Vorticity and Vortex Dynamics 2011 Part 8 pptx

7.2.2 Plane Prandtl–Batchelor FlowsThis subsection discusses the Euler limit of two-dimensional flow over a sta-tionary body with steady separated vortex bubble of area S bounded by C.4 B

Thus, in (7.21), for the term with t we still have (7.17a,b), while for the terms with e θ we have, noticing ∂ s t = κn with κ being the curvature of Cs,

dΓ

dψ = Ce

ψ g(η) dη

, g(ψ) =

4

r(κ4− ∂ n log u s ) ds

r2u s ds . (7.23)But, as Cs shrinks to the core center as assumed, since n points toward the

center there must be u s → 0+ with ∂ n u s < 0 and κ → +∞ Hence, Γ would

be singular if C = 0 The permissible solution is thus simply Γ = constant,

implying that ω s = 0 and v = C/r (C = 0 if the bubble flow extends to

the z-axis) On the other hand, this removes the e θ-component in (7.21b),making it the same as (7.18) and recovering (7.14) The proof of the theorem

is therefore completed

We make two remarks on the theorem and closed bubble flow First, theformation mechanism of these bubble flows is very different from that of con-centrated vortices by the rolling up of free vortex layers For the former theviscous effect has to take sufficient time to fully act on the motion, sending thevorticity from the sheet to the interior, which finally reaches an equilibriumsteady state The sheet vorticity is supplemented by the outer flow But forthe latter there is no sufficient time for diffusion to reach equilibrium state.Goldstein and Hultgren (1988) have pointed out that in this case the vorticitycan have variable distribution in closed streamlines

Second, the physical explanation of the theorem is simple For steady dimensional flow, the viscous vorticity equation

two-u · ∇ω = ν∇2ω

is a diffusion equation If ω changes across streamlines, there must be an

inward or outward vorticity diffusive flux But at the center of closed lines there is no vorticity source or sink (Sect 4.1); so in steady state thisdiffusion cannot exist The only possibility is, therefore, a constant vorticity,which implies no vorticity diffusion In contrast, for rotationally symmetric

stream-flow, in cylindrical coordinates (x, r, θ) by (7.15) one finds a uniform axial

viscous force

which is balanced by a uniform pressure gradient Thus, the inviscid velocitydistribution is not altered Both this physical interpretation and the proof

7.2 Steady Separated Bubble Flows in Euler Limit 345

procedure of the theorem stress the key role of vorticity diffusive flux σ =

ν∂ n ω defined by (4.17) in the formation of the bubble’s core region In fact,

that flux was precisely introduced from examining the line integral of ν ∇×ω,

and can well replace ν ∇ × ω in the invariant conditions To see this, we use

the notation of (4.23) so that on a surface S

en-(Chernyshenko 1998) This is why in the Euler limit H becomes constant along

the streamline The corresponding invariant condition for closed streamlines

is, evidently,

Cs

∂ω

which is an alternative form of the two-dimensional version of (7.16)

For axisymmetric flow, from (7.20) it follows that

obvious We just note that in the Euler limit with ω n= 0 (4.25) gives

7.2.2 Plane Prandtl–Batchelor Flows

This subsection discusses the Euler limit of two-dimensional flow over a

sta-tionary body with steady separated vortex bubble of area S bounded by C.4

By (7.13), outside and inside the bubble the vorticity is zero and constant,respectively We have Bernoulli integral not only outside the bubble but also

in the core region of the bubble, since in (7.12) there is ω × u = ∇(ω0ψ),

ρ + ω0ψc in the core region, (7.29)

where the suffix c denotes the bubble center where q2=|∇ψ|2= 0 Therefore,

let ψ = 0 along C with ψ > 0 and n > 0 in the bubble as in Fig 7.13, the

Euler limit of the flow is a solution of the following problem:

of the true viscous solution as  ≡ Re −1 → 0 and in which ω0 and [[H]] are

specially determined Here the Reynolds number is defined based on the body

size R and U ∞ As said in the beginning of this section, this is achieved byreturning to viscous analysis, where the boundary vortex sheet is replaced by

a cyclic vortex layer of finite thickness This issue will be addressed later afterdiscussing some general properties of the solution of (7.30)

Let U e = ψ ,n | ψ=0 − be the potential velocity at C, by (4.118a) the vortex sheet strength γ = u − U eis given by

γ = − [[H]]

U m

, U m= 1

The stream function can be written as ψ = U ∞ y + ψ1 + ψ2, where the first

term is the stream function of uniform oncoming flow, and by the Biot–Savartlaw

Analyses of axisymmetric Prandtl–Batchelor flow are relatively rare One example

is the Hill spherical vortex with swirl (Sect 6.3.2)

7.2 Steady Separated Bubble Flows in Euler Limit 347

are the contributions of ω0 and γ, respectively Moreover, since ω0 depends

on U ∞, we need a compatibility condition

U ∞=%

2[[H]] − ∂

∂y (ψ1+ ψ2) at the upstream end. (7.33)

Then the inviscid problem amounts to finding the shape and strength of the

sheet for given ω0and [[H]], including the separation and re-attachment points.

