Vorticity and Vortex Dynamics 2011 Part 6 pps

The consistency of the the-ory with both Prandtl’s theory for two-dimensional steady separation and theLagrangian theory unsteady boundary-layer separation has been confirmed.The theory h

242 5 Vorticity Dynamics in Flow Separation

x = ξ, y = η, u = u0(ξ, η), v = v0(ξ, η) at τ = 0. (5.85)The boundary conditions are essentially the same as in the Eulerian descrip-tion:

(x, u) = (ξ, 0), (y, v) = (0, 0) on η = 0, (5.86a)

x τ = u → U(x, t) as η → ∞. (5.86b)

Once the integration of (5.83) to (5.86) gives x = x(ξ, η, τ ) at a subsequent time, one obtains y(ξ, η, τ ) from (5.81b), and then velocities from (5.79).

An inspection of this Lagrangian formulation reveals a key simplification:

owing to the approximate nature of (5.73), the streamwise position x and velocity u can be solved independently from solving the normal position y and velocity v Moreover, although a rigorous proof is not available, there has been strong evidence that the dynamic system (5.83–5.86) remains regular even after the singularity is formed (but the solution for t > ts may not

be physically realistic) Accepting this as a hypothesis, then, the singularity develops solely from the continuity equation In this sense, the theory is entirely

within kinematics In particular, (5.81a) indicates that the mechanism for thesingularity to occur is similar to the formation of shock in gas dynamics due

to the coalescence of characteristics In fact, the fluid-element normal location

y can be found by integrating (5.81b) along the curves x = const in the (ξ, η) plane Let l be the arclength along such a curve with l = 0 at the wall η = 0,

then

y =

 l0

dl

|∇ ξ x| =

 s0

dl (x2

ξ + x2)1/2 (5.87)

Now, at the separation point u x should be unbounded; so if u ξ and u η is

bounded then (5.82) implies that y ξ and/or y η must be unbounded Thus, the

mapping between (x, y) and (ξ, η) is singular, which in (5.87) manifests as

∇ ξ x = 0 at (ξ, t) = (ξs, ts). (5.88)This singularity condition has two effects First, all infinitesimal deformations

δξ of fluid element do not cause any change of the streamwise position of the

element in physical space:

δx = δξ · ∇ ξ x = 0 at (ξ, t) = (ξs , ts). (5.89)Namely, as fluid elements move along their pathlines, they are blocked and

squashed at a vertical barrier at some x, and hence must extend

unbound-edly along the normal as schematically shown in Fig 5.18, resulting in theseparation

Second, by (5.83b) we see at once that (5.88) implies the first part of theMRS criterion, (5.72a) or (5.77b) Therefore, when the fluid-element squashingprocess reaches the singular state, it reaches zero-vorticity state too Becausethe Lagrangian description does not distinguish steady and unsteady flow,

5.4 Unsteady Separation 243Shen (1978) points out that the same mechanism as sketched in Fig 5.18

is also responsible for the Goldstein singularity in steady separation withinboundary-layer approximation, and the MRS version of the Prandtl condition

(5.1) is derivable from (5.87) that is “no more than a formalized expression

of the Prandtl concept — that the boundary layer must break away when a packet of fluid particles are stopped in their forward advance along the wall.”

The second part of the MRS criterion can also be derived from (5.88) In

fact, denote the Lagrangian coordinates of the singularity point by ξMRS, of

which the propagation speed is (a dot denotes d/dt), owing to (5.88),

d

dt x(ξMRS, t) = ˙x + ˙ξMRS· ∇ ξ x = ˙x, (5.90)which is indeed the local streamwise velocity of the element, in agreementwith (5.77a) Therefore, the MRS criterion is rationalized

Van Dommenlen and Shen (1982) conducted a numerical calculation based

on the earlier theory for flow over impulsively started circular cylinder As

sketched in Fig 5.19, the singular point was found to appear at θ = 111 ◦ and t = 3.0045, which moves upstream with u = −0.52U The separation

location differs from the full Navier–Stokes solution (Fig 5.15) since the former

is for Re → ∞ asymptotically rather than at a finite Reynolds number The

separation location is also different from that of the Goldstein singularity for

steady flow, θ = 104.5 ◦ After the singularity is formed, the upper part ofthe boundary layer turns to a free separated vortex layer On top of Fig 5.19are the profiles of velocity and vorticity (normalized by wall vorticity) close toseparation, from which it is evident that as the bifurcation tears the boundarylayer apart the irrotational region in between is enlarged

