Flow separation

All solid objects traveling through a fluid (or alternatively a stationary object exposed to a moving fluid) acquire a boundary layer of fluid around them where viscous forces occur in the layer of fluid close to the solid surface. Boundary layers can be either laminar or turbulent. A reasonable assessment of whether the boundary layer will be laminar or turbulent can be made by calculating the Reynolds number of the local flow conditions.

Flow separation occurs when the boundary layer travels far enough against an adverse pressure gradient that the speed of the boundary layer relative to the object falls almost to zero.^[1]^[2] The fluid flow becomes detached from the surface of the object, and instead takes the forms of eddies and vortices. In aerodynamics, flow separation can often result in increased drag, particularly pressure drag which is caused by the pressure differential between the front and rear surfaces of the object as it travels through the fluid. For this reason much effort and research has gone into the design of aerodynamic and hydrodynamic surfaces which delay flow separation and keep the local flow attached for as long as possible. Examples of this include the fur on a tennis ball, dimples on a golf ball, turbulators on a glider, which induce an early transition to turbulent flow regime; vortex generators on light aircraft, for controlling the separation pattern; and leading edge extensions for high angles of attack on the wings of aircraft such as the F/A-18 Hornet.

Boundary layer separation is the detachment of a boundary layer from the surface into a broader wake.^[3] Boundary layer separation occurs when the portion of the boundary layer closest to the wall or leading edge reverses in flow direction. The separation point is defined as the point between the forward and backward flow, where the shear stress is zero. The overall boundary layer initially thickens suddenly at the separation point and is then forced off the surface by the reversed flow at its bottom.^[4]

Adverse pressure gradient

The flow reversal is primarily caused by an adverse pressure gradient imposed on the boundary layer by the outer potential flow. The streamwise momentum equation inside the boundary layer is approximately stated as

u {\partial u \over \partial s} = -{1 \over \rho}{dp \over ds} + {\nu} {\partial^2 u \over \partial y^2}

where $s,y$ are streamwise and normal coordinates. An adverse pressure gradient is when $dp/ds > 0$ , which then can be seen to cause the velocity $u$ to decrease along $s$ and possibly go to zero if the adverse pressure gradient is strong enough.^[5]

Graphical representation of the velocity profile in the boundary layer. The last profile represents reverse flow which shows separated flow.

Influencing parameters

The tendency of a boundary layer to separate primarily depends on the distribution of the adverse or negative edge velocity gradient $du_o/ds (s) < 0$ along the surface, which in turn is directly related to the pressure and its gradient by the differential form of the Bernoulli relation, which is the same as the momentum equation for the outer inviscid flow.

\rho u_o {du_o \over ds} = -{dp \over ds}

But the general magnitudes of $du_o/ds$ required for separation are much greater for turbulent than for laminar flow, the former being able to tolerate nearly an order of magnitude stronger flow deceleration. A secondary influence is the Reynolds number. For a given adverse $du_o/ds$ distribution, the separation resistance of a turbulent boundary layer increases slightly with increasing Reynolds number. In contrast, the separation resistance of a laminar boundary layer is independent of Reynolds number — a somewhat counterintuitive fact.

Internal separation

Boundary layer separation can occur for internal flows. It can result from such causes such as a rapidly expanding duct of pipe. Separation occurs due to an adverse pressure gradient encountered as the flow expands, causing an extended region of separated flow. The part of the flow that separates the recirculating flow and the flow through the central region of the duct is called the dividing streamline.^[4] The point where the dividing streamline attaches to the wall again is called the reattachment point. As the flow goes farther downstream it eventually achieves an equilibrium state and has no reverse flow.

Effects of boundary layer separation

When the boundary layer separates, its displacement thickness increases sharply, which modifies the outside potential flow and pressure field. In the case of airfoils, the pressure field modification results in an increase in pressure drag, and if severe enough will also result in loss of lift and stall, all of which are undesirable. For internal flows, flow separation produces an increase in the flow losses, and stall-type phenomena such as compressor surge, both undesirable phenomena.^[6]

Another effect of boundary layer separation is shedding vortices, known as Kármán vortex street. When the vortices begin to shed off the bounded surface they do so at a certain frequency. The shedding of the vortices then could cause vibrations in the structure that they are shedding off. When the frequency of the shedding vortices reaches the resonance frequency of the structure, it could cause serious structural failures.

Footnotes

↑ Anderson, John D. (2004), Introduction to Flight, Section 4.20 (5th edition)
↑ Clancy, L.J., Aerodynamics, Section 4.14
↑ White (2010), "Fluid Mechanics", Section 7.1 (7th edition)
1 2 Wilcox, David C. Basic Fluid Mechanics. 3rd ed. Mill Valley: DCW Industries, Inc., 2007. 664-668.
↑ Balmer, David. "Separation of Boundary Layers." School of Engineering and Electronics. 2 December 2007. University of Edinburgh. 12 March 2008 <http://www.see.ed.ac.uk/~johnc/teaching/fluidmechanics4/2003-04/fluids14/separation.html>.
↑ Fielding, Suzanne. "Laminar Boundary Layer Separation." 27 October 2005. The University of Manchester. 12 March 2008 <http://www.maths.manchester.ac.uk/~suzanne/teaching/BLT/sec4c.pdf>.

References

Anderson, John D. (2004), Introduction to Flight, McGraw-Hill. ISBN 0-07-282569-3.
Clancy, L.J. (1975), Aerodynamics, Pitman Publishing Limited, London ISBN 0-273-01120-0.

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