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Basic bluff-body aerodynamics II. Wind loading and structural response Lecture 9 Dr. J.D. Holmes. -0.20. -0.10. -0.20. x. x. x. leeward wall. roof. -0 . 23. -0.18. -0.23. x. x. x. -0.20. -0.20. x. x. -0.2. side wall. -0.5. -0.5. Sym.about C L. -0.7. -0.8. -0.8.

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basic bluff body aerodynamics ii

Basic bluff-body aerodynamics II

Wind loading and structural response

Lecture 9 Dr. J.D. Holmes

basic bluff body aerodynamics

-0.20

-0.10

-0.20

x

x

x

leeward wall

roof

-0.23

-0.18

-0.23

x

x

x

-0.20

-0.20

x

x

-0.2

side wall

-0.5

-0.5

Sym.about

CL

-0.7

-0.8

-0.8

-0.6

-0.8

windward wall

0.7

0.5

0.0

Wind

Basic bluff-body aerodynamics
  • Pressures on prisms in turbulent boundary layer :
  • drag coefficient (based on Uh )  0.8
basic bluff body aerodynamics3

x-0.6

Roof

-0.5

-0.6

x-0.6

0.9 x

-0.56 to –0.59

-0.4 to –0.49

-0.6

x

x

Wind

-0.5

Leeward wall

x -0.6

Windward wall

Side wall

-0.6

Wind

-0.5

-0.5

0.5 x

-0.6

-0.7

x 0.4

0.3 x

Basic bluff-body aerodynamics
  • Pressures on prisms in turbulent boundary layer :

shows effect of velocity profile

nearly uniform

basic bluff body aerodynamics4

Laminar boundary layer

Separation

Sub-critical

Re < 2  105

Cd = 1.2

Basic bluff-body aerodynamics
  • Circular cylinders :

Complexity due to interacting effects of surface roughness, Reynolds Number and turbulence in the approach flow

Flow regimes in smooth flow :

Subcritical regime : most wind-tunnel tests - separation at about 90o from the windward generator

basic bluff body aerodynamics5

Turbulent

Laminar

Separation

Super-critical

Re  5  105

Cd 0.4

Basic bluff-body aerodynamics
  • Circular cylinders :

Complexity due to interacting effects of surface roughness, Reynolds Number and turbulence in the approach flow

Flow regimes in smooth flow :

Supercritical : flow in boundary layer becomes turbulent - separation at 140o - minimum drag coefficient

basic bluff body aerodynamics6

Separation

Turbulent

Post-critical

Re  107

Cd  0.7

Basic bluff-body aerodynamics
  • Circular cylinders :

Complexity due to interacting effects of surface roughness, Reynolds Number and turbulence in the approach flow

Flow regimes in smooth flow :

Post-critical : flow in boundary layer is turbulent - separation at about 120o

basic bluff body aerodynamics7

1.0

0.5

0

-0.5

-1.0

-1.5

-2.0

-2.5

q

q degrees

20 60 100 140

Cp

U

Drag coefficient mainly determined by pressure on leeward side (wake)

Basic bluff-body aerodynamics
  • Circular cylinders :

Pressure distributions at sub-critical and super-critical Reynolds Numbers

basic bluff body aerodynamics8

increasing surface roughness

1.2

0.8

0.4

k/b = 0.02

k/b = 0.007

Cd

k/b = 0.002

104 2 4 8 105 2 4 8 106 2 4 8 107

Re

U

b

Sanded surface

Smooth surface

Basic bluff-body aerodynamics
  • Circular cylinders :

Effect of surface roughness :

Increasing surface roughness : decreases critical Re - increases minimum Cd

basic bluff body aerodynamics9

b

Cp

h

Basic bluff-body aerodynamics
  • Circular cylinders :

Effect of aspect ratio on mean pressure distribution :

Silos, tanks in atmospheric boundary layer

Decreasing h/b : increases minimum Cp (less negative)

basic bluff body aerodynamics10
Basic bluff-body aerodynamics
  • Fluctuating forces and pressures on bluff bodies :

Sources of fluctuating pressures and forces :

  • Freestream turbulence (buffeting)
  • - associated with flow fluctuations in the approach flow
  • Vortex-shedding (wake-induced)
  • - unsteady flow generated by the bluff body itself
  • Aeroelastic forces
  • - forces due to the movement of the body (e.g. aerodynamic damping)
basic bluff body aerodynamics11
Basic bluff-body aerodynamics
  • Buffeting - the Quasi-steady assumption :

Fluctuating pressure on the body is assumed to follow the variations in wind velocity in the approach flow :

p(t) = Cpo (1/2) a [U(t)]2

Cpo is a quasi-steady pressure coefficient

Expanding :

p(t) = Cpo (1/2) a [U + u(t) ]2

= Cpo (1/2) a [U2 + 2U u(t) + u(t)2 ]

Taking mean values :

p = Cpo (1/2) a [U2 + u2]

basic bluff body aerodynamics12

Squaring and taking mean values :