A closed vortex bubble carried by a body will produce a lift Let S and

ω0 be the dimensionless area and vorticity of the bubble, respectively, scaled

by R and U ∞ Then by the well-known (dimensional) Kutta–Joukowski lift

formula L = ρU ∞ Γ (for more discussion see Chap 11), with Γ = ω0S being

the total vorticity or circulation of the bubble , the additional lift coefficient

due to the bubble (not including the lift caused by vortex sheet γ) simply

be [[H]] = 0 and the external potential flow must be tangent to the surface, see

Fig 7.14 later Thus, in a small neighborhood of A, in terms of local Cartesian coordinates (x, y) the sheet equation y = f (x) must satisfy f (0) = f
(0) = 0,

i.e., y = o(x) The flow inside the cusp varies mainly along the y-direction, so that (7.30b) is reduced to ∂2ψ/∂y2  −ω0, of which the solution satisfying

as we inferred from the Kutta condition in Sect 4.4.3 This being the case,

the local flow is nothing but a Kirchhoff free-streamline flow, which gives

Fig 7.15 Streamlines for airfoil with a trapped vortex ω0 =−20, [[H]] = 0.53.

Adapted from Bunyakin et al (1998)

Several Prandtl–Batchelor bubble flows with the above common featureshave been studied Sadovskii (1971) was the first to solve the problem for the

flow of Fig 7.14, where a pair of Prandtl–Batchelor vortices of length L are symmetrically in touch, known as the Sadovskii flow He derived a pair of integral equations for the sheet shape f (x) and strength γ(x) with a given constant [[H]] The equations were solved numerically The computation was

improved by Moore et al (1988) who gave a complete set of the solutions forSadovskii flow

Other investigated Prandtl–Batchelor flows include corner flows (e.g.,Chernyshenko 1984; Moore et al 1988) and flow over a flat plat with aforward-facing flap (Saffman and Tanveer 1984) The latter was motivated

by the concept that if at a large angle of attack a stationary vortex can becaptured, then the lift will be greatly enhanced The most interesting config-uration along this line is to capture a vortex by an airfoil with a cavity onits upper surface, studied by Bunyakin et al (1996) Owing to (7.34), such

an airfoil may have additional lift but avoid early separation under strongadverse pressure gradient on the upper surface, where the original solid wall

is replaced by a free shear layer like a flexible moving belt.5 The authorsfound that due to the structural instability 3 of closed bubble flow, the bubble

shape is very sensitive to the given ω0 and [[H]], and only in a certain range

of these parameters can a meaningful solution be obtained Figure 7.15 plotsthe configuration and flow pattern

We now turn to the cyclic viscous vortex-layer Squire (1956) was the first

to exemplify that a matched asymptotic expansion can be applied to fix the

flow within the layer as well as ω0 and [[H]] We use the same intrinsic frame

(t, n) as in Fig 7.13, and let quantities be made dimensionless by body size R,

density ρ, and U ∞ Assume now Fig 7.14 represents a bubble on a flat wall

Let s move from A (s = 0) to B (s = s B) along the separated vortex sheet5

As remarked in Sect 7.1.2, the vortex in the cavity is much weaker than thatformed by the rolling-up of a free vortex layer The latter was proposed by Wuand Wu (1992) Such a strong vortex must have axial flow and is expected to bemore stable as well

7.2 Steady Separated Bubble Flows in Euler Limit 349

(in free shear layer) and returns A (s = s A > s B) along the wall (in attached

boundary layer) Let u(s, ψ) be the streamwise velocity in the vortex layer such that dψ = udn Both free vortex layer and attached boundary layer are

governed by a boundary-layer type of equation:

and a rescaled stream function Ψ = Re 1/2 ψ, and denote the total enthalpy

inside the viscous vortex layer by

a known p(s) Then the upstream condition, periodic condition, and wall

condition for (7.36) are, respectively,

Ψ > 0 : g(0, Ψ ) = g(s A , Ψ ); (7.37b)

Ψ = 0, s B < s < s A: g = p(s). (7.37c)

Besides, since ψ = 0+ corresponds to Ψ → ∞ in the Euler limit, (7.30c) gives

the matching condition

Ψ → ∞ : g → H(ψ)| ψ=0+= H | ψ=0 − − [[H]] < ∞, (7.37d)which ensures the uniqueness of the solution

Generically, the boundary-layer approximation cannot be applied to

re-gions near A and B, referred to as turn rere-gions But when a turn region has a

cusp, only the normal variation is important, and the inviscid cusp flow awayfrom the viscous free vortex layer and boundary layer is stationary In fact,

the characteristic flow rate Q in the turn region must be of the same order as that in a boundary layer: Qturn∼ Re −1/2, so that locally there is

Return∼ Uturn LturnRe ∼ Qturn Re ∼ Re1

→ ∞.

