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TurbulenceAn Introduction for Scientists and Engineers$

Peter Davidson

Print publication date: 2015

Print ISBN-13: 9780198722588

Published to Oxford Scholarship Online: August 2015

DOI: 10.1093/acprof:oso/9780198722588.001.0001

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(p.611) Appendix 2 The properties of isolated vortices: invariants, far-field properties, and long-range interactions

(p.611) Appendix 2 The properties of isolated vortices: invariants, far-field properties, and long-range interactions

Source:
Turbulence
Author(s):

P. A. Davidson

Publisher:
Oxford University Press

We have defined an eddy as a coherent blob of vorticity. Such a blob retains its identity in a turbulent flow since vorticity can spread by material movement or by diffusion only. It is of interest to identify the kinematic and dynamic properties of an isolated eddy since they play an important role in the large-scale dynamics of homogeneous turbulence. For example, they are crucial to understanding the origins of Saffman’s invariant and Loitsyansky’s integral, as well as the distinction between the two (see Section 6.3). They are also central to understanding the nature of Batchelor’s long-range pressure forces (see Section 6.3.3). The topics we shall discuss are:

  • the far-field velocity induced by an isolated eddy;

  • the pressure distribution in the far field;

  • integral invariants of an isolated eddy;

  • long-range interactions between eddies.

A2.1 The far-field velocity induced by an isolated eddy

Consider an isolated blob of vorticity sitting near x=0 in an infinite fluid which is motionless at infinity (Figure A2.1). The vector potential, A, for the velocity field is defined by

(A2.1)
×A=u,A=0,
Appendix 2 The properties of isolated vortices: invariants, far-field properties, and long-range interactions

Figure A2.1 An isolated eddy located at x = 0.

and is related to the vorticity by the expression

(A2.2)
2A=ω.

This may be inverted using the Biot–Savart law to give

(A2.3)
A(x)=14πωxdxxx,

(p.612) where the integral is over all space. To find the vector potential some distance from the blob of vorticity we expand xx1 in a Taylor series in x1. This yields

(A2.4)
1xx=1rDi(r)xi+Bij(r)xixj+,

where r=x and

Di=xi1r,Bij=122xixj1r.

On substituting this expansion into integral (A2.3), we find

(A2.5)
4πA(x)=1rωdxDi(r)xiωdx+Bij(r)xixjωdx+.

However, the first term on the right integrates to zero since

(A2.6)
Vωidx=Vωxidx=SxiωdS=0,

the vorticity being confined to the region near x=0. (Here V is a large volume whose surface, S, recedes to infinity.) Also, the second integral can be transformed using the relationship

(A2.7)
V[ xiωj+xjωi ]dx=V(xixjω)dx=SxixjωdS=0,

from which,

(A2.8)
Dixiωjdx=12Di[ xiωjxjωi ]dx=12D×[x×ω]dx.

Our expansion for the vector potential now simplifies to

(A2.9)
4πA(x)=D(r)×L+Bij(r)xixjωdx+,

where

(A2.10)
L=12x×ωdx.

(p.613) We shall see shortly that L is a measure of the linear momentum introduced into the fluid by virtue of the presence of the eddy, and it is an invariant of the motion. It is called the linear impulse of the eddy. Substituting for D(r) and taking the curl gives the far-field velocity distribution:

(A2.11)
u(x)=14π(L)(1/r)+14πBij(r)×xixjωdx+.

Evidently the velocity in the far-field is O(r3) if L is finite and Or4 if L happens to be zero. We can find the corresponding far-field scalar potential for u by rewriting (A2.11) as

u=ϕ=14πL1/r+,

which tells us that the scalar potential for u in the far field has the form

(A2.12)
ϕ=14πL(1/r)+.

We shall return to these expressions shortly.

A2.2 The pressure distribution in the far field

The pressure field at large distances from the eddy may be found by taking the divergence of the Navier–Stokes equation to give

2p=ρuu,

and then inverting this using the Biot–Savart law:

(A2.13)
pρ=14πuudxxx.

Substituting for xx using (A2.4) and noting that

xiuu=xiuuuiu,

we find

4πpρ=1ruudxDi(r)xiuuuiudx+Bij(r)xixjuudx+.

The first two volume integrals convert to surface integrals which, since uO(r3), vanish on the surface of a sphere of infinite radius. The integrand of the third integral can be written as 2uiuj plus some divergences which also vanish, and so the leading-order contribution to the far-field pressure is simply

(A2.14)
pρ=14π2xixj1ruiujdx+.

We shall use this expression below.

(p.614) A2.3 Integral invariants of an isolated eddy: linear and angular impulse

We now consider those integral invariants of an isolated eddy which are related to the principles of conservation of linear and angular momentum. These invariants are called the linear impulse and angular impulse of the eddy. They are both integrals of the vorticity field.

The first point to note is that a comparison of (A2.11), (A2.12), and (A2.14) yields

(A2.15)
dLdt=0.