Cp2 (1/4) a2 [4U2]= Cp2a2U2 u2

Basic bluff-body aerodynamics
  • Buffeting - the Quasi-steady assumption :

Small turbulence intensities :

(e.g. for Iu = 0.15, u2 = 0.0225U2 )

p  Cpo (1/2) aU2 =Cp (1/2) aU2

i.e. Cpo is approximately equal to Cp

Fluctuating component :

p' (t) = Cpo (1/2) a [2U u'(t) + u'(t)2 ]

basic bluff body aerodynamics13

p(t)

Time

Basic bluff-body aerodynamics
  • Peak pressures by the Quasi-steady assumption :

Quasi-steady assumption gives predictions of either maximum or minimum pressure, depending on sign of Cp

basic bluff body aerodynamics14
Basic bluff-body aerodynamics
  • Vortex shedding :

On a long (two-dimensional) bluff body, the rolling up of separating shear layers generates vortices on each side alternately

  • Occurs in smooth or turbulent approach flow
  • may be enhanced by vibration of the body (‘lock-in’)
  • cross-wind force produced as each vortex is shed
basic bluff body aerodynamics15
Basic bluff-body aerodynamics
  • Vortex shedding :

Strouhal Number - non dimensional vortex shedding frequency, ns :

  • b = cross-wind dimension of body
  • St varies with shape of cross section
  • circular cylinder : varies with Reynolds Number
basic bluff body aerodynamics16
Basic bluff-body aerodynamics
  • Vortex shedding - circular cylinder :
  • vortex shedding not regular in the super-critical Reynolds Number range
basic bluff body aerodynamics17

0.12

2.5b

0.06

2b

0.14

~10b

0.08

Basic bluff-body aerodynamics
  • Vortex shedding - other cross-sections :
basic bluff body aerodynamics18
Basic bluff-body aerodynamics
  • fluctuating pressure coefficient :
  • fluctuating sectional force coefficient :
  • fluctuating (total) force coefficient :
basic bluff body aerodynamics19

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Fluctuating side force coefficient Cl

105 106 107

Reynolds number, Re

Basic bluff-body aerodynamics
  • fluctuating cross-wind sectional force coefficient for circular cylinder :

dependecy on Reynolds Number

basic bluff body aerodynamics20
Basic bluff-body aerodynamics
  • Quasi-steadyfluctuating pressure coefficient :
  • Quasi-steady drag coefficient :
basic bluff body aerodynamics21
Basic bluff-body aerodynamics
  • Correlation coefficient for fluctuating forces on a two-dimensional body :
  • Correlation length :

y is separation distance between sections

basic bluff body aerodynamics22

6

4

2

0

Correlation length / diameter

104 105 106

Reynoldsnumber, Re

Basic bluff-body aerodynamics
  • Correlation length for a stationary circular cylinder (smooth flow) :

cross-wind vibration at same frequency as vortex shedding increases correlation length

basic bluff body aerodynamics23

L

Basic bluff-body aerodynamics
  • Total fluctuating force on a slender body :

We require the total mean and fluctuating forces on the whole body

basic bluff body aerodynamics24
Basic bluff-body aerodynamics
  • Total fluctuating force on a slender body :

mean total force : F = fiyi

instantaneous total fluctuating force : F(t) =  fi (t) yi

= f1(t) y1 + f2(t) y2 + ……………….fN(t) yN

  • Squaring both sides : [F(t)]2 = [ f1(t) y1 + f2(t) y2 + ……………….fN(t) yN]2
  • = [f1(t) y1]2 + [f2(t) y2]2 ..+ [fN(t) yN]2 + f1(t) f2(t) y1y2 + f1(t) f3(t) y1y3 +...
basic bluff body aerodynamics25
Basic bluff-body aerodynamics
  • Total fluctuating force on a slender body :

Taking mean values :

As yi, yj tend to zero :

writing the integrand (covariance) as :

This relates the total mean square fluctuating force to the sectional force

basic bluff body aerodynamics26
Basic bluff-body aerodynamics
  • Total fluctuating force on a slender body :

Introduce a new variable (yi - yj) :

Special case (1) - full correlation,(yi-yj) = 1 :

fluctuating forces treated like static forces

Special case (2) - low correlation,correlation length l is much less than L :

mean square fluctuating force is proportional to the correlation length - applicable to slender towers

basic bluff body aerodynamics27

The double integral : is represented by the volume under the graph :

yj

yi

Basic bluff-body aerodynamics
  • Total fluctuating force on a slender body :

Symmetric about diagonal since (yj-yi) =(yi-yj ). Along the diagonal, the height is 1.0

On lines parallel to the diagonal, height is constant

basic bluff body aerodynamics28

Volume under slice = (z)(L-z)2

L

yi-yj=0

yi-yj= z

z/2

yj

z /2

yi

Total volume =

Basic bluff-body aerodynamics
  • Total fluctuating force on a slender body :

Consider the contribution from the slice as shown :

Length of slice = (L-z)2

(reduced to single integral)