We now introduce local scales in the neighborhood of (ξs, ts) so that the

singularity can be removed Assume tsis the first time for a singular

boundary-layer separation point to form Since x(ξ, t) is a regular function of ξ and t around (ξs, ts) one can perform a Taylor expansion of x and form the deck

structure thereby Meanwhile, (5.88) should also be expanded to a Taylorseries since it may not be satisfied anywhere forδt = t − ts < 0 To simplify

the expansion, let δξ = ξ − ξs, and make a proper shift and rotation of the

previous arbitrarily chosen Lagrangian coordinate system to a new system

(l1 , l2, t) The Jacobian J is invariant under the coordinate transformation,

and has characteristics

Although at t = ts the boundary-layer approximation blows up, at times

shortly before tsa rescaled asymptotic expansion can be conducted to describethe flow field After some algebra, it can be found that the proper scales are

l1=|δt| 1/2 L1, l2=|δt| 3/4 L2, (5.92a,b)

¯

x ≡ x − x(ξ , t) = |δt| 3/2 X, y = |δt| −1/4 Y, (5.93a,b)

244 5 Vorticity Dynamics in Flow Separation

xs

xs

xs

U y

S

S

w

Fig 5.19 Vorticity contours obtained from the Lagrangian boundary-layer

equa-tion for impulsively started circular cylinder t = 3.0045 On top are the profiles of

velocity and vorticity (normalized by wall vorticity) close to separation Reproducedfrom Van Dommenlen and Shen (1982)

O (|dt |3/2 )

O (|dt |-1/4 )

O (1) y

x

O (1)

Fig 5.20 Scales of unsteady boundary-layer bifurcation at δt before singularity is

formed Reproduced from Cowley et al (1990)

where L1 , L2, X, Y = O(1) These scales at a δt < 0 are shown schematically

 L ∗0

L ∗1

dL ∗ (2X ∗ − 3L ∗ − L ∗3)1/2 , (5.94)

where L ∗0 is the real root of the cubic polynomial in the square root of thedenominator The solution (5.94) can be cast to elliptic integrals of the firstkind The signs of the square roots and the limits of integration are determined

by the topology of the lines of constant X ∗ that consists of three segmentsshown in Fig 5.20 Leaving the mathematic details aside, the scaled vorticitycontours of the nearly separated boundary layer is shown in Fig 5.21, whichalso shows the sudden thickening of the boundary layer

Finally, similar to the steady case where the scaling is closed by

find-ing the relation of the lower-deck thickness δ and Re, we now need to close the theory by finding the relation of δt and Re Once again, since the MRS criterion implies the shearing is vanishingly small near S, the only possible

mechanism to balance the normal extension of fluid elements is the normal

pressure gradient ∆p y ∼ ∆p x in an irrotational upper deck In the separation

zone shown in Fig 5.20, as a fluid element moves past a streamwise extent

O( |δt| 3/2 ) but climbs up a thickness (in global scale) O(Re −1/2 |δt| −1/4), it

experiences a upwelling velocity v of O(Re −1/2 |δt| −7/4) The balance in the

normal momentum, ∂v/∂x = −∂p/∂y, together with the fact that x ∼ y in

the upper deck, indicates that the locally induced pressure reads

246 5 Vorticity Dynamics in Flow Separation

By (5.93b), at this time the scaled boundary-layer displacement thickness has

vari-on whether this situativari-on reflects the physical cascade process in tranasitivari-on

to turbulence associated with successive instabilities at a series of decreasingscales, or simply due to the limitation of the matched asymptotic theory itself

