Presentation on theme: "Separation in B.L.T. context"— Presentation transcript:
1 Separation in B.L.T. context Separated Flows Wakes and CavitiesSeparation in B.L.T. context
2 2.1 Boundary Layer Theory (Prandtl) Simplification of the Navier-Stokes equation for large Reynolds flowsinviscid solution (Euler)thin boundary layerpressure uniform accross the B.L.
3 2.1 Boundary Layer Theory (Prandtl) using the dimensionless quantities in the NS eq.In the limit of large Re:
4 2.1 Boundary Layer Theory (Prandtl) The pressure gradient is considered to be given by the inviscid flow, approximated by Bernoulli's theorem :
5 2.1 Boundary Layer Theory (Prandtl) Problem to solve : flow around a bodyCompute the inviscid flow around the body given by the potential flow theory to have the pressure gradient.Use this pressure gradient to compute the history of the BL on the body.Is it easy ?Inviscid solution is not unique.Lack of coupling: the inviscid flow cannot react to the BL dynamics.The BLT needs an initial velocity profile: what happens if back flow occurs for a given inviscid solution that do not have any back flow ?It is not so easy !, and even very limited but gives the theoretical description for the understanding of a large Re flow at separation.
6 2.2 Attached boundary layers In the external streamConstno pressure gradientyExternal streamadversethe lower the velocity the greatest the rate of change of the velocityfavourablethinningthickeningBL1u/Um(x)
7 2.2 Attached boundary layers In the BL, close to the wall and at the wallThe pressure gradient is observable in the curvature of the velocity profilethinningthicheningnegative zero positive
8 2.2 Attached boundary layers Both the BL and the inviscid flow behaves in the same way :Acceleration = thinning of the BL (favourable pressure gradient)Deceleration = thickening of the BL (adverse pressure gradient)
9 2.2 Attached boundary layers adversefavourablePotential flow theory for inviscid part of the flow:
10 2.2 Attached boundary layers Result of the BLT (matched asymptotic theory):m<0, non-uniquenessm< back-flow : BLT not physical
11 2.2 Attached boundary layers Result of the BLT:Show that the displacement thickness is :and that the wall shear stress is :Discussion:Which is the flow with a constant boundary layer thickness ? why ?Describe the flow with the constant wall shear stress
12 2.2 Attached boundary layers The BL starts to grow at the point when the dividing streamline coming from far upstream intersects the body.Two cases for initial velocity profile :Smooth edge : m=1, i.e. the BL of a stagnation pointSharp edge : one member of the family m>=0
13 2.3 Separated boundary layers BLT limitationsGoldstein singularity :BLT Integration for BLSThe solution blows when pressure gradient becomes adverseSomething is missing in the BLT !Starting pointzone of adverse pressure gradientPressure gradient from real pressure field
14 2.3 Separated boundary layers Origin of the Goldstein's singularityBLT = prime quantities are O(1) as Re near separation, due to fluid ejection :There is no possibility to balance this ejection (neither pressure gradient nor viscous diffusion in y-direction in BLT)'The singularity develops in a sublayer :
15 2.3 Separated boundary layers The triple deck theoryupper deckmain decklower deckThe flow ejection (lower deck) at separation introduces a pressure gradient in the external flow
16 2.4 Unsteady flow separation Initial condition such that at t=0 the flow is potential, i.e. inviscid everywhere :Dynamics :No steady solution, the BL will grow and saturate due to the action of pressure gradient and viscous diffusionInitial condition for a circular cylinder:After ?
17 2.4 Unsteady flow separation Circular cylinder impulsively started from rest at constant velocity.Dynamics :
18 2.4 Unsteady flow separation at t ~ 0, the equation can be approximated by u1 with :
20 2.4 Unsteady flow separation f/erf is max at =0 (worth 1/0.7 at the wall).the first back flow (u<0) occurs, if dU/dx < 0 at =0The time that makes the wall shear stress zero :
21 2.4 Unsteady flow separation Application to the circular cylinder
22 2.4 Unsteady flow separation High Reynolds : secondary separations
23 2.8 ConclusionIn the BLT context, the necessary condition for separation with the adverse pressure gradient corresponds to a deceleration of the potential flow (Bernoulli theorem).For instance, if strong enough, it gives a "location" for which the friction at the wall is zero that will announce a separation. However the BLT of Prandtl is not complete and unable to give a description of the BL at separation.At a separation the BL develops a triple layer structure (improvment of the Prandtl BLT)As far as the potential flow solution is not much modified by the separation, the boundary layer theory remains relevant to predict the onset of separationIt is still relevant if the potential flow of the separated flow (chapter Free Streamline Theory) is used to compute the BL solution.