0.2 0

Fig 7.16 Velocity profile as a function of ψ in the cyclic layer of Fig 7.15 for the

relevant Euler solution of flow over an airfoil with trapped vortex Adapted fromBunyakin et al (1998)

Thus, to the leading order the flow is inviscid and Bernoulli integral holds Inthis case it can be shown that when using (7.36) as the governing equation theturn region can be ignored (see Bunyakin et al (1998) for references) Thusthe formulation for the cyclic vortex layer is completed

For example, by solving (7.36) under condition (7.37), as well as the ventional boundary-layer equation for flow over a wall, Bunyakin et al (1998)extend their inviscid solution shown in Fig 7.15 to include the viscous attachedand free vortex layers Then the whole flow field is determined Figure 7.16shows the velocity profiles in the cyclic vortex layer The airfoil was carefullyselected to avoid any smooth-surface separation other than the fixed frontand rear points of the cavity To make the airfoil look more realistic, tangentblowing was introduced at three points of the cavity wall (arrows in Fig 7.16).This requires an extension of boundary conditions (7.37) and is omitted here

con-7.2.3 Steady Global Wake in Euler Limit

The preceding discussion on plane Prandtl–Batchelor flows is for the tion where the bubble size is of the same order of the body size A differentand more challenging problem relevant to the Prandtl–Batchelor flow is theasymptotic form of the entire vortical wake behind a bluff body It will be

situa-seen in Sect 7.4 that as the Reynolds number Re = U D/ν (based on ter D) increases to about 50 the wake behind a circular cylinder starts to be spontaneously unsteady and vortex shedding occurs However, the mathemat-

diame-ical existence of a steady but unstable wake cannot be excluded Numerdiame-ically,

careful Navier–Stokes calculations (Fornberg 1985) which specifically nate the possibility of unsteadiness and asymmetry have shown that steadywake is a Navier–Stokes solution Theoretically, such a mathematical solutionhas been obtained in the Euler limit Roshko (1993) remarks that, while this

elimi-7.2 Steady Separated Bubble Flows in Euler Limit 351solution is mainly of academic interest, “it is an intriguing and importantone for theoretical fluid mechanics and it provides perspective on the ‘realproblem’.”

The classic Kirchhoff free-streamline wake, which is open at downstreamend and the fluid therein is stationary, was criticized by Batchelor (1956b).The dilemma is: if the wake is open, how can the downstream boundary con-dition that the flow resumes uniform be satisfied? And, if the wake is closed,then the Prandtl–Batchelor theorem requires that the wake has a uniformvorticity rather than being stationary Thus, Batchelor (1956b) proposed that

the steady wake in the Euler limit is a closed bubble with ω0and [[H]] as meters It has been found that the wake length increases linearly as Re; and its width increases initially as O(Re 1/2 ), but after Re > 150 turns to be O(Re)

para-as well, see Fig 7.17 Moreover, in such a big pair of separated bubbles thevorticity is basically constant as predicted by the Prandtl–Batchelor theorem;and at the outer boundary of the bubbles there is a thin vortex layer, whichtends to vanish as the characteristic velocity increases toward downstream.After many researchers’ effort, a complete asymptotic theory of steadyseparated flow has been established and supported by numerical tests Forcomprehensive reviews see Sychev et al (1998, Chap 6) and Chernyshenko(1998); a few major points are briefly outlined here

First, in the global bubble scale the flow is a uniquely determined inviscid

Sadovski flow (where the cylinder shrinks to a point as Re → ∞), of which the

width-to-length ratio is h/L = 0.300 and the area is S = αL2 with α  0.44.

The vorticity ω0 is fixed such that, by the Bobyleff–Forsythe formula (2.159)and from (2.76), the total dimensionless dissipation rate and the total drag

coefficient Cd(nondimensionalized by ρU ∞2R) are, respectively,

Fig 7.17 Schematic flow pattern of steady global wake behind a circular cylinder

(the small semicircle at the left end of the plot) From Chernyshenko (1998)

where kd is drag coefficient in the Kirchhoff flow with the velocity tude on the free streamline equal to unity, depending on the body shape andseparation point Comparing (7.39) and (7.40) yields

1

Cω0

where D0 is a constant; for flow past an isolated body D0  0.235 The key

physics behind (7.42) is the vorticity balance The vorticity diffuses toward thesymmetry line where it vanishes, and also diffuses across the bubble boundary.This loss of vorticity must be compensated by that produced from the bodysurface and advected into the flow In this problem one only needs the neteffect of vorticity discharged from the body rather than the detailed diffusionand advection process; so it suffices to know the sum of vorticity diffusive

flux σ and advective flux u n ω across any line segment, which is nothing but

the end-point difference of the total enthalpy Indeed, as an easy extension of(7.26), by applying (7.25) to any line segment there is (Chernyshenko 1998)

H B − H A=

 B

A

This is why [[H]] enters (7.41), which also shows that the jump must vanish

in the Euler limit The four equations (7.38–7.40) and (7.42) then determine

the four unknowns ω0, S, Cd, and [[H]], with only kd depending on the body

shape Namely, in addition to (7.39) for the drag and (7.41) for [[H]], there is

7.3 Steady Free Vortex-Layer Separated Flow

Closed-bubble separated flows discussed in Sect 7.2 are relatively rare in

re-ality The common situation is free vortex-layer separated flow, in which a

separated vortex layer rolls into a vortex and the flows at both sides of thelayer come from the same main stream As said in Sect 7.1.2, the free vortex-

layer may come from both closed separation initiating at a saddle point of the

τ w-field, for example at the apex of a slender delta wing as shown in Fig 7.7b,

and open separation initiating at an ordinary point of the τ -field, as seen

7.3 Steady Free Vortex-Layer Separated Flow 353

in Fig 6.1 A prototype of free vortex-layer separated flow is a pair of vortexsheets shed from a slender wing, which roll into vortices above the wing andgreatly enhance the lift Being steady and stable in a range of parameters, this

kind of detached-vortex flow has become the second generation of aeronautical

flow type in practical use (after the attached flow type over streamlined body;e.g K¨uchemann (1978))

