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On the Taylor hypothesis corrections for measured energy spectra of turbulence
PHY$10A ELSEVIER
Physica D 104 (1997) 163-183
On the Taylor hypothesis corrections for measured energy spectra of turbulence E. Gledzer* Institute of Atmospheric Physics, 109017, Moscow, Russian Federation
Received 13 July 1995; revised 23 July 1996; accepted 25 November 1996 Communicated by U. Frisch
Abstract The acceleration terms of the Navier-Stokes equations are taken into account to obtain the corrections for the measured longitudinal spectrum of homogeneous isotropic turbulence. The viscous term correction gives approximately uniform relative deviation of the sign such as the Lumely correction for intertial subrange with no influence on the slope of the spectrum but can be significantly increased for scales less than the Kolmogorov one. The deviations for small wave numbers are also estimated. In addition, the different forms of the corrections in the case of advection velocity fluctuations are discussed and some simple one-dimensional models are presented. Keywords: Turbulence; Spectrum; Advection; Acceleration
1. Introduction The Taylor hypothesis or frozen turbulence hypothesis was used in a great number of experimental works. But only few papers examined the theoretical backgrounds of this hypothesis which enable us to express the spatial statistical characteristics corresponding to a fixed time in terms of the temporal series measured at one point. Maybe, one of the most important investigation of Taylor's approximation was made by Lumley [7] and by Heskestad [5]. Other early investigations devoted to this problem are also discussed in [4,6]. The main mechanism which prevents a direct use of this hypothesis is, according to Lumley, the fluctuations of the advection velocity. The appropriate correction obtained by Lumley gives the deviations between the measured and the valid one-dimensional spectra of the streamwise velocity fluctuations (see [2,3,12]). Taylor's approximation is commonly written as follows: O/Ot = - U I O / O x l ,
* Address for correspondence: LEGI/IMG, B.E 53, 38041 Grenoble Cedex 9, France. 0167-2789/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PH SO 167-2789(96)00300-4
(1)
164
E. Gledzer/Physica D 104 (1997) 163-183
where U1 is the mean velocity in the Xl direction and t is a time. The corresponding wave number form is kl U1 = w,
(2)
where kl is the xl-component of the wave number vector and w is a frequency. For velocity field equation (1) is written as follows:
Oui/Ot + UjOui/Oxj = 0,
(3)
where Uj is the mean velocity of the flow in the point x and ui is the advected velocity field. Another relation, namely (4)
OUi/Ot d- UjOUi/OXj = O,
where ui = Ui + u I is both the fluctuating advection velocity and the instanteneous velocity of flow, was used by Heskestad [5] for the investigation of the advection velocity fluctuations influence and by Tennekes I1 l] for the estimation of the Eulerian time microscale in isotropic turbulence. Eq. (4) is the Navier-Stokes equation without the accelartion terms. So the investigations of the corrections to the Taylor hypothesis usually disregard the influence of these terms. The main aim of the present study is to take into account the influence of the disregarded terms. We show that the viscous term correction does not give any significant contribution for measured spectra in inertial range of developed turbulence (where the Lumley correction also has no influence on the slope of an energy spectrum curve in log-log plot) and is increased in the dissipative range. The corresponding threshold of the wave numbers is presented for some model spectrum of the dissipative range. The correction for the pressure term acceleration is also obtained. In Section 2 we consider simple one-dimensional models of the corrections, in particular, some model of correction for acceleration. In Section 3 the corresponding equations for three-dimensional case are presented which give the different types of corrections. The spectral forms of these corrections are considered in Sections 4 and 5. The main result of this paper is Eq. (45) for valid one-dimensional spectrum of the velocity field when the spectrum measured in one point is known. In Section 6 we discuss the deviations from measured spectra for inertial and dissipative ranges.
2. One-dimensional models Here, we recall the main results which give the energy spectrum of one-dimensional processes on the basis of the Taylor hypothesis and the Lumley type correction. In particular, the case is considered when the mean advection velocity is absent. We present also some model of correction for acceleration.
2.1. Taylor's hypothesis Following Eq. (1) velocities u at point x ----0 at times t + r and t + rl are equal to the velocities at time t but at the points - U r and - U r l upstream: u ( - U r , t) = u(0, t + r),
u ( - U r l , t) = u(0, t q- t'l).
Multiplying left- and right-hand sides of these relations we obtain that the space correlation function B(Us) = ( u ( - U r , t)u(--Url, t)), s = r -- rl is equal to the measured time correlation function BinS(s) = (u(0, t + rl + s)u(O, t + rl)). Here the brackets (---) mean the ensemble averaging. The corresponding space spectrum B(k),
E. Gledzer/Physica D 104 (1997) 163-183
165
O0
B(r) = (u(x, t)u(x + r, t)) = [ e-ikrB(k)dk,
r = Us.
(5)
can be expressed in terms of the measured time spectrum B ms BinS(s) = (u(0, t + s + rl)U(0, t + rl)) = [
e-ikUsBms(kU)U dk.
(6)
--OQ Eq. (5) and (6) give
B(k) = UBmS(kU).
(7)
This is equivalent to (2).
2.2. Lumley's approximation When advection velocity U has random fluctuating component U' with (U') = 0, the corresponding distance r = (U + U')s in (5) is a fluctuating value. The measured spectrum includes averaging over U'. So, inverse Fourier-transformation of (6) gives
O@ BmS(kU) = ~1 f
oO
eikUs ds f
--00
e-~k . . .Us . . (e-~k u S)u,B(k, ) dk'
--00
with the averaging on U'. This formula should be considered as a result of two types of averaging. One type is the ensemble averaging for fixed U' which gives spectrum B(k). Another type is U'-averaging. With the approximation
(e-ik'U'S)u, ~ 1 -- l k'2s2(U'2), the previous relation leads to Lumely's type correction
B(k) = UBmS(kU)
(U '2) d2(k2B(k)) 2U 2
(8)
dk 2
(see also [ 12]). Here we suppose that the advection velocity fluctuations are small enough compared with the mean advection velocity U, U~2/U 2 << I. The sign of the correction to the measured value UB ms(kU) for spectrum B(k) ~ k -5/3 is positive for one-dimensional processes. Note, that the above equations also may be interpreted as following expression:
k
UBms(kU)=(B(I+~J'/U)
1+
u'/u)v, 1 '
which shows that the measured spectrum is the weighted averaged value of valid spectrum. To investigate the correction to the slope of the function X (x) = In B(k), x = In k (log-log plot) it is convenient to represent (8) for small u ' Z / u 2 in the form
dx(x ) d~
dxmS(x) --
dx
Ut2 [( 2U 2
where XmS(s) = ln(UBms(uk)).
dxmS(x)'~ d2xms(x) 3 +2
~
,]
dx 2
d3xmS(x)l +
~
j,
(9)
E. Gledzer/Physica D 104 (1997) 163-183
166
It is clear that this approximation gives the correction to the slope of the function X (x) only in the regions where X (x) has nonzero curvature (for small third derivative). For usual one-dimensional spectra the second derivative is negative (in log-log plot), so the sign of the correction to the slope depends on the value of the measured slope d x ms (x)/dx. If this derivative is less then - 3 (for example, - 5) then the correction to the measured slope is negative (the slope is larger then for the measured spectrum). We can also present within considered approximation the correction to the correlation function of one-dimensional process. Using above formulas we obtain BmS(s) = (u(0, t)u(O, t + s)) = (u(O, t ) u [ - ( U -t- U')s, t]) = (B(Us + U's))u', where brackets (...) mean both the ensemble averaging and U'-averging while (.. ")u' is Ur-averaging. For small U ' / U this equation gives (U'2) _2 OZB(r ) BmS(s)=B(r)+-~-~-r ~r 2 ,
r = Us.
The sign of the correction to measured value B ms also depends on the curvature of B(r) and gives the increasing value for B(r) ~ r 2/3 without changing the slope in log-log plot.
2.3. No mean advection velocity In the considered cases the advection velocity was taken constant (random or nonrandom). When this velocity is time-dependent the correlation function can be represented in the following form: T
I fu(--X,
t)u(0, t) dt =(u(O, t +
17)U(0,
t)) = Bms(r),
0 t+r
/ I
X ------X ( r l t ) = [
V(t')dt',
(10)
i /
t
where we have assumed for simplicity rl = 0 compared with above cases. Here the time-averging was used over a period T which is much greater than the correlection time. The form of advection velocity V (t t) may depend on concrete experiment. For example, Pinton and Labb6 [9] in the experiments with KS.rrn~in swirling flows used
,/
t+T/2
V(t) = ~
u(O, t') dt',
t-T/2 where T is a period of disc rotation. We shall consider the case of zero mean advection velocity, (V(t)) = 0, when the above theory becomes inapplicable. Let us divide all period T of averaging in (10) in two parts. One part contains all time-intervals {t}+ when the displacement X = X (r It) is positive, X (r It) > 0. Another part consists on the intervals {t }_ with negative X, X(rlt) < 0. Positive values of displacement X(rlt) may be interpreted as produced by some positive mean advection velocity U+ > 0 and by fluctuations U~_ : X(rlt) = (U+ + U~_)r, so that U+ + U~_ > 0. The same for negative X: X(rlt) = - ( U _ + Ut_)r, where U_ > 0, U_ + U t_ > 0. The values U+ and U_ are of the order (V2(t)) 1/2, Now for every set of intervals {t}+ and {t}_ we can use the approximations of Sections 2.1 and 2.2 where, of course, the fluctuations U~, U'_ are not small, but at least, IU~ I < U+, IU'_ I < U_.
E. Gledzer/Physica D 104 (1997) 163-183
167
This shows that for the case of zero advection velocity the Lumely approximation may be used with the advection of the order of the root-mean-square velocity value.
2.4. Correctionsfor periodic oscillations of acceleration In above cases the velocity field was considered as some conservative quantity of Lagrangian fluid particle. Generally, it is not valid, and the particle changes its velocity. Let us consider the case when Lagrangian particle is subjected by the advection with the constant velocity U together with the external time-periodic oscillations of acceleration, a cos(Ot) with a frequency o9. Therefore, the Lagrangian velocity and the position of fluid particle (which at time t was at point x) at time t + r are equal sin(Or)
V(t + rlx, t) = U + v'(t + rlx, t) + a - - ,
O
t+r
X =- X(t +
1 - cos(Or) f i x , t) = x + U r + a O2
+ f o'(~lx, t)d~,
(11)
t
where v' is the Lagrangian velocity fluctuations. For the Eulerian point of view, we have for velocity fluctuations near mean value U
t+z
f Ov'(~lx, t) d~. u[X, t + r] = u(x, t) + a--------~- + J O~ sin(Or)
t
Neglecting fluctuations v' compared with a-terms, we obtain sin(Or) u ( x + U r + a 1 - c o s (2O r ) o , t + r ) =u(x,t)+a-----~-~
(12)
Using the same relation for another starting point x + r and time delay rl we obtain for correlation function (5)
B(r) = (u(x, t)u(x + r, t)) =/3mS(s) - a 2 sin(Or) sin(Orl) .('22
r -~ Us - a
cos(Or) - cos(Orl)
~(-22
s = r - rl,
(13)
where/3mS(s) is the measured time correlation function (u(0, t + rl)u(0, t + r)). Fourier-transformation of Eq. (13) with the spectrum B(k) defined by (5) gives for the right-hand side of (13) the measured spectrum and the correction proportional to a 2. BmS(kU)
4 0a2 2U
( 6 ( k - ~ - ) + 3 ( k - ~+- )
- e-2iOrl' (k _ ~---)"_ e2it2rl • (k + ~-~)) •
(14)
In (14) we have the term describing the constant deviation from the measured spectrum, proporitional to 8(k (O/U)) + 3(k + (O/U)), and the terms which are oscillating (proportional to e -2is2rj , e 2i°rl ). The amplitude of the oscillations of acceleration may be considered as a random value with zero mean value, so we can make the a-averaging in (14). For the Fourier-transformation of the left-hand side of (13) we obtain
dk' --OO
~ ei(k-k')Us 2~---~--dsS(kl' s, rl)B(k/), --OO
E. Gledzer/PhysicaD 104(1997)163-183
168 where
S(U, s, rl)
= e x p ( i k ' a ( c o s ( 1 2 r ) - c o s ( ~ 2 r l ) / I 2 2 ) . In the case of small amplitude of oscillations,
Ikla/S221 <<
1, it follows
S(k', s, rl)
=
1 - --~[l
- c o s ( X 2 s ) + cos(2K2rl)Dl(s) - s i n ( 2 ~ r l ) D 2 ( s ) l
Dl(S) = l[(cos(~2s) - l) 2 - sin2(~2s)l,
,
D2(s) = [ c o s ( ~ s ) - 11 sinff2s).
