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Study of the anisotropy of a passive scalar field at the level of dissipation
Michel Gonzalez
To cite this version:
Michel Gonzalez. Study of the anisotropy of a passive scalar field at the level of dissipation. Physics
of Fluids, American Institute of Physics, 2000, 12 (9), pp.2302 - 2310. �10.1063/1.1287516�. �hal-
01437151�
Study of the anisotropy of a passive scalar field at the level of dissipation
Michel Gonzalez
CNRS UMR 6614/CORIA Universite´ de Rouen, 76821 Mont-Saint-Aignan Cedex, France
共Received 4 January 2000; accepted 5 June 2000
兲The anisotropy of a passive scalar field at the level of second-order moments of the scalar derivatives is studied starting from the exact equations for the components of the second-rank tensor E defined by E
i j⫽ 2D( ⬘ / x
i)( ⬘ / x
j) (D is the molecular diffusivity of scalar ). Analysis requires also the equations for the components of a mixed tensor defined by the correlations between scalar and velocity gradients. After the examination of this set of equations, it is conjectured that, in the case of forcing by a mean scalar gradient in isotropic turbulence, the anisotropy of tensor E is produced by cliffs of temperature occurring in the direction of the mean gradient. A model for this mechanism is proposed including possible indirect influence of shear through the large-scale anisotropy of the scalar field. Predictions in the situation where a mean scalar gradient is combined with homogeneous shear agree tolerably well with experimental data. This suggests that the proposed picture describing production of small-scale anisotropy implied by mean gradient forcing has to be completed. © 2000 American Institute of Physics.
关S1070-6631
共00
兲50609-6
兴I. INTRODUCTION
Investigating contamination of small scales by large- scale features such as anisotropic forcing is akin to the ef- forts made for testing the concepts introduced in Kolmogor- ov’s theory. In this respect, the passive scalar has become a subject of growing interest since local isotropy of scalar fields in shear flows has been seriously questioned.
1,3–6The relevance of this problem has furthermore been confirmed by experiments showing that, in isotropic turbulence, forcing by a mean scalar gradient results in a small-scale anisotropy of the scalar field which persists when Reynolds number is increased.
7,8A measure of the fine-scale anisotropy of the scalar field can be derived from various criteria among which are the value of the derivative skewness and refined tests such as checking to what extent the derivative spectra agree with their theoretical, isotropic expressions.
9As shown by Van Atta,
9anisotropy estimated through second-order moments of scalar derivatives is not inconsistent with isotropy at the smallest scales since departures from isotropy measured in this way may be explained by anisotropy at the low-wave- number end of the gradient spectra. Possible contamination of the scalar field at this level, however, deserves investiga- tion at least with regard to the question of the validity of isotropy arguments which are set forth in some scalar dissi- pation measurements and models.
The present work is focused on the anisotropy felt by a passive scalar field at the level of dissipation as a conse- quence of forcing by mean gradients. The set of exact equa- tions for the components of tensors E and F defined, respec- tively, by E
i j⫽ 2D( ⬘ / x
i)( ⬘ / x
j) and F
i jk⫽ D( ⬘ / x
i)( u
k⬘ / x
j) is considered with intent to bring out the influence of large-scale forcing. The task is delicate in that these equations include not only terms expressing direct action of forcing but also unclosed terms in the form
of higher order correlations between gradients which are a priori not free from anisotropic effects. Thereby, the study is based partly on the analysis of exact equations, partly on modeling. After the general equations are given
共Sec. II
兲, analysis in the case of forcing by a scalar gradient applied in isotropic turbulence is undertaken
共Sec. III
兲; a mechanism is put forward for explaining anisotropy of tensor E in this situation and a model is subsequently devised. An algebraic modeling of the components of F is also proposed
共Sec. IV
兲. Finally, the case of forcing by combined scalar gradient and shear is considered; comparisons of model predictions with experimental data are reported and discussed
共Sec. V
兲.
II. GENERAL EQUATIONS A. Scalar gradients correlations
A second-rank tensor, E , is defined as
E
i j⫽ 2D ⬘
x
i ⬘
x
j,
where D is the molecular diffusivity of the scalar quantity, . The overbar denotes Reynolds averaging and prime fluctuat- ing quantities. The components of E have the dimension of scalar variance dissipation; the mean dissipation rate of the scalar fluctuations energy, ⬘
2/2, is nothing but the half trace of E : ⑀
⫽ E
␣␣/2. The equation for E
i jis derived from the instantaneous convection-diffusion equation for with the assumptions of incompressibility and constant diffusivity.
The procedure is quite standard and is not reported here.
Dropping transport terms
共which is justified in the homoge- neous situations which will be considered in the following
兲and in the limit of large Reynolds and Pe´clet numbers:
2302
1070-6631/2000/12(9)/2302/9/$17.00 © 2000 American Institute of Physics
d E
i jdt ⫽⫺E
i␣ ¯ u
␣ x
j⫺ E
j␣ ¯ u
␣ x
i⫺ 2
共Fi j␣⫹F
ji␣兲 ¯
x
␣⫹ S
i j⫹D
i j.