No general theory is available for free vortex-layer separated flow even

in the Euler limit, because as seen in Sect 4.4.4 the self-induced rolling-upprocess of a vortex sheet is inherently nonlinear One has to appeal to approxi-mate theories or numerical simulation The simplest theory in the Eulerlimit is fully linearized, in which the vortex-sheet rolling up is completelyignored so that the sheet location is known This is the case in Prandtl’sclassic lifting-line theory for a thin wing of large aspect ratio (e.g., Prandtland Tietjens 1934; Glauert 1947; see also Chap 11) But here we need to ad-dress the nonlinearity of the self-induction, with the expense that in someother aspects significant simplification has to be made This is the case of

the slender-body theory to be used throughout this section.6 We consider theslender approximation of vortex-sheet conditions first, then review methodsfor computing the self-induced evolution of leading-edge vortex-sheet and freewake vortex sheet Finally, we analyze the stability of a class of slender freevortex-layer separated flow

7.3.1 Slender Approximation of Free Vortex Sheet

Consider a steady flow over a point-nose slender body shown in Fig 7.18 In

a body coordinate-system Oxyz with the body axis along the x-direction and

z-axis vertical up, let the local angle of attack at x be α(x) = O()

that the constant oncoming velocity U has (x, z) components

U = (U cos α, U sin α) = (U, U α) + O(2), (7.45)

and the disturbance velocity components are (u
, v, w) = O(U ) Due to

the slenderness, the x-wise disturbance of the body to the fluid is much

smaller than those in cross directions Consequently, a three-dimensional flow

problem is reduced to a cross-flow U α over two-dimensional sections of the body at different x, and away from the vortex sheet we only need to con- sider a two-dimensional disturbance velocity potential ϕ(y, z; x) The three- dimensionality of the flow lies in the x-dependent boundary condition, with

6

This section could be shifted to Chap 11 on aerodynamics We put it here forunderstanding the basic physics of free vortex-layer separated flow as the coun-terpart of closed-bubble separated flow Although in engineering applications theslender-body theory has now been replaced by more numerically oriented meth-ods, it provides an opportunity to demonstrate how the general theory of three-dimensional vortex sheet dynamics is specified to concrete problems of significantpractical value

Fig 7.18 Free vortex-layer separated flow from a slender body

x being a parameter Since U α

in its generality: for attached flow over a complicated configuration it is theonly analytical method, and for separated flow it is the simplest semianalyticalmethod

The cross-flow potential ϕ is to be superposed to potentials due to the

uniform oncoming flow, angle of attack, and the body volume Let the cross

area of the body be A(x) which for a remote observer appears as a source distribution b0(x) along the x-axis The relation between A(x) and b0(x) varies

from subsonic to supersonic oncoming flows The total potential and far-fieldboundary condition read

u+· n = u − · n = ¯u · n = 0, ¯u · [[u]] = 0, (7.48a,b)Equation (7.48a) tells not only the continuity of normal velocity across anyvortex sheet, but also the fact that the sheet must be a stream surface in steady

flow Equation (7.48b) is from the Kelvin circulation theorem D[[ϕ]]/Dt = 0,

7.3 Steady Free Vortex-Layer Separated Flow 355which by (4.132) and (4.133) is equivalent to the pressure continuity What

we need now is to find the slender-approximation of (7.48a,b), the steadyversion of the Kutta condition (4.141), and the Biot–Savart formula (3.31) inconvenient component form The following algebra is based on the work ofClark (1976) and J.H.B Smith (1978)

As shown in Fig 7.19a, let a slender vortex sheet Σ intersect a cross-plane

π with unit normal e x at a curve C, which has unit tangent vector t Then

ex × t = n c is the unit vector normal to C, and (e x , t, n c) form a local

orthonormal “C-frame”, where by (7.45) we write

u = (U + u x )e x + u s t + u n n c + O(2). (7.49)

Alternatively, one can use a polar coordinate system (r, θ), see Fig 7.19b,

which leads to an orthonormal “P -frame” (e x , e r , e θ), by which we can define

On vortex sheet Σ : r = r(x, θ) (7.50a)

Along separation line : θ = g(x), r = f (x, g(x)). (7.50b)

The two frames are related by a rotation about the x-axis by an angle φ:

Now, since (7.48) is expressed in the vortex-sheet intrinsic frame, we need to

use both C- and P -frames to construct that frame under slender

approxima-tion By (7.50a) we have

Fig 7.19 Local coordinate systems for slender-approximation analysis, (a) the

“C-frame”(e x , t, n c ), (b) the “P -frame” (e r , e θ) on a cross plane The unit normal

vector n of Σ is generally not aligned to n

is also a unit vector tangent to Σ, generally not perpendicular to t Then by

(7.49) and (7.51), the normal of Σ is given by

Thus, u n is approximately continuous across Σ, and hence

[[u]] = [[u x ]]e x + [[u s ]]t + O(2). (7.53)

Therefore, by (7.52) the slender stream-surface condition (7.48a) reads

thus, actually [[u n ]] = O(3) But [[u s]] has to be retained here since there is no