The argument goes as follows. In a potential flow the viscous forces are zero since ν2u=ν××u=0, and so Bernoulli’s theorem for unsteady potential flow demands, in the far field,

ϕt+pρ+u22=0.

Since pO(r3) and u2O(r6) for large r, this reduces to

ϕt=O(r3).

Equation (A2.15) follows immediately since, in the far field, ϕL1/r+O(r3). Thus the linear impulse, L, is an invariant of the motion.

The invariance of L is a direct consequence of the principle of conservation of linear momentum. This becomes clear if we note that

t12x×ω=uu+u2/2+12x×uωx×ωu12νx×ω+2×ω×ωx.

(This follows from the evolution equation for ω.) Integrating over a large sphere of radius R which encloses all the vorticity (Figure A2.2), and using Bernoulli’s theorem to evaluate u2/2 in the far field, we find

ddtVR[ 12(x×ω) ]dV=SRu(udS)SR(p/ρ)dSSRϕtdS.
Appendix 2 The properties of isolated vortices: invariants, far-field properties, and long-range interactions

Figure A2.2 Control volume VR used to evaluate the linear momentum.

Compare this with the linear momentum equation applied to VR,

ddtVRudV=SRuudSSRp/ρdS.

Evidently,

ddtVRudVVR12x×ωdV=SRϕtdS=OR1.

(p.615) In the limit R we have

L=VRudV+constant,

which shows that, to within a constant, the linear impulse L is equal to the linear momentum in a sphere of large radius.

The constant may be found by direct evaluation of udV. The integration is complicated by the fact that udVis only conditionally convergent, but the details are spelt out in Bachelor (1967). It turns out that, no matter how large we make VR, there is always some linear momentum outside VR. When the integrals are evaluated we find that, for R, the contribution to udV from within VR is 23L, while that from outside is 13L (Bachelor, 1967). Adding the two gives

(A2.16)
L=udV,Linear impulse=Linear momentum.

In fact, the relationship

VRudV=23L=13(x×ω)dx

turns out to hold for any spherical control volume that encloses the vorticity, not just VR in the limit of R. This may be confirmed as follows. We use (A2.3) to write

VRudx=VR×Adx=SRA×dS=14πSR[ VRω(x)dx| xx | ]×dS,

where x locates a point on the surface SR and x is a point within the interior of VR. Exchanging the order of integrations, we find

VRudx=14πVR[ SRdS| xx | ]×ω(x)dx=13(x×ω)dx,

since the surface integral is readily shown to be equal to (4π/3)x.

(p.616) We now turn our attention to the angular momentum. Again we must be careful as it is by no means clear that x×udV is a convergent integral, since uO(r3) at large r. This requires that we take a rather circuitous route to applying the principle of conservation of angular momentum. Let us introduce the angular momentum integral

H=VRx×udx,

where VR is a sphere of radius R which completely encloses the vorticity field (Figure A2.2). We do not know what happens to H as R, but it is certainly well defined for finite R. Now it is readily verified that

6x×u=2x×x×ω+3×r2uωr2x,

and so it follows that

(A2.17)
H=13VRx×x×ωdx.

(The second term on the right integrates to give 3R2ωdV, which is zero because of (A2.6), while the third term gives a surface integral which vanishes because ω is zero on the surface of VR.) This second measure of H is perfectly well behaved as R since there are no contributions to (A2.17) outside the vortex blob. Integral A2.17 is called the angular impulse of the eddy. Now the principle of conservation of angular momentum tells us that

ddtVRx×udx=SRx×uudS,

there being no viscous torque on SR because the flow outside SR has no vorticity.1 It follows from (A2.17) that

ddt[ 13VRx×(x×ω)dx ]=SR(x×u)udS.

We can now safely take the limit R since both integrals are convergent. We find

ddt[ 13VRx×(x×ω)dx ]=O(R3),

and so, in the limit R, the angular impulse is conserved:

(A2.18)
13VRx×(x×ω)dx=constant.

(p.617) In summary, then, an isolated eddy has two integral invariants, its linear impulse and its angular impulse:

L=12x×ωdV=constant,H=13x×(x×ω)dV=constant.(Linear impulse)(Angular impulse)

The invariance of these quantities corresponds to the conservation of linear momentum and angular momentum, respectively. If the eddy has a finite linear impulse then uO(r3), whereas an eddy with finite angular impulse but zero linear impulse has uO(r4). Thus eddies which possess linear impulse cast a longer shadow than those that do not. It is this which lies behind the potent long-range correlations in Saffman’s spectrum (see Section 6.3.4).

A2.4 Long-range interactions between eddies

We may use the results above to show how a Batchelor Ek4 or a Saffman Ek2 spectrum arises. To establish a Batchelor spectrum we assume that, at t=0, there are no long-range velocity correlations (they decay exponentially fast). We then ask, can long-range power-law velocity correlations emerge naturally via the dynamical equations? In particular, we shall derive a dynamic equation for the triple correlations of the form

(A2.19)
t[ u3K(r,t) ]=t ux2ux uuuu .

