Klyatskin V.I. Lectures on Dynamics of Stochastic Systems

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256 Lectures on Dynamics of Stochastic Systems

∂

∂t

(r, t) +

(r, t)u

(r, t)

∂

(r, t)

∂r

= 2

∂

∂r

(r, t)

∂

∂r

−1

(r, r

)

∂u

, t)

∂r

∂U

, t)

∂r

(r, t)

∂

∂r

−1

(r, r

)

∂u

, t)

∂r

∂U

, t)

∂r

!+#

. (10.7)

This system of equations is closed in U(r, t). The coefﬁcients in the left-hand side of

Eq. (10.7) and the integral operator in the right-hand side can be expressed in terms of

the correlation tensor of vortex ﬁeld of unperturbed turbulence u

(r, t)

(r, t)u

, t)

dq8

(q)e

(

r−r

)

. (10.8)

In view of the fact that ﬁeld u

(r, t) is the solenoidal ﬁeld, we have

(q) = 1

(q)F(q),

where

(q) = δ

−

We will assume the energy spectrum of turbulent pulsations F(q) known. From

Eq. (10.8) follows in particular that the energy of turbulent pulsations is expressed

through the energy spectrum by the relationship

T =



(r, t)



= 4π

∞

dq q

F(q),

and coefﬁcients in the left-hand side of Eq. (10.7) are given by the formula

(r, t)u

(r, t)

Tδ

kl.

(10.9)

Finally, using Eq. (10.8), the right-hand side of Eq. (10.7) can be represented in the

form

−

dqe

iqr

(q, t)g

(q), (10.10)

where U(q, t) is the spatial Fourier transform of average velocity ﬁeld U(r, t) and

tensor g

(q) is given by the formula

(p) = 2

(

q + p

)

[

(

+ p

)

(

)

(

+ p

)

(

)

]

F(q). (10.11)

Some Other Approximate Approaches to the Problems of Statistical Hydrodynamics 257

From the obvious fact that tensor g

(p) is invariant relative rotations in space, it

follows that it must have the form

(p) = A(p)δ

+ B(p)

. (10.12)

Moreover, the term proportional to B(p) disappears in Eq. (10.12) in view of Eq. (10.4)

(it expresses the property of incompressibility of average ﬁeld U(r, t)) and the iden-

tity pU( p, t) = 0 following from Eq. (10.4), so that the right-hand side (10.10) of

Eq. (10.7) assumes the form

−

dqe

iqr

(q, t)A(q).

Substituting this expression in the right-hand side of Eq. (10.7) and taking into

account Eq. (10.9), we rewrite Eq. (10.6), (10.7) in the form

∂

∂t

(r, t) + τ

(r, t) = −

∂

∂r

P(r, t),

∂

∂t

(r, t) +

T1U

(r, t) = −A(−i∇)U

(r, t).

(10.13)

The kernel of the integral operator appeared in the second equation can be obtained

from comparison of Eq. (10.11) with (10.12). The result is as follows

A( p) =

F(q)



+ p



−

(

)

−

(

)

− 2

(

)

. (10.14)

We can perform integration in Eq. (10.14) over angular coordinate to obtain the

expression

A(p) = 2πp

∞

dq F(q)

(

−

− p



− p



8qp



q + p

q − p



)

. (10.15)

We note additionally that Eq. (10.13) with allowance for Eq. (10.4) yields the identity

1P(r, t) = 0, so that P(r, t) = P

= const. Then, eliminating quantity τ

(r, t) from

system of equations (10.11) and (10.4), we arrive at a single equation in the vector

ﬁeld of average velocity of ﬂow U(r, t),

∂

∂t

U(r, t) −



T1 + A(−i∇)



U(r, t) = 0. (10.16)

The corresponding dispersion equation

(p) =

− A(p)

258 Lectures on Dynamics of Stochastic Systems

can be rewritten in the form

(p) = 2πp

∞

dq F(q)f





where

f (x) =

2 − x

− 1

−



− 1



16x



1 + x

1 − x



Function f (x) has the following asymptotics

f (x) =

(

2/3, x  1,

4/15 x  1.

Moreover, the inequalities

≥ f (x) ≥

hold for arbitrary x.