共1
兲In Eq.
共1
兲, u
iis the ith component of the velocity vector, F
i jkare the components of a mixed tensor:
F
i jk⫽ D ⬘
x
i u
k⬘
x
j, and
S
i j⫽⫺ 2D
冉 x ⬘
i x ⬘
␣ u
␣⬘
x
j⫹ ⬘
x
j ⬘
x
␣ u
␣⬘
x
i冊;
D
i j⫽⫺ 4D
2
x
␣冉 x ⬘
i冊 x
␣冉 x ⬘
j冊.
The terms of Eq.
共1
兲can be interpreted as follows. The first three terms on right-hand side
共rhs
兲are production terms and can be viewed as representing large-scale forcing by mean velocity and scalar gradients. The fourth and fifth terms represent, respectively, production by stretching and destruction by molecular dissipation. The order-of- magnitude analysis proves that although the stretching and dissipation terms are of order Re
larger than production terms (Re
being the Reynolds number based on the Taylor microscale,
), their difference is of the same order as the latter.
10,11An outstanding feature of the E
i jequation is the lack of terms to which a return to isotropy can explicitly be ascribed.
This sharply contrasts with the case of the Reynolds stresses dissipation tensor, ⑀
i j⫽ 2 ( u
i⬘ / x
␣)( u
j⬘ / x
␣) ( is the ki- nematic viscosity
兲the equation of which includes a pressure term ensuring redistribution.
12–14However, this does not mean that the components of E do not experience any return to isotropy. Indeed, if such a mechanism were not present then, experiments would reveal ever-increasing anisotropy of E , which is not the case. Now, examination of Eq.
共1
兲sug- gests that return to isotropy has to be ascribed to stretching and dissipation since other terms represent anisotropic forc- ing. Molecular dissipation appears, moreover, to be the best candidate. In this respect, note that Lumley
15has put forward the deviatoric part of viscous dissipation acts, in addition to the pressure term, as a redistribution mechanism in the equa- tion for the Reynolds stresses.
B. Scalar gradients–velocity gradients correlations It is worth noting, from Eq.
共1
兲, that forcing by mean shear involves the components of tensor E itself whereas forcing by mean scalar gradients set the mixed tensor, F , into play. Derivation of an evolution equation for the com- ponents F
i jkis similar to that of the E
i jequation. The only difference, this time, lies in the use of the Navier–Stokes equation in addition to the convection–diffusion equation for
. With the same previous assumptions
共no transport terms, large Reynolds, and Pe´clet numbers
兲, this equation is written as follows:
d F
i jkdt ⫽⫺ F
i j␣ u ¯
k x
␣⫺ 1
2 Sc
⫺1e
i j␣k ¯
x
␣⫺ F
␣jk ¯ u
␣ x
i⫺ F
i␣k ¯ u
␣ x
j⫹
⌺i jk⫹⌬
i jk⫹
⌸i jk.
共2
兲Sc is the Schmidt number, Sc ⫽ /D,
0, the density, and e
i jkl⫽ 2 ( u
k⬘ / x
i)( u
l⬘ / x
j). In addition,
⌺i jk
⫽⫺ D
冉 x
␣⬘
u
k⬘
x
j u
␣⬘
x
i⫹ ⬘
x
i u
k⬘
x
␣ u
␣⬘
x
j冊,
⌬i jk
⫽⫺ 2D
2
x
␣冉 x ⬘
i冊 x
␣冉 u x
k⬘
j冊⫺ D
2共1 ⫺ Sc
兲 ⬘
x
i
2 x
␣ x
␣冉 u x
k⬘
j冊,
⌸i jk
⫽⫺ D
0 ⬘
x
i
2p ⬘
x
j x
k.
The terms of Eq.
共2
兲can be interpreted, again, as pro- duction by mean gradients
共the first to fourth term on the rhs
兲, production by stretching and destruction by molecular dissipation
共respectively, the fifth and sixth terms
兲. In addi- tion, Eq.
共2
兲includes a pressure term
共the last term on the rhs
兲arising from the interaction between the scalar and the velocity fields in the form of a scalar gradient-pressure Hes- sian correlation. The order-of-magnitude analysis of Eq.
共2
兲can be undertaken in the same way as for Eq.
共1
兲.
Interestingly, production of F
i jkby mean scalar gradi- ents
共the second term on the rhs
兲involves the components of the fourth-rank tensor, e
i jkl. This implies that forcing by a mean scalar gradient affects scalar dissipation through the interaction between the large scales of the scalar field and the dissipative scales of the velocity field. Such a mechanism could thereby be a cause of contamination of the scalar dis- sipation field by the anisotropy
共if any
兲of the turbulence energy dissipation. The corresponding term is, however, Schmidt number dependent as it is proportional to Sc
⫺1. The previous interaction is thus ineffective at large Schmidt num- ber but is enhanced if the Schmidt number assumes small or moderate values.
