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Unformatted text preview: Isotropic Turbulence
A Summary of Tony Burden’s Lectures, Spring 2003 Strictly speaking, isotropy, i.e. independence of orientation (riktning), implies homo—
geneity, i.e. independence of position in space. In most situations, all the averaged
properties of isotropic turbulence can also be assumed to be invariant under reﬂection
in space. Notation A number of different notations are used for the mean velocity and the ﬂuctuating
part of the velocity in turbulent ﬂows. In this chapter, a ﬂuctuation, i.e. the difference
between an instantaneous physical quantity and its mean, is denoted by a lower—case
letter, 6.9. u. Some alternative notations are given in the following table. inst. mean ﬂuct.
previous chapters u U u’
this chapter 7 U u
Pope (2000) U <U> u
Tennekes & Lumley a U u The operation of forming an average will be denoted by angular brackets, = 0.
A. Singlepoint and TwoPoint Velocity Correlations Singlepoint correlations In isotropic turbulence, the Singlepoint correlations satisfy, <u2> : <v2> : <w2> while <uv> : <vw> : <wu> : 0, who so that we can deﬁne u?) : <u2> :
write, = = = u?) with <u1u2> — K. ( uT 2 uo. ) In Cartesian coordinates, we can I
/\
: [\D
:
0.3
v

/\
:
OJ
:
H
v

O i.e.
W W = US 5117‘ Twopoint correlations Tvvo—point velocity correlations retain more information on the structure of turbulence
than singlepoint correlations. In isotropic turbulence, which is also on average invariant under reﬂection in space,
there are only two independent, non—trivial, tvvo—point velocity correlations; the longi—
tudinal covariance, f, deﬁned by, ugﬂr) = <u(6,t)u(ré’x,t)> = <v(6,t)v(r5y,t)>, and the transverse covariance, 9, deﬁned by, —» ugg(r) = <v(0,t)v(ra,t)> = <u(0,t)u(r5y,t)>. The general tvvo—point correlation, t)uj(f’, t)>, must be a proper tensor func—
tion of u?) and F = i” — f. Dependence on any other combination of f and i” is ruled out
by spatial hornogeneity. When the turbulence is on average invariant under reﬂection
in space, the tvvo—point correlation can be written in the form, <ui<at>uj<az+m>> = u3{g<r>6zj + (M —g<r>>’””"‘}. T2 The continuity condition; aim<ui<awuj<5+m>>= 0 : 9(7”) 2 5r———f(r). Integral length scales Using the tvvo—point correlations, we can give the large length scale, ZT in Ch. 4, a precise
deﬁnition. The integral length scales are large length scales, A N ZT, deﬁned by, A, = /Ooof(r)dr 2 W12, Am<u(6)u(xax)> dx, and Taylor rnicroscales The Taylor rnicroscales are quite small length scales deﬁned by, A’ : {43(0)}1/2 and At 2 {—gianll/Z' In isotropic turbulence the Taylor rnicroscales have the same order of magnitude; gm : pawn) :> A; 2 Via. The Taylor rnicroscales are intermediate length scales, ZT N A > /\ > 77 = ZK. The Taylor rnicroscale can be deﬁned more generally, and more loosely i.e in an
order—of—rnagnitude fashion, in terms of the correlation of velocity gradients vvhich dorn—
inates the dissipation, au‘ au‘ u2 1/112
1 1 N —T <—> 5 N T
81] 81“, A2 A; ' The expressions for the dissipation which are given in the next subsection show how
these relations are satisﬁed in isotropic turbulence. (A physical understanding of /\ in
terms of a particular scale of structures in the turbulence is still diﬁcult to achieve since
the velocity scale uT in 5 N l/UZT/Ag is ‘vvrong’ in the sense that it characterizes the large
energy—bearing eddies while 5 and 1/ characterize the small dissipative eddies.) The order—of—rnagnitude estimate for 5 in high—ReT turbulence yields the scale re—
lations: 2 3 2
uT uT , vuT /\ /\ _1/2
5 lT/UT ZT W1 5 A2 —> ZT A eT In terms of the Kolornogorov rnicroscales, 2 2
vuK , vuT uT 77 _1/4
5 _ — Wlth 5 N — —> /\ N — —> — N RC .
