Basic bluff-body aerodynamics II

1. Basic bluff-body aerodynamics II Wind loading and structural response Lecture 9 Dr. J.D. Holmes

2. -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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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)

10. 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)

11. 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]

12. 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 ]

13. 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

14. 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

15. 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

16. Basic bluff-body aerodynamics • Vortex shedding - circular cylinder : • vortex shedding not regular in the super-critical Reynolds Number range

17. 0.12 2.5b 0.06 2b 0.14 ~10b 0.08 Basic bluff-body aerodynamics • Vortex shedding - other cross-sections :

18. Basic bluff-body aerodynamics • fluctuating pressure coefficient : • fluctuating sectional force coefficient : • fluctuating (total) force coefficient :

19. 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

20. Basic bluff-body aerodynamics • Quasi-steadyfluctuating pressure coefficient : • Quasi-steady drag coefficient :

21. Basic bluff-body aerodynamics • Correlation coefficient for fluctuating forces on a two-dimensional body : • Correlation length : y is separation distance between sections

22. 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

23. L Basic bluff-body aerodynamics • Total fluctuating force on a slender body : We require the total mean and fluctuating forces on the whole body

24. 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 +...

25. 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

26. 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

27. 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

28. 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)

29. End of Lecture 9John Holmes225-405-3789 JHolmes@lsu.edu