δ ν τ ρ δ ⎧ = < ⎪ ⎛ ⎞ = _{⎜ ⎟} = _⎨ ⎝ ⎠ _{⎪ = >} ⎩ ⎛ ⎞ = _{⎜ ⎟} ⎝ ⎠ (1)

( )

2 1 1 1 1 7 7 0 0 0 1 4 2 2 7 1 1 1 72 where is still unknown.

To determine , we combine Eq. 1 with Eq. 2 to get the ODE:

7 0.0225 72 w d U dx u u u u dy dY Y Y dY U U U U d U U U dx θ τ ρ θ δ δ δ δ δ ν δ ρ ρ δ ∞ = ⎛ ⎞ ⎛ ⎞ = _⎜ − _⎟ = _⎜ − _⎟ = − = ⎝ ⎠ ⎝ ⎠ ⎛ _{⎞ =} ⎜ ⎟ ⎝ ⎠

∫

∫

∫

1 4 1 4 1 5 ₄ 5 1 5 or 0.231 with BC: 0 0, we obtain 0.370 0.370 , or Re_x d dx U at x x U x ν δ δ δ ν δ δ ⎛ ⎞ = _{⎜ ⎟} ⎝ ⎠ = = ⎛ ⎞ = _{⎜ ⎟} = ⎝ ⎠ (2)

( )

1 1 1 ₅ 1 4 * _{7 5} 0 0 0 1 5 ₄ * 5 1 1 1 0.0463 8 0.036 , 72 u u dy dY Y dY x U U U x U δ ν δ δ δ δ ν θ θ δ δ ∞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = _⎜ − _⎟ = _⎜ − _⎟ = − = = _{⎜ ⎟} ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ = = _{⎜ ⎟} < < ⎝ ⎠

∫

∫

∫

(

)

1 4 2 2 1 4 1 5 5 5 1 5 2 2 1 5 0 0 1 5 2 0.0288 0.0225 0.37 / Re 0.0288 0.036 , Re 0.072 1 _Re 2

: The above results are valid for smooth flat p

w x f w f Df U U U U x A D b dx b U dx U A b Ux D C U A ν ρ τ ρ ν ν τ ρ ρ ρ ⎡ ⎤ ⎢ ⎥ = = ⎢ ⎥ ⎣ ⎦ ⎛ ⎞ = = _{⎜ ⎟} = = ⎝ ⎠ = =

∫

∫

Note l l l l l 5 7 x 4 1 5 5 1 1 2 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

z 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

Empirical equations for C

_Df

9.2.6 Effects of Pressure Gradient

Inviscid flow over a cylinder

Viscous flow over a cylinder

Viscous flow over cylinder: velocity-pressure

z Separated flow

z 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

z Outside the boundary layer

z Momentum integral boundary layer equation

2 constant 2 fs fs fs U p dU dp U dx dx ρ ρ + = = −

(

2

)

* 2 (for , ) fs w fs fs w dU d d U U U C U dx dx dx θ τ = −ρ θ + ρδ ⇔ = τ = ρ

z This equation represents a balance between viscous forces

( ), pressure forces( ), and the fluid momentum( ).

z Eq. 9.35 can be used to provide information about the

boundary layer thickness, wall shear stress, etc,

w τ

θ

fs fs dU dp U dx = −ρ dx (9.34) (9.35)

Derivation of equation (9.35)

( )

(

)

( ) ( ) ( ) ( ) ( ) 2 0 0 0 0 0 (a) 1 (b) (b)- (a) 1 1 u v x y u u U u v U x y x y u U U uU u U u vU vu y x x y U dy u U u dy U u dy vU vu dy y x x y δ δ δ δ τ ρ τ ρ τ ρ ∂ ₊ ∂ ₌ ∂ ∂ ∂ ₊ ∂ ₌ ∂ ₊ ∂ ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ − = − + − + − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ − = − + − + − ∂ ∂ ∂ ∂

