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Enstrophy and momentum fluxes in two-dimensional shear flow turbulence
Physica D 51 (1991) 569-578 North-Holland
Enstrophy and momentum fluxes in two-dimensional shear flow turbulence S a d a y o s h i T o h a, Koji O h k i t a n i b a n d M i c h i o Y a m a d a c aDepartment of Physics, Faculty of Science, Kyoto University, Kyoto 606-01, Japan bResearch Institute for Mathematical Sciences, Kyoto University, Kyoto 606-01, Japan CDisaster Prevention Research Institute, Kyoto University, Uji 611, Japan Two-dimensional turbulence with a mean linear shear is numerically studied under periodic boundary conditions. Enstrophy is transferred to higher wavenumbers both through nonlinear interaction between turbulent fluctuations and mean linear shear. When the mean shear rate is sufficiently large, enstrophy is transferred exclusively by the latter mechanism and then the energy spectrum and the momentum flux spectrum show k-3-form. Coherent vortices grow when its vorticity has the same sign as that of the background shear.
I. Introduction
Two-dimensional turbulence has been studied extensively in numerical simulations over years [1-3]. One major motivation is that fundamental aspects of its dynamics may help us to understand various phenomena occurring in the geophysical context. In these simulations substantial understanding has been obtained regarding the enstrophy cascade and its relevant phenomena. Among them spontaneous formation of coherent vortices out of a random vorticity field, first noticed by McWilliams [4], is of particular interest. This is because such coherent vortices are expected to be closely related with more realistic vortices that appear in geophysical circumstances. Actually, the majority of the simulations conducted so far concern homogeneous and isotropic turbulence, that is, turbulence without a mean flow. Recall, however that almost all the systems of geophysical interest are associated with a mean flow. It is then natural to make the numerical model more realistic by subjecting it to a mean flow. A notable numerical simulation of Jupiter's Great red spot has been done by Marcus [5], who found that
large spots of vorticity are stable if their vorticity has the same sign as the background shear. In this paper we study fundamental aspects of dynamics of two-dimensional shear flow turbulence, confining ourselves to the case of a linear mean flow, i.e. a mean flow with a linear velocity profiles. We first study how the vortex selection takes place in the physical space for several different strength of the background shear. Then we turn our attention to the spectral properties, such as energy spectrum, enstrophy flux and momentum flux, in order to clarify the interaction between turbulent fluctuations and the mean flow. A distinctive feature of two-dimensional turbulence is that the linear mean flow cannot sustain turbulence. Therefore in order to study statistical properties we introduce an external forcing to make the system statistically stationary. In section 2, the governing equations are described together with the numerical methods. In section 3, the selection of the vortices is treated as one of the physical space characteristics. As spectral characterizations, the enstrophy flux and the spectrum of the momentum are elaborated in sections 4 and 5, respectively. Finally, section 6 is devoted to conclusion.
0167-2789/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)
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S. Toh et al. / Two-dimensional shear flow turbulence
2. Formulation and numerical method In this paper, we employ a linear shear flow as the mean flow. Thus the turbulent fluctuation is statistically homogeneous and then governed by the following equation: 0to
Ow
U=Ty,
o~=Aq,,
~--- ~-
+ ~
at - - U ~ - J ( O , w ) - u 4 A 2 w + f
+ds,
a~o
a~o c,=0--~-,
u=-~,
(1)
,
where w and 0 are the vorticity and the stream function of the turbulent fluctuations, respectively, and J denotes the Jacobian operator. We denote the linear mean flow in the x-direction by U and the shear rate by y. We use the superviscosity instead of Newtonian viscosity to obtain effectively higher Reynolds number flows. Note that the hyperviscous term does not alter significantly the emergence of the coherent vortices [4]. The forcing function f is introduced to make turbulence stationary. The last term in the r.h.s. is the energy suction function, which is introduced in the forced case to avoid a limitless accumulation of the energy by inverse cascade. The reason why the forcing is needed to sustain the turbulence is that the mean flow does not supply the enstrophy to the turbulent fluctuations. Actually the total energy and enstrophy are respectively governed by the following equations: d~ d f f D l (V~O)2d x
=
-,ffDuv dx
+
,f~DUU d x - ~ 4 f f D u " AZudx, (2a)
~TffD½oo2dx=-~4ffDo,A2,odx ( -
-n(t)), (2b)
when the forcing and suction terms are absent.