(15)
The relations (14) and (15) contain the oscillating terms which are proportional to e -2i~rj and e 2i~rt , respectively. The appearence of these oscillating terms has the same reason which exists for a spectrum of the velocity differences field 8r v (t) = v (t + r) - v (t) which is considered for fixed delay r [ 11. The Fourier-transformation of the correlation function of 8~ v(t) is
/ dseiWS(SrV(t)Srv(t+ s))= [ dsei~°s(2B(s)- B(s + r ) - B(s-
F~(~o)
tl --00
r)),
(16)
--~
where B(s) is the correlation function of velocity, B(s) = (v(t)v(t + s)). Therefore, in terms of the velocity spectrum ~b(w), we obtain instead of (16) OG
Fr(w) = 24~(w)(1 - cos(wr)),
~O(w) = /
ei~°SB(s) ds.
(17)
--OQ
Expression (17) contains like in (15) and (14) the oscillations near some mean value 2~b(w) which depend on the delay quantity r. Hence, it is necessary to make the r-averaging in order to find the velocity spectrum 4~(09) on the basis of the velocity increments spectrum Fr (09). But we cannot obtain the spectrum Fr (w) for w ~ 0 after making this r-averaging, because in this case we have Fr (0) = 2q~ (0) instead of the exact result Fr (0) = 0. The same procedure should be adopted for Eq. (15) and (14). In this case, we get
dk' jei(k-k')Us(U2(a2)[1--cos(~s)])B(k')2~ ds 1 ~f_24 --0~
--00
=BmS(kU,
4~22 U(a2)
(,(k-~)-k-~(kq--~)).
(18,
This relation permits us to calculate B(k) for a given measured spectrum BmS(kU). The left-hand side may be transformed in the form closed to that of Lumely's with the help of the following approximation, 1 - cos(~2s) 1.Q2s2: 1
-~B(k) +
(a 2) 2122U-----~
02(k2B(k)) Ok2
Without any approximation the calculation of the left-hand side of (18) leads to
-uB(k)
(a2) {kZB(k)_ 1
k+-~-) 2 B ( k + ~ ) + ( k - - ~ ) Z B ( k - ~ - ) ] } .
(19)
For 12/U << k this gives the previous result. The right-hand side of (18) shows that the acceleration terms give another type of corrections for measured spectrum B ms (k U). In particular, for a given one-frequency oscillation the measured amplitude with wave number k = 12/U should be reduced. Also, if there is no turbulence, B(k) = 0, then
E. Gledzer/Physica D 104 (1997) 163-183
169
the measured spectrum B ms consists only with one harmonic, k : ~ / U . The term B(k + (I2/U)) in (19), which corresponds to Lumley's type correction, describes the Doppler-effect of the fluctuations of advection velocity for a given wave number. According to (18) for fixed (a 2) the reduction of frequency S2 increases the values of the corrections because the amplitudes of the velocity and the displacement fluctuations (1 1) are increased at J2 ~ 0. But note that Eq. (18) cannot be valid for $-2 ~ 0. This is the case of a velocity with a constant trend, U + a t , for which the correction must be finite. Singularity on I2 appears after rl-averaging in (14) and (15) which is made over some period T > zr/12. We shall see that this is the general case of corrections for acceleration.
3. Lagrangian particles dynamics Now we shall take into account the acceleration terms of the hydrodynamical equations. Let us consider threedimensional Lagrangian dynamics of fluid particles. Denoting V(t + r lx, t) and X(t + f i x , t) the velocity and the coordinate at time t + r of a liquid particle, which at time t was at point x, obviously, we have the equations: t+r
V(t+rlx, t)=V(tlx, t)+ f
OV(flx, Of t) d~,
t
t+r
X(t + f i x , t) = x + f
(20)
V(~lx, t)d~.
t
The Eulerian and Lagrangian dynamics are linked by relation
V(t + rlx, t ) = u ( X , t + r),
u ( x , t ) = V(tlx, t).
(21)
From (20) and (21) we get for the Eulerian velocity at point x:
u(x,t)=u
+
V(flx, t)df, t + r
-
t
Of t
Now let us suppose that the velocity of each liquid particle can be expanded with the uniform advection velocity and the fluctuating parts, namely
V(flx, t ) : U + V ' ( f l x ,
t),
U : const.
Then, (22) can be written as
u(x,t)=u
V'(flx, t ) d ~ , t + r
+Ur+
-
Of
t
It the measurements are carried out a point x = 0 at time t + r, then the velocity at point x at time t may be obtained from (23) with the condition t+r
x + ur + f v'(flx, t
t ) d f = 0.
(24)
E. Gledzer/Physica D 104 (1997) 163-183
170
0
On +
+ r. t)
Fig. 1. Lagrangian particles dynamics.
Now, let us consider velocites at two points x and x + r at the same time t. From (23) and (24), we have the relations with the velocities at point x ----0 at two times t + r and t + rl (Fig. 1). For correlation function we obtain now
Bij(r) = (ui(x, t)uj(x + r, t)) = B~jS(s) - I1 - 12 - 13
(25)
in which
B~jS(s) = (ui(O, t + z)uj(0,
t + l~1)),
S =Z'--X'I,
\]t/r OVi,(~lx, t) I1 = { uj(x+r,t) d~ ,
12 =
u i (x, t) (t/r
13 =
° Ox+rt' I O~ d~ ,
~r d~
d~l
OVi,(~lx, t)OVj(~llx +r,t) ) ~1
(26)
t and r satisfies the equations t+r
r = U s + p ( r , rllt),
p(r, rllt) =
t+r
f W(~lx, t)d~- f V'(~lx+r,t)d~. t
(27)
t
4. The Lumley type corrections By multiplying the left-hand side and the first term of the right-hand side of (25) by e -ik~ us, and integrating, we obtain for the first term of the right-hand side
a~jS(kl e ) =
1 / e_iklUsB~S(s ) ds,
(28)
E. Gledzer/PhysicaD 104 (1997) 163-183
171
which is the measured spectrum. Here we have made the ensemble averaging only. The rl-averaging, like that in Section 2.4 (transition from (14) and (15) to (18)), will be fulfilled later. For the left-hand side (using (27)), one obtains
e-'k~USds eiktgs(eik'P(r'~11t))Bji(k')d[¢ldk2 dk3,
= (lq, k2, k3),
(29)
nji (k t) is the three-dimensional spectrum of velocity field
where
]3ji (r) = f eikrBji (k) dk.
(30)
Now we assume that the Lagrangian fluctuation field (see also [12]) is valid. Therefore, we have (eikp(r'r'lt))
1 --
lkak ~
/(;
U' is small enough, and the approximation of Lumley [7]
V~(~lx, t) d~ - R ~ 21 (rl) '
x,t+rl
)(Z
Vtfl(~lx, t) d ~ - R ~ l(rl)
,
(31)
t
where we have introduced the random fluctuating dispersions of fluid particles t+r
t+r
Rl(rl) = f v'(elx, l)d~,
R2(rl)= f v'(,Ix +r,t)d,,
t
t
21
Ra (rl) = R2a(rl) - Rlc~(rl).
(32)
In (31) the Lagrangian term, namely t+r
f
t+r
d~l
tq-r I
f d~2(V~(~llx,t)V;(~21x,t))
(33)
t+r I
is equal, for stationary homogeneous turbulence field (see [8, p. 528]), to O<)
4 f ELt~(o9)
sin 2 0 ) ( S / 2 ) 092 dog, s = r - r l ,
(34)
0
where EL~ (09) is the Lagrangian spectra of the velocity field. In Appendix A, it is shown that the term (33) gives the main contribution in Eq. (31). Therefore, (29) leads to oo ~ L -
t, sin2
1 f e_i(k,_~l)u s ds (1 -- 2k 2 f t~ (09)
W'(S/2)
~
dof
)
Bji (k') d~:l dk2 dk3,
0
k2 =
+
(35)
+
L where for simplicity we have excepted the isotropic condition E,~# (w)
=
EL(o))~ufl).
E. Gledzer/Physica D 104 (1997) 163-183
172
For small s, s << 1/o9o, where ~ is the frequency which corresponds to the maximum value of EL(w), the integral with E L in (35) is roughly equal to Is2BL, B L = f o EL(°f) dw' and hence, from (28) and (35) we can deduce
1(/
B~jS(klU) = -~ k2 = k2 +
B L 02
Bji(k)dk2dk3 q- -2-U- -2 0 k ~
f
kZBji(k) dk2 dk3
)
,
(36)
+
Eq. (36) is exactly the relation of Lumley [7] and Wyngaard and Clifford [ 12] (with B L estimated by mean square of advection velocity fluctuations (U'2)). We shall transform (36) to one-dimensional spectrum after considering the terms with acceleration in (25).
5. Acceleration term corrections Now, it is necessary to estimate the contribution of the terms I1, 12,/3 on the right-hand side of (25) which includes the acceleration. We consider these terms for fixed x and r. For the acceleration 3 Vi ~Of we except as some approximate value the acceleration of fluid particle in point x at time ~:
3V[(~lx, t)
.~
Of
1 3pl(x,~) - + vAui(x,f). p
(37)
OXi
A valid value should be taken at position X ( f [ x , t) of particle, but we do not know a dependence of X from x and consider that for small fluctuations of V' an averaged distance between X and x is small enough. Besides, in the conditions of homogeneous turbulence this acceleration statistically does not depend on the position of fluid particle. In any case, we may consider Eq. (37) as zero term into series on powers of X - x. The Taylor hypothesis which is restored in (23) by neglecting the terms with V ~may be considered as zero-order approximation for some disturbance theory. In particular, this means that at this order, Eq. (23) with V' = 0 is approximately valid for any field T~, P(x, f ) ~ 79(X -- U(f - t), t).
(38)
Therefore, from (37) and (38) the second term I1 of the right-hand side of (25) can be written as follows t+r
=- f
uj(x + r , t )
! t+r
=-f (uj(x+r't)lap'(x-u(f-t)'t)) Oxi t t+r
+ v f <.j(x + r, t)Au~(x, f)) dr,
(39)
t
where u'(x, t) = u(x, t) - U. Only in (39) the ensemble averaging on velocity field has been made (the rl-averaging like in Section 2.4 will be made later (Appendix B)).
