chapter 3 flows around submerged bodies

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FLOWS AROUND SUBMERGED BODIES KKKJ3123 FLUID MECHANICS

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Characteristics of External Flows Characteristics of External Flows Boundary Layers Boundary Layers Boundary Layer Theory for Flat Plate Boundary Layer Theory for Flat Plate Drag Drag Lift Lift


3.1 CHARACTERISTICS OF EXTERNAL FLOWS

External flow refers to flows around a closed solid body, not constrained by boundaries and without free surfaces.

Figure 3.1 External flow for 2D, axisymmetric and 3D cases

The characteristics of the flow is determined by the local Reynolds Number:

UxUx

x ==Re (3.1)

where U is the upstream velocity and x the characteristic length.

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Figure 3.2 Pressure and wall shear stress distribution of an airfoil

The fluid-body interaction on the surface of the body leads to wall shear stresses w due to viscous effects and pressure p.

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Figure3.3 Shear and pressure force distribution on area dA

The force analysis on an elemental area dA in the above figure yields: ( ) ( )( ) ( )

cossin

sincosdAdApdF

dAdApdF

wy

wx

+=+=

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Therefore, drag D and lift L could be obtained:

+== dAdApdFD wx sincos (3.2) +== dAdApdFL wy cossin (3.3) Forces D and L are known as aerodynamic (for air/gas) or

hydrodynamic (for water/liquid) forces.

Experimentally, p, w, D and L can be made related to dynamic pressure 221 U : Local Pressure Coefficient:

221 U

ppC p = (3.4)

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Local Friction Coefficient:

221 U

c wf = (3.5)

Drag Coefficient:

AUDCD 2

21 = (3.6)

Lift Coefficient:

AULCL 2

21 = (3.7)

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EXAMPLE 3.1

Water flows past an equal triangular bar as shown in the figure below. Using the given pressure distribution and neglecting shear forces, calculate the drag and lift on the bar.

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Drag (w = 0):

( )

( )( ) ( ) ( )[ ]( )( ) ( ) N 0.4603.02.16.19985.0

5.025.025.0

5.0225.01

60cos2180cos

60cos60cos180cos

sincos

2

222

221

212

21

321

===+=

+=+=

++=+=

AUAUAU

AUAU

dApdAp

dApdApdAp

dAdApD w

Since body is symmetry:

0=L

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3.2 BOUNDARY LAYERS

Figure 3.4 Boundary layers for 3 different values of Rel

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Boundary layer is the region around the surface of a body where viscous effects are dominant ( )UV 99.0< .

Outside of the boundary layer, the flow could be assumed inviscid. Flows with high velocity (high Re) could lead to circulation or wakes

downstream of the body.

For a blunt body, such as cylinders, flow separation could occur starting from a separation point or location.

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Figure 3.5 Boundary layers around a cylinder for 3 different ReD values

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There are 3 parameters used to represent boundary layer: 1. Boundary layer thickness value of y where V , U99.0===

ll

U

Hence, flow is turbulent. Boundary layer thickness:

( )( )

mm 50.6m 006498.036.001805.0

01805.010298.4

16.0Re

16.071671

====

==

ll

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(b) Drag needs to be calculated for both surfaces of the hydrofoil:

Case (i) Consider the smooth turbulent curve:

( )0034970

10298.4

031.0Re

031.071671

.

C fD

=

==l

( )( )( )( )( ) ( )

N 6.3218.036.01299800349702

22

21

221

==

=.

AUCD Df

Case (ii) Consider the transition curve Rel = 5 105:

003162010298.4

14400034970Re1440

Re031.0

671

.

.C fD

=== ll

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( )( )( )( )( ) ( )

N 9.4218.036.01299800316202

22

21

221

==

=.

AUCD Df

Case (iii) Consider the relative roughness:

0064670360122.0log62.189.1log62.189.1

5.25.2

.

C fD

=

=

=

l

( )( )( )( )( ) ( )

N 0.2618.036.01299800646702

22

21

221

==

=.

AUCD Df

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3.4 DRAG

In actual situations, distribution of pressure p and shear stress w in Eq. (3.2) below, are difficult to obtain analytically:

+= dAdApD w sincos Alternatively, D is assumed proportional to 221 U with the

proportionality coefficient being the drag coefficient CD obtained from graphs:

AUDCD 2

21 = (3.23)

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From Eq. (3.2), drag could be divided into two components: > Pressure drag

AUCdApD Dpp2

21cos ==

> Friction drag AUCdAD Dfwf

221sin ==

Thus,

DfDpD CCC +=

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In the equation above, area A could be either one of the following: > Frontal area projected area of the body as seen from the stream

(for stubby bodies with large shape ratio),

> Planform area projected area of the body as seen from above (suitable for thin bodies with large surface area or flat-shaped),

> Wetted area for bodies that float and move on free surfaces.

Factors that influence CD is given by the relationship: ( )lFr,Ma,Re,,shape=DC

For shape factors, blunt bodies have larger CD compared to streamlined shaped bodies.