5.4.3 Unsteady Flow Separation

We now turn to generic unsteady separation Although some of the results

of Sect 5.2 are equally applicable to unsteady flow, a complete, general, andlocal unsteady separation theory had not been available until a very recentwork of Haller (2004), who obtained an exact two-dimensional theory for bothincompressible and compressible unsteady flow with general time dependence,applicable to arbitrary stationary or moving wall The theory is essentially ofkinematic nature, in which the separation point (to be defined later) can beeither fixed on the wall or moving along the wall The consistency of the the-ory with both Prandtl’s theory for two-dimensional steady separation and theLagrangian theory unsteady boundary-layer separation has been confirmed.The theory has been further improved by Haller and coworkers, and extended

to three dimensions (Kilic et al 2005; Surana et al 2005b,c) Therefore, we vote this subsection to an introduction to Haller’s unsteady separation theorybased on Haller (2004) and Kilic et al (2005), focusing on the simplest case

de-Namely, we assume the flow is incompressible with ρ = 1, and the separation point is fixed to a no-slip wall ∂B at y = 0, referred to as fixed separation.

Its results turn out to apply to any unsteady flow with a mean component,including turbulent boundary layers and flows dominated by vortex shedding.The assertion made in Sect 5.4.2, that flow separation is essentially a ma-terial evolution process, can be clearly demonstrated by the time evolution

of a fixed separation and reattachment for an analytical periodic flow modelshown in Fig 5.22 (see (5.111) later) Similar to Fig 5.19, a set of mater-ial lines initially aligned to the wall evolves to form an upwelling, then asingular-looking tip, and then a sharp spike More crucially, there appears

a distinguished material line, which attracts fluid particles released from itsboth sides and ejects them into the main stream This special material line

signifies the separation profile, of which a rational identification is the key

5.4 Unsteady Separation 247

Fig 5.22 Time evolution of material lines and streamlines for a periodic separation

bubble model (5.109) with circular frequency 2π (a) t = 0, (b) t = 8.2, (c) t = 9.95, (d) t = 1 5.0, (e) t = 1 8.65, (f ) t = 25 The time-dependent curve initially cutting the

material lines but then serving as their approximate asymptotic line is the separationprofile (up to quadratic order) to be identified later From Haller (2004)

of a general separation theory Note that Fig 5.22 shows that instantaneousstreamlines are irrelevant when the separation is unsteady

This being the case, we start from the dynamic system (5.5):

˙x = u(x, y, t), y = v(x, y, t),˙ (5.96)which due to the no-slip condition and continuity can be cast to, similar to(5.31),

˙x = yA(x, y, t), y = y˙ 2C(x, y, t), (5.97)where

A(x, y, t) =

 1 0

u y (x, sy, t)ds, C(x, y, t) =

 1 0

 1 0

v yy (x, sqy, t)q dq ds.

(5.98)

The incompressibility further requires

Now, denote the material line signifying the separation profile by M(t), which

as seen in Fig 5.22 is “anchored” to the fixed separation point (x, y) = (γ, 0) for all t by the no-slip condition In dynamic system terms, M(t) is an unstable manifold for a fixed point on the wall, locally described by a time-dependent

path

248 5 Vorticity Dynamics in Flow Separation

While generic material lines emanating from the wall converge to the wall as

t → −∞, M(t) is an exception, with the following properties:

1 it is unique, i.e., no other separation profile emerges from the same

bound-ary point;

2 it is transverse, i.e., does not become asymptotically tangent to the wall

in backward time;13 and

3 it is regular up to nth order (n ≥ 1), i.e., M(t) admits n derivatives that are uniformly bounded at the wall for all t.

Then, substituting (5.100) into (5.97), one finds thatM(t) satisfies a tial differential equation (the separation equation)

par-F t = A(γ + yF, y, t) − yC(γ + yF, y, t)(F + yF y ), (5.101)from which unsteady separation criteria can be deduced By (5.100), approx-imate separation profile can be expressed by series expansion

Consider separation criteria first Setting y = 0 in (5.101) yields a linear

equation ˙f0(t) = a(t), thus (t0 is an arbitrary reference time)

f0(t) = f0(t0) +

 t

t0

A(γ, 0, τ )dτ. (5.103)

Since by the earlier property (2)M(t) cannot become asymptotically tangent

to the wall, f0(t) must be uniformly bounded By (5.98) and u y=−ω on the

wall, therefore, a necessary separation criterion is

lim

t →−∞sup

Then, as the generalization of (5.1b), by using v yy =−u yx = ω xon the walland after some algebra, it can be proved that the second necessary separationcriterion is  −∞