Here 37(t) is the enstrophy dissipation function, and D denotes some region with aD being its boundary. Eq. (2b) shows the total enstrophy is not sustained by the linear mean flow but monotonically dissipated by the viscosity. Here, we focus our attention on the interaction between vortices and the linear shear in the inertial range. The forcing, the suction and the dissipation terms are therefore taken in order that the substantial inertial subrange be maintained. We expect that the qualitative features of the inertial subrange do not depend on the forms of these terms, although the details of the flow field may. There are some difficulties to perform numerical simulations in the Cartecian coordinate system because of the inhomogeneous term in eq. (1). Then we introduce the new coordinate system (X, Y) = (x - yyt, y) which becomes skewed as time increases [6]. In terms of this new coordinate system, eq. (1) is expressed as follows: 0to'
0 '
at - - U ~ - x - J ( 0 , ~ o ' ) -
u4A'2w' + f + d ~ , (3)
oJ'=A'~o,
A r
u'-
oO OY'
L,'- aO OX '
+ -~-yt~-- R
This equation is solved as an initial-value problem in a doubly periodic square domain of width 2w in the new coordinate system ( X , Y ) by the dealiased pseudospectral method and the modified R u n g e - K u t t a scheme. We use 256 modes for the Y-direction and 128 for the X-direction. The value of v4 is 2.4 × 10 -6. To allow the simulation to progress for a substantial time, it is necessary to redistribute modes at regular intervals [7]. It should be noted that the wavenumbers in the original coordinates are related to those in the new coordinates as (kx, k y ) = (Kx, K y - ytKx). The details of the numerical scheme will be explained in the forthcoming papers. In the following sections, all the results of numerical
S. Toh et al. / Two-dimensional shear flow turbulence
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simulations are represented in the original coordinates (x, y). 3. Vortex selection
In this section, we examine the emergence of coherent vortices, the selection of vortices by the mean flow and the dependence of flow patterns on the relative vorticity of turbulent fluctuations to that of the mean flow. We use a random initial condition where the amplitudes of Fourier modes of vorticity obey a scalar function, {ck/[1 + (k/ko)4]} 1/2 and their phases are randomized.
First, we treat freely decaying case. In fig. 1, the contour plots of vorticity for y = 1 are shown. From the random initial condition, many small vortices emerge. Vortices with the opposite sign of vorticity to that of the mean flow are elongated by the shear. Because of the periodic boundary conditions, the elongated vortices continue to fill the domain, and are homogenized by the viscosity. On the other hand, vortices with the same sign of vorticity as the mean flow repeat merging and grow into larger ones. These large vortices are reminiscent of the Great Red Spot on Jupiter
[5].
S. Toh et a L / Two-dimensional shear flow turbulence
572
Next we study the forced cases. We use the following forcing and suction to make turbulence stationary:
f=foexp(iO)
for3
= 0
otherwise,
d~=uok -2
(4)
for k < 3,
= 0
otherwise,
(5)
where the phase ~ is a random variable for each 100
a I
50
'
~
time
0
b
i
time
"'v ;i
50
~\tIU-'\\,j,L/.t'lk.l~t.jr'tjlt\'%\.l
k/
100
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,
50
1
50
Ji
100
50
.t.¢
100
^ . \JC] -'1\~ '~'\ i'~\J'~\it~l\.
