Chapter 9 2006 02user.engineering.uiowa.edu/~fluids/Archive/lecture_notes/Chapter_9...Chapter 9 Flow...

57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 1

Chapter 9 Flow over Immersed Bodies Basic Considerations Recall separation of drag components into form and skin-friction

( )

⎭⎬⎫

⎩⎨⎧

∫ ⋅τ+∫ ⋅−ρ

= ∞S

wS2

D dAitdAinppAV

21

1C

CDp Cf

( )⎭⎬⎫

⎩⎨⎧ ⋅∫ −

ρ= ∞ dAjnpp

AV21

1CS2

L

ct << 1 Cf > > CDp streamlined body

ct ∼ 1 CDp > > Cf bluff body


Streamlining: One way to reduce the drag Make a body streamlined: reduce the flow separation reduce the pressure drag increase the surface area increase the friction drag

Trade-off relationship between pressure drag and friction drag

Trade-off relationship between pressure drag and friction drag

Benefit of streamlining: reducing vibration and noise


Qualitative Description of the Boundary Layer Recall our previous description of the flow-field regions for high Re flow about slender bodies


τw = shear stress τw ∝ rate of strain (velocity gradient)

=0yy

u

=∂∂

µ

large near the surface where

fluid undergoes large changes to satisfy the no-slip condition

Boundary layer theory is a simplified form of the complete NS equations and provides τw as well as a means of estimating Cform. Formally, boundary-layer theory represents the asymptotic form of the Navier-Stokes equations for high Re flow about slender bodies. As mentioned before, the NS equations are 2nd order nonlinear PDE and their solutions represent a formidable challenge. Thus, simplified forms have proven to be very useful. Near the turn of the century (1904), Prandtl put forth boundary-layer theory, which resolved D’Alembert’s paradox. As mentioned previously, boundary-layer theory represents the asymptotic form of the NS equations for high Re flow about slender bodies. The latter requirement is necessary since the theory is restricted to unseparated flow. In fact, the boundary-layer equations are singular at


separation, and thus, provide no information at or beyond separation. However, the requirements of the theory are met in many practical situations and the theory has many times over proven to be invaluable to modern engineering. The assumptions of the theory are as follows: Variable order of magnitude u U O(1) v δ<<L O(ε) ε = δ/L

x∂∂ L O(1)

y∂∂ 1/δ O(ε-1)

ν δ2 ε2 The theory assumes that viscous effects are confined to a thin layer close to the surface within which there is a dominant flow direction (x) such that u ∼ U and v << u. However, gradients across δ are very large in order to satisfy the no slip condition. Next, we apply the above order of magnitude estimates to the NS equations.


⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+∂∂

µ+∂∂

−=∂∂

+∂∂

2

2

2

2

yu

xu

xp

yuv

xuu

1 1 ε ε-1 ε2 1 ε-2

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+∂∂

µ+∂∂

−=∂∂

+∂∂

2

2

2

2

xv

xv

yp

yvv

xvu

1 ε ε 1 ε2 1 ε-1

0yv

xu

=∂∂

+∂∂

1 1

Retaining terms of O(1) only results in the celebrated boundary-layer equations

2

2

yu

xp

yuv

xuu

∂∂

µ+∂∂

−=∂∂

+∂∂

0yp=

∂∂

0yv

xu

=∂∂

+∂∂

Some important aspects of the boundary-layer equations:

1) the y-momentum equation reduces to

0yp=

∂∂

elliptic

parabolic


i.e., p = pe = constant across the boundary layer

from the Bernoulli equation:

=ρ+ 2ee U

21p constant

i.e., x

UUxp e

ee

∂∂

ρ−=∂∂

Thus, the boundary-layer equations are solved subject to a specified inviscid pressure distribution

2) continuity equation is unaffected 3) Although NS equations are fully elliptic, the

boundary-layer equations are parabolic and can be solved using marching techniques