E. Gledzer/Physica D 104 (1997) 163-183
173
The first term in (39) and all the other terms containing first power of pressure and velocity will vanish, because a scalar isotropic field cannot be correlated with a solenoidal isotropic vector field (see [8, p. 57]). By using the zero-order approximation (38) in the second term for u~ (x, ~), I
ui(x, ~) ~ ui(x + U(t - ~), t) _ --/ OVi'(~l Ix, s~) dsel '
(40)
we obtain for (39) with the help of (40), (37) and (38): t+r
t se
t+r
+v 2
f
d~ f
t
d~l(Uj(X
+r,t)A2ui(x + U ( t - ~ l ) , t ) ) .
(41)
t
Note that in (4 l) all correlations are taken at the same time t. The third term of the right-hand side of (25), I2, may be got from (41) by substitution x ~ x + r, x + r --~ x, r ~ rl, rl ~ r , i --+ j , j --~ i: t+r
12 = v f (u: (x, t)Auj (x + r + U(t - ~1 ), t))) ds~l t t+rl q-V 2 f
~1 d~l
f d~(u:(x,t)A2u~(x+r+U(t-~),t)).
t
(42)
t
By using the correlation function (25) Eqs. (41) and (42) may be written as
t+Zl
t+r
f A]3ji(r-FU(~--t))d~q-v f
./1 q-12 ---V
t
Al3ji(rq-U(t-~l))d~l
t
t+r
se
-}-192/ d~f t
d~lA2]3ji(r q - U ( ~ I - t))
t ~]
t+rl -{-v 2 f
d~l
t
f d~A2~ji(r t
+U(t-~)),
(43)
For the last term in (25) and (26) we can use Eq. (37) with x --> x + r, ~ ~ ~l and obtain t+rl
t+r
13=v2 f t
d~
f d~l(Axu~(x-U(~--t),t)Aru:.(x-~-r--U(~l--l),t)) t
t+r
+
t
t+rl
d, f t
d~l
( O ( p [ x - U ( ~ - l ) , l ] - - f i O ( p [ x q - r - U ( ~ l - t )dxj ,t]--fi))_d:q
"
E. Gledzer/Physica D 104 (1997) 163-183
174
Here the zero-order approximation (38) transfers all fields to the same time t. In terms of correlation functions, we have tq-r
tq-'r1
tq-r
td-rl
t
t
1
02Bpp(r - U(~! - ~)) Ori Orj
t
t
(44)
where 13pp(r) is the correlation function of pressure deviations (from some mean value ~). Further, we consider the case of longitudinal (L) mean convection velocity, U = (U, 0, 0). Also in (43) and (44) r = (Us, 0, 0) and i, j correspond to longitudinal correlation function. For the spectral form of Eq. (25) with (26), (35), (43) and (44) the following equation can be obtained (Appendix B):
FI(kl)=UB~d(klU)+ 1L 2
8
~
::1(
k2Fl(kl)+4
k 3k 2 -
kFl(k)dk
Fl(k) d k + ~
v2k2
- 8---~-
__ k 3k2 -
FI (kl)
kl8 dFl(kl)~ ~ ]
Fl(k) dk
kl/2 Fpp(kl) +
;Fpp
(45)
where the left-hand side FI (kl) is the valid one-dimensional spectrum, the first term of the right-hand side is the measured longitudinal spectrum, the operator L (according to (35)) is defined for any function G by L G ( k l ) - 4/-1 2Jr
f
e t(kl
ds
f
EL(a;') sin2(°fs/2) dofG(kl) dkl Of2
(46)
0
(see Appendix B). According to (36) we can take the approximation L = --(BL/U2)(O2/Ok~), therefore in the second term of the right-hand side of (45) we recognize the correction written by Lumley [7] (see also [12]): BE kl2 2U 2
dkl2
2fl(kl)
•
(47)
6. Some applications and discussion We consider, as an example, the application of Eq. (45) and (47) for measured one-dimensional spectrum
F°(k ) = U B~(kl U) = k-5/3 e -~k°
(48)
in which 1/is the Kolmogorov scale. The constant in (48) is omitted (Eq. (45) is linear). Following experimental data the constant/~ = 5.2 (see, for example [10]). We also neglect the pressure-term correction in (45) (for large wave numbers the spectrum of pressure decreases faster than that of the velocity (k-7/3)). The corrections for measured spectrum F°(k) on the right-hand side of (45) are both the Lumley type correction (the term with L) and the acceleration velocity correction. The first type of corrections describes the fluctuations of the distance from which the fluid particles come to the point of observation. The second correction describes the
E. Gledzer/PhysicaD 104 (1997) 163-183
175
fluctuations of the velocity from the starting point to the point of observation. We assume these two corrections to be small, so, on the right-hand side of (45) we shall use the approximation FI (k) = F°(k). With (48), Lumley's correction in (47) can be estimated as: 2U 2BL (~
+ ~ k ~ + (13k~)2)F°(k),
(49)
that corresponds to a "negative" correction when the slope (in log-log plot) increases at large k. We can also represent this correction in a form similar to (9) (for the log-log plot). For the derivative d In x ----In k the correction to the measured value d In F°(x)/dx is equal to BE ( d In F°(x) ~ d 2 In F°(x) 2U 2 \ 1 - 2 ~-x J dx 2
F(x)/dx,
B E d 3 In F°(x) 2U 2 dx 3
This gives the increasing slope in the region of negative second derivative of the function In F°(x). The correction (49) has to be clearly separated from the correction for velocity fluctuations of one-dimensional process (8). For measured spectrum B°(k) of the form (48) the correction on the right-hand side of (8) is equal to
(U'2) (2 ~_ 2 flkrl _ (flkyl)2)BO(k),
(50)
2U 2 and so, the opposite of Lumley's correction (49), is "positive", for the wave numbers k < 1/3fl7 with a decreasing slope. For k > 1/3/37 the slope increases (for fl "-~ 5, k > 1/157). This is the main difference between the corrections for data considered as one-dimensional process or as the longitudinal component of three- dimensional velocity field. For the three-dimensional case the correction described by the second term of the right-hand side of (45) like for (18) and (19) may be represented in more general form than (47). From (46) after integrating by s we have oo
LG(kl) -- - f EL(w') G(kl + (w'/U)) q- G(k109 ,2 - (w'/U)) -
2G(kl)
dw',
(51)
0 where G(kl) = k~ Fl (kl) + 4 fk7 kF1 (k) dk. This, of course, gives (47) for w r ---> 0 but, more generally, describes the Doppler-effect influence of the velocity fluctuations for the spectrum of frequencies. The correction containing the terms with v 2 in (45) is negative like Lumley's. The sense of this correction is close enough to the Lumley correction, because the value vZk 2 in (45) is some squared velocity. For kl ~ 1/0 it is the velocity of dissipation range, which is increasing with kl. So, this term is significant in the dissipative interval of wave numbers. From another side, this correction generalizes the one-frequency acceleration correction of the right-hand side of (18) and contains the spectrum of frequencies which are larger than ½kl. Note, that the leading term of correction in the case of kl > 1/0 comes from the term with the second integral in (45) and for the model spectrum (48) has the order o f ( v / ~ r l U ) 2 (flrlkl)exP[½flrlkl]F°(kl). This can be larger than the corresponding leading term of Lumley's correction (48), (BL/2u2)(flklT)2FO(kl), for wave numbers k which satisfy the inequality exp[l~ok]/~krl > BL(flrl)2/2V 2. So the threshold of wave numbers for which the viscous term correction can be significant is reduced by reducing the scale of Kolmogorov r/and energy ½B E of turbulence. As an example for the considered case below r / = 0.15 mm, v = 0.15 cm2/s this threshold corresponds to the value ~7if the Lagrangian velocity fluctuations W ~ (B E) 1/2 ~ 20 c m / s .
E. Gledzer/Physica D 104 (1997) 163-183
176
-1
-4
xx •
%
% % % i
Fig. 2. Deviationof the calculatedspectrafrom the measuredspectrumin near Kolmogorovscale range in log E - log(krl/2rr) coordinates, E is the energy spectrum: solid line, BL/u 2 = 0.01; short-dashed line, BL/u 2 = 0.1; long-dashed line, model spectrum (48).
Fig. 2 shows, in log-log plot, the spectrum with summarized Lumley's and viscous acceleration corrections calculated on the basis of Eqs. (45) and (47) (without pressure term). Values of parameters are equal to: B L / U 2 = 0.01 for solid line and 0.1 for short-dashed line (U = 10 m/s, 1/= 0.15 mm, v = 0.15 cm2/s). For comparison, the model spectrum (48) is presented (long-dashed line). Note that the summarized correction does not influence the spectrum in the inertial interval of wave numbers where Lumley's correction is equal to zero and viscous acceleration correction is negligible. The Lumley correction (47) and the correction for acceleration may be comparable for dissipative range. In Figs. 3(a) and (b) the model spectrum (48), the spectrum with the Lumley correction (47) and the spectrum with the total correction (Lumley's and for the viscous acceleration) are presented separately for two values of advection velocity U = 10m/s and U = 2 m / s with the same level of turbulence B L / u 2 -----0.01. Fig. 3 shows that the viscous acceleration term has small influence in the case of large velocity transfer. But when the advection velocity is reduced (laboratory conditions) the correction for viscous acceleration is of the same order as Lumley's. Generally speaking, this result can be expected, because it is known (see [8, p. 259]) that in the developed turbulence local acceleration Ou/Ot is usually much greater than substantive acceleration du/dt. This means that local and inertial (advective) accelerations are approximately balanced (from this it follows Eq. (4)). But Lumley's correction is obtained from the inertial acceleration term and, hence, could be larger (or much larger) than the corrections from the substantive acceleration of the Navier-Stokes equations. The influence of the viscous acceleration correction in (45) may be significant also for small wave numbers because of multiplier k~-2 in some terms of (45). For the model spectrum (48) if kl --~ 0 the leading term of correction for acceleration is (v/floU)2(~rlkl) -1/3 72 F ( ~ ) Fl°(kl), (ff is gamma function). Note that for these wave numbers the evaluation of this correction looses the precision. Here we have the same effect like the one-dimensional model of oscillation of acceleration (18). The singularity on kl appears as a result of additional r-averaging (see Appendix B) which is not valid for kl ~ 0. In Fig. 4 the compensated spectrum for (48) and for spectra obtained with the help of Eq. (45) are shown for parameters of Fig. 2 (coordinates y = kS/3Fl(k), x = logkr/). But we should note that in this region the correction for pressure term becomes more important (last term in (45)). This correction may be significant for the energy containing scales but contains the measured quantity other than the energy spectrum.
E. Gledzer/Physica D 104 (1997) 163-183
177
-4.25 ~'~._ -4.5 •
~,.,,.
_~, . . . . . . . . . . . . . ~[ 0.22 0.24
'~¢~.. . . . . . . .
~:
0.2~5"~,~,, 0.28
0.3
-5.25f
-5.s[
" ":"'<-..
a
a ~ ~
b
-1.5
-2.5
-0.2 -0.15 -0.I -0.05
0.05 ,,
0..1
Fig. 3. The model spectrum (48) (solid line), the spectrum with the Lumley correction (47) (short-dashed line) and the spectrum with the total correction (Lumley's and for the viscous acceleration) (long-dashed line) in near Kolmogorov scale range for (a) U = 10 m/s, (b) U -----2m/s; the coordinates are the same as in Fig. 2 and B L / u 2 = 0.01.