O() term Note that [[u]] · e x cannot be simply identified as e x · ∇[[ϕ]] = [[ϕ]] ,x,

because e x is not a tangent vector and the resulting gradient will no longer

be tangent to Σ Rather, one has to express e x by e via (7.52), which yields

Then, the Kutta condition (4.143) requires that in the “downstream side”

of the vortex sheet Σ (the cusp side is denoted by superscript bar, see

Fig 7.19b) the flow must be along the separation line Since by (7.50b) thedirectional ratios of the separation line are

: f dg

dx ,

7.3 Steady Free Vortex-Layer Separated Flow 357

the same should be U + u − x : u − r : u − θ This yields, ignoring O(2) terms,

where the positive square root is taken to ensure u s is toward downstream

This is the slender Kutta condition, which determines the total circulation of

free vortex sheets shed from the separation line The two terms represent thespeeds needed for turning the free stream to the separation-line direction andfeeding the circulation into the vortex sheet, respectively

Finally, by (7.52) and (7.53), and recall that [[u x ]] = O(2), the slendervortex-sheet strength reads

γ = n × [[u]] = −e x [[u s ]] + O(2),

which causes an induced velocity u π on the cross plane only, in consistencywith the slender approximation (7.46) Therefore, it suffices to use the two-dimensional complex-variable form of the Biot–Savart formula:

where Γ0and Γ eare the values of the circulation at initial and terminal points

of the vortex sheet When Z → Z
, the Cauchy principal value is implied in

the integral and we return to the Birkhoff–Rott equation (4.143)

The above slender-body formulation can be simplified when the flow hasconical similarity, which will be so when the body model is conical and ex-tends infinitely long In the body-axis frame (Fig 7.18), a steady conical flow

(cf Sect 6.2.3) depends on only (y/x, z/x) and keeps the same along each

ray from the point nose The body shape and separated vortices are all alongrays The lateral size of a slender conical body is characterized by its semiapex

angle  or semispan s = kx with k

niently described by conical coordinates (y
, z
) = (y/s, z/s) in the cross-flow plane and it suffices to examine the flow in a single (y
, z
) plane Then in the

preceding vortex-sheet conditions we simply have ∂r/∂x = k, dg/dx = 0 At

a spatial point x, the ray has length R = |x| and unit vector e R = x/R given

A slender conical vortex can be approximated by a quasi two-dimensional

point vortex only on a spherical surface with normal e R

As indicated by (7.45–7.47), on a cross plane with normal e x, a slender

conical-flow problem is the superposition of an angle-of-attack problem u α

caused by a cross-flow U n  Uα, and a thickness problem u a = us(y
, z
) +

U e x caused by an axial flow U x  U at α = 0, where us represents a dimensional source distribution due to the body thickness Here, an important

two-consequence of e R = e x is: the axial flow U e x should be further decomposed

being a conical similarity parameter that measures the relative magnitude of

α Obviously, the linear velocity field u c exists in any slender conical flow,

and represents a uniform sink everywhere in the (y
, z
) plane The boundary

condition for the thickness problem has to be jointly satisfied by us and u c:

n · us=−n · u c=U n

K (y

n

y + z
n z ) on C, (7.62)

where n now denotes the unit normal vector of the cross-flow contour C.

Consequently, what matters is the total cross-flow with free-stream velocity

U n (Cai et al 2003):

e y v(y
, z
) + e z w(y
, z
) = u α (y
, z
) + us(y
, z
) + u c (y
, z
). (7.63)

7.3 Steady Free Vortex-Layer Separated Flow 359

7.3.2 Vortex Sheets Shed from Slender Wing

The quasi two-dimensional feature of the slender approximation naturallysuggests extending the conformal-mapping method from body contour to the

contour C of a separated vortex sheet in a cross-flow plane Z = y + iz, where C is treated as a cut This approach was mainly developed at Royal

Aeronautical Establishment (RAE) of the United Kingdom in 1960s and 1970s

so we call it the RAE method In this method (7.54), (7.55), (7.57), and (7.58)

are all necessary input A major difficulty is that once the vortex sheet rollsinto a vortex with distributed vorticity and spiral arms (Chap 8), inside thevortex core the conformal mapping can no longer be used In the RAE method(Smith 1968) the tightly rolled-up part of the sheet is replaced by a single line

vortex of circulation Γ v (x) at Z v on each cross plane, which is connected tothe unrolled sheet by a cut that can satisfy (7.53) but not (7.54) The bestone can do is to impose (7.54) in averaged sense, i.e., the total force acting on

the point vortex plus the cut is zero This condition determines Z v

Smith (1968) applied the RAE method to a slender flat-plate delta wing atincidence The wing has infinite downstream extension so the flow is conical

On a cross plane the wing semispan is s = kx Figure 7.20 is the calculated cation of vortex sheet and line vortex (appearing as a point) on the Z-plane at different α The agreement of computed spanwise pressure distribution with

lo-experimental data is reasonably good The main discrepancy is due to theincapability of computing the secondary separation from the upper surface

of the wing, induced by the primary leading-edge vortex (Fig 7.7a) The ondary vortex has opposite circulation and weakens the suction peak caused

sec-by the primary vortex

By using the RAE method, Fiddes (1980) calculated the symmetricallyseparated vortex sheets from a cone of elliptic cross section at incidence,which happens at the nose of aircrafts and missiles In the Euler limit the

location θ s of separation line is indeterminate, which in Fiddes’ calculationwas taken from experimentally observed values Fiddes (1980) also imbeddedthe triple-deck structure (Sect 5.2) into the calculation to iteratively deter-

mine the separation line from an initially assumed θ s The results are inreasonable agreement with experiments

Most of the role of the RAE method has now been replaced by moreefficient numerical methods.7 Numerical computations can also be greatlysimplified within the slender approximation (7.45) and (7.46) Write the local

axial velocity as U = x/t, a steady three-dimensional flow problem at different

x-stations is reduced to an unsteady two-dimensional cross-flow problem at

different t.