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Figure 3.10 Effect of geometry on drag coefficient

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Figure 3.11 Evolution in design of cars with aerodynamic characteristics

For Re factors, D is dependant on upstream velocity U, characteristic length l and fluid viscosity .

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For low Re, it could derived: ( )

Re22

,,

222221

Analysis lDimensiona

CUCU

UDC

CUDUfD

D ==== =

ll

l

ll

Figure 3.12 Drag coefficient CD for low Re flows

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Figure 3.13 Relation between the drag coefficient of cylinder/sphere and

Reynolds number Re

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Figure 3.13 (continued)

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Figure 3.14 Drag coefficient as a function of Re for other bodies

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For Ma effects, greater change in density (proportional to Ma refer chapter 9) increases CD.

Figure 3.15 Drag coefficient as a function of Ma for subsonic flow

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Figure 3.16 Drag coefficient as a function of Ma for supersonic flow

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For Fr factors, it only applies to bodies floating on free surfaces.

Figure 3.17 Drag coefficent as a function of Fr

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For the effect of wall surface roughness, roughness increases CD.

Figure 3.18 Drag coefficient for smooth and rough sphere

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Figure 3.19 Drag coefficient of some 2D objects for Re 104

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Figure 3.20 Drag coefficient of some 3D objects for Re 104

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EXAMPLE 3.4

Wind at 100 km/hr blows past a smooth surfaced water tower as shown in the figure below. Estimate the reacting moment required to avoid the tower from tipping over. Use air density = 1.23 kg/m3 and viscosity of air = 1.79 105 kg/m3.

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Reynolds No. for the sphere and cylinder: ( )

( )6

5

7

5

10589.836001000100

1079.15.423.1Re

10386.236001000100

1079.15.1223.1Re

=

===

==

cc

ss

Ud

Ud

Calculation of drag for the sphere and cylinder:

9.0,35.0 cDsD CC

( )( ) ( )N 20382

5.1243600

100010023.135.0 22

21

221

=

=

= ssDs AUCD

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( )( ) ( )( )N 28828

5.4153600100010023.19.0

2

21

221

=

=

= ccDc AUCD

From the free body diagram:

mkN 649mN 10493.6

21528828

25.121520382

22

5

==

+

+=

+

+= bDdbDM css

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3.5 LIFT

Similar to drag D, the distribution of pressure p and shear stress w in Eq. (3.3) shown below, are difficult to obtain analytically:

+= dAdApL w cossin Similar to drag D, the lift L could also be assumed proportional to

221 U with the proportionality constant being the lift coefficient CL:

AULCL 2

21 = (3.24)

The equation to determine L was developed by Kutta (1902) and Joukowski (1906), known as Kutta-Joukowski lift theorem:

= UbL (3.25)

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with being the circulation as described in Chapter 2: ( ) ++== CC dzwdyvdxudsV (3.26)

The factors that influence CL is similar as those factors effecting CD: ( )lFr,Ma,Re,,shape=LC

Figure 3.22 Symetrical and nonsymmterical aerofoil

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A symmetrical body produces L = 0, and L 0 could only happen if angle of attack > 0.

Most applications involved with the principle of lift operates at high Re ( 104) flows where w effects could be neglected.

Therefore, lift L is more dependant on the distribution of surface pressure p.


Figure 3.23 Distribution of surface pressure on a car

In aerofoils, the reference area is the projected plan area A = bc, where c is the chord dimension and b is the span dimension.

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Figure 3.24 Configuration of an aerofoil

In general, increase in angle of attack increases CL and CD. ( ) DL , However, that is too large generates turbulent wakes

causing aerofoils to stall.

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Figure 3.25 Relationship of CL and CD with angle of attack

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Figure 3.26 Relationship between CL and CD

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Addition of wing structures (flap/aileron) could alter the lift and drag characteristics of an aerofoil.

Figure 3.27 Effect of wing structures to an aerofoil

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EXAMPLE 3.5

Given below is the data for a glider flying through the atmospheric space:

Glider velocity U = 5 m/s Wing size b = 30 m, c = 2.5 m (average) Weight (including pilot) W = 950 N Drag coefficient CD = 0.046 (based on plan area) Efficiency of glide power = 0.8 (for drag resistance)

Determine the lift coefficient and required power of the glider to maintain the velocity at similar altitude. Use density of air = 1.23 kg/m3.


Equilibrium at the same altitude level: AUCLW L

221 ==

Hence, the lift coefficient:

( )( ) ( )( )

824.05.230523.1

9502222

===

bcUWCL

Power/efficiency relation for glider:

( )UAUCDUP D 221 == ( )( ) ( )( )

( ) W332

8.025.230523.1046.0

2

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===

AUCP D

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Chapter 3 Flows Around Submerged Bodies3.1 Characteristics of External FlowsExample 3.1

3.2 Boundary Layers3.3 Boundary Layer Theory for Flat PlateExample 3.2Example 3.3

3.4 DragExample 3.4

3.5 LiftExample 3.5

chapter 3 flows around submerged bodies

Documents

water flows

boundary layers figure

fluidbody interaction

local pressure coefficient

lift coefficient

forces d

blunt body

flow separation