0 0
time
0
time
25
50
50
10
20
25
Fig. 2. Time development of the total energy (solid line), the enstrophy (dashed line) and the palinstrophy (dash-dotted line), for (a) 3' = 0, (b) 3" = 1, (c) 3' = 2 and (d) ~ = 4. T h e energy, the enstrophy and the palinstrophy are respectively multiplied by a factor of 10, 5, 1/50.
m o d e and f0 = 4.0 and u 0 = 1.5. Because wavenumbers are time-dependent in our numerical scheme, we force the modes in the shell so that the forcing is kept almost isotropic. In fig. 2, the time series of the palinstrophy, the enstrophy and the energy after transient periods are shown for 3' = 0, 1,2,4. It is noted that the fast variations of the energy and the enstrophy are due to the random forcing. For 7 = 1, all three quantities have their maximum average values, and their variations are most intense. This suggests the strength of turbulence is enhanced by the shear as far as the shear rate 7 is not so large. F o r shearless t w o - d i m e n s i o n a l t u r b u l e n c e , McWilliams inferred that turbulence fluctuations are produced in the stretched regions surrounding the coherent vortices. It is then natural to expect that the elongation of vortices by the shear may also contribute to the production of turbulence. In fig. 3, the contour plot of vorticity is shown for 7 = 1. This figure is a snapshot and shows a typical pattern. As in the decaying case, most localized vortices have the same sign of the vorticity as the mean flow. Contrary to the decaying case, their sizes do not grow without limit and seem to be of the scales in inertial range like vortices seen in shearless forced two-dimensional turbulence. Vortices with the positive vorticity are elongated, while those with positive but relatively large vorticity survive. The larger the values of 3' become, the more vortices are elongated by the mean flow (see fig. 3). This asymmetry in vortex parity is qualitatively explained by the following ideas for the inviscid cases. Kida found a class of solutions of an elliptic vortex with uniform vorticity in a uniform straining and vorticity [8]. Particularly for the linear shear (our case), it is pointed out that vortices with the value of vorticity between 0 and (3 + 2v/2-)3' ( = w c) are elongated. Otherwise stationary elliptic vortices exist and their axes are parallel to the x or y axis. Though these results are exactly true only for uniform vortices and for inviscid cases, some of the dynamics of vortices
S. Toh et al. / Two-dimensional shear flow turbulence
573
to separate coherent structures from turbulent regions in shearless two-dimensional turbulence. In Weiss' method [9], the relative magnitudes of strain and vorticity, Q = S 2 + S 2 - w 2 is defined where S 1 = Ol,l / O x - Ov/Oy, S 2 = Ov/Ox + Ou/Oy and ~o = Ov/Ox - Ou/Oy. The region with Q > 0 is regarded as turbulent, and one with Q < 0 is neutral, which then corresponds roughly to coherent vortices. As a matter of fact, f Q is related to the eigenvalue of the time variation of the vorticity gradient on a fluid percel. That is, a region with Q > 0 is elongated and that with
are qualitatively explained by means of Kida's vortices. Actually, the typical positive values of vorticity of vortices are larger than toc for y = 1, of the same order as toc for 3' = 2 and smaller than toc for 3' = 4. These are consistent with the observation that the larger the shear rate becomes, the less the vortices with positive vorticity can be observed for the same forcing. It is noted that the value of vorticity itself decreases with the shear rate. We can also explain the asymmetry by extending Weiss' method, which was used in refs. [2, 4]
a 27r
Y
0
X
27r
b
c
27r
27r
Y
Y
0 0
X
27r
0
X
21r
Fig. 3. Contours of vorticity in the forced case for (a) y = 1, (b) y = 2 and (c) 3' = 4. The c o n t o u r intervals are (a) 3, (b) 2, and (c) 1. Regions with positive vorticity are shaded.