The pressure term,
⌸i jk, raises a special problem for it includes the pressure Hessian which plays an important role in vortex dynamics.
16,17The difficulty in modeling the pres- sure Hessian lies in its nonlocal nature. As shown by Ohki- tani and Kishiba,
16it can be expressed as
2p
x
i x
j⫽ 1 3
2p
x
␣ x
␣␦
i j⫹ K
i j,
in which K
i jis a nonlocal term in the form of an integral over space. The restricted Euler model proposed by Vieillefosse
18keeps only the isotropic, local part of the pres- sure Hessian. It has been shown
19that this approximation retains the main features of the velocity gradients behavior, at least in isotropic or homogeneous sheared turbulence.
Martı´n et al.
20recently devised a model for the evolution of the velocity gradients neglecting the anisotropic part of the
2303
Phys. Fluids, Vol. 12, No. 9, September 2000 Study of the anisotropy of a passive scalar field at the . . .
pressure Hessian and retrieved most of the known properties of the small scales of turbulence. Restricting the pressure Hessian to its isotropic part implies, for the pressure term of the F
i jkequation,
⌸i jk
⫽⫺ D 3
0 ⬘
x
i
2p ⬘
x
␣ x
␣␦
jk.
In an incompressible flow, the pressure Laplacian is easily derived from the equation for the fluctuating velocity which finally leads to
⌸i jk
⫽ 1
3
冉D x ⬘
i u x
␣⬘
 u
⬘
x
␣⫹ 2 F
i␣ ¯ u
 x
␣冊␦
jk.
共3
兲III. FORCING BY A MEAN SCALAR GRADIENT IN ISOTROPIC TURBULENCE
A. Analysis
The effect of forcing by a scalar gradient can be studied in the basic configuration of the uniform, transverse, mean scalar gradient in isotropic turbulence as first suggested by Corrsin.
21Further experiments in decaying grid turbulence
22–24have confirmed the steadiness of the scalar gradient with downstream distance as well as the fact that reasonable cross-stream homogeneity of the scalar field can be ensured provided that the scalar profile is generated by a device independent of the grid.
In the following, it is assumed that a cross-stream, pas- sive, scalar gradient
⌫is applied in isotropic turbulence in such a way that
¯ ⫽
⌫x
2,
⌫⫽constant,
⌫⬎0.
Assuming unity Schmidt number, Eqs.
共1
兲and
共2
兲can then be written as
d E
11dt ⫽⫺ 4
⌫F112⫹ S
11⫹ D
11,
共4
兲d E
22dt ⫽⫺ 4
⌫F222⫹ S
22⫹ D
22,
共5
兲d E
33dt ⫽⫺ 4
⌫F332⫹ S
33⫹ D
33,
共6
兲d F
112dt ⫽⫺ 1
2
⌫e
1122⫹
⌺112⫹
⌬112,
共7
兲d F
222dt ⫽⫺ 1
2
⌫e
2222⫹
⌺222⫹
⌬222⫹
⌸222,
共8
兲d F
332dt ⫽⫺ 1
2
⌫e
3322⫹
⌺332⫹
⌬332.
共9
兲The pressure term,
⌸222, is approximated as shown in Sec. II
关Eq.
共3
兲兴and, in the present case of zero mean shear, reduces to
⌸222
⫽ D 3
⬘
x
2 u
␣⬘
x
 u
⬘
x
␣.
The anisotropy of tensor E is described by Eqs.
共4
兲–
共9
兲. Isotropy of the velocity field
共which implies e
i j22⫽ 0 and hence F
i j2⫽ 0 when i
⫽j ) precludes off-diagonal compo- nents. It is worth noticing that the mean scalar gradient af- fects all three diagonal components of E through interaction with tensor F
关Eqs.
共4
兲–
共6
兲兴. In passing, components F
112and F
332which play, respectively, on E
11and E
33are pro- moted in regions of the flow where vorticity has one compo- nent normal to the mean scalar gradient, whereas F
222共
which plays on E
22) is promoted in regions where strain acts along the mean gradient.
Although all three diagonal components of E are influ- enced by large-scale forcing, it is experimentally proved that, in the present case, E
22/ E
11lies between 1.2 and 1.6
共and that E
11⯝ E
33) independently of Reynolds number.