Thus, the time-dependent development of disturbances in average ﬂow is governed

by the hyperbolic equation (10.16). As a result, the turbulent medium possesses cer-

tain quasi-elastic properties; namely, disturbances diffuse in the turbulent medium as

transverse waves showing dispersion property. The phase and group velocities of these

waves vary in the limits

T ≥ c ≥

10.2 Sound Radiation by Vortex Motions

In the previous section, we considered the physical effect immediately related to sta-

tistical averaging of the nonlinear system of hydrodynamical equations in the case

of noncompressible ﬂow. Here, we consider an effect related to weakly compressible

media; namely, we consider the problem on sound radiation by a weakly compressible

medium. This problem corresponds to the inclusion of random ﬁelds in the linearized

equations of hydrodynamics. Note that, within the framework of linear equations,

parametric action of turbulent medium yields the equations of acoustics with random

refractive index.

Turbulent motion of a ﬂow in certain ﬁnite spatial region excites acoustic waves

outside this region. In the case of a weakly compressible ﬂow, this sound ﬁeld is as if it

Some Other Approximate Approaches to the Problems of Statistical Hydrodynamics 259

were generated by the static distribution of acoustic quadrupoles whose instantaneous

power per unit volume is given by the relationship

(r, t) = ρ

(r, t)v

(r, t),

where ρ

is the average density and v

(r, t) are the components of the velocity of

ﬂow in turbulent region V inside which the turbulence is assumed homogeneous and

isotropic in space and stationary in time (we use the coordinate system in which the

whole of the ﬂow is quiescent). Turbulent motions cause the ﬂuctuating waves of

density ρ(r, t) that satisfy the wave equation



∂

∂t

−c



ρ(r, t) =

∂

∂r

(r, t), (10.17)

where c

is the sound velocity in the homogeneous portion of the medium, i.e., outside

the region of turbulent motions.

The solution to this equation has the form of the retarded solution

ρ(r, t) =

4πc

∂

∂r

dyT



y, t−

|r − y|



For distances signiﬁcantly exceeding linear sizes of turbulent region V, this solution

can be reduced to the asymptotic expression

ρ(r, t) =

4πc



y, t −

|r − y|



, (10.18)

where

(

y, t

)

∂

∂t

(

y, t

)

Correspondingly, average energy ﬂux density of acoustic waves excited by turbulent

motions

q(r, t) =



(r, t)



is given by the expression

q(r, t) =

16π





y, t−

|r − y|





z, t −

|r − z|



. (10.19)

260 Lectures on Dynamics of Stochastic Systems

Further calculations require the knowledge of the space–time correlators of the

velocity ﬁeld. The current theory of strong turbulence is incapable of yielding the

corresponding relationships. For this reason, investigators limit themselves to vari-

ous plausible hypotheses that allow complete calculations to be performed (see, e.g.,

[58–60]). In particular, it was shown that, with allowance for incompressibility of ﬂow

in volume V and the use of the Kolmogorov–Obukhov hypothesis and a number of

simplifying hypotheses for splitting space-time correlators, from Eq. (10.19) follows

that both average energy ﬂux density and acoustic power are proportional to ∼ M

where

M =



(r, t)



is the Mach number (signiﬁcantly smaller than unity).

We note that this result can be explained purely hydrodynamically, by analyz-

ing vortex interactions in weakly compressible medium [61]. The simplest sound-

radiating vortex systems are the pair of vortex lines (radiating cylindrical waves) and

the pair of vortex rings (radiating spherical waves).

10.2.1 Sound Radiation by Vortex Lines

Consider two parallel vortex lines separated by distance 2h and characterized by equal

intensities

κ =

πξ σ,

where ξ is the vorticity (the size of the vortex uniformly distributed over the area of

inﬁnitely small section σ ), so that the circulation about each of vortex line is

0 = 2πκ.

We will call these vortex lines simply vortices. In a noncompressible ﬂow, these

vortices revolve with angular velocity

ω =

around the center of the line connecting these vortices (see Fig. 10.1a and, e.g., [62]).

Select the coordinate system with the origin at a ﬁxed point and z-axis along

the vortex line. In this coordinate system, velocity potential ϕ

(r, t) (v

(r, t) =

−Re∇ϕ

(r, t)) and squared velocity v

(r, t) assume the forms

(r, t) = ik ln

2iθ

−h

2iωt

(r, t) =

4κ

+ h

−2r

cos 2(ωt−θ)

(10.20)

Here re

iθ

is the radius-vector of the observation point.