574
S. Toh et a L / Two-dimensional shear flow turbulence
two kinds of cascade processes are due to nonlinear interaction between different wavenumber components. In our case, however, the linear mean flow makes the turbulent flow anisotropic and the cascade process is somewhat deformed. The enstrophy flux in the turbulence with linear mean flow is given by the sum of two kinds of terms,
27r
eg Y •
r
D
FNL(k) = fflkl>k R e [ w ( - k )
q
FL(k ) 0
X
=
J(g~, to)] d k ,
- fflkl=k3"kxkylto(k)12 dk,
(6)
(7)
27r
Fig. 4. The black regions have large negative Q~ - Q ( _< - 20) and the shaded ones have positive Q s - Q (>-20). In region A, Qs < 0 while Q < 0, because of large positive value of the additional term (see fig. 3a).
Q < 0 rotates. It is easy to include the m e a n flow effect in this method. The extended Q is Qs = $2 + $22 _ to2 + 2 3 ' ( $ 2 + to) and this indicates that a region with to > 0 tends to be elongated and vise versa when 3' > 0 (the vorticity of the mean flow is - 3 ' ) . This result suggests that vortices with the same sign of vorticity as the mean flow tend to survive. Comparison of Qs with Q in the case of fig. 3a shows that Qs is smaller (larger) than Q in the regions of negative (positive) vorticity. In some regions, Qs and Q have opposite signs due to the shear effect (fig. 4), which indicates that the linear shear has a stabilizing or destabilizing effect depending on the sign of the vorticity. It is also noteworthy that the maximum value of vorticity of the region A is over 15, much larger than toc estimated from Kida's model.
4. Enstrophy flux It is well known that in shearless two-dimensional turbulence, the enstrophy is transferred to a higher wavenumber region while the energy is transferred to a lower wavenumber region. These
the first of which denotes the nonlinear enstrophy transfer which has the same form as that in the shearless case, and the second is the enstrophy transfer arising from the interaction between the linear shear flow and the turbulent fluctuations. Throughout the following, we call the latter the linear enstrophy transfer term, because it is linear with respect to the turbulent fluctuations. For stationary state, the total enstrophy flux, i.e. the sum of FNL(k) and Fe(k), is equal to the enstrophy dissipation rate ~ in the inertial subrange. The two-dimensional spectrum of the enstrophy is given in fig. 5 for 3' = 0, 1, 2, 4. An isotropic distribution of the enstrophy is observed in the shearless case 3' = 0. This isotropy reflects the fact that the nonlinear transfer term is isotropic. In contrast with this, the linear transfer term is anisotropic. An inspection of the expression of FL(k) shows that the linear transfer term carries the enstrophy in wavenumber space along lines parallel to the ky-aXis from ky = ~ to ky = - o , . The stationary enstrophy distribution in a sheared case, therefore, results from a balance between this anisotropic transfer by the linear term and the nonlinear returning to isotropy. Fig. 5 for nonzero values of 3' shows this balance, with the spectrum being more anisotropic for stronger shear rate. In the case of 3' = 4, the spectrum is much elongated along the ky-aXis, indicating that
S. Toh et al. / Two-dimensional shear flow turbulence
the anisotropic linear transfer dominates over the nonlinear effect. The enstrophy fluxes by the linear and the nonlinear terms, FNL(k)and FL(k), are shown in fig. 6 together with the enstrophy dissipation term,
(8)
F,(k) = fgk
For the check of the numerical integration, the sum of FNL(k), FL(k) and Fn(k) is also shown,
b
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which should coincide with ~/ everywhere except at forced wavenumbers. In the shearless case (y = 0 ) a short inertial subrange is observed, where the nonlinear flux FNL(k) is constant and the enstrophy spectrum takes k - l - f o r m (see section 5). In the weakly sheared case (y = 1), the enstrophy is transferred by both the linear and the nonlinear mechanisms, but the nonlinear flux is larger than the linear one at k > 5. We note that the total enstrophy flux is larger than that in the shearless case. This is due not only to the participation of the linear enstrophy flux, but also to the enhancement of the nonlinear enstrophy transfer. It is interesting that the nonlinear flux still takes a constant value definitely less than -q in a wavenumber range slightly higher than the shearless case. For 7 = 2, the linear flux becomes larger than the nonlinear one. The nonlinear flux is strongly suppressed in comparison with the case of 7 = 1, while the linear flux takes a value slightly larger than that for y = 1. Thus the reversal of the order of the fluxes comes mainly from the suppression of the nonlinear enstrophy flux. This also brings about the decrease of the total enstrophy flux. Therefore the total enstrophy flux takes the maximum value for 3' = 1. For more strongly sheared case (3' = 4), the nonlinear enstrophy transfer is almost perfectly suppressed, and the linear flux dominates in the enstrophy transfer to higher wavenumbers. Note that the total enstrophy transfer is smaller than that in the case of y = 2. We also remark that as the linear flux dominates over the nonlinear one, the turbulent fluctuations of the velocity become weak and the turbulent field becomes strongly anisotropic (see section 5).