lg A2 UK ’7 /\ T Example: The earth’s planetary boundary layer (ReT N 108); llern, /\N0,1rn, 77N1rnrn. Viscous dissipation In terms of the two—point velocity correlations, the mean rate of viscous dissipation of
the kinetic energy of the turbulence is, 2 _ + 8111‘ _ V 81182:, — V 8.1] 8x, — V 8x, 8x, m
  2 2 15w 2 30—
A; A; In the last line the Taylor microscales, /\l and /\t, have been introduced. Exercise (inldrnmgsévnmg)
Check that the two—point velocity correlation <ui(f7t)uj(f+ F,t)> : u?) {g(7”)6ij — — 90)) mm} r2 reduces to <ui(f,t)uj(f,t)) : 1136,, in the singlepoint limit, 7“ —> 0. Recommended course reading Sec. 6.3 ‘Tvvoipoint correlation7 in “Turbulent ﬂows77 by S. B. Pope, C.U.P., 2000. Alternative recommended reading
“Turbulence”, J.O. Hinze, McGravv—Hill (1st edition, 1959; 2nd edition, 1975).
pp 53—56 & 65—66 in Landahl & Mollo—Christensen (1986). Secs 5.1 & 5.2 in “Introduction to Turbulence77 by RA. Libby (1996). Sec. 3.1, ‘Velocity correlations and spatial scales’, in ‘An introduction to turbulent ﬂovv’, J. Mathieu & J. Scott, C.U.P., (2000). Sections 19.4 and 20.2 in “Physical Fluid Dynamics77 by D.J. Tritton, O.U.P., 1988. B. The wavenumber spectrum of isotropic turbulence TWO—point velocity correlations do not readily yield information on length scales which
are smaller than the Taylor micro—scales. This information is contained in correlations
of two—point velocity differences, 17(f—l— it) — Wit), and in correlations of velocity
derivatives. The Fourier transform ﬁlters this information out of the two—point velocity
correlations. This is one reason for studying the vvave—number spectrum. Another reason is that the dynamical equation governing the two—point correlation
is a little bit less difﬁcult to solve in vvave—number space, i.e. after Fourier—transforming
with respect to the separation between the points, F. This is particularly true of terms
which contain the pressure fluctuations, p. (The derivation of the dynamical equation
is not included in the examination for this course.) The Fourier vvave—number spectrum of homogeneous turbulence is defined by, A 1 00
(Fiﬁ/at) : / <ui(f,t) uj(f——F,t)> €Xp(—i/§.F) d377, —00
so that the inverse formula yields the spectral representation,
w /\
<ui(f,t) u,(5+m)> = / CI),,(/%’,t) expo/3F) (13/3.
—00 Remember that small lengthscales correspond to large wave numbers and, vice versa,
large lengthscales to small wave numbers.
General basic properties The velocity is real so : ui(f,t) Where . denotes complex conjugation. NOW, using the definition of EISZ‘ﬂ/at) above, [$i1(’¥vt)] * (271,3 /00 <Ui(57t) uj(flF,t)>* eXpﬁH/KF) d3? (2717)?) /00 <ui(fvt)uj(f+77,t)> exp(—i(_,§) .F)d3,:’ : $ij(_gjt)‘ —00 The ‘usual’ relation in fourier analysis. In statistically homogeneous turbulence the tvvoipoint correlation remains unchanged
when the position f is shifted to another point i"; <ui(f,t)uj(f—l—F,t)> = <uj(f——F,t)ui(f,t)> = <uj(f’,t)ui(f’—F,t)>, Where 5’ : f—l—F so that f : f’—F. The deﬁnition of the Fourier wave—number spectrum
implies now that7 A _} 1 00 _} (PM/W) = (2mg/ <uj($'at)
1 00 r
277 wows/cat) >3/ A —» ‘13ng E337: (—E).(—T) = (—H).F’ Where F” = —r, In isotropic turbulence the spectrum is independent of the distinction between I? and
—/3 so7 crux—lat) = (EA/at), @jpat) = (EA/at), and [chm/2,0] = (Tux/at). The continuity condition a A
—<ui(f,t)uj(f—l—F,t)> = 0 : MCDME’J) = 0.
