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

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

- 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

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

d' L Two characteristic dimensions : L, d'

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

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

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

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

Around Circular cylinders (SMOOTH SEPARATION)

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.

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

Drag on a circular cylinder No asymptotics behavior before 107

3.1.4 Transition from sub to supercitical flow : Drag Crisis CD Before the drag crisis subcritical flow After the drag crisis supercritical flow

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.

#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)

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

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

3.2.1 Strouhal number

(if inviscid dynamics) 3.2.2 2D 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 !

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

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

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

Mean pressure Lowest pressure = vorticies

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

The more bluff the body the lower the strouhal number

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

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

3.2.2 Recirculation bubble size

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

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

3.2.3 3D Dynamics S universal q Sq=S cosq

3.1.4 Drag Crisis

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