7In numerical approaches the vortex-sheet conditions (7.48a) and (7.48b) are matically satisfied, but the Kutta condition and Biot-Savart formula or its equiv-alence remain necessary

Fig 7.20 Location of vortex sheet (solid curve) and line vortex (circle) over a

slender delta wing for different K = α/k (marked by numbers), calculated by the

RAE method From Smith (1968)

Figure 7.21a shows the pattern of a leading-edge vortex sheet from a der delta wing of finite chord length (so the flow does not have conical simi-larity) at a typical cross-plane, computed by Ma and Jin (1991) using the

slen-viscous vortex-in-cell method According to the Kutta condition, the vortex

sheet is set to leave the leading edge tangentially By carefully adjusting thegrid, Ma and Jin were able to capture not only the global pattern but alsothe tendency of the sheet to break into discrete vortices due to the Kelvin–Helmholtz instability (Chap 9)

Then, as the leading-edge vortex layer travels downstream to the wake,

it must meet the trailing-edge vortex layer and merge to a single andcomplicated structure The rolling-up of wake vortex-sheet alone has been

beautifully computed by Krasny (1987) also within the same U = x/t

approx-imation, as already exemplified by Fig 4.21; but the evolution of the mergedleading- and trailing-edge vortex sheets is of particular interest This has alsobeen computed by Ma and Jin (1991), see Fig 7.21b at a typical wake plane.Figure 7.21c is the sketch of the vortex-sheet pattern by K¨uchemann (1978)for comparison

It should be stressed that although numerically one can compute thetightly rolled-up part of a vortex sheet, this part cannot be accurately simu-lated on a cross plane In the preceding analysis we have neglected the con-

tribution of [[u x ]] = e x · ∇Γ to the vortex sheet strength γ, which for an

isolated vortex sheet is of O(2) However, when many layers of the sheet

squeeze together, [[u x ]] may have an O(1) integrated effect and cause a strong

7.3 Steady Free Vortex-Layer Separated Flow 361

(a)

(c)

(b)

Fig 7.21 The evolution of vortex sheets shed from a slender delta wing at α = 20.5 ◦

and Re = 1 06 based on root-chord length, computed by Ma and Jin (1990), (a) a leading-edge vortex sheet upstream the trailing edge, (b) a merged leading- and trailing-edge vortex sheet in the wake, (c) is reproduced from K¨uchemann (1978)

axial velocity inside the vortex core (for an example see Sect 11.5.4) This willmake the flow inevitably three-dimensional, beyond the ability of any slenderapproximation

7.3.3 Stability of Vortex Pairs Over Slender Conical Body

We now shift our focus from the rolling up process of vortex sheets shed from aslender body to the slender vortex pair formed thereby, which is the strongest

structure in a free vortex-layer separated flow As the angle of attack α

in-creases to a critical value, an originally steady, stable, and symmetric vortexpair above a slender wing or body may become unstable, leading to asymmet-ric or unsteady structures, or both The asymmetric vortices will cause a largerolling moment in the case of slender wing, or a large side force in the case ofslender smooth body, even at zero roll and yawing angles, respectively Theunderlying physical mechanism has been a controversial issue for long time.But, within the slender conical-flow approximation, simple inviscid analysesmay explain the phenomenon quite well at least qualitatively, without theneed for systematic knowledge of hydrodynamic stability

The occurrence of the asymmetric vortex-flow solution besides the metric one over a circular cone at high enough angles of attack has been

sym-found by Dyer et al (1982) using the simplest cross-flow point-vortex model.

They observed that at a critical similarity parameter K defined by (7.61)

the vortex flow has a bifurcation to a symmetric solution and an ric solution even when the separation lines are postulated as symmetric Asystematic analysis of the instability of the symmetric and asymmetric vortexpair under small conical disturbances has been carried out by Cai et al (2003,2004) for various slender conical flows In their analysis the feeding sheet isignored which, although important in estimating the total force (e.g., Fiddesand Smith 1982), has negligible effect on vortex stability due to much weakervorticity concentration

asymmet-In what follows we present the main analysis of Cai et al (2003) for thestability of symmetric vortex pairs Our concern is not the incompressible and

potential flow field but the motion of point vortices in the (y
, z
) = (y/x, z/x) plane, which is a discrete dynamic system constrained by the flow boundary

conditions (for a general discussion of point-vortex system see Sect 8.3.1) For

example, with vortices 1 and 2 behind a slender cone of radius a = kx, the

system has 4 degrees of freedom By (7.62) and (7.63), the velocity of one such

vortex is determined by the sum of the vortex-induced velocity u α(excluding

the self-induction), the source-caused velocity us, and the velocity u c caused

by the sink In terms of complex variable Z = y
+ iz
, therefore, for vortex 1

2π

1

The point-vortex velocity defined in this way represents the velocity at which

the vortex would move, and will be called vortex velocity to be distinguished

from the flow-field velocity Note that the uniform sink flow is not an analytic

function of Z In general dZ/dt = F (Z, Z) may not be two-dimensionally

divergence-free although the flow is incompressible This is not only due to theexistence of variable axial velocity in three-dimensional slender-body theory,but also because we are dealing with a discrete system even for truly two-dimensional point vortices

