## 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

Source:
Turbulence
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)
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Figure A2.1 An isolated eddy located at x = 0.

and is related to the vorticity by the expression

(A2.2)
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This may be inverted using the Biot–Savart law to give

(A2.3)
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(p.612) where the integral is over all space. To find the vector potential some distance from the blob of vorticity we expand $x′−x−1$ in a Taylor series in $x−1$. This yields

(A2.4)
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where $r=x$ and

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

(A2.5)
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However, the first term on the right integrates to zero since

(A2.6)
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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)
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from which,

(A2.8)
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Our expansion for the vector potential now simplifies to

(A2.9)
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where

(A2.10)
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(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)
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Evidently the velocity in the far-field is $O(r−3)$ if L is finite and $Or−4$ if L happens to be zero. We can find the corresponding far-field scalar potential for u by rewriting (A2.11) as

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

(A2.12)
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# 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

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and then inverting this using the Biot–Savart law:

(A2.13)
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Substituting for $x′−x$ using (A2.4) and noting that

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we find

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The first two volume integrals convert to surface integrals which, since $u∼O(r−3)$, 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)
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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)
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The argument goes as follows. In a potential flow the viscous forces are zero since $ν∇2u=−ν∇×∇×u=0$, and so Bernoulli’s theorem for unsteady potential flow demands, in the far field,

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Since $p∼O(r−3)$ and $u2∼O(r−6)$ for large r, this reduces to

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Equation (A2.15) follows immediately since, in the far field, $ϕ∼L⋅∇1/r+O(r−3)$. 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

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(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

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

Compare this with the linear momentum equation applied to $VR$,

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Evidently,

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(p.615) In the limit $R→∞$ we have

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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 $∫udV$is 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)
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In fact, the relationship

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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

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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

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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 $u∼O(r−3)$ 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

$Display mathematics$

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

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and so it follows that

(A2.17)
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(The second term on the right integrates to give $3R2∫ωdV$, 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

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

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

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

(A2.18)
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(p.617) In summary, then, an isolated eddy has two integral invariants, its linear impulse and its angular impulse:

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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 $u∞∼O(r−3)$, whereas an eddy with finite angular impulse but zero linear impulse has $u∞∼O(r−4)$. 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 $E∼k4$ or a Saffman $E∼k2$ 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)
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It turns out that the term involving fourth-order correlations decays as $r−4$ 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 $K∞∼r−4$ implies $uu′∞∼r−6$ in isotropic turbulence. (In anisotropic turbulence we find $uu′∞∼r−5$.) As shown in Chapter 6, an $r−6$ 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 $uu′∞∼r−6$ 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 $E∼k2$ spectrum can arise only if $uu′∞∼r−3$. 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 $uu′∞∼r−3$ at $t=0$. Again we can use the results above to ask, under what conditions is the assumption $uu′∞∼r−3$ 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 $E∼k4$ 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 $E∼k4$ 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

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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)
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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)
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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,

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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 $O(r−4)$ term in expansion (A2.11) gives rise directly to long-range triple correlations of order $r−4$. These, in turn, lead to a $O(r−6)$ contribution to $uiuj′$ in isotropic turbulence ($r−5$ 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 $r−1$, a larger velocity at $x′=reˆx$:

(A2.22)
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where L is the net linear impulse of the eddies near $x=0$. This suggests the possibility of a $r−3$ contribution to $uxux′$. It is this which lies behind the Saffman spectrum

(A2.23)
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(A2.24)
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(p.619) where $L=∫​〈u⋅u′〉dr$ is Saffman’s integral. Note that $L$ is an invariant of the Saffman spectrum so that, if $L=0$ at $t=0$, then $E∼k4$ for all time. Thus, whether or not we have a Batchelor ($E∼k4$) or Saffman ($E∼k2$) 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

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

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is finite, which is consistent with the notion that $L$ is non-zero in Saffman turbulence.

In summary, then, whether we have a $k2$or $k4$ spectrum depends on the mechanism which generates the turbulence. If enough linear impulse is injected into the fluid at $t=0$, then we expect $E∼k2$ 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 $E∼k4$ 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 $∫∂τij/∂xjdV=ρν∫∇2udV=−ρν∫∇×ωdV$. Converting back to a surface integral we find that the net viscous force is $ρν∮ω×dS$, which is zero since $ω=0$ on $SR$. In a similar way the net viscous torque on $SR$ can be written as $ρν∮x×(ω×dS)−2ρν∮x(ω⋅dS)$, which is also zero.