Separated Flows Wakes and Cavities Generalities 1.1 What is separation ? A streamline leaves the body and turns into the interior of the fluid 2D separation 3D separation 1.1 What is separation ? Separation is intimitaley related to the no-slip condition for instance: stagnation point flow y=0 1.2 The mechanism of smooth 2D separation Vorticity is an intrinsic local ingredient of the flow dynamics. Vorticity at the wall : Separation = reversed vorticity flow region 1.2 The mechanism of smooth 2D separation The key to understanding when separation may occur is : How is reversed vorticity introduced in the flow ? Before separation After separation 1.2 The mechanism of smooth 2D separation Steady 2D flow - x-momentum equation At the wall (exact) The pressure gradient at the wall creates a vorticity gradient at the wall responsible for vorticity transport (diffusion) in the flow. and this is the mechanism for the reversed vorticity introduction 1.2 The mechanism of smooth 2D separation U vorticity transport by viscous diffusion + vorticity gradient Only negative vorticity at the wall Need to introduce positive vorticity by viscous diffusion 1.2 The mechanism of smooth 2D separation since can be realized if : >0 U need to have a positive or adverse pressure gradient at the wall 1.2 The mechanism of smooth 2D separation >0 If If >0 is strong enough : is not strong enough the vorticity magnitude is reduced but the vorticity not reversed = no separation 1.2 The mechanism of smooth 2D separation At the wall: relationship between slope of vorticity, curvature of velocity and pressure gradient = negative zero (Blasius) positive 1.2 The mechanism of smooth 2D separation Separation may occur as long as the flow develops a strong adverse pressure gradient (introducing reversed vorticity by viscous diffusion in the flow) Adverse pressure gradient at the wall is a necessary condition for separation, but not sufficient. Nothing has been said so far about the flow Reynolds number ! Actually, the mechanism for separation applies whatever the Reynolds number is. 1.3 local criteria : on-wall signature shear at the wall (or skin friction) 1.3 local criteria : on-wall signature For 2D flows, the shear is a scalar and : (with boundary convention) At S, the wall shear stress It is zero at S. (or wall vorticity) changes sign, Prandtl criteria 1.3 local criteria : on-wall signature For 3D flows, it is more complicated ... skin friction lines roll-up into an eddy h streamlines surface • Skin friction lines convergence • Zero skin friction On the separation line S, the skin friction is generally different from zero (shear along the line) Prandtl criteria not applicable Lighthill criteria 1.4 Low Re separation an example at (Re=0.01)... Very low Re: no convection : upstream-downstream symmetry Where does the adverse pressure gradient come from ? 1.5 Intermediate Re separation - cylinder A bit of convection : upstream-downstream symmetry is broken 1.5 Intermediate Re separation - cylinder if Re= Ud/ > 4 eddies recirculation region L reattachment separation angle S 1.5 Intermediate Re separation - cylinder L ~ d Re where Re= Ud/ Re = 10 S Streamlines Vorticity Re = 40 S Viscous diffusion + advection : S ~ Cte +Re -1/2 1.5 Intermediate Re separation - step flow L L ~ d Re are the result of : • horizontal advection by U • vertical diffusion by viscosity 1.5 Intermediate Re separation - step flow Re = 100 Steady Re = 230 Rc = 350 threshold Re = 400 Unsteady Re = 500 6h 1.5 Intermediate Re separation - step flow Re = 630 Re = 850 Unsteady Re = 1050 Re = 1200 6h 1.5 Intermediate Re separation - step flow Fixed separation points (separation at edge) L varies as : L/h L h Re steady 14 12 10 8 6 4 2 0 0 500 1000 1500 2000 2500 3000 3500 Re 1.5 Large Re separation Boundary layer separation and reattachment THEORETICAL FRAME Vorticity is only confined to the solid boundary in a layer <<d. ~dRe-1/2 d • Inviscid motion outside the layer • Boundary layer equation inside the layer (Boundary Layer Theory, BLT) • Matched asymptotic theory y/h 1.6 Large Re separation The separated boundary layer Re = 60; 100; 160; 210; 270; 2600 Re = 200 5 Laminar Turbulent Mixing layer profile 4,5 4 3,5 3 2,5 2 1,5 1 0,5 0 0 0,2 0,4 0,6 0,8 1 x/xR 1,2 1,4 1,6 1,8 Re = 1000 2 1.6 Large Re separation Sketch of a separated boundary layer Laminar Turbulent 1.6 Large Re separation Stability of the laminar separated boundary layer increases downstream inertial instability x>xt Kelvin-Helmholtz instability xt -xS: transition point moves upstream as Re increases xt S xt-xS ~d Re -1/2 xt 1.6 Large Re separation Stability of the separated boundary layer Re=100 Re=10000 1.6 Large Re separation Stability of the separated boundary layer Re=10000 1.7 Conclusion An adverse pressure gradient at the wall is a necessary condition for separation, but not sufficient. The adverse pressure gradient can be either created by friction (creeping flows) or of inertia (Euler flows) The separated boundary layer is similar to a mixing layer which entrains the flow from low speed region toward the ML. How strong the adverse pressure gradient should be ? We are going to study the case of large Reynolds number flows.