7. Concluding remarks In this paper the different forms of the Taylor hypothesis corrections were presented. In the beginning we discussed the corrections for one-dimensional processes. In this case the corrections are given for a spectrum and for correlation function. The case o f zero mean advection velocity was also considered. The simple one-dimensional model was presented which explains the sense of the correction for acceleration. When the value of acceleration is taken from the Navier-Stokes equations the evaluation of the correction becomes more complex, but the final equation for the corrections is simple enough and may be used for the estimations in the dissipative range of turbulence. Concerning structure or correlation functions we can obtain the corresponding corrections using formula BLL (r) = f-~oo eik~r F1 (k I) dk] and Eq. (45). But now the integrals on the right-hand side of this equation give the terms which cannot be expressed by the terms of function/~LL(r) o r its derivatives and integrals. So the corrections in this case should contain some other measured quantities. More simple is the case of the advection velocity fluctuations described by L u m l e y ' s correction (36), (47) or (51 ). In this case the corrections may be obtained for correlations and structure functions. Detailed investigations of these
178
E. Gledzer/Physica D 104 (1997) 163-183
-i0
-8
-6
-4
-2
0
Fig. 4, Compensated spectrum (48) (long-dashed line) and compensated spectra obtained by using Eq. (45): solid line, BL/U 2 = 0.01; short-dashed line, BL / u 2 = O.1.
corrections together with the effect of nonisotropic conditions for tensors up to fourth rank have been presented by Hill [13].
Acknowledgements The author wishes to thank Y. Gagne, B. Castaing, E.J. Hopfinger, A.M. Yaglom, A. Praskovsky and E. Villermaux for many discussions during the research. I am grateful to Dr. R. Hill for having communicated some of his results. This research was supported by DRET (contracts 93-1100 and 94-2591 A).
Appendix A If vector l(r) is the relative position of two particles with cordinates RI (r) and R2(r), then (32) gives {R21 (rl)R~ 1(rl)) = (/a(rl)//~(rl)} - rart~,
(A.1)
where r = / ( 0 ) , r = Us. Another quantity in (31) t+r
"/
t+r
V~(~lx, t)d~ + R~ (rl) t+rl
V~(~lx, t)d~
>
(A.2)
t+rl
can be estimated as s d 2 drl
- - - ( l a (rl)l# (r 1)).
(A.3)
E. Gledzer/Physica D 104 (1997) 163-183
179
For not very large time Z-l, the right-hand side of (A.1) is roughly equal (see [8, p. 543]) to rl2Dat~(r), where D~t~ is structure function of velocity field (Eulerian). So, if the time of averaging 7~ for ri in (32) is less than s ( B L / D L L (r)) i/2 the approximate value (34) of term (33) is greater than (A. 1) and (A.2).
Appendix B We shall use the following formula for isotropic correlation and spectral functions: oo
f
~LL(r) = BLL(r) =
eikirF1 (kl) dkl,
--(30 O0
k2/
kl
ABLL(r) = --2rr
e iklr dkl -oo
FNN(k)
1 --
k 3 dk,
kl
AeI3LL(r)=2Zr f eikirdkif FNN(k)( l-k~'~k'dk'k2] -oo
(B.I)
kI
•~LL(k) = FNN(k) {1 - kl2"~ \ k2 7 '
4zrk2FNN(k) = E(k),
oO
f
/3LL(k) dk2 dk3 ki
(B.2) kl
where FI (kl) is one-dimensional spectrum and FNN (k) is lateral three-dimensional spectrum, E(k) is the energy spectrum of isotropic velocity field. We multipy (43) and (44) by (1/2zr)e -ikt Us and integrate by s from - c ¢ to oo, using (B. 1). For terms of the order v in (43), we obtain
v
i
-co
e-ik~S~ds
ii
do
0
eiSh(~s~+u~)dk'l -oo
i(dkl 3
1 - ~2'~ FNN(k)
ki
+.ie-"'sd.i..ie"''-'".',i..' l--k27FNN(k),
E. Gledzer/Physica D 104 (1997) 163-183
180
which, if we except (1/2Jr)
2~v iklU 2
f e iyx dy
= 3(x), gives
eik,u~fdkk3 , -- k,Z] kt k27 FNN(k)
22zr1) r -- -iklU L
f
(
] a ~ 2 /
kll2
(B.3) In the same way, for the terms of the order oo
t+r
se--t
t
0
I) 2
in (43) we have
c~
v2fe-iklVsdsf d~ f dr/f e~'(vs+V"'d?,l~(?q) -c~
--~
t+r--s
~--t
+v2fe-iklV'ds f defdof ei~l(U'-U",°i'~°(?'l) -oo
t OG
0 OO
as
--oo
(e i~lUr +
e - i / ~ l U ( r - s ) -- 2
el(kl -kl)Us,
+
--OO
--O<)
~(~l)=f dkk'
1- k-f-~FNN(k). k2]
(B.4)
Now we can use the averaging on r because the terms of the order v 2 contribute to the last order of this disturbance theory. So, the oscillating terms proportional to e ikl ur vanish in (B.4) and the integration by s gives
(B.5)
Similarly, for the term of the order
V2
f
e -iklUs
-~
ds
o~
dr/1
dr/ 0
0
f
d/q~o(/q) ]
in (44), we obtain
e iil(Us+U(O-ol)) d~'l~p(k'l) -oo
oo
~__1)2 f
V2
e ifqUs
ds
--
e i/q U ( r - s ) -- e i/qUr + 1
-
ei(k-kj )t Us
(klU) 2
which after r-averaging gives 2Jr v2
(B.6)
E. Gledzer/Physica D 104 (1997) 163-183
181
Eqs. (B.5) and (B.6) are the full contribution of the terms of the order v2 to the averaged Fourier-transformed right-hand side of (25):
27r v 2 k5 k2U3 4 dk / L k~
1-
dkk 5 1 -
FNN(k)+2
(B.7)
FNN(k)
k~/2
The contribution of the term with pressure correlation may be found identical to (B.6) (by using the onedimensional pressure spectrum Fop(k1)). It has the form (sign minus in (44) is included)
1 (
U3p2
1
(~))
(B.8)
Fpp(kl) + ~Fpp
Now, using (B.1), (B.3), (B.7), (B.8) and (35) we obtain the equation that describes both Lumley's type and acceleration corrections to the measured spectrum BLn~(kl U):
-~
-~L
dkk
1 - k2 ]
kl
Jl-
= B~(k1U) + -iZklU2 LeiklVrk' dkk
1
Jt
kZ~k2] E ( k ) - e -iklur/2k,/2 dkk
1 - 4k2] E(k)
]
i)2 ~:2~ k3 ~2~ dk 1 - 4k2] E(k) k2U 3 2 dkk 3 1 - 2 k 2 / E ( k ) + [. k~ k~/2
U3fi2
(B.9)
Fpp(kl) + ~Fpp
where the operator L (according to (35)) is defined by OG
LG(kl)-
4U2rrf ei(kl-~l)Usdsf
EL(wt) sin2 °f 2
dJG(~:l) d/~l.
(B.10)
0 In Eqs. (B.7) and (B.9) the values of k'l are equal to the lower limits of the corresponding integrals. Eq. (B.9) gives E(k) for a given B~d(klU). The term of the order v is of oscillatory type and vanishes when averaged on r. But, this oscillating correction of the order v is also included in the function E(k) which is contained in both integrals in the term of the order v. So after r-averaging this term gives contribution of the order v2. To find the value of this contribution we start from the relation between ~0(kl) = dk k(1 - (k2/k2)) E(k) and one-dimensional spectrum FI (kl):
fff
(B.11)
k;
k2]
kl
E. Gledzer/Physica D 104 (1997) 163-183
182
So the term of the order v (B.9) has the form P
i2klU2 (eiklUr ~o(kl) -- e-iklUr/2~o ( ~ ) )
(B.12)
.
This gives the ~ v correction of FI (kl) (if zero-order term is equal to F°l(ki) = U B ~ ( k ! U))
v eiklUr ~FI (kl) -----i2klU
/
2k2FO(kl ) + 8
J ) dkkF°(k)
kl
v e_ikiUr/2 i2ki U
[()
2 kl 2FO 2
-+-8
i
dkkFO(k)
(B.13)
kl/2
Now we can use (B. 13) in (B. 11) and (B. 12) to obtain the corrections of the order v2. Note that only one contribution remains after r-averaging: from the second term in (B. 12) when for ~0(½kl ) to use the first-term correction in (B. 13). So we obtain
v i2klU 2
--e-ik'Ur/22(kl'] 2 v---Leikiur/2 \ 2 ] iklU
2
2F°
dkkFp(k)
-t-8 kl/2
3v2 = 4U
[ (_~)2 (~_~.l)f 2 --
F°
+8
dkkF°(k)
]
.
(B.14)
kj/2
This term is of the same order as that of the terms of the order v2 in (B.9). Expressing them with the help of El (kl), oo
E(k) = 2k4Fl(kl) + kl
k2 7
8f
dkk(3k 2 - k2)Fl(k),
kj oo
f k d k E ( k ) = 4k2Fl(kl) - k~ dF1 (kl)+~f~F,~ dk-------~ kl
(B.15)
kl
we obtain from (B. 15), (B. 14) and (B.9) for terms of the order v 2
"2{ t12
4U-----~ 2 - -
FI
+8
i
kl/2
dkkFl(k)
] 21 -- -k~U3 -
2~(kl) + ~p
+k~
k~
v2k 2 31 (8Fl (ki) - ki F~ (ki))
~ L~Z167k F,,~,~+~f~,,~~3k 2 -
i
F1 (k) dk
kdkE(k)
1
..]
E. Gledzer/PhysicaD 104(1997)163-183
183
So, Eq. (B.9) can be expressed in the following form:
Fl (kl) : UB~(kl U) + 1L2 kZFl(kl)+4
dkkFl(k) -8--~-
Fj(kl)
kl d F l ( k l ) ]
kl - 8 -v2k~ --~-
,(
U2p2
f 'k ~14k
3k2 -
, Fpp(kl) + ~Fpp
F(k)dk + -~1 f k 3k2kl/2
Fl(k)
dk (B.16)
References [1] R.A. Antonia and D. Britz, A note on the spectrum of the time structure function, University of Newcastle, T.N.-FM 83/2 (1983). [2] R.A. Antonia, N. Phan-Yhien and A.J. Chambers, Taylor's hypothesis and the probability density functions of temporal velocity and temperature derivatives in a turbulent flow, J. Fluid Mech. 100 (1980) 193. [3] F.H. Champagne, The fine-scale structure of the turbulent velocity field, J. Fluid Mech. 86 (1978) 67. [4] M.J. Fisher and P.O.A.L. Davies, Correlation measurements in a non-frozen pattern of turbulence, J. Fluid Mech. 18 (1964) 97. [5] G. Heskestad, A generalized Taylor hypothesis application for high Reynolds number turbulent shear flows, J. Appl. Mech. 87 (1965) 735. [6] C.C. Lin, On Taylor's hypothesis in wind tunnel turbulence, Quart. Appl. Math. 10 (1953) 295. [7] J.L. Lumley, Interpretation of time spectra measured in high-intensity shear flows, Phys. Fluids 8 (1965) 1056. [8] A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics, Vol. 2 (MIT Press, Cambridge, MA, 1975). [9] J.-E Pinton and R. Labb~, Correction to the Taylor hypothesis in swirling flows, J. Phys. II 4 (1994) 1461. [10] S.G. Saddoughi and S.V. Veeravalli, Local isotropy in turbulent boundary layers at high Reynolds number, J. Fluid Mech. 268 (1994) 333. [11] H. Tennekes, Eulerian and Lagrangian time microscales in isotropic turbulence, J. Fluid Mech. 67 (1975) 561. [ 12] J. Wyngaard and S.E Clifford, Taylor's hypothesis and high-frequency turbulence spectra, J. Atomospheric Sci. 34 (1977) 922. 1131 R.J. Hill, Corrections to Taylor's frozen turbulence approximation, Atmos. Res. 40 (1996) 153.