5. Momentum flux and energy spectrum -40
.
0~:
. I0
. . 20
40
30
40
0 ,u /am I 0
20
30
40
Fig. 5. C o n t o u r s of t h e t i m e - a v e r a g e d t w o - d i m e n s i o n a l ens t r o p h y s p e c t r a ~(kx, ky) for (a) 3' = 0, (b) 3" = 1, (c) 3" = 2 a n d (d) 3' = 4. T h e levels a r e t a k e n l o g a r i t h m i c a l l y with e q u a l intervals.
In this section we study the spectral characteristics of the Reynolds stress (the momentum flux in the physical space) together with the energy spectrum. The Fourier spectrum of the Reynolds
S. Toh et al. / Two-dimensional shear flow turbulence
576
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C ~++
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Fig. 6. Time average of enstrophy spectral fluxes for (a) 3' = 0, (b) 3' = 1, (c) 3' = 2 and (d) 3' = 4. Circles denote the nonlinear flux FNL(k) and plusses the linear flux FL(k). The dash-dotted curve represents FNL(k)+FI(k), the dashed curve F,7(k) and the triangles denote F N L ( k ) + F L ( k ) + F,(k). The solid line denotes the time average of rt(t). I0 °
t0°
~
10 -I
~-
0-2
lO -~ i 0 -2
0-2
10-3
0-4
10 -4
0-5
i 0 -s
0-6
i0-5
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i 0 -r
10 ° 10-t
0-g 0-~o
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+÷+
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,0, ,0 , t ,05 t
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r
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io
C
10 -g F
....... 30
r
k-3
i0-1o ~ oo
I
,
j I , ,,,,i 3 k IO
.... 30
,,,, ioo
Fig. 7. Time average of the energy spectrum (solid curve) and ur-spectrum (circles denoting positive values and plusses negative ones) for (a) 3' = 1, (b) 3" = 2 and (c) 3' = 4.
S. Toh et al. / Two-dimensional shear flow turbulence
stress (uv-spectrum) is given by
ff (Ikl-k)TFk~:ky-I.,(k) 12dk, and the energy spectrum
E(k) = ff~(Ikl-
(9)
E(k) by
k ) ]u__u_((~)]2d k ,
-yk3uv(k).
negative values at k > 5, consistent with an intuition that the m o m e n t u m should be transferred downward in physical space.