arj Together With the symmetry property Which comes from spatial homogeneity this yields7 K : §<ui<m>ui<f+ow>> = / %<’I3n<l%3t>1d3ﬁ —OO / %%$ii(ﬁ,t)a2dﬂﬁda,
0 When the three—dimensional integral over wave—number space is expressed in spherical coordinates. The mean kinetic energy of the turbulence can be written in the form7 K 2/ E(H,t)d/£,
0 Where7 is the scalar spectral energy density of the turbulence, given here by its most general
deﬁnition. For simplicity we will use the scalar spectrum EM, t) rather than the tensor spectrum EMA/at) whenever possible. Isotropic turbulence The vvave—number spectrum, EMA/at), must be a proper tensor function of I? that sat—
isﬁes the continuity condition. In terms of the scalar spectral energy density, EM, t), it
can be written in the form, A _, E(H,t) Hiﬁ‘
(Fiﬁ/ﬁt) = 47TH2 {6ij— J}7 H2 for turbulence that is invariant under spatial reﬂections. In isotropic turbulence, the
scalar spectral energy density is given by, mm) = 2M? Eng/at). Exercise (inldrnmgsévnmg)
Derive a relation between and (This requires some familiarity with
three—dimensional, spherically symmetric integral transforms.) Dissipation rate In terms of the scalar spectral energy density, the viscous dissipation of kinetic energy
of homogeneous turbulence is given by, 2
6 = 1/ [—838Tj <ui(:1:,t)ui(:1:—l—r,t)> 2/ 2VH2E(H,t)dH
0 The Equilibrium range of wave numbers (High ReT) When ReT >> 1, the small scales are in quasi—eqiuilibrium with the large energy—bearing
scales 7 see Ch. 4. In terms of the scalar vvave—number, H, the small scales can be deﬁned by, 2
—7T<<A => HA>>27T>1,
H 7.6. H is large. This range of wave numbers is characterized by the mean rate of transfer
of energy from the large scales to the small scales, 7.6. by E(t). Mathematically, for the
scalar spectral energy density, this can be expressed by, E(H,t) : Feq.(€(t),/£,l/> When HA >> 1. (Where ‘eq.’ stands for ‘equilibrium’). (The way in Which time, t, appears in this
equation gives a concrete example of What is meant by quasi—equilibrium.) The Inertial subrange of the Equilibrium range (High ReT) When ReT >> 1, A N Re3T/477 >> 77 Which means that there Will be a substantial
range of wave numbers satisfying, 271' 1 1
A>>—>>77 0r —<<H<<—.
H A 77 In this range, 27T/H >> 77 , i.e. H77 << 1, implies that Viscous processes are negligible
(hence the name inertial). Mathematically, for the scalar spectral energy density, this
means that, E(H, t) : FKO (E(t), H), Where ‘Ko’ stands for the Russian scientist Kolmogorov. Now, dimensional analysis
yields,
E(H,t) : a52/3H_5/3 7 Which is often refered to as the Kolmogorov spectrum. The Equilibrium range including the Dissipation subrange (High ReT)
Using FKO : a52/3H_5/3 to express F601,,
E(Hvt) = 0662/3/i—5/3feqlm7), in the Whole equilibrium range, HA >> 1, Which can be divided into sub—ranges according to;. the inertial sub—range H << Hd H77 2 0 and feq.(/£77) 2 1
the dissipation sub—range H 2 Hd 21/H2E has a maximum at H77 < 1
the far dissipation sub—range H >> Hd feq. —> 0 exponentially When H77 —> 00 Where Hd : 277/77. Recommended course reading
Sec. 6.5 ‘Velocity spectra7 in “Turbulent ﬂows77 by S. B. Pope, C.U.P., 2000. Secs 8.1—8.4 in Tennekes & Lumley (1972). Alternative recommended reading
“A Model of Turbulence”, Leo P. Kadanoff, Physics Today, September 1995
pp 59—64 in Landahl & Mollo—Christensen (1986). Sec. 5.2 in “Introduction to Turbulence77 by PA. Libby (1996). Ch. 6, ‘Spectral analysis of homogeneous turbulence’, pp 2397245, in ‘An introduction
to turbulent ﬂow’, J. Mathieu & J. Scott, C.U.P., (2000). Sec. 6.4 ‘Conse uences of isotro 7 in ‘An introduction to turbulent ﬂow7 J. Mathieu
7 q py 7 7 & J. Scott, C.U.P., (2000). Sections 19.5 and 20.3 in “Physical Fluid Dynamics77 by D.J. Tritton, O.U.P., 1988. C. The Dynamical Equations The dynamical equation for EM, t) in isotropic turbulence can be derived in a straight—
forward way from the dynamical equation for the two—point velocity correlation via
the dynamical equation for (Fiji/3, t). (The derivation of the dynamical equation is not
included in the examination for this course.) The twopoint dynamical equation In the absence of a mean ﬂow, the ﬂuctuating instantaneous velocity satisﬁes, , 2 i
am : a mu; I a — lap  a uz at 891:; 81:; p896, delaxl‘ Together with, 8 _, _, 8m _, _, _, 8w _,
a<ui($1,t)uj(x2,t)> 2 <8, ($1,t)uj(:1:2,t)> + <ui(:1:1,t)a—t‘7(:1:2,t)>
this leads straightforwardly to, g<ui(f1,t)uj(f2,t)> = a —» —» —» a —» —» —»
— <ui(:1:1,t) uj(:1:2,t) ul(:1:1,t)> — am I <ui(:1:1,t) uj(:1:2,t) ul(:1:2,t)>
2 891:2, 82 a? a a
I V<8$118x11 I 8$218$21)<ui($17t)uj(x27t)>. This equation is subjected to a coordinate transformation, {51,172} —> {77, :17}, where, —» —» —» —» —» $1 1' 7” $2 —$1 —» $2 —» —» f—l—F :1: 91:1 so that f is a ‘position’ and F is the separation between the two original points. The
spatial derivatives are given by 10 In spatially homogeneous turbulence, averaged quantities depend on F but not on f Calm : 0’) so that the tvvo—point dynamical equation becomes, g<m<m>uj<£+m>> = 82 _, _, _,
+21/8rlarl <ui(:1:,t) uj(:1:——r,t)> The presence of third—order velocity correlations in the dynamical equation for the
second—order correlation is an example of the general closure problem. In the singlepoint limit, i.e. when 7“ —> 0,
a —» —» —» —» a —» —» —» —» —»
— <ui(:1:,t) uj(:1:——r,t) ul(:1:,t)> — — <ui(:1:,t) uj(:1:——r,t) ul(:1:——r,t)> —> 0.