Physica D 104 (1997) 163-183
On the Taylor hypothesis corrections for measured energy spectra of turbulence E. Gledzer* Institute of Atmospheric Physics, 109017, Moscow, Russian Federation
Received 13 July 1995; revised 23 July 1996; accepted 25 November 1996 Communicated by U. Frisch
Abstract The acceleration terms of the Navier-Stokes equations are taken into account to obtain the corrections for the measured longitudinal spectrum of homogeneous isotropic turbulence. The viscous term correction gives approximately uniform relative deviation of the sign such as the Lumely correction for intertial subrange with no influence on the slope of the spectrum but can be significantly increased for scales less than the Kolmogorov one. The deviations for small wave numbers are also estimated. In addition, the different forms of the corrections in the case of advection velocity fluctuations are discussed and some simple one-dimensional models are presented. Keywords: Turbulence; Spectrum; Advection; Acceleration
1. Introduction The Taylor hypothesis or frozen turbulence hypothesis was used in a great number of experimental works. But only few papers examined the theoretical backgrounds of this hypothesis which enable us to express the spatial statistical characteristics corresponding to a fixed time in terms of the temporal series measured at one point. Maybe, one of the most important investigation of Taylor's approximation was made by Lumley [7] and by Heskestad [5]. Other early investigations devoted to this problem are also discussed in [4,6]. The main mechanism which prevents a direct use of this hypothesis is, according to Lumley, the fluctuations of the advection velocity. The appropriate correction obtained by Lumley gives the deviations between the measured and the valid one-dimensional spectra of the streamwise velocity fluctuations (see [2,3,12]). Taylor's approximation is commonly written as follows: O/Ot = - U I O / O x l ,
* Address for correspondence: LEGI/IMG, B.E 53, 38041 Grenoble Cedex 9, France. 0167-2789/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PH SO 167-2789(96)00300-4
(1)
164
E. Gledzer/Physica D 104 (1997) 163-183
where U1 is the mean velocity in the Xl direction and t is a time. The corresponding wave number form is kl U1 = w,
(2)
where kl is the xl-component of the wave number vector and w is a frequency. For velocity field equation (1) is written as follows:
Oui/Ot + UjOui/Oxj = 0,
(3)
where Uj is the mean velocity of the flow in the point x and ui is the advected velocity field. Another relation, namely (4)
OUi/Ot d- UjOUi/OXj = O,
where ui = Ui + u I is both the fluctuating advection velocity and the instanteneous velocity of flow, was used by Heskestad [5] for the investigation of the advection velocity fluctuations influence and by Tennekes I1 l] for the estimation of the Eulerian time microscale in isotropic turbulence. Eq. (4) is the Navier-Stokes equation without the accelartion terms. So the investigations of the corrections to the Taylor hypothesis usually disregard the influence of these terms. The main aim of the present study is to take into account the influence of the disregarded terms. We show that the viscous term correction does not give any significant contribution for measured spectra in inertial range of developed turbulence (where the Lumley correction also has no influence on the slope of an energy spectrum curve in log-log plot) and is increased in the dissipative range. The corresponding threshold of the wave numbers is presented for some model spectrum of the dissipative range. The correction for the pressure term acceleration is also obtained. In Section 2 we consider simple one-dimensional models of the corrections, in particular, some model of correction for acceleration. In Section 3 the corresponding equations for three-dimensional case are presented which give the different types of corrections. The spectral forms of these corrections are considered in Sections 4 and 5. The main result of this paper is Eq. (45) for valid one-dimensional spectrum of the velocity field when the spectrum measured in one point is known. In Section 6 we discuss the deviations from measured spectra for inertial and dissipative ranges.
2. One-dimensional models Here, we recall the main results which give the energy spectrum of one-dimensional processes on the basis of the Taylor hypothesis and the Lumley type correction. In particular, the case is considered when the mean advection velocity is absent. We present also some model of correction for acceleration.
2.1. Taylor's hypothesis Following Eq. (1) velocities u at point x ----0 at times t + r and t + rl are equal to the velocities at time t but at the points - U r and - U r l upstream: u ( - U r , t) = u(0, t + r),
u ( - U r l , t) = u(0, t q- t'l).
Multiplying left- and right-hand sides of these relations we obtain that the space correlation function B(Us) = ( u ( - U r , t)u(--Url, t)), s = r -- rl is equal to the measured time correlation function BinS(s) = (u(0, t + rl + s)u(O, t + rl)). Here the brackets (---) mean the ensemble averaging. The corresponding space spectrum B(k),
E. Gledzer/Physica D 104 (1997) 163-183
165
O0
B(r) = (u(x, t)u(x + r, t)) = [ e-ikrB(k)dk,
r = Us.
(5)
can be expressed in terms of the measured time spectrum B ms BinS(s) = (u(0, t + s + rl)U(0, t + rl)) = [
e-ikUsBms(kU)U dk.
(6)
--OQ Eq. (5) and (6) give
B(k) = UBmS(kU).
(7)
This is equivalent to (2).
2.2. Lumley's approximation When advection velocity U has random fluctuating component U' with (U') = 0, the corresponding distance r = (U + U')s in (5) is a fluctuating value. The measured spectrum includes averaging over U'. So, inverse Fourier-transformation of (6) gives
O@ BmS(kU) = ~1 f
oO
eikUs ds f
--00
e-~k . . .Us . . (e-~k u S)u,B(k, ) dk'
--00
with the averaging on U'. This formula should be considered as a result of two types of averaging. One type is the ensemble averaging for fixed U' which gives spectrum B(k). Another type is U'-averaging. With the approximation
(e-ik'U'S)u, ~ 1 -- l k'2s2(U'2), the previous relation leads to Lumely's type correction
B(k) = UBmS(kU)
(U '2) d2(k2B(k)) 2U 2
(8)
dk 2
(see also [ 12]). Here we suppose that the advection velocity fluctuations are small enough compared with the mean advection velocity U, U~2/U 2 << I. The sign of the correction to the measured value UB ms(kU) for spectrum B(k) ~ k -5/3 is positive for one-dimensional processes. Note, that the above equations also may be interpreted as following expression:
k
UBms(kU)=(B(I+~J'/U)
1+
u'/u)v, 1 '
which shows that the measured spectrum is the weighted averaged value of valid spectrum. To investigate the correction to the slope of the function X (x) = In B(k), x = In k (log-log plot) it is convenient to represent (8) for small u ' Z / u 2 in the form
dx(x ) d~
dxmS(x) --
dx
Ut2 [( 2U 2
where XmS(s) = ln(UBms(uk)).
dxmS(x)'~ d2xms(x) 3 +2
~
,]
dx 2
d3xmS(x)l +
~
j,
(9)
E. Gledzer/Physica D 104 (1997) 163-183
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It is clear that this approximation gives the correction to the slope of the function X (x) only in the regions where X (x) has nonzero curvature (for small third derivative). For usual one-dimensional spectra the second derivative is negative (in log-log plot), so the sign of the correction to the slope depends on the value of the measured slope d x ms (x)/dx. If this derivative is less then - 3 (for example, - 5) then the correction to the measured slope is negative (the slope is larger then for the measured spectrum). We can also present within considered approximation the correction to the correlation function of one-dimensional process. Using above formulas we obtain BmS(s) = (u(0, t)u(O, t + s)) = (u(O, t ) u [ - ( U -t- U')s, t]) = (B(Us + U's))u', where brackets (...) mean both the ensemble averaging and U'-averging while (.. ")u' is Ur-averaging. For small U ' / U this equation gives (U'2) _2 OZB(r ) BmS(s)=B(r)+-~-~-r ~r 2 ,
r = Us.
The sign of the correction to measured value B ms also depends on the curvature of B(r) and gives the increasing value for B(r) ~ r 2/3 without changing the slope in log-log plot.
2.3. No mean advection velocity In the considered cases the advection velocity was taken constant (random or nonrandom). When this velocity is time-dependent the correlation function can be represented in the following form: T
I fu(--X,
t)u(0, t) dt =(u(O, t +
17)U(0,
t)) = Bms(r),
0 t+r
/ I
X ------X ( r l t ) = [
V(t')dt',
(10)
i /
t
where we have assumed for simplicity rl = 0 compared with above cases. Here the time-averging was used over a period T which is much greater than the correlection time. The form of advection velocity V (t t) may depend on concrete experiment. For example, Pinton and Labb6 [9] in the experiments with KS.rrn~in swirling flows used
,/
t+T/2
V(t) = ~
u(O, t') dt',
t-T/2 where T is a period of disc rotation. We shall consider the case of zero mean advection velocity, (V(t)) = 0, when the above theory becomes inapplicable. Let us divide all period T of averaging in (10) in two parts. One part contains all time-intervals {t}+ when the displacement X = X (r It) is positive, X (r It) > 0. Another part consists on the intervals {t }_ with negative X, X(rlt) < 0. Positive values of displacement X(rlt) may be interpreted as produced by some positive mean advection velocity U+ > 0 and by fluctuations U~_ : X(rlt) = (U+ + U~_)r, so that U+ + U~_ > 0. The same for negative X: X(rlt) = - ( U _ + Ut_)r, where U_ > 0, U_ + U t_ > 0. The values U+ and U_ are of the order (V2(t)) 1/2, Now for every set of intervals {t}+ and {t}_ we can use the approximations of Sections 2.1 and 2.2 where, of course, the fluctuations U~, U'_ are not small, but at least, IU~ I < U+, IU'_ I < U_.
E. Gledzer/Physica D 104 (1997) 163-183
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This shows that for the case of zero advection velocity the Lumely approximation may be used with the advection of the order of the root-mean-square velocity value.
2.4. Correctionsfor periodic oscillations of acceleration In above cases the velocity field was considered as some conservative quantity of Lagrangian fluid particle. Generally, it is not valid, and the particle changes its velocity. Let us consider the case when Lagrangian particle is subjected by the advection with the constant velocity U together with the external time-periodic oscillations of acceleration, a cos(Ot) with a frequency o9. Therefore, the Lagrangian velocity and the position of fluid particle (which at time t was at point x) at time t + r are equal sin(Or)
V(t + rlx, t) = U + v'(t + rlx, t) + a - - ,
O
t+r
X =- X(t +
1 - cos(Or) f i x , t) = x + U r + a O2
+ f o'(~lx, t)d~,
(11)
t
where v' is the Lagrangian velocity fluctuations. For the Eulerian point of view, we have for velocity fluctuations near mean value U
t+z
f Ov'(~lx, t) d~. u[X, t + r] = u(x, t) + a--------~- + J O~ sin(Or)
t
Neglecting fluctuations v' compared with a-terms, we obtain sin(Or) u ( x + U r + a 1 - c o s (2O r ) o , t + r ) =u(x,t)+a-----~-~
(12)
Using the same relation for another starting point x + r and time delay rl we obtain for correlation function (5)
B(r) = (u(x, t)u(x + r, t)) =/3mS(s) - a 2 sin(Or) sin(Orl) .('22
r -~ Us - a
cos(Or) - cos(Orl)
~(-22
s = r - rl,
(13)
where/3mS(s) is the measured time correlation function (u(0, t + rl)u(0, t + r)). Fourier-transformation of Eq. (13) with the spectrum B(k) defined by (5) gives for the right-hand side of (13) the measured spectrum and the correction proportional to a 2. BmS(kU)
4 0a2 2U
( 6 ( k - ~ - ) + 3 ( k - ~+- )
- e-2iOrl' (k _ ~---)"_ e2it2rl • (k + ~-~)) •
(14)
In (14) we have the term describing the constant deviation from the measured spectrum, proporitional to 8(k (O/U)) + 3(k + (O/U)), and the terms which are oscillating (proportional to e -2is2rj , e 2i°rl ). The amplitude of the oscillations of acceleration may be considered as a random value with zero mean value, so we can make the a-averaging in (14). For the Fourier-transformation of the left-hand side of (13) we obtain
dk' --OO
~ ei(k-k')Us 2~---~--dsS(kl' s, rl)B(k/), --OO
E. Gledzer/PhysicaD 104(1997)163-183
168 where
S(U, s, rl)
= e x p ( i k ' a ( c o s ( 1 2 r ) - c o s ( ~ 2 r l ) / I 2 2 ) . In the case of small amplitude of oscillations,
Ikla/S221 <<
1, it follows
S(k', s, rl)
=
1 - --~[l
- c o s ( X 2 s ) + cos(2K2rl)Dl(s) - s i n ( 2 ~ r l ) D 2 ( s ) l
Dl(S) = l[(cos(~2s) - l) 2 - sin2(~2s)l,
,
D2(s) = [ c o s ( ~ s ) - 11 sinff2s).