6. Conclusion
(10)
where 6 denotes Dirac's delta function. The uvspectrum vanishes in the statistical sense for shearless turbulence. It should be noted that the uv spectrum is directly related to the linear enstrophy flux F L ( k ) b y the following relation: FL(k ) =
577
(11)
Therefore if the linear enstrophy flux is constant over some wavenumber range, then the uv spectrum shows a k-3-1aw in the same range. We show in fig. 7 the uv-spectra and the energy spectra. For the weakly sheared case (y = 1), both the uv-spectrum and the energy spectrum shows a power law with the exponent close to - 4 . At present we have no explanation for this power law of the uv-spectrum. As for the energy spectrum, we note that there are some coherent vortices in the vorticity field owing to the weakness of the shear rate (fig. 3a). The energy spectrum in this case, whose exponent is stepper than - 3 , can be explained by the effect of these vortices just as in the case of shearless two-dimensional turbulence [10]. As the shear rate becomes larger, both spectra show less steep power laws. For the strongly sheared case (y = 4) the uv-spectrum shows a power law k -3. This is consistent with eq. (11), which indicates that when the linear transfer term is dominant in the enstrophy transfer process, i.e. Fe(k) ~ ~7, the Fourier spectrum of the Reynolds stress has a power laws, uv(k) ~ k -3. The energy spectrum also behaves nearly as k -3, and the coherent vortices are no longer observed (fig. 3c). We note that in each case the uv-spectrum takes
The nonlinear development of vortices and turbulence characteristics in a uniform shear flow are studied numerically. The vortices with the same sign of vorticity as that of the background shear flow survives against the shearing effect, while the vortices with the opposite sign are stretched and finally disappear. These results are consistent with Marcus's result on vortex selection [5], and also with the Kida's result [8] for an elliptic vortex in a strained flow. In the turbulence with the background uniform shear flow, there are two kinds of enstrophy fluxes in wavenumber space; one is due to the nonlinear interaction among turbulent fluctuations, and the other to the interaction between turbulent fluctuations and the background shear flow. The latter interaction is a linear one with respect to the turbulent fluctuation and may be called the linear flux. This linear flux has a direct relation to the spectrum of the Reynolds stress. As the shear rate increases, the nonlinear flux gets suppressed, which results in the dominance of the linear flux in the enstrophy transfer. The magnitude of the turbulent fluctuations get also weak with the increase of the shear rate. The Fourier spectrum of the Reynolds stress appears to have a power-law form (uv(k) ~ k -a) for weakly sheared case. As the shear rate increases, the dominance of the linear flux leads to a power form of k - 3 of the Reynolds stress.
Acknowledgements One of the authors (S.T.) would like to express his sincere thanks to the staffs of CNLS for their hospitality and kind help given to him at an unexpected accident. This work was supported in
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s. Toh et al. / Two-dimensional shear flow turbulence
part by a Grant-in-Aid
(no. 02740192) from the
Ministry of Education,
Science, and Culture
of
Japan.
References [1] A. Babiano, C. Basdevant, B. Legras and R. Sadourny, J. Fluid Mech. 194 (1987) 379. [2] M.E. Brachet, M. Meneguzzi, H. Politano and P.L. Sulem, J. Fluid Mech. 194 (1988) 333.
[3] S. Kida, M. Yamada and K. Ohkitani, Fluid Dyn. Res. 4 (1988) 271. [4] J.C. McWilliams, J. Fluid Mech. 146 (1984) 21. [5] P.S. Marcus, Nature 331 (1988) 693; J. Fluid Mech. 215 (1990) 393. [6] Y. Kaneda, T. Gotoh and N. Bekki, Phys. Fluids A 1 (1989) 1225. [7] R.S. Rogallo, NASA Tech. Mere. 81315 (1981). [8] S. Kida, J. Phys. Soc. Japan 50 (1981) 3517. [9] J. Weiss, Physica D 48 (1991) 273. [10] R. Benzi, P. Patarnello and P. Santangelo, J. Phys. A. 21 (1988) 1221.
Enstrophy and momentum fluxes in two-dimensional shear flow turbulence S a d a y o s h i T o h a, Koji O h k i t a n i b a n d M i c h i o Y a m a d a c aDepartment of Physics, Faculty of Science, Kyoto University, Kyoto 606-01, Japan bResearch Institute for Mathematical Sciences, Kyoto University, Kyoto 606-01, Japan CDisaster Prevention Research Institute, Kyoto University, Uji 611, Japan Two-dimensional turbulence with a mean linear shear is numerically studied under periodic boundary conditions. Enstrophy is transferred to higher wavenumbers both through nonlinear interaction between turbulent fluctuations and mean linear shear. When the mean shear rate is sufficiently large, enstrophy is transferred exclusively by the latter mechanism and then the energy spectrum and the momentum flux spectrum show k-3-form. Coherent vortices grow when its vorticity has the same sign as that of the background shear.