4) Boundary conditions

u = v = 0 y = 0 u = Ue y = δ

+ appropriate initial conditions @ xi There are quite a few analytic solutions to the boundary-layer equations. Also numerical techniques are available for arbitrary geometries, including both two- and three-dimensional flows. Here, as an example, we consider the

edge value, i.e., inviscid flow value!


simple, but extremely important case of the boundary layer development over a flat plate. Quantitative Relations for the Laminar Boundary Layer Laminar boundary-layer over a flat plate: Blasius solution (1908) student of Prandtl

0yv

xu

=∂∂

+∂∂

2

2

yu

yuv

xuu

∂∂

ν=∂∂

+∂∂

u = v = 0 @ y = 0 u = U∞ @ y = δ We now introduce a dimensionless transverse coordinate and a stream function, i.e.,

δ

∝ν

=η ∞ yx

Uy

( )ην=ψ ∞ fxU

( )η′=∂η∂

η∂ψ∂

=∂ψ∂

= ∞fUyy

u ∞=′ U/uf

Note: xp∂

∂ = 0

for a flat plate


( )ffxU

21

xv −′η

ν=

∂ψ∂

−= ∞

substitution into the boundary-layer equations yields 0f2ff =′′′+′′ Blasius Equation 0ff =′= @ η = 0 1f =′ @ η = 1 The Blasius equation is a 3rd order ODE which can be solved by standard methods (Runge-Kutta). Also, series solutions are possible. Interestingly, although simple in appearance no analytic solution has yet been found. Finally, it should be recognized that the Blasius solution is a similarity solution, i.e., the non-dimensional velocity profile f′ vs. η is independent of x. That is, by suitably scaling all the velocity profiles have neatly collapsed onto a single curve. Now, lets consider the characteristics of the Blasius solution:

∞U

u vs. y

VU

Uv ∞

∞

vs. y

Rex5

=δ value of y where u/U∞ = .99

ν∞=

xUxRe


∞ν

′′∞µ=τ

U/x2

)0(fUw

i.e., xRe

664.0U

2cx

2w

fθ

==ρτ

=∞

see below

∫ ==L

0fff )L(c2dxc

L1C


=LRe

328.1

ν∞LU

Other:

∫ =⎟⎟⎠

⎞⎜⎜⎝

⎛−=δ

δ

∞0 x

*

Rex7208.1dy

Uu1 displacement thickness

measure of displacement of inviscid flow to due boundary layer

∫ =⎟⎟⎠

⎞⎜⎜⎝

⎛−=θ

δ

∞∞0 xRex664.0dy

Uu

Uu1 momentum thickness

measure of loss of momentum due to boundary layer

H = shape parameter = θδ*

=2.5916

Quantitative Relations for the Turbulent Boundary Layer 2-D Boundary-layer Form of RANS equations

0yv

xu

=∂∂

+∂∂


For flat plate or δ for general case

( )vuyy

upxy

uvxuu 2

2e ′′

∂∂

−∂∂

ν+⎟⎠

⎞⎜⎝

⎛ρ∂

∂−=

∂∂

+∂∂

requires modeling Momentum Integral Analysis Background: History and Modern Approach: FD To obtain general momentum integral relation which is valid for both laminar and turbulent flow

( )dycontinuity)vu(equationmomentum0y∫ −+∞

=

( )dxdU

UH2

dxdc

21

U f2w θ

++θ

==ρτ

dxdUU

dxdp

ρ=−

flat plate equation 0dxdU

=

∫ ⎟⎠⎞

⎜⎝⎛ −=θ

δ

0dy

Uu1

Uu momentum thickness

θδ

=*

H shape parameter

dyUu1

0

* ∫ ⎟⎠⎞

⎜⎝⎛ −=δ

δ displacement thickness


y = h + δ*= streamline starts in uniform flow

merges with δ at 3

Steady ρ = constant neglect g v << u = uo ⇒ p = constant i.e., -∇p = 0

Can also be derived by CV analysis as shown next for flat plate boundary layer. Momentum Equation Applied to the Boundary Layer CV = 1, 2, 3, 4