7,8Starting from an isotropic situation, the predominance of E
22can be explained only by some mechanism promoting this compo- nent. However, direct production of E
22occurs through F
222which, itself, is produced by ⫺ e
2222⌫/2
共note that, here, F
112, F
222, and F
332are negative
兲. Now, in isotropic turbu- lence, this latter term is only half the production terms of F
112and F
332since e
2222⫽ e
1122/2 ⫽ e
3322/2. In addition, it will be shown in the following that, in those strain regions where F
222is promoted, the pressure term
⌸222is likely to act as a destruction mechanism. Hence, predominance of component E
22over E
11and E
33cannot be understood by direct production. Most probably, values of E
22/ E
11larger than unity have to be explained through an influence of the mean gradient on stretching and dissipation of F
i jkresulting in
⌺222⫹⌬
222⬍
⌺112⫹⌬
112, possibly implying F
222⬍F
112and/or a similar effect felt by the stretching and dissipation of E
i j, both of these mechanisms leading to enhancement of E
22with respect to E
11and E
33.
B. Proposed mechanism for the anisotropy of E There is now ample evidence of a persistent skewness of the scalar derivative / x
2of the same sign as the one of the imposed mean gradient
7,8,23,25–27共positive in the case un- der study
兲. Besides, recent experiments and simulations
7,26,27have confirmed that, even in unsheared situations, forcing by a mean scalar gradient induces ramp-cliff type events in the scalar time signal. The latter have been invoked to explain the skewness of / x
2and can be understood as resulting from the transit, along the gradient, of fluid lumps moving from the low- region toward the high- region and vice versa
共Fig. 1
兲.
23,25Jumps in / x
2are connected with the upstream front of these fluid structures. If, following for in- stance Budwig et al.,
23the existence of a local stagnation region upstream of each lump is assumed then jumps in
/ x
2are strongly correlated with negative jumps in
u
2/ x
2.
An inference regarding the pressure term
⌸222can be drawn from the above-mentioned picture. This term can be written in function of vorticity and strain as
⌸222
⫽⫺ D 3
⬘
x
2冉1 2
2⫺
2冊,
with
2⫽ 1 2
冉 u x
␣⬘
 u
␣⬘
x
⫹ u
␣⬘
x
 u
⬘
x
␣冊,
2⫽ u
␣⬘
x
 u
␣⬘
x
⫺ u
␣⬘
x
 u
⬘
x
␣.
It follows that
⌸222is weak in shear regions
共where
2/2
⯝
2) and does not affect F
222in the latter. Furthermore, in strain regions, where F
222is mainly produced,
⌸222⯝ D/3 ( ⬘ / x
2)
2, that is,
⌸222⬎ 0, which implies, as previously mentioned, that
⌸222acts on F
222as a destruction mechanism. The positivity of
⌸222is reinforced by the term
⬘ / x
2( u
2⬘ / x
2)
2as a result of the correlation of jumps in
/ x
2 共positive in this situation
兲with those in u
2/ x
2. Another consequence of the previous phenomenological model is the enhancement of the stretching of E
22via the term ⫺ D( ⬘ / x
2)
2 u
2⬘ / x
2. The correlation of negative jumps in u
2/ x
2with jumps in / x
2caused by the fronts of the fluid lumps moving along x
2direction indeed implies a positive contribution to this strain term. The order of mag- nitude of such a contribution can be estimated as follows.
Taking for granted that the typical size of the fluid lumps is the integral scale,
⌳, and that the thickness of their front is of the order of the Taylor microscale,
, the magnitude of a jump in the scalar derivative is
⌫⌳/
.
7The resulting contri- bution to the stretching of E
22is therefore of order
D u
2⬘
冉⌫⌳冊2⬃
D u
2⬘
⬘
22,
and is larger by Re
than the production term in the E
22equation
共which scales as Du
2⬘ ⬘
2/
2⌳). However, the above-mentioned contribution is to be weighted by the frac- tion of ramp-cliff events which Tong and Warhaft,
7from their measurements of the probability density function of
/ x
2, show to vary approximately as Re
⫺1. On the whole, the effect of cliff events on the stretching of E
22thereby appears to be only slightly Reynolds number depen- dent. Nevertheless, this contribution cannot be neglected in the budget of E
22since it is of the same order as other terms
共including the sum of stretching and dissipation
兲. It is likely that, via the same mechanism, the component F
222is rein- forced as well.
Finally, the above-mentioned scenario can be summa- rized in a different, although equivalent, way. Isotropy of the velocity field precludes any preferential direction of strain.
However, the compressive strain events occurring along the mean scalar gradient / x
2 共corresponding, for instance, to the stagnation region upstream of the fronts moving in x
2direction
兲make the small scales of the scalar feel the large- scale asymmetry imposed by the mean gradient. This results, here, in enhancement of E
22with respect to E
11and E
33. In the following, a model for this mechanism is proposed.
C. Consequence for modeling
Accounting for anisotropy at the level of stretching and dissipation of E implies that the components of the latter can be written as
S
i j⫽ S
i j 0⫹ S
i jA
, D
i j⫽D
i j0
⫹ D
i j A. S
i j0
and D
i j0
represent the isotropic parts of S
i jand D
i j. S
i j Aand D
i jA
are the components of the anisotropic parts, repre- sented by traceless tensors ( S
␣␣A⫽ 0, D
␣␣A⫽ 0).