Some Other Approximate Approaches to the Problems of Statistical Hydrodynamics 261

h (t )

(a) (b)

2h (t )

(t )

Figure 10.1 Schematics of dynamics of (a) vortex lines and (b) vortex rings.

According to the Bernoulli equation, local velocity pulsations described by

Eq. (10.20) must produce the corresponding pressure pulsations; in the case of weakly

compressible medium, namely, under the condition that M  1, where M =

2hc

the Mach number and c

is the sound velocity, these pressure pulsations will propagate

at large distances as sound waves.

Encircle the origin with a circle of radius R such that h  R  λ, where λ is the

sound wavelength. This is possible because

 1 for M  1.

In region r < R, dynamics of ﬂow approximately coincides with the dynamics

of noncompressible ﬂow. In other words, the dynamics of ﬂow is described by

Eqs. (10.20) in this region.

In region r > R, equations of motion coincide with the standard equations of

acoustics (see, e.g., [63])



∂

∂t

−c



ϕ(r, t) = 0, p

(r, t) = ρ

∂

∂t

ϕ(r, t). (10.21)

Here, ϕ(r, t) is the potential of velocity in the acoustic wave, p

(r, t) is the pressure

in the wave, and ρ

is the density of the medium.

Taking into account the fact that velocity v

(r, t) depends on t and θ only in com-

bination 2

(

ωt − θ

)

, we will seek the solution to Eq. (10.21) in the form

ϕ(r, t) = f (r)e

2i(ωt−θ)

, (10.22)

Substituting Eq. (10.22) in Eq. (10.21) and solving the resulting equation in function

f (r) with allowance for the radiation condition, we obtain

ϕ(r, t) = AH

(

2ωr/c

)

2i(ωt−θ)

, (10.23)

where H

(z) is the Hankel function of the second kind and A is a constant.

262 Lectures on Dynamics of Stochastic Systems

For r  c

/2ω we have the standard divergent cylindrical wave with wavelength

λ = πc

/ω

ϕ(r, t) = A

πωr

exp{2i(ωt − ωr/c

− θ − 3π/8)}.

In the opposite case r  λ, we obtain

ϕ(r, t) = i



ωr



2i(ωt−θ)

Potential ϕ(r, t) in region h  r  λ must coincide with the oscillating portion of

potential ϕ

(r, t) (see, e.g., [63]), i.e., with

(1)

(r, t) = −iκ

2i(ωt−θ)

This condition yields the following expression for constant A

A = −πκM

= −

πκ

Consequently, potential ϕ(r, t) in the wave zone assumes the form

ϕ(r, t) = −κM

3/2

πh

exp{2i(ωt − ωr/c

− θ − 3π/8)}. (10.24)

The pressure in sound waves can be determined from potential (10.24) using

Eq. (10.21). The result is as follows

(r, t) = −2κωρ

3/2

πh

exp{2i(ωt − ωr/c

− θ − 3π/8)}.

The sound intensity (energy radiated per unit time) can be obtained by integrating

along the circle of radius R  λ

I =



(r, t)



= 2π

. (10.25)

The radiated energy must coincide with the interaction energy of vortices located

in region r < R. Total energy in region r < R is

E =

dSv

(r, t). (10.26)

Some Other Approximate Approaches to the Problems of Statistical Hydrodynamics 263

Substituting Eq. (10.20) in (10.26) and discarding inﬁnite terms corresponding to the

energy of motion of vortices themselves (we assume vortices the point vortices), we

obtain the interaction energy in the form

= 4πκ

ln(R/h). (10.27)

The interaction energy can vary only at the expense of varying the distance between

vortices (h = h(t)), because the circulation remains intact due to the fact that we

consider nonviscous medium.

Differentiating Eq. (10.27) with respect to time, we obtain energy variation rate

I(t) = −4πρ

h(t)

h(t), (10.28)

which is just transferred into the energy of acoustic waves. Using Eq. (10.25) and

(10.28), we obtain that the distance between vortices satisﬁes the equation

h(t) =

πκM

2h(t)

. (10.29)

Integrating Eq. (10.29) with allowance for the fact that

M = M(t) =

2h(t)c

we obtain

h(t) = h

1 + 6πM

1/6

Thus, the intensity of radiated sound is proportional to M

, I ∼ M

. It is obvious

that, in the case of statistically distributed system of vortex line pairs, this estimate

will remain valid for certain portion of the plane.