∫

∫

∫

∫

( ) ( ) ( ) ( )

( )

0 0 0 0 0 2 0 0 2 * 1 1 1 U dy u U u dy U u dy y x x U u U u dy U u dy x x u u U u U dy U dy x U U x U U U U x x δ δ δ δ δ δ δ τ ρ τ ρ θ δ ∂ ∂ ∂ − = − + − ∂ ∂ ∂ ∂ ∂ = − + − ∂ ∂ ⎡ ⎤ ∂ ⎛ ⎞ ∂ ⎛ ⎞ = _{⎢ ⎜} − _⎟ + _⎜ − _{⎟ ⎥} ∂ _{⎣ ⎝ ⎠} ∂ _{⎝ ⎠ ⎦} ∂ ∂ = + ∂ ∂

∫

∫

∫

∫

∫

∫

∫

(9.35) v=0 at y=0,δ

9.3 Drag

z

Any 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.

(

)

2 1

2

Drag coefficient:

shape, Re, Ma, Fr, /

D

D

C

U A

φ

ε

ρ

=

=

_l

These are determined experimentally, and very

few can be obtained analytically.

9.3.1 Friction Drag

z

Drag due to the shear stress , on the object.

τ

_w z

For large Re, the friction drag is generally smaller

than pressure drag.

z

However, for highly streamlined bodies or for low

Reynolds number flow, most of the drag may be

due to friction drag.

z

Drag on a plate of width b and length l

2

1

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 C D C D C ε ε

Friction Drag

9.3.2 Pressure Drag

z Drag due to pressure on the object.

z Pressure drag – also called form drag because of its

strong dependency on the shape or form of the object.

0 2

0

2 2

cos : pressure coefficient 1 2 cos cos : reference pressure 1 1 2 2 p p p p Dp p p D p dA C U p dA C dA D C p A U A U A θ ρ θ θ ρ ρ − = = = = =

∫

∫

∫

Pressure drag--Large Reynolds number

(

)

(

)

2 0 0 2 2 2

For large Reynolds number, inertial effect viscous effect 1

dynamic pressure

2

cos : pressure coefficient

/ 2 cos cos 1 1 2 2 Therefore, i p p p p Dp p p p U p p D p dA C U p dA C dA D C A U A U A C ρ θ ρ θ θ ρ ρ − ∝ − = = = = =

∫

∫

∫

(

)

s independent of Reynolds number,

is also independent of Reynolds number Re 1

Dp

(

)

(

)

2 2

For very small Reynolds number inertial effect viscous effect

viscous stress 1 1 1 _Re 2 2 Comparisons: 1 For laminar pipe flow

Re

For large Reynolds number constant

p Dp w U _U p D l _l C U Ul U A U l f f μ _μ μ ρ ρ ρ τ μ ⎧ _∝ ⎪⎪ ₌ ⎨ ⎪ ∝ ⎪⎩

(

)

pipe flow

Pressure drag--low Reynolds number

For 0, the pressure drag on any shaped object (symmetrical or 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 Dependence

9.3.3 Drag Coefficient Data and

Examples

shape, Re, Ma, Fr,

D

C

= ⎜

φ

⎛

ε

⎞

_⎟

⎝

_l

⎠

Figure 9.20

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

Shape Dependence

Figure 9.19

Drag 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 2 2 2

viscous force pressure force ( , , )

From dimensional analysis

[ ( / ) ] 2 2 , Re 1 _Re 2 D D f U l D C lU C U l l D C lU C Ul C U l U l μ μ μ μ ρ ρ μ ρ ≈ = = = ⋅ = = = =

Note: For Re<1, streamlining

increases the drag due to an increase in the area on

which shear force act.

