# Generalities Separated Flows Wakes and Cavities. 1.1 What is separation ? A streamline leaves the body and turns into the interior of the fluid 2D separation.

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Generalities Separated Flows Wakes and Cavities

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

How is reversed vorticity introduced in the flow ? After separation Before separation The key to understanding when separation may occur is : 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 - + Only negative vorticity at the wall Need to introduce positive vorticity by viscous diffusion vorticity gradient vorticity transport by viscous diffusion 1.2 The mechanism of smooth 2D separation

can be realized if : >0 U - + need to have a positive or adverse pressure gradient at the wall since 1.2 The mechanism of smooth 2D separation

If is strong enough : If is not strong enough the vorticity magnitude is reduced but the vorticity not reversed = no separation >0 1.2 The mechanism of smooth 2D separation

negative zero (Blasius) positive = =  1.2 The mechanism of smooth 2D separation At the wall: relationship between slope of vorticity, curvature of velocity and pressure gradient

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. Separation may occur as long as the flow develops a strong adverse pressure gradient (introducing reversed vorticity by viscous diffusion in the flow) 1.2 The mechanism of smooth 2D separation

1.3 local criteria : on-wall signature shear at the wall (or skin friction)

1.3 local criteria : on-wall signature At S, the wall shear stress (or wall vorticity) changes sign, It is zero at S. (with boundary convention) Prandtl criteria For 2D flows, the shear is a scalar and :

1.3 local criteria : on-wall signature Lighthill criteria For 3D flows, it is more complicated... streamlines surface roll-up into an eddy On the separation line S, the skin friction is generally different from zero (shear along the line) Prandtl criteria not applicable skin friction lines Skin friction lines convergence Zero skin friction h  

1.4 Low Re separation Very low Re: no convection : upstream-downstream symmetry an example at (Re=0.01)... 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 eddies recirculation region L reattachment separation angle  S if Re= Ud/ > 4

1.5 Intermediate Re separation - cylinder L ~ d Re where Re= Ud/

Re = 10 Re = 40 SS SS Viscous diffusion + advection :  S ~ Cte +  Re -1/2 Streamlines Vorticity

1.5 Intermediate Re separation - step flow L ~ d Re are the result of : horizontal advection by U vertical diffusion by viscosity L

1.5 Intermediate Re separation - step flow Re = 100 Re = 230 Re = 400 Re = 500 Steady Unsteady Rc = 350 threshold 6h

1.5 Intermediate Re separation - step flow Re = 630 Re = 850 Re = 1050 Re = 1200 6h Unsteady

1.5 Intermediate Re separation - step flow Fixed separation points (separation at edge) L varies as : steady

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 Inviscid motion outside the layer Boundary layer equation inside the layer (Boundary Layer Theory, BLT) Matched asymptotic theory  d

1.6 Large Re separation Re = 200 Re = 1000 The separated boundary layer Mixing layer profile Laminar Turbulent

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 x t -x S : transition point moves upstream as Re increases x t -x S ~d Re -1/2 x>x t Kelvin-Helmholtz instability S xtxt xtxt increases downstream  inertial instability

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.

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