(15)
The relations (14) and (15) contain the oscillating terms which are proportional to e -2i~rj and e 2i~rt , respectively. The appearence of these oscillating terms has the same reason which exists for a spectrum of the velocity differences field 8r v (t) = v (t + r) - v (t) which is considered for fixed delay r [ 11. The Fourier-transformation of the correlation function of 8~ v(t) is
/ dseiWS(SrV(t)Srv(t+ s))= [ dsei~°s(2B(s)- B(s + r ) - B(s-
F~(~o)
tl --00
r)),
(16)
--~
where B(s) is the correlation function of velocity, B(s) = (v(t)v(t + s)). Therefore, in terms of the velocity spectrum ~b(w), we obtain instead of (16) OG
Fr(w) = 24~(w)(1 - cos(wr)),
~O(w) = /
ei~°SB(s) ds.
(17)
--OQ
Expression (17) contains like in (15) and (14) the oscillations near some mean value 2~b(w) which depend on the delay quantity r. Hence, it is necessary to make the r-averaging in order to find the velocity spectrum 4~(09) on the basis of the velocity increments spectrum Fr (09). But we cannot obtain the spectrum Fr (w) for w ~ 0 after making this r-averaging, because in this case we have Fr (0) = 2q~ (0) instead of the exact result Fr (0) = 0. The same procedure should be adopted for Eq. (15) and (14). In this case, we get
dk' jei(k-k')Us(U2(a2)[1--cos(~s)])B(k')2~ ds 1 ~f_24 --0~
--00
=BmS(kU,
4~22 U(a2)
(,(k-~)-k-~(kq--~)).
(18,
This relation permits us to calculate B(k) for a given measured spectrum BmS(kU). The left-hand side may be transformed in the form closed to that of Lumely's with the help of the following approximation, 1 - cos(~2s) 1.Q2s2: 1
-~B(k) +
(a 2) 2122U-----~
02(k2B(k)) Ok2
Without any approximation the calculation of the left-hand side of (18) leads to
-uB(k)
(a2) {kZB(k)_ 1
k+-~-) 2 B ( k + ~ ) + ( k - - ~ ) Z B ( k - ~ - ) ] } .
(19)
For 12/U << k this gives the previous result. The right-hand side of (18) shows that the acceleration terms give another type of corrections for measured spectrum B ms (k U). In particular, for a given one-frequency oscillation the measured amplitude with wave number k = 12/U should be reduced. Also, if there is no turbulence, B(k) = 0, then
E. Gledzer/Physica D 104 (1997) 163-183
169
the measured spectrum B ms consists only with one harmonic, k : ~ / U . The term B(k + (I2/U)) in (19), which corresponds to Lumley's type correction, describes the Doppler-effect of the fluctuations of advection velocity for a given wave number. According to (18) for fixed (a 2) the reduction of frequency S2 increases the values of the corrections because the amplitudes of the velocity and the displacement fluctuations (1 1) are increased at J2 ~ 0. But note that Eq. (18) cannot be valid for $-2 ~ 0. This is the case of a velocity with a constant trend, U + a t , for which the correction must be finite. Singularity on I2 appears after rl-averaging in (14) and (15) which is made over some period T > zr/12. We shall see that this is the general case of corrections for acceleration.
3. Lagrangian particles dynamics Now we shall take into account the acceleration terms of the hydrodynamical equations. Let us consider threedimensional Lagrangian dynamics of fluid particles. Denoting V(t + r lx, t) and X(t + f i x , t) the velocity and the coordinate at time t + r of a liquid particle, which at time t was at point x, obviously, we have the equations: t+r
V(t+rlx, t)=V(tlx, t)+ f
OV(flx, Of t) d~,
t
t+r
X(t + f i x , t) = x + f
(20)
V(~lx, t)d~.
t
The Eulerian and Lagrangian dynamics are linked by relation
V(t + rlx, t ) = u ( X , t + r),
u ( x , t ) = V(tlx, t).
(21)
From (20) and (21) we get for the Eulerian velocity at point x:
u(x,t)=u
+
V(flx, t)df, t + r
-
t
Of t
Now let us suppose that the velocity of each liquid particle can be expanded with the uniform advection velocity and the fluctuating parts, namely
V(flx, t ) : U + V ' ( f l x ,
t),
U : const.
Then, (22) can be written as
u(x,t)=u
V'(flx, t ) d ~ , t + r
+Ur+
-
Of
t
It the measurements are carried out a point x = 0 at time t + r, then the velocity at point x at time t may be obtained from (23) with the condition t+r
x + ur + f v'(flx, t
t ) d f = 0.
(24)
E. Gledzer/Physica D 104 (1997) 163-183
170
0
On +
+ r. t)
Fig. 1. Lagrangian particles dynamics.
Now, let us consider velocites at two points x and x + r at the same time t. From (23) and (24), we have the relations with the velocities at point x ----0 at two times t + r and t + rl (Fig. 1). For correlation function we obtain now
Bij(r) = (ui(x, t)uj(x + r, t)) = B~jS(s) - I1 - 12 - 13
(25)
in which
B~jS(s) = (ui(O, t + z)uj(0,
t + l~1)),
S =Z'--X'I,
\]t/r OVi,(~lx, t) I1 = { uj(x+r,t) d~ ,
12 =
u i (x, t) (t/r
13 =
° Ox+rt' I O~ d~ ,
~r d~
d~l
OVi,(~lx, t)OVj(~llx +r,t) ) ~1
(26)
t and r satisfies the equations t+r
r = U s + p ( r , rllt),
p(r, rllt) =
t+r
f W(~lx, t)d~- f V'(~lx+r,t)d~. t
(27)
t
4. The Lumley type corrections By multiplying the left-hand side and the first term of the right-hand side of (25) by e -ik~ us, and integrating, we obtain for the first term of the right-hand side
a~jS(kl e ) =
1 / e_iklUsB~S(s ) ds,
(28)
E. Gledzer/PhysicaD 104 (1997) 163-183
171
which is the measured spectrum. Here we have made the ensemble averaging only. The rl-averaging, like that in Section 2.4 (transition from (14) and (15) to (18)), will be fulfilled later. For the left-hand side (using (27)), one obtains
e-'k~USds eiktgs(eik'P(r'~11t))Bji(k')d[¢ldk2 dk3,
= (lq, k2, k3),
(29)
nji (k t) is the three-dimensional spectrum of velocity field
where
]3ji (r) = f eikrBji (k) dk.
(30)
Now we assume that the Lagrangian fluctuation field (see also [12]) is valid. Therefore, we have (eikp(r'r'lt))
1 --
lkak ~
/(;
U' is small enough, and the approximation of Lumley [7]
V~(~lx, t) d~ - R ~ 21 (rl) '
x,t+rl
)(Z
Vtfl(~lx, t) d ~ - R ~ l(rl)
,
(31)
t
where we have introduced the random fluctuating dispersions of fluid particles t+r
t+r
Rl(rl) = f v'(elx, l)d~,
R2(rl)= f v'(,Ix +r,t)d,,
t
t
21
Ra (rl) = R2a(rl) - Rlc~(rl).
(32)
In (31) the Lagrangian term, namely t+r
f
t+r
d~l
tq-r I
f d~2(V~(~llx,t)V;(~21x,t))
(33)
t+r I
is equal, for stationary homogeneous turbulence field (see [8, p. 528]), to O<)
4 f ELt~(o9)
sin 2 0 ) ( S / 2 ) 092 dog, s = r - r l ,
(34)
0
where EL~ (09) is the Lagrangian spectra of the velocity field. In Appendix A, it is shown that the term (33) gives the main contribution in Eq. (31). Therefore, (29) leads to oo ~ L -
t, sin2
1 f e_i(k,_~l)u s ds (1 -- 2k 2 f t~ (09)
W'(S/2)
~
dof
)
Bji (k') d~:l dk2 dk3,
0
k2 =
+
(35)
+
L where for simplicity we have excepted the isotropic condition E,~# (w)
=
EL(o))~ufl).
E. Gledzer/Physica D 104 (1997) 163-183
172
For small s, s << 1/o9o, where ~ is the frequency which corresponds to the maximum value of EL(w), the integral with E L in (35) is roughly equal to Is2BL, B L = f o EL(°f) dw' and hence, from (28) and (35) we can deduce
1(/
B~jS(klU) = -~ k2 = k2 +
B L 02
Bji(k)dk2dk3 q- -2-U- -2 0 k ~
f
kZBji(k) dk2 dk3
)
,
(36)
+
Eq. (36) is exactly the relation of Lumley [7] and Wyngaard and Clifford [ 12] (with B L estimated by mean square of advection velocity fluctuations (U'2)). We shall transform (36) to one-dimensional spectrum after considering the terms with acceleration in (25).
5. Acceleration term corrections Now, it is necessary to estimate the contribution of the terms I1, 12,/3 on the right-hand side of (25) which includes the acceleration. We consider these terms for fixed x and r. For the acceleration 3 Vi ~Of we except as some approximate value the acceleration of fluid particle in point x at time ~:
3V[(~lx, t)
.~
Of
1 3pl(x,~) - + vAui(x,f). p
(37)
OXi
A valid value should be taken at position X ( f [ x , t) of particle, but we do not know a dependence of X from x and consider that for small fluctuations of V' an averaged distance between X and x is small enough. Besides, in the conditions of homogeneous turbulence this acceleration statistically does not depend on the position of fluid particle. In any case, we may consider Eq. (37) as zero term into series on powers of X - x. The Taylor hypothesis which is restored in (23) by neglecting the terms with V ~may be considered as zero-order approximation for some disturbance theory. In particular, this means that at this order, Eq. (23) with V' = 0 is approximately valid for any field T~, P(x, f ) ~ 79(X -- U(f - t), t).
(38)
Therefore, from (37) and (38) the second term I1 of the right-hand side of (25) can be written as follows t+r
=- f
uj(x + r , t )
! t+r
=-f (uj(x+r't)lap'(x-u(f-t)'t)) Oxi t t+r
+ v f <.j(x + r, t)Au~(x, f)) dr,
(39)
t
where u'(x, t) = u(x, t) - U. Only in (39) the ensemble averaging on velocity field has been made (the rl-averaging like in Section 2.4 will be made later (Appendix B)).