13All other material lines that start to be transverse remain so for any finite time,

but become tangent to the wall as t → −∞ (G Haller, 2005, private

communi-cation)

5.4 Unsteady Separation 249

which for steady flow becomes (t0+∞)ω x=∞ and hence ω x > 0, equivalent

to (5.1b) In particular, for periodic flow with period T , the integration interval

in (5.104) can be replaced by (0, T ); while in (5.105) one splits the integrand

into a mean and an oscillating part, with the former having to be negative.Thus, the two necessary separation criteria are simply reduced to

 T

0

ω(s, t)dt = 0,

 T0

ω x (s, t)dt > 0. (5.106a,b)

In general, criterion (5.104) can be expressed in a form more suitable forcomputations Recall that any material lines emanating from any wall points

near s will align with the wall as t → −∞, which by (5.103) is possible only

if, for sufficiently small|x − γ|,

 −∞

t0

u y (x, 0, τ )dτ =

+∞ if x > γ,

 t

t0

u y (γeff, 0, τ )dτ = 0 such that γ = lim

t →−∞ γeff(t, t0), (5.108)

see Fig 5.23 The reattachment point can be similarly defined

Moreover, while criteria (5.104) and (5.105) permit weak separation by which particles near s may turn back towards the wall for a finite period of

time, a slight revision of (5.105) can give a sufficient condition for stronger

monotonic separation by which particles near s move away monotonically

from the wall without turning back Haller (2004) proves that this is simplyensured by

−u xy (s, t) = ω x (s, t) > c0 > 0, (5.109)

of which the physical implication is obvious (cf Fig 4.12)

Haller (2004) has used the earlier theory to derive explicit general formulas

for the time-dependent coefficients f0(t), f1(t), of (5.106) up to quadratic

order In particular, for steady flow the slope ofM reduces to

f0=− u yy (s) 3u (s) =− p x (s)

3τ (s) ,

250 5 Vorticity Dynamics in Flow Separation

Fig 5.23 The convergence of γeff to γ

in agreement with (5.30) where φ is the angle of the separation line relative

to the x-axis The second equality uses the Navier–Stokes equation as we did

in Sect 5.2, except which all the earlier results are evidently kinematic; usewas made of only the continuity equation

The fixed separation conditions (5.104) and (5.105) have been improved

by Kilic et al (2005), assuming that the unsteady velocity fields under

con-sideration admit a finite time asymptotic average in time After some lengthy

algebra, the authors show that (5.104) and (5.105) can be replaced by

which are a direct generalization of (5.106) to aperiodic flow

As an analytic example, consider a periodic separation bubble model rived by Ghosh et al (1998),

de-u(x, y, t) = −y + 3y2+ x2y −2

3y3+ βxy sin nt, v(x, y, t) = −xy2−1

Substituting this model into (5.108) yields (γ2 − 1)T = 0 and 2γT > 0,

T = 2π/n Thus, the fixed separation point is at γ = −1 and the ment point at γ = +1, as shown in Fig 5.22 for n = 2π and β = 3, which

reattach-is in agreement with the numerical observation of Ghosh et al (1998) The

expansion coefficients f0 (t), , f3(t) can also be derived, which gives the

ap-proximate separation profile also shown in Fig 5.22

The preceding unsteady separation theory indicates that the final results

on the separation definition and criteria are fully Eulerian that do not require

5.4 Unsteady Separation 251the advection of fluid particles This is a unique advantage of the theory Likethe general three-dimensional steady separation theory of Sect 5.2, this un-steady theory meets three highly desired requirements proposed, respectively,

by Sears and Telionis (1975), Cowley et al (1990), and Wu et al (2000),

and summarized by Haller: independent of our ability to solve the layer equations accurately; independent of the coordinate system selected; and expressible solely by quantities measured or computed along the wall.