Now return to real variables, assume at t = 0 the vortex at x0is stationary

with u0= 0 (a fixed point), and a disturbance shifts it to x0+ δx at t = 0+(δx is constrained by boundary conditions), such that similar to (2.13) or

(7.3) but now for discrete vortices, the linearized system and solution read

7.3 Steady Free Vortex-Layer Separated Flow 363for nonlinearly stable), neutrally stable, and unstable (sufficient for nonlin-

early unstable) if both λ1and λ2have negative real parts, are imaginary, and

at least one of λ1and λ2has positive real part, respectively It is easily verifiedthat

λ 1,2 =12

In complex variable the disturbance displacement reads (δZ1, δZ2) for vortices

1 and 2 Since for any disturbance there is

U n = U and Z2= Z1, see Fig 7.22 The stationary condition for undisturbedvortices is obtained by letting the right-hand side of (7.64) vanish, which

leads to the famous F¨ oppl vortices (F¨oppl 1913)8: stationary vortex pair must

be located on a pair of special curves (F¨ oppl line, see Fig 7.22) with special

two-The F¨oppl vortices are the first theoretical model of closed-bubble separated flow,which we now see is very different from the corresponding Euler-limit

Foppl line

x

Fig 7.22 Stationary vortex pair behind a circular cylinder and the F¨oppl line

series of conformal mappings, if the vortex system is initially placed

symmet-rically then there must be D0= 0, and hence (7.65a) is reduced to

λ 1,2=±%−J0

It is then easily seen that for antisymmetric disturbance there is J0< 0, and

hence the flow is unstable (first proved by F¨oppl (1913)); while for symmetric

disturbance there is J0 > 0, and hence the flow is linear neutrally stable

(first proved by Smith (1973)) Therefore, for an arbitrary small disturbance,

a symmetric point-vortex pair in any two-dimensional incompressible flow

cannot be stable

In contrast to truly two-dimensional flow, the uniform sink (7.60) plays

a crucial role in the slender-conical vortex stability, since solely by this termthere is

D0=− 2U n

no matter where the vortices locate By (7.66b), the appearance of this sink isnecessary for a pair of vortices to be stable It reflects a basic stabilizing mech-

anism (in a highly simplified manner, of course): t he vorticity continuously

generated from the body surface and entering slender free vortices can be anced by its continuous axial advection, so the over-saturation and shedding

bal-of vorticity, typical in truly two-dimensional flow, can be avoided or delayed.Topologically, a vortex in a background sink-flow appears as a stable spiral on

the (y
, z
) plane rather than a center (Sect 7.1.1) Note that since by (7.68)

D0 is inversely proportional to K, for fixed  an increase of α always tends

to make the flow less stable or more unstable So does the thickness effect,which pushes the vortices away from the body The thickness effect overrides

the sink effect when J0< 0, for which the vortex will appear as a saddle point

in the (y
, z
) plane and is unstable

Cai et al (2003) have analyzed the stability property of several typical

con-ical configurations as function of K and the separation point (the semisaddle

on the contour C) For slender circular cones the stationary symmetric vortex

7.3 Steady Free Vortex-Layer Separated Flow 365flow is stable under small symmetric disturbances, but unstable under smallantisymmetric disturbances as demonstrated by Fig 7.23 Hence, an initiallysymmetric vortex flow tends to become asymmetric The instability is mainlyfrom the thickness effect.

In contrast, for slender flat-plate delta wing with us = 0 in (7.63), the

angle-of-attack problem is obtainable from (7.64) by conformal mapping It is

found that J0 > 0 for both symmetric and antisymmetric disturbances with

0 < K ≤ 10, see Fig 7.24, and hence the symmetric vortex pair is stable This

confirms and extends an earlier result of Huang and Chow (1996) In between

2 -2

-1

0 1 2

Fig 7.23 D0 and J0 for symmetric vortices above slender circular cones of

differ-ent K, under symmetric and antisymmetric disturbances The separation angle is

θ = 34 ◦(counted from the real total point) Reproduced from Cai et al (2003)

2 -2

-1

0 1 2

Fig 7.24 D0 and J0 for symmetric vortices above slender delta wing of different

K, under symmetric and antisymmetric disturbances Reproduced from Cai et al.