As already mentioned, although terms S
i j0
and D
i j 0are of order Re
when compared to other terms, their sum is of order Re
0. In the absence of forcing, Newman et al.
28model S
␣␣0⫹ D
␣␣0as a dissipation mechanism
共of order Re
0):
S
␣␣0⫹D
␣␣0 ⬅2 N
0⫽⫺ 2
⑀
2 ⬘
2,
where
, which is a functional of tensorial invariants, is restricted to a linear function of R, the scalar-to-velocity time scale ratio (R ⫽ ⬘
2⑀ /q
2⑀
, with q
2/2 and ⑀ being, respec- tively, the kinetic energy of turbulence and its dissipation rate
兲:
⫽⫺共 C
SR ⫹ C
D兲.
Constants C
Sand C
Dhave already been determined.
28–32Lumley
15previously suggested that, in anisotropic situations, the decay of ⑀
should depend on the anisotropy of the scalar field. Following this idea, Zeman and Lumley
32include a production term stemming from the anisotropy of the scalar field in their modeled ⑀
equation. Generalizing this concept to the components S
i j0
⫹ D
i j0
leads to S
i j0
⫹ D
i j 0⬅2
3
共N0⫹N
A兲␦
i j,
FIG. 1. Phenomenological model of fluid lumps explaining ramp-cliff events in the scalar signal and connected jumps in the instantaneous scalar derivative under influence of a mean scalar gradient关taken from Thoroddsen and Van Atta共1992兲 共Ref. 25兲in which the scalar is temperature兴.
2305 Phys. Fluids, Vol. 12, No. 9, September 2000 Study of the anisotropy of a passive scalar field at the . . .
where N
Ais a term resulting from large-scale anisotropy which, here, is written as
N
A⫽⫺ C
Au
␣⬘ ⬘ x ¯
␣
⑀
⬘
2.
C
Ais a constant which will be estimated later on.
Now, the model for S
i j⫹ D
i jhas to include the effect of large-scale anisotropy on S
i jdue to mean scalar gradient.
Following the discussion of Sec. II B, it is proposed to model this mechanism by a term of order Re
0representing produc- tion of E
i jas a result of the transit of fluid lumps along the mean gradient. In this respect, a general expression could be
P
i jA
⫽⫺ 2C
Au
␣⬘ ⬘ x ¯
␣冉
b b
ib b
j⫺ 1 3 ␦
i j冊 ⑀ ⬘
2with
b
i⫽ u
i⬘ ⬘
共q
2 ⬘
2兲1/2.
The nonlinear term b
ib
j/b
b
represents large-scale anisot- ropy of the scalar field and is introduced for including indi- rect effect of shear. The term in ␦
i j/3 ensures that P
␣␣A⫽ 0. In the modeled expression for S
i j⫹ D
i j, it will cancel with 2 N
A␦
i j/3.
It is to be stressed that there is apparently neither theo- retical nor experimental information regarding the return to isotropy of the scalar field at the level of dissipation. At this stage, the only term of the E
i jequation to which such a mechanism could be ascribed is the anisotropic part of the dissipative term D
i j, as already discussed in Sec. II A. In passing, note that the ramp-cliff events also cause jumps in the second-order spatial derivative along the mean scalar gra- dient, which enhances dissipation and most likely tends to balance their above-mentioned production effect. Recent ex- perimental results
7,8show that anisotropy at the smallest scales of the scalar field does not seem to relax with increas- ing Reynolds number suggesting that return to isotropy could be represented by a term which does not depend on the latter.
Using a relaxation-type expression, return to isotropy could be modeled as
R
i jA
⫽⫺
冉E
i j⫺ 2 3 ⑀
␦
i j冊
R⫺1.
In the above-mentioned expression,
Ris a characteristic time scale. In the absence of firm knowledge of this time scale and following the statement that return to isotropy is ensured by molecular dissipation, an obvious model consists in assuming that
R⫺1is proportional to the scalar frequency:
R⫺1⫽ ␣
R⑀
⬘
2.
An upper bound of ␣
Rcan be estimated by noticing that some experimental and numerical results
6,33,34display a stronger anisotropy for scalar gradients than for velocity gra- dients. This, indeed, leads one to surmise that return to isot- ropy is slower for the scalar gradients. Now, in the model of
Speziale and Gatski
14for the anisotropy of Reynolds stresses dissipation, the return-to-isotropy frequency is approxi- mately 12 ⑀ /q
2. It is therefore postulated that ␣
R⬍ 12. Fur- thermore, the relaxation frequency has to be larger than the dissipation one to ensure that, in the decaying case, return to isotropy effectively acts before complete small-scale mixing is achieved. With C
S⯝ C
D⯝ 2
共Refs. 28–32
兲and R of order one, this implies ␣
R⬎ 4. In Sec. V, model predictions in the situation of the Tavoularis and Corrsin’s experiment
2,3will be obtained with ␣
R⫽ 7.