boundary-Summary

1 Phenomenologically, flow separation is a local process in which fluid ments adjacent to a wall no longer move along the wall but turn to theinterior of the fluid In its strong form and at large Reynolds numbers, theprocess may evolve to boundary-layer separation where the whole layerbreaks away and thereby significantly alters the global flow field Physi-cally, flow separation is due to the boundary coupling of the two funda-mental dynamic processes A near-wall adverse pressure gradient yields a

ele-boundary vorticity flux σ p, which creates new vorticity with direction ferent from that of existing one, so the accumulation of the former in spaceand time causes a transition of the near-wall vorticity from being carriedalong by the wall to shedding off Thus, a vorticity-dynamics description

dif-of separation is especially illuminating, which can be obtained from theconventional momentum considerations owing to the on-wall equivalence

between the τ w -field and its orthogonal ωB-field, and that between the

∇ π p-field and its orthogonal σ p-field

2 A general flow-separation process without any specification to its strengthcan be studies in an infinitesimal neighborhood of a separation point orseparation line, by using a Taylor expansion of the continuity and Navier–Stokes equations The criteria for separation zone and separation line atlarge Reynolds numbers can be formulated in terms of the earlier twopairs of orthogonal on-wall vector fields For steady separation and in twodimensions, the criteria amount to those well-known ones due to Prandtl

In three dimensions, the separation zone is characterized by the strong

converging of τ -lines or large positive on-wall curvature of ω-lines If the separation starts at a fixed point of the τ -field (“closed separation”), a

generic separation line can be uniquely determined But at large Re a

significant separated free shear layer may start to form and/or cease to

shed off at ordinary points of a τ -line; for which the separation line may

be approximately identified as the line with maximum ω-line curvature in

the separation zone

3 Boundary-layer separation at large Re involves the flow behavior in the

whole layer and its interaction with external flow in a small but finitezone Although this process is governed by the Navier–Stokes equation,the matched asymptotic expansion has contributed an elegant triple-deck

252 5 Vorticity Dynamics in Flow Separation

theory that clarifies the underlying physics and represents the second eration of the boundary-layer theory The triple-deck theory has been fullydeveloped for steady separation, but becomes difficult for unsteady sep-aration due to the involvement of different time scales at different stages

gen-of separation process So far the only successful theory for this situation

is based on the Lagrangian description, which confirms the MRS criterionand extends it to three dimensions

4 Both generic flow separation and boundary-layer separation are essentiallymaterial evolution processes, and hence favors the use of Lagrangian de-scription when the flow is unsteady This explains the success of the La-grangian approach and difficulty of the Eulerian approach to unsteadyboundary-layer separation For generic unsteady flow separation, a com-plete local theory can be developed also by starting from the Lagrangiandescription, of which however the final results on the separation criteriaand separation profile can still be expressed by on-wall Eulerian variables.The on-wall signatures of separation can be used as a convenient tool incomplex flow diagnosis and separation control

Part II

Vortex Dynamics

Typical Vortex Solutions

In Chap 4 we have studied attached and free vortex layers, and seen thatthe rolling up of a free vortex layer forms a vortex which has the highestpossible vorticity concentration as mentioned in the beginning of Sect 1.3(the formation process of vortices will be further discussed in Sect 8.1) In this

chapter we start the dynamics of vortices by presenting a number of typical

exact viscous and inviscid vortex solutions, and asymptotically approximatevortex solutions, followed by a basic open issue on how to rationally define avortex

Exact solutions can provide a thorough physical understanding, serve asthe testing bed of the accuracy of approximate approaches and as the basicflow in their stability analyses (Chap 9) However, exact Navier–Stokes vortexsolutions, mostly confined to incompressible flow (see reviews of Wang 1989,1991), are obtainable only under highly idealized conditions In certain aspectsthey behave quite unrealistic, and some solutions may correspond to real flowsonly in a local region and/or a finite period of time

In reality there is no single isolated straight vortex with nonzero totalcirculation Vortices always appear as loops (in three dimensions) or in pairs(in two dimensions), and hence each vortex is in the strain field caused byother vortices and boundary conditions For a thin-core strained vortex onemay find asymptotic solutions analytically, which complement the shortage ofexact solutions and may also play similar roles as the exact solutions.Unless stated otherwise, throughout this chapter we assume the flow is

incompressible flow with ρ = 1.