(2003)

the circular cones and flat-plate delta wing, slender elliptic cones with different

thickness ratio τ and separation points have also been examined For example, for antisymmetric disturbance, as τ increases from zero, the flow changes from stable to unstable The stable spiral and saddle patterns in the (y
, z
) planefor a stable and unstable vortex, respectively, are also illustrated

Cai et al (2003) have also studied the stabilizing effect of fins on cone anddelta wing Their predictions based on this simple model have been comparedwith available experimental results with reasonable agreement Using the sameapproach, Cai et al (2004) have further studied the stability of asymmetricvortex pair

Needless to say, the conical-flow assumption employed in the precedinganalysis cannot follow the development of disturbances along a vortex axis as

in the case of convective instability (Chap 9), which has been considered by

some authors a preferred mechanism for certain vortex asymmetry problems(see the review of Cai et al (2003)) Another possible mechanism for theappearance of asymmetric vortices having been argued is the asymmetry offlow separation/reattachment on both sides of the body (Ericsson 1992)

7.4 Unsteady Bluff-Body Separated Flow

Unlike steady separated bubble flow behind a bluff body discussed in Sect 7.2,unsteady separated flow from bluff bodies is a very common existence and ofgreat significance in engineering applications It causes fluctuating drag andlateral force to the body and is a major source of flow-induced structuralvibration and noise Of various transient or periodic separated flows, the in-compressible flow past a stationary and nominally two-dimensional bluff cylin-

der of cross-flow length D is most important It already possesses almost the

entire complexity of shearing process, such as flow separation, free shear layerand its rolling up, vortex interactions, various shear instabilities, transition tothree-dimensional flow and to turbulence, and unsteady turbulent separatedflow After over a century of effort since Strouhal (1878) observed that the fre-

quency f of vortex shedding is proportional to U/D with the proportionality constant now being known as the Strouhal number St = f D/U , and K´arm´an(1912) constructed the vortex street model (Sect 6.4.3) and estimated thedrag, “the problem of bluff body flow remains almost entirely in the empir-ical, descriptive realm of knowledge.” (Roshko 1993) The great complexityand importance in applications of bluff-body flows are well demonstrated bythe comprehensive two-volume manograph of Zdravkovich (1997, 2002).Nevertheless, the formation mechanism of vortex shedding has been clar-ified, and some of the mechanisms that cause sudden changes of the flow

patterns, Strouhal number St, and drag coefficient CD at different Reynolds

numbers Re = U D/ν have been identified For the latest comprehensive

review see Williamson (1996) In this section we review the basic ena, discuss the formation process of the vortex shedding, and introduce a

phenom-7.4 Unsteady Bluff-Body Separated Flow 367

model to predict the basic Re-dependence of St and CD in which the grated vorticity and energy balances are incorporated

inte-7.4.1 Basic Flow Phenomena

Bluff bodies with smooth-surface separation and fixed separation are typified

by circular cylinder and flat plate normal to the oncoming flow, respectively In

both cases the flow is characterized by high CDand periodic vortex shedding,

of which the quantitative behavior depends on the Reynolds number The key

issue of bluff-body flow is the Re-dependence of St and CD, and the underlying

mechanisms In Fig 7.25 we plot the curves of St, the time-averaged CD,

and the time-averaged base suction coefficient −Cpb versus Re for circular

cylinder, based on experimentally measured data.9 The −Cpb is defined as

negative of the pressure coefficient at the downstream end b of the body:

−Cpb=− pb1− p ∞

which reflects the sensitivity of the flow pattern to Re more adequately than that of CD The−Cpb for Navier–Stokes solutions of the steady attached and separated flow (Dennis and Chang 1970; Fornberg 1985) at Re < 700 are also

shown for comparison, of which the trend is opposite to the realistic unsteadyflow

Figure 7.25 reveals that there are different regimes divided by some

crit-ical Re marked by A, B, , J Each regime has its special flow pattern, as

summarized schematically in Fig 7.26 for side view and top view A few cal visualization photos are shown in Fig 10.42 of Chap 10; for more see Van

typi-Dyke (1982) In each regime, both St and −Cpb have corresponding specialfeatures The physical events behind Figs 7.25 and 7.26 are briefly outlinedbelow (Roshko 1993; Williamson 1996; Noack 1999)

1.Steady flow (regime before A, Re < 49).

Similar to the separated bubble flow over a sphere seen in Sect 4.2.2, at Re =

4 a pair of standing vortices appears behind the circular cylinder due to alocal topological bifurcation at the rear stagnation point, characterized bythe appearance of new fixed points (Bakker 1991) But, no hydrodynamic

instability occurs (Yin and Sun 2003); so we say a kinematic bifurcation Then,

in the steady separated-bubble regime (4 < Re < 49) the flow is globally stable

with respect to all three-dimensional disturbances

2.Laminar parallel and oblique shedding (regime A–B, 49 < Re < 140–194).

At the first critical Reynolds number Recr1  49the flow becomes linearly

9The real measured data have certain diversity due to the difference of tal conditions such as cylinder roughness, amplitude and spectra of free-streamturbulence, aspect ratio of the cylinder, end conditions, and blockage ratio, etc

unstable with respect to two-dimensional disturbances and experiences a

(dy-namic) supercritical Hopf bifurcation, leading to laminar and parallel vortex

shedding, which forms a K´arm´an vortex street that is linearly stable to dimensional disturbances

three-The experimental conditions at the spanwise ends of the cylinder always

cause some three-dimensional disturbances, which when Re > 64 will gate to the midspan region of the cylinder and lead to oblique shedding (vor- ticity lines make an angle θ = 0 to the cylinder axis) with lower frequency

propa-St θ = St0cos θ In different spanwise regions the oblique shedding may have different St θ , as sketched in Fig 7.26, so as Re increases at the measurement

point the measured signal may jump from one mode to another to manifest as

a discontinuity in the St-curve This was observed at Re  75 (not shown in

Fig 7.25) The end-conditions can be carefully controlled to resume parallel