The model for stretching and dissipation can finally be written in the following form:
S
i j⫹ D
i j⬅2
3
共N
0⫹ N
A兲␦
i j⫹P
i j A⫹ R
i jA
, and thus
S
i j⫹ D
i j⬅⫺2
3
共C
SR ⫹ C
D兲⑀
2 ⬘
2␦
i j⫺ 2C
Au
␣⬘ ⬘ ¯ x
␣b
ib
jb
b
⑀
⬘
2⫺ ␣
R冉E
i j⫺ 2 3 ⑀
␦
i j冊 ⑀ ⬘
2.
共10
兲In the present case, only E
22is affected through the produc- tion term ⫺ 2C
Au
2⬘ ⬘
⌫⑀
/ ⬘
2(b
1⫽ b
3⫽ 0). If a mean shear
共say, ¯ u
1/ x
2) were superposed upon the mean scalar gra- dient, then, since b
1would be nonzero, the nonlinear term b
ib
j/b
b
would make E
11feel, as well, the effect of cliffs.
Finally, using the above-mentioned model for S
i j⫹ D
i jin Eq.
共1
兲and contracting the latter results in the following
⑀
equation:
d ⑀
dt ⫽⫺E
␣ u
␣ x
⫺ 2 F
␣␣
x
⫺ C
Au
␣⬘ ⬘ x
␣
⑀
⬘
2⫺
共C
SR ⫹ C
D兲⑀
2 ⬘
2.
The third term on the rhs, although under the form of pro-
duction by mean scalar gradient, has a different origin than
the second one since it stems from the modeling of the an-
isotropy of stretching. It can be compared, in fact, with the
additional term that Zeman and Lumley
32have introduced in
their ⑀
model as a measure of anisotropy of the scalar field
共but have written as u
␣⬘ ⬘ u
␣⬘ ⬘ ). Current models include
only one term representing production by mean scalar gradi-
ents; this is equivalent to model production due to interaction
between F and the scalar gradient and production resulting
from anisotropy at the level of stretching as a whole. This
fact is used in Sec. IV for connecting the constants of the
present model to the production constant of usual ⑀
models
关Eq.
共14
兲兴.
IV. SCALAR GRADIENTS–VELOCITY GRADIENTS CORRELATIONS
A. Algebraic modeling
Complete description of the anisotropy of tensor E re- quires modeling the F
i jkcomponents. Note that models for the molecular dissipation of the scalar fluxes u
k⬘ ⬘ that is, the sum (1 ⫹ Sc) F
␣␣k, have recently been proposed.
35–37In the present case, knowledge of the individual components of F is needed and the model has to be more general. If, instead of modeling the F
i jkequation, an algebraic model is devised, then, a natural way consists in writing F in function of the anisotropy of the scalar field. Nondimensional components of F defined as
F
i jk* ⫽ F
i jk共⑀⑀
兲⫺1/2,
共11
兲are written as follows:
F
i jk* ⫽ ␣
1冉␦
i j⫺ b b
␣ib b
j␣冊b
k⫹ ␣
3共⫺3 ␦
ikb
j⫹ ␦
jkb
i兲.
共12
兲This is tantamount to connect anisotropy at the level of dis- sipation with large-scale anisotropy.
Equation
共12
兲ensures that F
␣␣k*
⬀b
kand that, in isotro- pic conditions, F
i jk* ⫽ 0. Continuity is also satisfied since F
i*
␣␣⫽ 0.
In passing, note that in the case of forcing by a mean scalar gradient
⌫( ¯ ⫽
⌫x
2) in isotropic turbulence: F
112*
⫽ F
332* ⫽ ␣
1b
2and F
222* ⫽⫺ 2 ␣
3b
2. This obviously implies
␣
1⬎ 0 and ␣
3⬍ 0.
B. The problem of unknown parameters
At this stage, ␣
1, ␣
3, and C
Aare still to be determined.
Experimental data on the F
i jkcomponents would help in determining ␣
1and ␣
3but these are unfortunately rather scarce.
2,11First, it is attempted to get an estimate of ␣
1con- sidering the case of forcing by a transverse mean scalar gra- dient in isotropic turbulence. Then, the measurements of An- tonia and Browne
11in a plane jet are used for approximating the ratio ␣
3/ ␣
1.
Using the model given by Eq.
共10
兲in the equations for E
11and E
22as well as Eqs.