6.1 Governing Equations

The geometric characters of columnar vortices and vortex rings makes it often

convenient to use a cylindrical coordinate system (r, θ, z) with u = (u, v, w) and ω = (ω , ω , ω ) By the general formula for any vector A

256 6 Typical Vortex Solutions

∇ × A = 1

r

er re θ ez

∂ r ∂ θ ∂ z

A r rA θ A z

where H = q2/2 + p is the total enthalpy Whenever needed, substituting

(6.2) to (6.4) yields the common component momentum equations in terms ofvelocity and pressure One of the component forms of the vorticity transportequations is

6.1Governing Equations 257

component This kind of vortices are called pure vortices, with all vorticity

lines being along the axis and all streamlines are closed circles centered at the

z-axis Then, if w is nonzero and r-dependent, ω θ will appear too such thatthe velocity and vorticity lines become helical This kind of vortices are called

swirling vortices, having nonzero helicity density ω · u.

We have seen in Sect 3.3.1 that a Stokes stream function ψ can be troduced to ensure the continuity, which expresses u and w by (3.57) but not v:

in-u = −1r

Thus, the velocity and vorticity, and hence their governing equations, can be

expressed in terms of two scalar functions, ψ and Γ = rv (differing from the circulation around a circle centered at r = 0 by a factor 1/2π) Consequently,

(6.2) and (6.8) yield

ω r=−1r

r

∂ψ

∂r

+1

r

∂2ψ

The role of Γ for ω r and ω z is exactly the same as that of ψ for u and

w Contours of Γ and ψ on an (r, z)-plane are the intersections of vorticity

surfaces and stream surfaces with the plane, respectively Then (6.4b) and(6.5b) can be cast to

r

∂Γ

∂r

+∂

r4

∂Γ2

which govern the azimuthal and meridional motions, respectively Γ and ω θ

may serve as the basic variables to be solved, all other quantities can be

inferred therefrom They are coupled solely through the z-dependence of v, which happens, e.g., if the vortex hits a boundary at z = 0 as sketched in

Fig 3.5a

Most of existing exact vortex solutions, either viscous or effectively viscid, were found when (6.5) can be linearized That is, when the flow is

in-generalized Beltramian satisfying (3.63) It is therefore appropriate here to

examine when this happens in general (not confined to two-dimensional or

ro-tationally symmetric flows where ω or ω/r is a function f (ψ, t)) We consider inviscid steady flow and viscous unsteady flow separately.

258 6 Typical Vortex Solutions

Any incompressible, effectively inviscid, and steady flow must be

general-ized Beltramian, since then (2.163) is reduced to

ψ = constant around the z-axis Therefore, by (6.12a) there is Γ = C(ψ),

from (6.9a,b) and (6.12b) it follows that:

ω r=− 1

r

dC dψ

then since by (6.11) u · ∇H = 0, we also have H = H(ψ), and hence (6.13c)

comes from (6.8) Therefore, there remains only a single differential equation

to be solved for steady inviscid axisymmetric flows:

r ∂

∂r

1

r

∂ψ

∂r

+∂

In passing, we note that (6.14) can be extended to nonaxisymmetric case

by using the transformation (2.112), a special form of the Helmholtz position:

decom-u = ∇φ + ∇ψ × ∇χ.

Since the two stream functions ψ and χ define two families of stream surfaces, their intersections are streamlines along which H is constant Thus, we have

of which the projection to the directions of∇ψ and ∇χ yields a pair of

sym-metric equations (Keller 1996)

ω · ∇χ + ∂H

∂ψ = 0, ω · ∇ψ − ∂H

along with ∇2φ = 0 This set of equations are applicable to any

three-dimensional steady inviscid flows (still generalized Beltramian) and has beenused by Keller (1996) to study axisymmetric vortices with helical waves and

other relevant flows When the flow is nonaxisymmetric, Γ = rv is no longer

an integral of the motion and the flow cannot be expressed by ψ alone.

On the other hand, by directly looking at the Lamb-vector components in

(6.4), we find that a viscous axisymmetric vortex with ω z = 0 will be ized Beltramian if and only if

But now (6.4a) can be reduced to ∂p/∂r = v2/r with H = p + (v2+ w2)/2

even for viscous flow Conversely, after rewriting (6.5) to make ∇ × (ω × u)

appear explicitly, an inspection of its component form indicates that with

ω z = 0, for the flow to be generalized Beltramian it is necessary that u, v, w are independent of z Then (6.3) implies that u = C(t)/r, which would lead

to a singularity at the vortex axis if C = 0 Hence (6.16) follows.