∫ τ==−x

0wdxbdragD pressure force = 0 for v << Uo

force on CV wall shear stress u ∼ Uo

( ) ( )∫ ⋅ρ+∑ ∫ ⋅ρ=−=31

x dAVudAVuDF

= ( ) ∫ρ+−ρ

3

22o dyubbhU

∫ρ−ρ=δ

0

22o dyubbhU)x(D


next eliminate h using continuity

∫=

∫ρ=ρ

∫ ⋅ρ+∫ ⋅ρ=

δ

δ

0o

0o

31

udyhU

udybbhU

dAVdAV0

( ) ∫ρ−∫ρ=δδ

0

2

0o dyubudybUxD

= ( )∫ −ρδ

0o dyuUub

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

ρ=

δ

0 oo2o

D dyUu1

Uu

L2

bLU21

DC

θ = momentum thickness

L2CDθ

=

L2

bLU21

dxb

AU21

DC2o

x

0w

2o

Dθ

=ρ

∫ τ=

ρ=

depends on u(y)


( ) ( )∫ θ=ρ

τx

0 2o

w x2dxxU

21

dxd

U212

12o

w θ=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

ρ

τ

dxd

2cf θ

= cf = local skin friction coefficient

momentum integral relation for

flat plate boundary layer

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−=θ

δ

0 oody

uu1

uu

Approximate solution for a laminar boundary-layer Assume cubic polynomial for u(y)

32 DyCyByAUu

+++=∞

0yuu 2

2=

∂∂

= y = 0 A = 0 B = δ23

0yu;Uu =∂∂

= ∞ y = δ C = 0 D = 3

21δ−


i.e., 3y

21y

23

Uu

⎟⎠⎞

⎜⎝⎛δ

+δ

=

dxdc

21

U f2w θ

==ρτ momentum integral equation for 0

dxdp

=

dxd139.

23U

U1

2δ

=⎥⎦⎤

⎢⎣⎡

δµ

ρ dy

Uu1

Uu

0∫ ⎟

⎠⎞

⎜⎝⎛ −=θ

δ

dydu

w µ=τ

Compare with Exact Blassius

i.e., xRex65.4

=δ xRe

x5 7% ↓

x

2

w ReV323. ρ

=τ x

2

ReU332. ρ 3%↓

x

f Re646.c =

xRe664.

L

f Re29.1C =

LRe33.1

δ=⎟⎟

⎠

⎞⎜⎜⎝

⎛δ

+δ

==

23Uy

23

23Uu

0y

2

y


( )∫ τρ

=L

0w

2f dxx

bLU21

1C

span length total skin-friction drag coefficient Approximate solution Turbulent Boundary-Layer Ret ∼ 3 X 106 for a flat plate boundary layer Recrit ∼ 500,000

dxd

2cf θ

=

as was done for the approximate laminar flat plate boundary-layer analysis, solve by expressing cf = cf (δ) and θ = θ(δ) and integrate, i.e. assume log-law valid across entire turbulent boundary-layer

Byuln1uu *

* +νκ

=

at y = δ, u = U

Buln1uU *

* +νδ

κ=

2/1

f

2cRe ⎟⎠⎞

⎜⎝⎛

δ

neglect laminar sub layer and velocity defect region


or 52cReln44.2

c2 2/1

f2/1

f+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛δ

6/1

f Re02.c −δ≅ power-law fit

Next, evaluate

∫ ⎟⎠⎞

⎜⎝⎛ −=

θ δ

0dy

Uu1

Uu

dxd

dxd

can use log-law or more simply a power law fit

7/1yUu

⎟⎠⎞

⎜⎝⎛δ

=

( )δθ=δ=θ727

⇒ dxdU

727

dxdUU

21c 222

fwδ

ρ=θ

ρ=ρ=τ

dxd72.9Re 6/1 δ

=−δ

or 7/1xRe16.

x−=

δ

7/6x∝δ almost linear

7/1x

f Re027.c =

( )LC67

Re031.C f7/1

Lf ==

cf (δ)