共11
兲and
共12
兲for expressing F
112and F
222, the following equation is derived:
R
eq1/2冉␣
1⫹ 2 ␣
3E E
1122冊⫽ C 2
A冉E E
1122冊⫺ ␣
R⫺共 6
共PC
SR / ⑀
eq⫹
兲eqC
D兲⫻
冉1 ⫺ E E
1122冊,
共13
兲where E
11/ E
22is the ratio of E
11to E
22at equilibrium. P
/ ⑀
is the production/dissipation ratio of the scalar variance ( P
/ ⑀
⫽⫺ u
2⬘ ⬘
⌫/ ⑀
), which, in the case of forcing by a uniform, transverse, mean scalar gradient, has experimen- tally been proved to tend to a constant value, as well as the scalar-to-velocity time scale ratio R.
22,24The equilibrium val- ues of these quantities are, respectively, denoted by ( P
/ ⑀
)
eqand R
eq.
If it is furthermore required that the equation for E
␣␣/2 provides the right level of production of ⑀
, then identifica- tion of its production term with the one of the standard mod- eled ⑀
equation
28,30gives
2
⌫F␣␣2⫹ C
Au
2⬘ ⬘
⌫⑀
⬘
2⫽ C
Pu
2⬘ ⬘
⌫⑀
⬘
2, which can be written as
4
共␣
1⫺ ␣
3兲R
eq1/2⫽ C
P⫺ C
A,
共14
兲where C
Pis an already determined constant,
28,30C
P⯝ 2. An expression for ␣
1is derived by using Eq.
共14
兲in Eq.
共13
兲:
␣
1R
eq1/2⫽ C
P2
E
11/ E
221 ⫹ 2
共E
11/ E
22兲⫺ ␣
R⫺共 C
SR
eq⫹ C
D兲6
共P/ ⑀
兲eq1 ⫺共E
11/ E
22兲1 ⫹ 2
共E
11/ E
22兲.
共15
兲The assumption of weak anisotropy yields an approximation of Eq.
共15
兲namely, that ␣
1R
eq1/2should be slightly smaller than C
P/6. However, it is known that R
eqis not universal even in the simpler case of an homogeneous scalar field de- caying in isotropic turbulence. In the situation of an imposed scalar gradient, the experimental results of Sirivat and Warhaft
22show that R
eqis, with some scatter, close to 2/3;
the same value can be inferred from the measurements of Mydlarski and Warhaft.
8R
eqis three times larger in Gibson and Dakos’s experiment.
24In passing, the experiment of Siri- vat and Warhaft is such that ( P
/ ⑀
)
eq⯝ 1.5 whereas this ratio is around 2.2 in the case investigated by Gibson and Dakos. The nonuniversality of R
eqprecludes estimating reli- ably ␣
1as a constant. For this reason, it is stated that ␣
1and
␣
3depend on R in such a way that ␣
1R
1/2⫽ ␣
1⬘ and ␣
3R
1/2⫽ ␣
3⬘ , ␣
1⬘ and ␣
3⬘ being constants.
Antonia and Browne
11have measured
( ⬘ / x)( u ⬘ / x) (
⬀F
111) and ( ⬘ / x)( v ⬘ / x) (
⬀F112) in a turbulent plane jet. It is straightforward to show that, in this case, using the model expressed by Eq.
共12
兲:
␣
3␣
1⫽ 1
2 冉 1 ⫺ b b
21F F
111*
112* 冊冉 1 ⫹ b b
1222冊
⫺1.
The ratio of the scalar fluxes, b
1/b
2⬅u ⬘ ⬘ / v ⬘ ⬘ , on a sec- tion of the jet can be derived from a study by Antonia
38on a heated plane jet. Combining these data with those of Antonia and Browne
11on F
111and F
112, ␣
3/ ␣
1is indeed found to be approximately constant on a significant portion of a trans- versal section, ranging from ⫺ 0.6 at y /L
u⫽ 0.3 (L
uis the half-maximum velocity width
兲to ⫺ 0.4 at y /L
u⫽ 0.9, with a mean value close to ⫺ 0.5. This suggests that ␣
3⯝⫺ 0.5 ␣
1. The model constants are finally estimated as
␣
3⬘ ⯝⫺ 0.5 ␣
1⬘ , C
A⫽ C
P⫺ 4
共␣
1⬘ ⫺ ␣
3⬘
兲, and ␣
1⬘ slightly smaller than C
P/6.
2307
Phys. Fluids, Vol. 12, No. 9, September 2000 Study of the anisotropy of a passive scalar field at the . . .
V. COMBINED EFFECTS OF SCALAR GRADIENT AND SHEAR
A. Equations for E in homogeneous sheared turbulence
Measurements as well as simulations
4,6,39,40have re- vealed that, in passively heated shear flows, significant anisotropies occur at the level of temperature variance dissi- pation. This feature has also been emphasized by the experi- mental study of Tavoularis and Corrsin
2,3in homogeneous sheared turbulence. In this situation, the flow direction is x
1and the scalar and velocity mean gradients, respectively,
⌫and S, are applied in the x
2direction. The mean scalar and velocity fields are thus defined as
¯ ⫽
⌫x
2,
⌫⫽constant,
⌫⬎0,
共16
兲¯ u
i
⫽ Sx
2␦
i1, S ⫽ constant, S ⬎ 0.