Once (6.16) holds, in (6.5) for ω = (0, ω θ , ω z), the viscous terms are solelybalanced by the unsteady terms

∂ω z

∂t =

ν r

260 6 Typical Vortex Solutions

6.2 Axisymmetric Columnar Vortices

6.2.1 Stretch-Free Columnar Vortices

We start from the simplest stretch-free vortex solutions of the form (6.16) In

this case (6.4) is reduced to

Unlike (6.12b), v and w are now decoupled If the flow is effectively inviscid,

the only equation we can use is (6.18a) in which the pressure can automatically

adjust itself to balance whatever centrifugal acceleration Thus, a stretch-free inviscid vortex can have arbitrary radial dependence, providing a big freedom

for constructing various inviscid vortex models The most familiar example is

the q-vortex, which fits many experimental data pretty well

In contrast, for viscous flow, if ω θ = 0 as in the case of pure vortices, we

have (6.18a) plus (6.17a) If the flow is steady, (6.17a) implies that ω z must

be a constant (can be zero), which by (6.2) leads to

6.2 Axisymmetric Columnar Vortices 261

implies either the well-known line vortex with A = 0 and constant Γ = 2 πrv or

a solid rotation with B = 0 There is no smooth and steady stretch-free viscous

solution in an unbounded domain, because to maintain a steady viscous flow

a constant driving force is necessary The best one can do is to artificially

combine a solid core of radius a and a potential outer flow, to form a Rankine vortex

(6.22)

where ω and a are the constant core vorticity and core radius, respectively.

Obviously, this is also an inviscid solution

As we allow the flow to decay freely, (6.17a) permits uniformly effectiveviscous solutions A complete set of similarity solutions has been given byNeufville (1957), who sets

The exponential decay of ω as η indicates that the vorticity is concentrated

in a region with η n (η) has n zeros and can be inferred from

recursive formulas:

L0(η) = 1, L1(η) = 1− η,

L n+1 (η) = (2n + 1 − η)L n (η) − n2L n−1 (η).

262 6 Typical Vortex Solutions

Two special modes of (6.24) are well known The mode n = 0 is the Oseen–Lamb vortex (Oseen 1912; Lamb 1932):

4νt0.

The Oseen–Lamb vortex may also be viewed as the axisymmetric part of the Stokes first problem analyzed in Sect 4.1.4 But, it is easily verifiedthat in unbounded domain an isolated Oseen–Lamb vortex has infinite totalkinetic energy and angular momentum

counter-Second, the mode n = 1 in (6.24) leads to the Taylor vortex (Taylor 1918):

v(r, t) = M r

8πνt2exp



− r24νt

exp



− r24νt

This solution has zero total circulation (because ω changes sign once) and finite M Note that (6.26) is nothing but the time derivative of (6.25) The

velocity profiles of Oseen–Lamb vortex and Taylor vortex are compared in

Fig 6.1c All higher modes with n > 1 in (6.24) have zero total circulation

and zero total angular momentum (Neufville 1957)

Any pure vortices are two-dimensional generalized Beltrami flow, for which

by (3.64) ω z (r, t) = f (ψ, t) While for the Rankine vortex we simply have f = constant, for Neufville’s vortex family f (ψ, t) is nonlinear.

6.2 Axisymmetric Columnar Vortices 263

t=0 t=0.5

t=1

t=8 t=2 t=0.5

V*

(c)

Fig 6.1 The decay of the circumferential velocity (a) and vorticity (b) of the

Oseen–Lamb vortex, and its comparison with the Taylor vortex in similarity

vari-ables (c) In (c) V ∗ ∝ v/t −1/2 and v/t −3/2 for the Oseen–Lamb vortex and Taylor

vortex, respectively Reproduced from Panton (1984)

6.2.2 Viscous Vortices with Axial Stretching

Vortex stretching occurs if the axial velocity w(r, z, t) is z dependent The simplest z-dependence of w is linear and uniform

w(r, t) = γ(t)z, u(r, t) = −1

2γ(t)r, γ > 0, (6.28)

where u(r, t) is derived from (6.3) The vorticity has only a z-component The flow can take this form only locally (r < ∞, |z| < ∞) From (6.5c) and

264 6 Typical Vortex Solutions

(6.28), the vorticity equation reads:

∂ω

∂t =

ν r

2γr

∂ω

On the right-hand side, the second term is a radial advection, while the third

term is a uniform stretching If at t = 0 a vortex element has unit length, then at time t its length will be

S(t) = exp

 t0

γ(t
)dt

(6.30)

or eγt if γ is a constant Once again, we seek similarity solutions of (6.29).