It turns out that the term involving fourth-order correlations decays as r4 at large r. Thus, even if K is exponentially small at t=0, it acquires an algebraic tail for t>0. We can then use the Karman–Howarth equation to show that Kr4 implies uur6 in isotropic turbulence. (In anisotropic turbulence we find uur5.) As shown in Chapter 6, an r6 decline in uu ensures that the leading-order term in Ek is O(k4), i.e. a Batchelor spectrum. In Section 6.3.3 we established the correlation uur6 through a consideration of the long-range pressure forces. Here we shall show that it also follows directly from the far-field properties of an isolated vortex, i.e. from the Biot–Savart law.

To establish a Saffman spectrum we must take a different route. We have seen in Section 6.3.4 that an Ek2 spectrum can arise only if uur3. Such strong long-range correlations cannot arise spontaneously in homogeneous turbulence. The long-range pressure forces are not potent enough. Thus, to obtain a Saffman spectrum, we must ensure that uur3 at t=0. Again we can use the results above to ask, under what conditions is the assumption uur3 kinematically admissible? We shall see that the key distinction between Batchelor and Saffman spectra is that, in the former, the leading-order term in (A2.11) is zero (or near zero) for a typical eddy. That is, the turbulence has very little linear impulse.

Let us start with Ek4 spectra. Consider a turbulent flow in which a typical eddy has negligible linear impulse. In such cases we can use expansion (A2.11) to explain the origin of Batchelor’s long-range correlations in Ek4 turbulence. We start by using (A2.11) to evaluate the x-component of u induced at x=reˆx by a remote cloud of eddies located near x=0 (Figure A2.3). It is not hard to show that

ux=ux(re^x)=14π3r4(xyωzxzωy)dx+.
Appendix 2 The properties of isolated vortices: invariants, far-field properties, and long-range interactions

Figure A2.3 The velocity component ux induced at x=reˆx a long distance from a cloud of eddies located near x=0.

(p.618) Differentiating with respect to time we find, after a little algebra,

(A2.20)
uxt=34πr42ux2uy2uz2dx+.

As we shall see, it is this expression which lies behind the dynamic equation for the long-range triple correlations discussed in Section 6.3.3.

Now suppose we have a homogeneous sea of eddies (blobs of vorticity) including our cloud centred at x=0. The Biot–Savart law then tells us that the velocity at x has many contributions (from the different vortex blobs), but that at least part of ux/t is given by (A2.20) and hence correlated to events in the vicinity of x=0. We might surmise, correctly as it turns out, that

(A2.21)
t ux2ux =34πr4 ux2(2ux2uy2uz2)dx,

where ux is evaluated at x=0. This might be compared with the pressure–velocity correlation derived in Section 6.3.3 in Example 6.5,

ux2p=ρ4πr3ux22ux2uy2uz2dx,

whose gradient in r appears as a source term in the evolution equation for ux2ux. In any event, the important point to note is that the O(r4) term in expansion (A2.11) gives rise directly to long-range triple correlations of order r4. These, in turn, lead to a O(r6) contribution to uiuj in isotropic turbulence (r5 in anisotropic turbulence).

Returning to Figure A2.3, when the linear momentum of a typical eddy is non-zero we have, to leading order in r1, a larger velocity at x=reˆx:

(A2.22)
ux=2Lx4πr3,

where L is the net linear impulse of the eddies near x=0. This suggests the possibility of a r3 contribution to uxux. It is this which lies behind the Saffman spectrum

(A2.23)
uxux=L4πr3+Or6,
(A2.24)
E(k)=Lk24π2+Ok4,

(p.619) where L=uudr is Saffman’s integral. Note that L is an invariant of the Saffman spectrum so that, if L=0 at t=0, then Ek4 for all time. Thus, whether or not we have a Batchelor (Ek4) or Saffman (Ek2) spectrum depends on the initial value of L. Equation (A2.22) suggests that a non-zero L requires a typical eddy to have a finite linear impulse L. The central limit theorem then suggests that, if the turbulent eddies are randomly orientated in a frame of reference in which u=0, we will have

VudVV1/2.

This, in turn, tells us that the normalized product of integrals of the type

limV1VVudVVudV=limV1VuudrdV= uu dr
is finite, which is consistent with the notion that L is non-zero in Saffman turbulence.

In summary, then, whether we have a k2or k4 spectrum depends on the mechanism which generates the turbulence. If enough linear impulse is injected into the fluid at t=0, then we expect Ek2 for all time. On the other hand, if the linear momentum udV is less than O(V1/2) in a frame of reference in which u=0, then a Ek4 spectrum will emerge. Both situations are readily generated on the computer. However, researchers still cannot agree as to which category real turbulence, say grid turbulence, belongs.

Notes:

(1) The net viscous force on the surface SR is τijdSj, where τij is the viscous stress. This can be rewritten as the volume integral τij/xjdV=ρν2udV=ρν×ωdV. Converting back to a surface integral we find that the net viscous force is ρνω×dS, which is zero since ω=0 on SR. In a similar way the net viscous torque on SR can be written as ρνx×(ω×dS)2ρνx(ωdS), which is also zero.