10.2.2 Sound Radiation by Vortex Rings

In a noncompressible ﬂow, a vortex ring of intensity κ causes the ﬂow to move with a

velocity

v(r, t) =

2π

dφ

s×r

which follows from the the Biot–Savart law (see, e.g. [62]). Here, s is the unit vector

tangent to the vortex ring (it is directed along the vortex vector), a is the radius of the

264 Lectures on Dynamics of Stochastic Systems

ring, and r is the vector specifying observation point position relative to points lying

on the ring.

In the cylindrical coordinate system with origin at the center of ring and z-axis

directed along the ring axis, we have

2π

dφ

cos φ

, v

= 0, v

2π

dφ

a−Rcos φ

, (10.30)

where

r =



−2Racos φ



1/2

Here

(

R, θ, z

)

are the coordinates of the radius-vector of the observation point.

We now have two vortex rings of equal intensities and equal radii a

at distance

. In this case, the front ring will increase in size, while the rear ring will decrease

and pursue the front ring (Fig. 10.1b). At a certain instant, it will penetrate through the

front ring, and the rings switch places. This phenomenon is called the game of vortex

rings. In a weakly compressible ﬂow, these motions of rings produce local regions

of compression and rarefaction; these regions propagate in the medium and, for large

distances assume the form of spherical acoustic waves. To determine the structure of

radiated sound, we must know relative motions of rings in a noncompressible ﬂow.

Assume rings have radii a

(t) and a

(t) and are separated by distance 2h(t) at

certain instant t. The rates of variations of ring radii are equal to radial velocities

that rings induce at each other, and the rate of variation of the distance between the

rings is equal to the difference of the z-components of velocities induced by the rings.

Consequently, we have

(t) = 2κa

(t)h(t)

dφ

cos φ

(t) − a

(t)|

(t) = −2κa

(t)h(t)

dφ

cos φ

(t) − a

(t)|

h(t) = −

(t)−a

(t)

dφ

(t) − a

(t)|

(10.31)

where

(t) − a

(t)| =

(t)+a

(t) + 4h

(t) − 2a

(t)a

(t) cos φ

1/2

Equations (10.31) should be solved with the initial conditions at t = 0

(0) = a

, h(0) = h

Some Other Approximate Approaches to the Problems of Statistical Hydrodynamics 265

The ﬁrst pair of equations immediately yields the relationship between a

(t) and

(t); namely,

(t) + a

(t) = 2a

. (10.32)

This relationship shows that the moment of inertia of rings relative to the z-axis is

conserved.

The integrals in the right-hand side of Eq. (10.31) can be expressed in terms of

elliptic functions.

If rings are far from each other

(

γ = h

 1

)

, they interact only slightly, and

we can assume that, in the ﬁrst approximation, they move independently with the

velocities determined by the ring areas. In the opposite limiting case γ  1 (namely

this case will be considered in what follows), the rings interact actively. The integrals

in the right-hand side of Eq.(10.31) are mainly contributed by the neighborhood of

point φ = 0. For this reason, we can replace cos φ in the numerators of the ﬁrst pair

of equations by unity. Thus we obtain the second integral of motion

(t) +

[

(t)−a

(t)

]

= 4h

. (10.33)

Integral (10.33) means that the distance between points on different rings at the

same polar angle is the conserved quantity.

In view of the existence of integrals (10.32) and (10.33), we can reduce system

(10.31) to a single equation in variable θ(t) determined by the equalities

(t) =

√

cos



− γ sin θ(t)



, a

(t) =

√

sin



− γ sin θ(t)



h(t) = h

cos θ(t). (10.34)

With this deﬁnition, conservation laws (10.32) and (10.33) are satisﬁed automat-

ically (in the ﬁrst order with respect to γ ). Substituting Eq. (10.34) in Eq. (10.31),

expanding the result in the series, and calculating the integral, we obtain

θ(t) =

Consequently, the ring radii and the distance between rings are given by the

expression

(t) = a

(

1 + γ sin(ωt)

)

, a

(t) = a

(

1 − γ sin(ωt)

)

h(t) = h

cos(ωt), (10.35)

where

ω =

, (10.36)

as in the above case of vortex lines.