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 1 Re 1 1 24 3 2 4 2 4 / B sand H O B H O H O D H O D H O H O H O H O D mm SG U W D F W SG D F D C For D U D C U D UD UD π π γ γ γ γ π π ρ ρ πμ ρ μ = = = + ∀ = = ∀ = = < ⎛ ⎞ = = ⎜_⎜ ⎟_⎟ = ⎝ ⎠

(

)

( ) 2 2 2 2 2 2 2 3 3 2 3 3 3 2 3 6 6 6.32 10 18 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 m U s kg _{N S} C m m UD π π γ πμ γ γ ρ ρ ρ μ ρ μ ρ μ − − ∴ = + = − = = × ° = = × = = < V9.7 Skydiving practice

Moderate Reynolds Number

C

_D

of cylinder and sphere

3 1/ 2 3 5 Re 10 Re [Re , (not )] 10 Re 10 constant D 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

Flow over cylinder at various Reynolds number

4

Re ≈ ×5 10 Re ≈ ×4 105 from M. Van Dyke, An

Laminar and Turbulent Boundary Layer

Separation

Drag of streamlined and blunt bodies

Drag coefficients

The different relations of C_D 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 C_D) decreases with increasing speed.

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

V9.9 Oscillating sign

V9.10 Flow past a flat plate V9.11 Flow past an ellipse

Example 9.11

Terminal velocity of a falling object 2 2 2 2 1 2 , 1 2 4 1 2 4 4 3 B ice B air air D B ice air B air D ice ice air D W D F W V F V U D C W F W F U D C W V gD U C γ γ π ρ γ γ π ρ γ ρ ρ = + = = = − ∴ = = ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠

Compressibility Effects

When the compressibility effects become important C_D = φ(Re, Ma)

Notes: 1. Compressibility effect is negligible for Ma<0.5. 2. C_D increases dramatically near Ma=1, due to

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 C D C D ε ε ↑→ ↑ ↑→ ag drops. (Fig. 9.25)

z Surface roughness influences drag when the boundary

layer is turbulent, because it protrudes through the laminar sublayer and alters the wall shear stress.

z In addition, surface roughness can alter the transitional

Surface Roughness

5

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 dimples D C × → × → ,rough ,smooth 0.25 0.5 0.5 D D C C = = 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

C_D =

φ

(Re, Fr)

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 drag

z Approximate 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.

z 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)

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

2

, shape, Re, Ma, Fr, 1 2 L L L C C l U A ε φ ρ ⎛ ⎞ = _{= ⎜ ⎟} ⎝ ⎠ For aircraft,

For car, for better traction and cornering ability

L L

↑ ↓

9.4.1 Surface Pressure Distribution

Froude no. – free surface present

ε

is relatively unimportant Ma is important for Ma > 0.8 Re effect is not great

The most important parameter that affects the lift coefficient is the shape of the object.

Figure 9.31

Pressure distribution on the surface of an automobile.

Surface Pressure Distribution

z Common lift-generating devices (airfoils, fans,…)

operate at large Re. Most of the lift comes from the surface pressure distribution.

z For Re<1, viscous effect and pressure are equally

important to the lift (for minute insects and the swimming of microscopic organisms).

Airfoil

z For airfoils, the characteristic area A is the planform

area in the definition of both C_Land C_D. For a

rectangular planform wing, A=bc, where c is the chord length, b is the length of the airfoil.

z Typical C_L ~O(1), i.e., L~(

ρ

U2/2)A,

and the wing loading L/A~(

ρ

U2_)/2:

z Aspect ratio

A

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

2 2 Wright Flyer: 1.5 / Boeing 747: 150 / lb ft lb ft In general, A ↑⇒ C_L ↑, C_D ↓

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

z 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.

Lift of airfoil with flap

z Lift 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

9.4.2 Circulation

2-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.

Circulation

Flow 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.

z Vee-formation of bird -- 25 birds fly 70% farther than 1 bird V9.19 Wing tip vortex

Circulation

Flow past a circular cylinder

z 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 9.38

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 and a lift.