1. Flow past bluff-bodies
Separated Flows Wakes and Cavities
Flow past bluff-bodies

2. Some appropriate definitions
3.1 Mean properties
Some appropriate definitions
Salient bodies & Smooth bodies U0 d
2D bluff-bodies = cylinders of varying cross section
Reynolds number
Pressure coefficient

3. - recirculation region or - separated region or - cavity
3.1.1 Mean Wake Topology
region of slow motion :
- bubble or
- recirculation region or
- separated region or
- cavity
circular cylinder

4. Time averaged flow visualisation Instantaneous flow visualisation
3.1.1 Mean Wake Topology
Time averaged flow visualisation
Instantaneous flow visualisation
Circular
Plate
~Steady Fully unsteady motion

5. d' L
Two characteristic dimensions : L, d'

6. Closure with a stagnation point u changes sign at the closure
Mean flow-Streamlines
Mean flow-u component
L = 0.9d d’= d
Closure with a stagnation point
u changes sign at the closure
Two counter rotating eddies inside

7. The bubble is a low velocity region with back flow
Velocity modulus
Pressure coefficient
The bubble is a low pression region (negative Cp) with small variations
Circular
Plate
Base pressure Cpb

8. shear stress on the body
3.1.2 Stress around the body
On the body surface  :
Viscous term : normal and tangent (shear) stress.
Pressure term : isotropic stress.
shear stress on the body
order of

9. Around Circular cylinders (SMOOTH SEPARATION)
Zero shear stress = condition for separation
adverse pressure gradient consistent with the following separation
base pressure

10. Around Circular cylinders (SMOOTH SEPARATION)

11. Around a plate (SHARP SEPARATION)
NO ADVERSE PRESSURE GRADIENT !
Different mechanism from smooth separation.
A salient edge imposes to the flow a tangential separation at the edge.

12. For bluff bodies with separation and Re>>1
3.1.3 Drag
If the flow remains attached only viscous contribution, the drag tends to zero as viscosity goes to zero : D'Alembert paradox.
With separation, region of low pressure region = non zero drag even as viscosity goes to zero. Form drag.
Drag Coefficient
For bluff bodies with separation and Re>>1 f

13. Drag on a circular cylinder
No asymptotics behavior before 107

15. 3.1.4 Transition from sub to supercitical flow : Drag CrisisCD
Before the drag crisis
subcritical flow
After the drag crisis
supercritical flow

16. Drag crisis : the end of a story that begun at Re=2000
At Re=2000, the laminar to turbulence transition appears in the mixing layers before the vortex roll-up. The transitional point get closer to the separation point as Re increases.
xt-xS ~d Re -1/2 S xt
The drag crisis is the consequence of the vicinity of the transitional point with the body wall.

17. #0 Strong symmetry breaking #1 (one side reattachment) #0
During the drag Crisis #0
Strong symmetry breaking
#1 (one side reattachment) #0
#2 (two sides reattachment)

18. flow reattachment : a Coanda effect
Laminar vs. Turbulent separation
flow reattachment : a Coanda effect
Wall shear stress on a plate

19. 3.2 Wake dynamic Synchronized frequency : f
→ Origin ? : Vorticity dynamics
Flow filling and emptying in the recirculating bubble
→ A key concept that affect the bubble length

20. 3.2.1 Strouhal number

21. (if inviscid dynamics)
D Dynamic of vorticity
with Vortex Method :
Discretization of the initial vorticity field
: Sum of particles having a circulation  (vortex points)
(if inviscid dynamics)
Let's look at the simulation !

22. axisymetric configuration = vortex ring
Pair of vortices
axisymetric configuration = vortex ring

23. Interaction (Biot et Savart) between two vorticity sheets having opposite circulation

24. Pressure dynamic
Color : pressure coefficient
Lines : iso-contour of vorticity
Low pressure = vorticies

25. Mean pressure
Lowest pressure = vorticies

26. d' Physical model for global frequency selection in a wake
Universal Strouhal number (Roshko 1954)

27. The more bluff the body the lower the strouhal number

28. xt-xS ~d Re -1/2 Physical mechanism (Gerrard 1966) Shear thickness 
For d' given, the thicker the shears the lower the frequency
Shear thickness 
Laminar
Formation length L
Turbulent
With :
xt-xS ~d Re -1/2

29. The change of the shear thickness at the formation length is at the origin of the Strouhal variation with Re

30. 3.2.2 Recirculation bubble size

31. Re   L  What governs the size L ?
Flux b has to be compensated by the entrained velocity by the mixing layer
Laminar
Re   L 

32. Re   L  For Re>2000, transition in VE xt
Consequence for the drag :

33. D Dynamics
S universal q
Sq=S cosq

34. 3.1.4 Drag Crisis

35. Drag crisis : the end of a story that begun at Re=2000
Base pressure coefficient