87”] 87”] (The Fourier transform of this expression is the rate of transfer of energy through
spectral space, T(H,t), and the fact that it vanishes in the singlepoint limit leads to fooo T (1% = 0.)
Exercises (inldrnmgsévnmgar)
Show that <p_1p(f,t) uj(f——F,t)> : 0 in isotropic turbulence. Show that, in homogeneous turbulence, the contracted forms of the pressure terms
vanish; a 1 _, _, _, _ 8,, <5pw>ul<x+m>> — o,
and, a 1 _, _, _, _ an <;p(:1:——r,t) u,(:1:,t)> — 0. 11 The dynamical equation for the energy spectrum In isotropic turbulence, the energy spectrum, E, can be written in terms of the two—point
velocity correlation in the form, A 2 2 00
mm) = 2m2q>,,(/z,t) = (27:33 / (ui(f,t)ui(f——F,t)) eXp(—iH.F)d3H, —00 so the dynamical equation governing E can be obtained by ﬁrst contracting = :>
sum) and then Fourier transforming the equation for the two—point velocity correlation, a 27TH2 00 a _, _. _. ._. 3_.
EE — (2F)3/ a (ui(:1:,t)u,(:1:——r,t)) eXp(—1H.F)dH. —00 The resulting equation is written in the form, 8E
— = T — 21/H2E.
at
21/H2E(H,t) is the rate of viscous dissipation of energy at scalar wave number H 7 see the end of sec. B. TM, t) is the Fourier transform of the terms containing third—order correlations in
the two—point equation. The detailed expression for T(H,t) is a bit complicated and
contains the third—order spectrum which has not been deﬁned in these notes. Since
the terms containing third—order correlations in the two—point equation vanish in the
singlepoint limit, i.e. when r —> 0, T(H,t) satisﬁes, / T(H,t) dH : 0.
0 TM, t) is the net effect on EM, t) of the non—linear inertial spectral transfer of the mean
kinetic energy of the turbulence from large scales (small H) to small scales (large H).
Consequently, T(H,t) is negative for small H and positive for large H. In terms of the
spectral transfer of energy, 5f, discussed in Chapter 4 of these notes, the transfer out
of the energy—bearing eddies (small H) is, 6f 2 —/ T(H,t)dH,
0 and the transfer into the dissipating eddies (large H) is, 6f 2 / T(H,t) dH, inert where Hinert is a wave—number, any wave—number, in the inertial sub—range of the equi—
librium range. 12 The equation for EM, t) is not closed. TM, t) has to be modelled in terms of EM, t)
before E(H,t) can be calculated. The singlepoint limit The singlepoint limit7 7“ —> 07 is achieved by integrating over all wave numbers; 31:: 1/ Eda /
0 0 dt dt
2/ TdH—/ Zl/HZEdH
0 0 20—5. 8_Ed,ﬁ
t 8 This is just the K—equation that was obtained for homogeneous turbulence in the absence
of mean shear in Chapter 3 of these notes. Recommended course reading Sec. 6.6 ‘The spectral View of the energy cascade7 in “Turbulent ﬂows77 by S. B. Pope7
C.U.P.7 2000. Alternative recommended reading
pp 59—64 in Landahl & Mollo—Christensen (1986). Sec. 5.2 in “Introduction to Turbulence77 by RA. Libby (1996). Sec. 6.37 ‘Spectral equations Via correlations in physical space’7 in ‘An introduction to turbulent ﬂovv’7 J. Mathieu & J. Scott7 C.U.P.7 (2000). Sections 20.2 and 20.3 in “Physical Fluid Dynamics77 by D.J. Tritton7 O.U.P.7 1988. 13 ...
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