flow separation and secondary flow.pdf

8/14/2019 flow separation and secondary flow.pdf

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190 CHAPTER 9. FLOW SEPARATION AND SECONDARY FLOW

E.S. Taylors movie: Secondary flows

All dependent on vorticity (potential flow have uniquesolutions).

Flow separation

vorticity from boundary, requires pressure gradients (see lec-tures)

No flow separation

Flow in a bend: material lines and vortex lines

Is the bathtub vortex really 2-dimensional? Necklace vortex

Ekman layers

Figure 9.1: Secondary flows: the role of vorticity, movie by E.S. Taylor


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9.1. CURVED CHANNEL 191

Figure 9.2: Secondary flow: creeping flow in a wedge, forward- and backward-facing steps, diffuser, wake of a sphere, necklace vortex around an obstaclein a boundary layer.


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Figure 9.3: Irrotational flow in a curved channel

The simple version, sometimes found in undergraduate texts, is possible ifwe ignore friction at the walls.

The presence of a secondary flow, invalidating the simple irrotationalmodel, can be explained by the presence of a mean shear in the profile (fasterflow at the surface, no-slip at the bottom). The upstream conditions should

include vorticity in the spanwise direction and this is key to the next level ofanalysis.

It is assumed that viscous effects are not dominant over the short traveltime through the curved portion of channel. One can rationalize that viscousdiffusion, though present, does not qualitatively affect the reasoning. Underthis assumption, Helmholtz theorem on vortex lines being material linesdoes apply as an approximation, and we track the progression of a materialline through the bend. Since the inside of the curve moves faster than theoutside, the material line will rotate relaive to the mean streamline at thecenter of the channel. Since the material line is also a vortex line, there will

be a vorticity component in the streamwise direction! (Fig. 9.4)This streamwise vorticity introduced by the bend is responsible for the

circulation of water outward at the free surface and inward along the bottomof the channel. This meachanism may be responsible for the meandering ofrivers, and for the erosion patterns (steep banks outside, sediment depositinside the curve).


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9.2. VORTICITY REVERSAL IN 2D SEPARATION 193

Figure 9.4: Creation of axial vorticity in a curved channel

A similar situation arises for laminar flow in a circular helical pipe (or ina bend) (Fig. 9.5). Two counter-rotating axial vortices arise, recirculating

the fluid across the mid-plane of the pipe. This is a simple mechanism ofmixing enhancement, used in some heat exchangers. Unlike the case of theopen channel, here we see streamlines actually losing contact with a solidsurface: that is flow separation.

9.2 Vorticity reversal in 2D separation

Flow separation is possible in potential flow (e.g. flow toward a stagnation

point, with separation streamline, Fig. 9.6) But, with the presence of vortic-ity, the range of possible phenomena expands enormously. We limit ourselvesto 2D flows, and start from the vorticity balance (without vortex stretching,of course)

t+U = 2 (9.1)


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Figure 9.5: Secondary flow in a helical pipe

3 2 1 0 1 2 30

0.5

1

1.5

2

2.5

3Potential flow at a stagnation point

x

y

Figure 9.6: Potential flow separation at a stagnation point


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9.2. VORTICITY REVERSAL IN 2D SEPARATION 195

Figure 9.7: Separation in a 2D boundary layer

In a 2D BL-type configuration (Fig. 9.7), the only component of vorticity is normal to the plane of the sketch:

= xv yu (9.2)

implying

y=2

xyv 2

yyu. (9.3)

Then, we look at continuity:

xu+yv= 0 (9.4)

Right along the wall, the ucomponent is zero (no-slip), as is thev component(impermeable wall) regardless ofx, which implies that

yv |y=0=xv |y=0= 0 (9.5)

It follows that

2xyv|y=0= 0 (9.6)

So, without any dynamics yet, we conclude that

y= 2

yyu (9.7)


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Figure 9.8: Vorticity reversal in front of a step

at the wall. This implies that changes in vorticity, including changes in sign,away from the wall is necessarily related to changes in stress (viscosity notpart of the kinematics, of course).

So the question is: how is reversed vorticity introduced? This needs to bethought of in the broader context of the introduction of vorticity of any signinto a flow: from Kelvins theorem, we know this to be a viscous phenomenon.But the ZPGBL was not all that enlightening in this regard, with all thevorticity introduced at the leading edge. The following analysis represents amore detailed look at the effect of pressure (surprise!) on vorticity1

In fact, taking the BL equation (with pressure gradient leading to thestep), and looking a the immediate vicinity of the wall (y 0), we have

0 = 1

xp+

2

yyu (9.8)

At the wall1

xp= y (9.9)

1So, when someone makes a statement to the effect that there is no pressure in the

vorticity equation, youll know the pressure effects are hidden but possibly present!


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9.3. INTRODUCTION OF VORTICITY 197

so that the conduction flux of vorticity from the wall is proportional to the

streamwise pressure gradient. When pressure increases, (positive) vorticitydiffuses from the wall into the flow.

9.3 Introduction of vorticity

Kelvins theorem (Ch. 5) gives us conditions under which an irrotational flowremains irrotational (although the theorem is not just about irrotational flow,of course!). It is complemented by an understanding of how vorticity may beintroduced. (A parallel with Bernoullis equation (Ch. 5), complemented bylosses (Ch. 3) and Croccos theorem (Ch. 5) for mechanisms of change for

B, is in order.) We are finally in position to do this.Of the five mechanisms listed below, three involve the effects of walls

(boundaries), while two operate in the bulk of the fluid.

1. In the Shapiro movies, we saw how, in the resence of density variations,the vorticity equation has a source term proportional to p, sothat a misalignment of the pressure and density gradients does in-troduce vorticity. This mechanism is inviscid, and can work at fluidinterfaces (remember the sloshing fluid in a container).

2. More generally, rotational forces (Coriolis force in rotating systems

see Ch. 10 ; Lorentz force in conducting fluids subject to magneticfields; etc.) can introduce vorticity in the flow.

3. Then, in Ch. 8, we saw the singularity at the plates leading edge intro-duces vorticity at one point. For ZPG, no other vorticity is introduced.Because the flow is not differentiable at sharp corners and such, onemay think of these singular points as a crude representation of regionsof intense pressure gradients, under the following item.

4. But, in the presence of pressure gradients, vorticity diffuses in from thewall. This is an important mechanism of flow separation in adverse

pressure gradients, with many complicating factors in 3D flows.

5. Finally, unsteady flow near a wall can also lead to diffusion of vorticityfrom the wall: at the wall the momentum equation

tu+1

xp= y (9.10)


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Figure 9.9: Vorticity introduction: singular points (Ch 8), pressure gradientsnear walls, density gradient, rotational forces (Ch 10), unsteady motion nearwalls (Ch 6.


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9.4. ADVANCED TOPICS AND IDEAS FOR FURTHER READING 199

applies to Stokes second problem (no pressure gradient) (see Ch. 7).

Once present somewhere in the flow, trace amounts of vorticity spread ev-erywhere by diffusion: think about the effect Greens function. The decisionto be made by the analyst is about the importance of trace vorticity in eachgiven situation.

9.4 Advanced topics and ideas for further read-

ing

Because flow separation affects dramatically the performance of aerodynamicsurfaces and wake formation, it is a very active field of current research. Thecontrol of separation is particularly important.

Problems

1. Describe changes in vorticity profiles in pictures of flow separation.

2. Construct a mind map about the role of vorticity in flow separation.

3. Discuss the differences between secondary flow separation, and the sep-

aration occuring in potential flows (e.g. flow around a cylinder).


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