共17
兲Whereas in the case of the pure scalar gradient ( ⬘ / x
1)
2and ( ⬘ / x
3)
2are equal and slightly smaller than ( ⬘ / x
2)
2,
8Tavoularis and Corrsin’s measurements show that ( ⬘ / x
2)
2⯝ ( ⬘ / x
3)
2⯝ 1.8( ⬘ / x
1)
2.
3Such a hier- archy of the diagonal terms of ( ⬘ / x
i)( ⬘ / x
j) has also been found in nonhomogeneous shear flows.
4,6,11In a homogeneous situation as studied by Tavoularis and Corrsin,
2,3the equations for the components of E derived from Eq.
共1
兲are written as follows:
d E
11dt ⫽⫺ 4
⌫F112⫹ S
11⫹ D
11, d E
22dt ⫽⫺ 2S E
21⫺ 4
⌫F222⫹ S
22⫹D
22, d E
33dt ⫽⫺ 4
⌫F332⫹ S
33⫹ D
33, d E
21dt ⫽⫺ S E
11⫺ 2
⌫共F212⫹F
122兲⫹S
21⫹ D
21.
Analysis of the previous equations reveals complex mecha- nisms. Mean shear affects E
22through interaction with the off-diagonal component, E
21. The off-diagonal component itself results from the interaction of shear with E
11. Equa- tions for E
11and E
33are free from direct shear effects.
As in the case of the pure scalar gradient, components F
112, F
222, and F
332interact with the latter to produce, re- spectively, E
11, E
22, and E
33. Equations for the components of F
共not displayed here
兲show that interaction of shear with F
112results in the production of F
212and F
122. The latter, interacting with the mean scalar gradient, play on E
21. F
222is directly affected by mean shear but F
112and F
332are not.
The equations for F also reveal intricate interactions due to the effect on F
222and F
122of the respective pressure terms
⌸222
and
⌸122, which, in the present case, include the mean velocity gradient
关Eq.
共3
兲兴.
The direct influence of shear on E
22could explain the predominance of this component in shear flows. However, understanding the reason why, in this type of flow, E
33⯝ E
22is more delicate because E
33does not directly experi-
ence the effect of shear. Now, the algebraic model for F
i jkgiven by Eq.
共12
兲includes an effect of shear through the large-scale anisotropy of the scalar field measured by vector b. As a matter of fact, in the case defined by Eqs.
共16
兲and
共17
兲, Eq.
共12
兲implies
F
332* F
112* ⫽ 1 ⫹
b
12b
22,
which, in the conditions of Tavoularis and Corrsin’s experiment
2(b
1⯝ 0.43, b
2⯝⫺ 0.19), leads to F
332* ⯝ 6 F
112* . Production by the mean scalar gradient is thus larger in the E
33equation than in the E
11equation as a consequence of indirect effect of shear on F
i jk.
In the following, predictions obtained from the equations for E
11, E
22, E
33, and E
21are compared with the experimen- tal data of Tavoularis and Corrsin
2,3using the models ex- pressed by Eqs.
共10
兲and
共12
兲for, respectively, S
i j⫹ D
i jand F
i jk. In this manner, the effect of shear is taken into account directly in E
21and E
22equations and, indirectly, through its effects on the F
i jkcomponents as well as on S
i j⫹ D
i jvia P
i j A关
Eq.
共10
兲兴.
B. Predictions in a homogeneous shear flow 1. Experimental conditions
Tavoularis and Corrsin
2,3have investigated the evolution of a temperature field under influence of a passive, mean transverse gradient (
⌫⫽9.5 °C m
⫺1) in a homogeneous shear flow (S ⫽ 46.8 s
⫺1). The streamwise, transverse, and spanwise directions are denoted, respectively, by x
1, x
2, and x
3. The mean centerline velocity is u ¯
1
(x
2/h ⫽ 0.5) ⫽ U
c⫽ 12.4 m s
⫺1. The height of the shear generator is h
⫽ 0.276 m. It can be checked, from the experimental data, that at x
1/h ⫽ 11, Sq
2/2 ⑀ ⯝ 6.1. Homogeneity of the fluctu- ating velocity and temperature fields as well as of their cross correlations are rather well ensured on a significant trans- verse portion of the flow. Detailed measurements including the fine-scale structure are provided at x
1/h ⫽ 7.5, x
1/h
⫽ 9.5, and x
1/h ⫽ 11. At x
1/h ⫽ 11, the Reynolds and Pe´clet numbers based on the longitudinal microscales are, respec- tively, Re
11
⫽ 266 and Pe
1