Note: can not be used to obtain cf (δ) since τw → ∞

i.e., much faster growth rate than laminar boundary layer


Alternate forms given in text depending on experimental information and power-law fit used, etc. (i.e., dependent on Re range.) Some additional relations given in texts for larger Re are as follows:

( )2.5810

.455 1700Relog Ref

LL

C = − Re > 107

( )732.Relog98.cL Lf −=δ

( ) 3.2

xf 65.Relog2c −−= Finally, a composite formula that takes into account both the initial laminar boundary-layer (with translation at ReCR = 500,000) and subsequent turbulent boundary layer

is L

5/1L

f Re1700

Re074.C −= 105 < Re < 107

Total shear-stress coefficient

Local shear-stress coefficient


Drag of 2-D Bodies First consider a flat plate both parallel and normal to the flow

( ) 0inppAV

21

1CS2

Dp =∫ ⋅−ρ

= ∞

∫ ⋅τρ

=S

w2

f dAitAV

21

1C

= 2/1LRe33.1 laminar flow

= 5/1LRe

074. turbulent flow

where Cp based on experimental data

vortex wake typical of bluff body flow

flow pattern


( )∫ ⋅−

ρ= ∞

S2Dp dAinpp

AV21

1C

= ∫S

pdACA1

= 2 using numerical integration of experimental data


Cf = 0 For bluff body flow experimental data used for cD. In general, Drag = f(V, L, ρ, µ, c, t, ε, T, etc.) from dimensional analysis c/L

⎟⎠⎞

⎜⎝⎛ ε

=ρ

= .etc,T,L

,Lt,ArRe,f

AV21

DragC2

D

scale factor


Potential Flow Solution: θ⎟⎟⎠

⎞⎜⎜⎝

⎛−−=ψ ∞ sin

rarU

2

22 U21pV

21p ∞∞ ρ+=ρ+

2

22r

2p U

uu1U

21

ppC∞

θ

∞

∞ +−=

ρ

−=

( ) θ−== 2p sin41arC surface pressure

Flow Separation Flow separation:

The fluid stream detaches itself from the surface of the body at sufficiently high velocities. Only appeared in viscous flow!! Flow separation forms the region called ‘separated region’

ru

r1ur

∂ψ∂

−=

θ∂ψ∂

=

θ


Inside the separation region: low-pressure, existence of recirculating/backflows viscous and rotational effects are the most significant!

Important physics related to flow separation:

’Stall’ for airplane (Recall the movie you saw at CFD-PreLab2!) Vortex shedding

(Recall your work at CFD-Lab2, AOA=16°! What did you see in your velocity-vector plot at the trailing edge of the air foil?)


Magnus effect: Lift generation by spinning Breaking the symmetry causes the lift!

Effect of the rate of rotation on the lift and drag coefficients of a smooth sphere:


Lift acting on the airfoil Lift force: the component of the net force (viscous+pressure) that is perpendicular to the flow direction

Variation of the lift-to-drag ratio with angle of attack:

The minimum flight velocity:

Total weight W of the aircraft be equal to the lift

ACWVAVCFW

LLL

max,min

2minmax,

221

ρρ =→==


< 0.3 flow is incompressible, i.e., ρ ∼ constant

Effect of Compressibility on Drag: CD = CD(Re, Ma)

aUMa ∞=

speed of sound = rate at which infinitesimal disturbances are propagated from their source into undisturbed medium

Ma < 1 subsonic Ma ∼ 1 transonic (=1 sonic flow) Ma > 1 supersonic Ma >> 1 hypersonic CD increases for Ma ∼ 1 due to shock waves and wave drag

Macritical(sphere) ∼ .6 Macritical(slender bodies) ∼ 1 For U > a: upstream flow is not warned of approaching

disturbance which results in the formation of shock waves across which flow properties and streamlines change discontinuously

Chapter 9 2006 02user.engineering.uiowa.edu/~fluids/Archive/lecture_notes/Chapter_9...Chapter 9 Flow...

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