Following Lundgren (1982; see also Kambe 1984), we introduce new stretchedvariables:

Therefore, from any pure vortex one can generate a uniformly stretched vortex

by the Lundgren transformation

ω(r, t) = S(t)ω ∗



S 1/2 (t)r,

 t0

S(t
)dt

The two flows before and after transformation have similar behavior

Because S(t) > 1 implies r < ρ, t < τ , and ω > ω ∗, the velocity and city of stretched vortex flow are enhanced, with shorter distances and fasterrotation time, in agreement with the kinematics discussed in Sect 3.5.3 For

vorti-example, from the Oseen–Lamb vortex and a constant γ, we find a new

6.2 Axisymmetric Columnar Vortices 265

Then, let t → ∞ in (6.33), we obtain an asymptotic steady solution of stretched vortex of radius δ ∼ (ν/γ) 1/2 , which exists only if γ > 0:

will be the same as that of the Oseen–Lamb vortex This is because the radialflow −γr/2 brings the far-field vorticity to the vortex core, which exactly compensates the viscous diffusion But we have ∂u/∂r | r=0=−γ/2 and hence

2πrω2(r)dr = ργΓ

2

Thus, the dissipation remains finite as ν → 0, which is a fundamental

as-sumption of turbulence theory (e.g., Frisch 1995) This vortex has served as

a building block of various vortex models for fine-scale turbulent structuresand starting point of searching for more complex vortex solutions (Sect 6.5).The Lamb vector of the Burgers vortex is

266 6 Typical Vortex Solutions

and using (6.4c), we can obtain l  and then l ⊥ from (6.36)

l  =−v(r)ω(r)e r=−∇



H −1

2w2

While l  is independent of γ and has the same form as nonstretched vortices,

l ⊥ is completely caused by the stretching as it should Remarkably, l  , l ⊥,

and ω are geometrically orthogonal.

Moreover, the strain-rate tensor of the Burgers vortex is

stretches at a rate γ, l ⊥ shrinks at rate γ/2 Based on the transport

equa-tion for l, Wu et al (1999b) have shown that vortex stretching is generically

associated with the Lamb-vector shrinking.

In this simplest model of stretched vortices, the nonlinearity caused by

l ⊥ = 0 can be made disappear by the Lundgren transformation (6.32) When

the axial velocity is nonuniform, one may consider more general families ofsemi-similarity solutions, in which if the nonlinear coupling between different

components can be artificially removed then the solutions may have closed

form To this end, we seek steady solutions of the form

equations Recall that v and ψ determine the motion along the azimuthal

direction and on the axial plane, respectively The equations are decoupled,

of which the ones for g and f can be integrated once, and the one for Γ can

6.2 Axisymmetric Columnar Vortices 267

where C1, C2, C3, and Γ0 are integration constants Then, if f ≡ 0, (6.40a) permits a simple solution g
/r = constant, which includes the Burgers vortex.

If f (r) = 0, a combination of the simple solution g
/r = const of (6.40a) and

(6.40b) makes the latter have solution (f
/r)
= C4 r, and hence

u(r) = A1r + A2r −1 , f (r) = B1r4+ B2 r2. (6.41)

By different choices of the constants, one finds a linear superposition of someelementary vortices and axial flows, including (Xiong and Wei 1999): a super-position of a circular-pipe Poiseuille flow and a forced vortex and/or a linevortex; that of a circular-pipe Poiseuille flow and a Burgers vortex; and a

swirling flow with singular sink (a simple model for bathtub vortex ) or source Now, assume instead an r-dependent axial velocity with an exponential

decay, similar to that in (6.19)

Sullivan (1959) wrote down this solution without giving derivation The

vor-ticity components of this Sullivan vortex are