E. Gledzer/Physica D 104 (1997) 163-183
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The first term in (39) and all the other terms containing first power of pressure and velocity will vanish, because a scalar isotropic field cannot be correlated with a solenoidal isotropic vector field (see [8, p. 57]). By using the zero-order approximation (38) in the second term for u~ (x, ~), I
ui(x, ~) ~ ui(x + U(t - ~), t) _ --/ OVi'(~l Ix, s~) dsel '
(40)
we obtain for (39) with the help of (40), (37) and (38): t+r
t se
t+r
+v 2
f
d~ f
t
d~l(Uj(X
+r,t)A2ui(x + U ( t - ~ l ) , t ) ) .
(41)
t
Note that in (4 l) all correlations are taken at the same time t. The third term of the right-hand side of (25), I2, may be got from (41) by substitution x ~ x + r, x + r --~ x, r ~ rl, rl ~ r , i --+ j , j --~ i: t+r
12 = v f (u: (x, t)Auj (x + r + U(t - ~1 ), t))) ds~l t t+rl q-V 2 f
~1 d~l
f d~(u:(x,t)A2u~(x+r+U(t-~),t)).
t
(42)
t
By using the correlation function (25) Eqs. (41) and (42) may be written as
t+Zl
t+r
f A]3ji(r-FU(~--t))d~q-v f
./1 q-12 ---V
t
Al3ji(rq-U(t-~l))d~l
t
t+r
se
-}-192/ d~f t
d~lA2]3ji(r q - U ( ~ I - t))
t ~]
t+rl -{-v 2 f
d~l
t
f d~A2~ji(r t
+U(t-~)),
(43)
For the last term in (25) and (26) we can use Eq. (37) with x --> x + r, ~ ~ ~l and obtain t+rl
t+r
13=v2 f t
d~
f d~l(Axu~(x-U(~--t),t)Aru:.(x-~-r--U(~l--l),t)) t
t+r
+
t
t+rl
d, f t
d~l
( O ( p [ x - U ( ~ - l ) , l ] - - f i O ( p [ x q - r - U ( ~ l - t )dxj ,t]--fi))_d:q
"
E. Gledzer/Physica D 104 (1997) 163-183
174
Here the zero-order approximation (38) transfers all fields to the same time t. In terms of correlation functions, we have tq-r
tq-'r1
tq-r
td-rl
t
t
1
02Bpp(r - U(~! - ~)) Ori Orj
t
t
(44)
where 13pp(r) is the correlation function of pressure deviations (from some mean value ~). Further, we consider the case of longitudinal (L) mean convection velocity, U = (U, 0, 0). Also in (43) and (44) r = (Us, 0, 0) and i, j correspond to longitudinal correlation function. For the spectral form of Eq. (25) with (26), (35), (43) and (44) the following equation can be obtained (Appendix B):
FI(kl)=UB~d(klU)+ 1L 2
8
~
::1(
k2Fl(kl)+4
k 3k 2 -
kFl(k)dk
Fl(k) d k + ~
v2k2
- 8---~-
__ k 3k2 -
FI (kl)
kl8 dFl(kl)~ ~ ]
Fl(k) dk
kl/2 Fpp(kl) +
;Fpp
(45)
where the left-hand side FI (kl) is the valid one-dimensional spectrum, the first term of the right-hand side is the measured longitudinal spectrum, the operator L (according to (35)) is defined for any function G by L G ( k l ) - 4/-1 2Jr
f
e t(kl
ds
f
EL(a;') sin2(°fs/2) dofG(kl) dkl Of2
(46)
0
(see Appendix B). According to (36) we can take the approximation L = --(BL/U2)(O2/Ok~), therefore in the second term of the right-hand side of (45) we recognize the correction written by Lumley [7] (see also [12]): BE kl2 2U 2
dkl2
2fl(kl)
•
(47)
6. Some applications and discussion We consider, as an example, the application of Eq. (45) and (47) for measured one-dimensional spectrum
F°(k ) = U B~(kl U) = k-5/3 e -~k°
(48)
in which 1/is the Kolmogorov scale. The constant in (48) is omitted (Eq. (45) is linear). Following experimental data the constant/~ = 5.2 (see, for example [10]). We also neglect the pressure-term correction in (45) (for large wave numbers the spectrum of pressure decreases faster than that of the velocity (k-7/3)). The corrections for measured spectrum F°(k) on the right-hand side of (45) are both the Lumley type correction (the term with L) and the acceleration velocity correction. The first type of corrections describes the fluctuations of the distance from which the fluid particles come to the point of observation. The second correction describes the
E. Gledzer/PhysicaD 104 (1997) 163-183
175
fluctuations of the velocity from the starting point to the point of observation. We assume these two corrections to be small, so, on the right-hand side of (45) we shall use the approximation FI (k) = F°(k). With (48), Lumley's correction in (47) can be estimated as: 2U 2BL (~
+ ~ k ~ + (13k~)2)F°(k),
(49)
that corresponds to a "negative" correction when the slope (in log-log plot) increases at large k. We can also represent this correction in a form similar to (9) (for the log-log plot). For the derivative d In x ----In k the correction to the measured value d In F°(x)/dx is equal to BE ( d In F°(x) ~ d 2 In F°(x) 2U 2 \ 1 - 2 ~-x J dx 2
F(x)/dx,
B E d 3 In F°(x) 2U 2 dx 3
This gives the increasing slope in the region of negative second derivative of the function In F°(x). The correction (49) has to be clearly separated from the correction for velocity fluctuations of one-dimensional process (8). For measured spectrum B°(k) of the form (48) the correction on the right-hand side of (8) is equal to
(U'2) (2 ~_ 2 flkrl _ (flkyl)2)BO(k),
(50)
2U 2 and so, the opposite of Lumley's correction (49), is "positive", for the wave numbers k < 1/3fl7 with a decreasing slope. For k > 1/3/37 the slope increases (for fl "-~ 5, k > 1/157). This is the main difference between the corrections for data considered as one-dimensional process or as the longitudinal component of three- dimensional velocity field. For the three-dimensional case the correction described by the second term of the right-hand side of (45) like for (18) and (19) may be represented in more general form than (47). From (46) after integrating by s we have oo
LG(kl) -- - f EL(w') G(kl + (w'/U)) q- G(k109 ,2 - (w'/U)) -
2G(kl)
dw',
(51)
0 where G(kl) = k~ Fl (kl) + 4 fk7 kF1 (k) dk. This, of course, gives (47) for w r ---> 0 but, more generally, describes the Doppler-effect influence of the velocity fluctuations for the spectrum of frequencies. The correction containing the terms with v 2 in (45) is negative like Lumley's. The sense of this correction is close enough to the Lumley correction, because the value vZk 2 in (45) is some squared velocity. For kl ~ 1/0 it is the velocity of dissipation range, which is increasing with kl. So, this term is significant in the dissipative interval of wave numbers. From another side, this correction generalizes the one-frequency acceleration correction of the right-hand side of (18) and contains the spectrum of frequencies which are larger than ½kl. Note, that the leading term of correction in the case of kl > 1/0 comes from the term with the second integral in (45) and for the model spectrum (48) has the order o f ( v / ~ r l U ) 2 (flrlkl)exP[½flrlkl]F°(kl). This can be larger than the corresponding leading term of Lumley's correction (48), (BL/2u2)(flklT)2FO(kl), for wave numbers k which satisfy the inequality exp[l~ok]/~krl > BL(flrl)2/2V 2. So the threshold of wave numbers for which the viscous term correction can be significant is reduced by reducing the scale of Kolmogorov r/and energy ½B E of turbulence. As an example for the considered case below r / = 0.15 mm, v = 0.15 cm2/s this threshold corresponds to the value ~7if the Lagrangian velocity fluctuations W ~ (B E) 1/2 ~ 20 c m / s .
E. Gledzer/Physica D 104 (1997) 163-183
176
-1
-4
xx •
%
% % % i
Fig. 2. Deviationof the calculatedspectrafrom the measuredspectrumin near Kolmogorovscale range in log E - log(krl/2rr) coordinates, E is the energy spectrum: solid line, BL/u 2 = 0.01; short-dashed line, BL/u 2 = 0.1; long-dashed line, model spectrum (48).
Fig. 2 shows, in log-log plot, the spectrum with summarized Lumley's and viscous acceleration corrections calculated on the basis of Eqs. (45) and (47) (without pressure term). Values of parameters are equal to: B L / U 2 = 0.01 for solid line and 0.1 for short-dashed line (U = 10 m/s, 1/= 0.15 mm, v = 0.15 cm2/s). For comparison, the model spectrum (48) is presented (long-dashed line). Note that the summarized correction does not influence the spectrum in the inertial interval of wave numbers where Lumley's correction is equal to zero and viscous acceleration correction is negligible. The Lumley correction (47) and the correction for acceleration may be comparable for dissipative range. In Figs. 3(a) and (b) the model spectrum (48), the spectrum with the Lumley correction (47) and the spectrum with the total correction (Lumley's and for the viscous acceleration) are presented separately for two values of advection velocity U = 10m/s and U = 2 m / s with the same level of turbulence B L / u 2 -----0.01. Fig. 3 shows that the viscous acceleration term has small influence in the case of large velocity transfer. But when the advection velocity is reduced (laboratory conditions) the correction for viscous acceleration is of the same order as Lumley's. Generally speaking, this result can be expected, because it is known (see [8, p. 259]) that in the developed turbulence local acceleration Ou/Ot is usually much greater than substantive acceleration du/dt. This means that local and inertial (advective) accelerations are approximately balanced (from this it follows Eq. (4)). But Lumley's correction is obtained from the inertial acceleration term and, hence, could be larger (or much larger) than the corrections from the substantive acceleration of the Navier-Stokes equations. The influence of the viscous acceleration correction in (45) may be significant also for small wave numbers because of multiplier k~-2 in some terms of (45). For the model spectrum (48) if kl --~ 0 the leading term of correction for acceleration is (v/floU)2(~rlkl) -1/3 72 F ( ~ ) Fl°(kl), (ff is gamma function). Note that for these wave numbers the evaluation of this correction looses the precision. Here we have the same effect like the one-dimensional model of oscillation of acceleration (18). The singularity on kl appears as a result of additional r-averaging (see Appendix B) which is not valid for kl ~ 0. In Fig. 4 the compensated spectrum for (48) and for spectra obtained with the help of Eq. (45) are shown for parameters of Fig. 2 (coordinates y = kS/3Fl(k), x = logkr/). But we should note that in this region the correction for pressure term becomes more important (last term in (45)). This correction may be significant for the energy containing scales but contains the measured quantity other than the energy spectrum.
E. Gledzer/Physica D 104 (1997) 163-183
177
-4.25 ~'~._ -4.5 •
~,.,,.
_~, . . . . . . . . . . . . . ~[ 0.22 0.24
'~¢~.. . . . . . . .
~:
0.2~5"~,~,, 0.28
0.3
-5.25f
-5.s[
" ":"'<-..
a
a ~ ~
b
-1.5
-2.5
-0.2 -0.15 -0.I -0.05
0.05 ,,
0..1
Fig. 3. The model spectrum (48) (solid line), the spectrum with the Lumley correction (47) (short-dashed line) and the spectrum with the total correction (Lumley's and for the viscous acceleration) (long-dashed line) in near Kolmogorov scale range for (a) U = 10 m/s, (b) U -----2m/s; the coordinates are the same as in Fig. 2 and B L / u 2 = 0.01.
7. Concluding remarks In this paper the different forms of the Taylor hypothesis corrections were presented. In the beginning we discussed the corrections for one-dimensional processes. In this case the corrections are given for a spectrum and for correlation function. The case o f zero mean advection velocity was also considered. The simple one-dimensional model was presented which explains the sense of the correction for acceleration. When the value of acceleration is taken from the Navier-Stokes equations the evaluation of the correction becomes more complex, but the final equation for the corrections is simple enough and may be used for the estimations in the dissipative range of turbulence. Concerning structure or correlation functions we can obtain the corresponding corrections using formula BLL (r) = f-~oo eik~r F1 (k I) dk] and Eq. (45). But now the integrals on the right-hand side of this equation give the terms which cannot be expressed by the terms of function/~LL(r) o r its derivatives and integrals. So the corrections in this case should contain some other measured quantities. More simple is the case of the advection velocity fluctuations described by L u m l e y ' s correction (36), (47) or (51 ). In this case the corrections may be obtained for correlations and structure functions. Detailed investigations of these
178
E. Gledzer/Physica D 104 (1997) 163-183
-i0
-8
-6
-4
-2
0
Fig. 4, Compensated spectrum (48) (long-dashed line) and compensated spectra obtained by using Eq. (45): solid line, BL/U 2 = 0.01; short-dashed line, BL / u 2 = O.1.
corrections together with the effect of nonisotropic conditions for tensors up to fourth rank have been presented by Hill [13].
Acknowledgements The author wishes to thank Y. Gagne, B. Castaing, E.J. Hopfinger, A.M. Yaglom, A. Praskovsky and E. Villermaux for many discussions during the research. I am grateful to Dr. R. Hill for having communicated some of his results. This research was supported by DRET (contracts 93-1100 and 94-2591 A).
Appendix A If vector l(r) is the relative position of two particles with cordinates RI (r) and R2(r), then (32) gives {R21 (rl)R~ 1(rl)) = (/a(rl)//~(rl)} - rart~,
(A.1)
where r = / ( 0 ) , r = Us. Another quantity in (31) t+r
"/
t+r
V~(~lx, t)d~ + R~ (rl) t+rl
V~(~lx, t)d~
>
(A.2)
t+rl
can be estimated as s d 2 drl
- - - ( l a (rl)l# (r 1)).
(A.3)
E. Gledzer/Physica D 104 (1997) 163-183
179
For not very large time Z-l, the right-hand side of (A.1) is roughly equal (see [8, p. 543]) to rl2Dat~(r), where D~t~ is structure function of velocity field (Eulerian). So, if the time of averaging 7~ for ri in (32) is less than s ( B L / D L L (r)) i/2 the approximate value (34) of term (33) is greater than (A. 1) and (A.2).
Appendix B We shall use the following formula for isotropic correlation and spectral functions: oo
f
~LL(r) = BLL(r) =
eikirF1 (kl) dkl,
--(30 O0
k2/
kl
ABLL(r) = --2rr
e iklr dkl -oo
FNN(k)
1 --
k 3 dk,
kl
AeI3LL(r)=2Zr f eikirdkif FNN(k)( l-k~'~k'dk'k2] -oo
(B.I)
kI
•~LL(k) = FNN(k) {1 - kl2"~ \ k2 7 '
4zrk2FNN(k) = E(k),
oO
f
/3LL(k) dk2 dk3 ki
(B.2) kl
where FI (kl) is one-dimensional spectrum and FNN (k) is lateral three-dimensional spectrum, E(k) is the energy spectrum of isotropic velocity field. We multipy (43) and (44) by (1/2zr)e -ikt Us and integrate by s from - c ¢ to oo, using (B. 1). For terms of the order v in (43), we obtain
v
i
-co
e-ik~S~ds
ii
do
0
eiSh(~s~+u~)dk'l -oo
i(dkl 3
1 - ~2'~ FNN(k)
ki
+.ie-"'sd.i..ie"''-'".',i..' l--k27FNN(k),
E. Gledzer/Physica D 104 (1997) 163-183
180
which, if we except (1/2Jr)
2~v iklU 2
f e iyx dy
= 3(x), gives
eik,u~fdkk3 , -- k,Z] kt k27 FNN(k)
22zr1) r -- -iklU L
f
(
] a ~ 2 /
kll2
(B.3) In the same way, for the terms of the order oo
t+r
se--t
t
0
I) 2
in (43) we have
c~
v2fe-iklVsdsf d~ f dr/f e~'(vs+V"'d?,l~(?q) -c~
--~
t+r--s
~--t
+v2fe-iklV'ds f defdof ei~l(U'-U",°i'~°(?'l) -oo
t OG
0 OO
as
--oo
(e i~lUr +
e - i / ~ l U ( r - s ) -- 2
el(kl -kl)Us,
+
--OO
--O<)
~(~l)=f dkk'
1- k-f-~FNN(k). k2]
(B.4)
Now we can use the averaging on r because the terms of the order v 2 contribute to the last order of this disturbance theory. So, the oscillating terms proportional to e ikl ur vanish in (B.4) and the integration by s gives
(B.5)
Similarly, for the term of the order
V2
f
e -iklUs
-~
ds
o~
dr/1
dr/ 0
0
f
d/q~o(/q) ]
in (44), we obtain
e iil(Us+U(O-ol)) d~'l~p(k'l) -oo
oo
~__1)2 f
V2
e ifqUs
ds
--
e i/q U ( r - s ) -- e i/qUr + 1
-
ei(k-kj )t Us
(klU) 2
which after r-averaging gives 2Jr v2
(B.6)
E. Gledzer/Physica D 104 (1997) 163-183
181
Eqs. (B.5) and (B.6) are the full contribution of the terms of the order v2 to the averaged Fourier-transformed right-hand side of (25):
27r v 2 k5 k2U3 4 dk / L k~
1-
dkk 5 1 -
FNN(k)+2
(B.7)
FNN(k)
k~/2
The contribution of the term with pressure correlation may be found identical to (B.6) (by using the onedimensional pressure spectrum Fop(k1)). It has the form (sign minus in (44) is included)
1 (
U3p2
1
(~))
(B.8)
Fpp(kl) + ~Fpp
Now, using (B.1), (B.3), (B.7), (B.8) and (35) we obtain the equation that describes both Lumley's type and acceleration corrections to the measured spectrum BLn~(kl U):
-~
-~L
dkk
1 - k2 ]
kl
Jl-
= B~(k1U) + -iZklU2 LeiklVrk' dkk
1
Jt
kZ~k2] E ( k ) - e -iklur/2k,/2 dkk
1 - 4k2] E(k)
]
i)2 ~:2~ k3 ~2~ dk 1 - 4k2] E(k) k2U 3 2 dkk 3 1 - 2 k 2 / E ( k ) + [. k~ k~/2
U3fi2
(B.9)
Fpp(kl) + ~Fpp
where the operator L (according to (35)) is defined by OG
LG(kl)-
4U2rrf ei(kl-~l)Usdsf
EL(wt) sin2 °f 2
dJG(~:l) d/~l.
(B.10)
0 In Eqs. (B.7) and (B.9) the values of k'l are equal to the lower limits of the corresponding integrals. Eq. (B.9) gives E(k) for a given B~d(klU). The term of the order v is of oscillatory type and vanishes when averaged on r. But, this oscillating correction of the order v is also included in the function E(k) which is contained in both integrals in the term of the order v. So after r-averaging this term gives contribution of the order v2. To find the value of this contribution we start from the relation between ~0(kl) = dk k(1 - (k2/k2)) E(k) and one-dimensional spectrum FI (kl):
fff
(B.11)
k;
k2]
kl
E. Gledzer/Physica D 104 (1997) 163-183
182
So the term of the order v (B.9) has the form P
i2klU2 (eiklUr ~o(kl) -- e-iklUr/2~o ( ~ ) )
(B.12)
.
This gives the ~ v correction of FI (kl) (if zero-order term is equal to F°l(ki) = U B ~ ( k ! U))
v eiklUr ~FI (kl) -----i2klU
/
2k2FO(kl ) + 8
J ) dkkF°(k)
kl
v e_ikiUr/2 i2ki U
[()
2 kl 2FO 2
-+-8
i
dkkFO(k)
(B.13)
kl/2
Now we can use (B. 13) in (B. 11) and (B. 12) to obtain the corrections of the order v2. Note that only one contribution remains after r-averaging: from the second term in (B. 12) when for ~0(½kl ) to use the first-term correction in (B. 13). So we obtain
v i2klU 2
--e-ik'Ur/22(kl'] 2 v---Leikiur/2 \ 2 ] iklU
2
2F°
dkkFp(k)
-t-8 kl/2
3v2 = 4U
[ (_~)2 (~_~.l)f 2 --
F°
+8
dkkF°(k)
]
.
(B.14)
kj/2
This term is of the same order as that of the terms of the order v2 in (B.9). Expressing them with the help of El (kl), oo
E(k) = 2k4Fl(kl) + kl
k2 7
8f
dkk(3k 2 - k2)Fl(k),
kj oo
f k d k E ( k ) = 4k2Fl(kl) - k~ dF1 (kl)+~f~F,~ dk-------~ kl
(B.15)
kl
we obtain from (B. 15), (B. 14) and (B.9) for terms of the order v 2
"2{ t12
4U-----~ 2 - -
FI
+8
i
kl/2
dkkFl(k)
] 21 -- -k~U3 -
2~(kl) + ~p
+k~
k~
v2k 2 31 (8Fl (ki) - ki F~ (ki))
~ L~Z167k F,,~,~+~f~,,~~3k 2 -
i
F1 (k) dk
kdkE(k)
1
..]
E. Gledzer/PhysicaD 104(1997)163-183
183
So, Eq. (B.9) can be expressed in the following form:
Fl (kl) : UB~(kl U) + 1L2 kZFl(kl)+4
dkkFl(k) -8--~-
Fj(kl)
kl d F l ( k l ) ]
kl - 8 -v2k~ --~-
,(
U2p2
f 'k ~14k
3k2 -
, Fpp(kl) + ~Fpp
F(k)dk + -~1 f k 3k2kl/2
Fl(k)
dk (B.16)
References [1] R.A. Antonia and D. Britz, A note on the spectrum of the time structure function, University of Newcastle, T.N.-FM 83/2 (1983). [2] R.A. Antonia, N. Phan-Yhien and A.J. Chambers, Taylor's hypothesis and the probability density functions of temporal velocity and temperature derivatives in a turbulent flow, J. Fluid Mech. 100 (1980) 193. [3] F.H. Champagne, The fine-scale structure of the turbulent velocity field, J. Fluid Mech. 86 (1978) 67. [4] M.J. Fisher and P.O.A.L. Davies, Correlation measurements in a non-frozen pattern of turbulence, J. Fluid Mech. 18 (1964) 97. [5] G. Heskestad, A generalized Taylor hypothesis application for high Reynolds number turbulent shear flows, J. Appl. Mech. 87 (1965) 735. [6] C.C. Lin, On Taylor's hypothesis in wind tunnel turbulence, Quart. Appl. Math. 10 (1953) 295. [7] J.L. Lumley, Interpretation of time spectra measured in high-intensity shear flows, Phys. Fluids 8 (1965) 1056. [8] A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics, Vol. 2 (MIT Press, Cambridge, MA, 1975). [9] J.-E Pinton and R. Labb~, Correction to the Taylor hypothesis in swirling flows, J. Phys. II 4 (1994) 1461. [10] S.G. Saddoughi and S.V. Veeravalli, Local isotropy in turbulent boundary layers at high Reynolds number, J. Fluid Mech. 268 (1994) 333. [11] H. Tennekes, Eulerian and Lagrangian time microscales in isotropic turbulence, J. Fluid Mech. 67 (1975) 561. [ 12] J. Wyngaard and S.E Clifford, Taylor's hypothesis and high-frequency turbulence spectra, J. Atomospheric Sci. 34 (1977) 922. 1131 R.J. Hill, Corrections to Taylor's frozen turbulence approximation, Atmos. Res. 40 (1996) 153.