I. Introduction
Two-dimensional turbulence has been studied extensively in numerical simulations over years [1-3]. One major motivation is that fundamental aspects of its dynamics may help us to understand various phenomena occurring in the geophysical context. In these simulations substantial understanding has been obtained regarding the enstrophy cascade and its relevant phenomena. Among them spontaneous formation of coherent vortices out of a random vorticity field, first noticed by McWilliams [4], is of particular interest. This is because such coherent vortices are expected to be closely related with more realistic vortices that appear in geophysical circumstances. Actually, the majority of the simulations conducted so far concern homogeneous and isotropic turbulence, that is, turbulence without a mean flow. Recall, however that almost all the systems of geophysical interest are associated with a mean flow. It is then natural to make the numerical model more realistic by subjecting it to a mean flow. A notable numerical simulation of Jupiter's Great red spot has been done by Marcus [5], who found that
large spots of vorticity are stable if their vorticity has the same sign as the background shear. In this paper we study fundamental aspects of dynamics of two-dimensional shear flow turbulence, confining ourselves to the case of a linear mean flow, i.e. a mean flow with a linear velocity profiles. We first study how the vortex selection takes place in the physical space for several different strength of the background shear. Then we turn our attention to the spectral properties, such as energy spectrum, enstrophy flux and momentum flux, in order to clarify the interaction between turbulent fluctuations and the mean flow. A distinctive feature of two-dimensional turbulence is that the linear mean flow cannot sustain turbulence. Therefore in order to study statistical properties we introduce an external forcing to make the system statistically stationary. In section 2, the governing equations are described together with the numerical methods. In section 3, the selection of the vortices is treated as one of the physical space characteristics. As spectral characterizations, the enstrophy flux and the spectrum of the momentum are elaborated in sections 4 and 5, respectively. Finally, section 6 is devoted to conclusion.
0167-2789/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)
570
S. Toh et al. / Two-dimensional shear flow turbulence
2. Formulation and numerical method In this paper, we employ a linear shear flow as the mean flow. Thus the turbulent fluctuation is statistically homogeneous and then governed by the following equation: 0to
Ow
U=Ty,
o~=Aq,,
~--- ~-
+ ~
at - - U ~ - J ( O , w ) - u 4 A 2 w + f
+ds,
a~o
a~o c,=0--~-,
u=-~,
(1)
,
where w and 0 are the vorticity and the stream function of the turbulent fluctuations, respectively, and J denotes the Jacobian operator. We denote the linear mean flow in the x-direction by U and the shear rate by y. We use the superviscosity instead of Newtonian viscosity to obtain effectively higher Reynolds number flows. Note that the hyperviscous term does not alter significantly the emergence of the coherent vortices [4]. The forcing function f is introduced to make turbulence stationary. The last term in the r.h.s. is the energy suction function, which is introduced in the forced case to avoid a limitless accumulation of the energy by inverse cascade. The reason why the forcing is needed to sustain the turbulence is that the mean flow does not supply the enstrophy to the turbulent fluctuations. Actually the total energy and enstrophy are respectively governed by the following equations: d~ d f f D l (V~O)2d x
=
-,ffDuv dx
+
,f~DUU d x - ~ 4 f f D u " AZudx, (2a)
~TffD½oo2dx=-~4ffDo,A2,odx ( -
-n(t)), (2b)
when the forcing and suction terms are absent.
Here 37(t) is the enstrophy dissipation function, and D denotes some region with aD being its boundary. Eq. (2b) shows the total enstrophy is not sustained by the linear mean flow but monotonically dissipated by the viscosity. Here, we focus our attention on the interaction between vortices and the linear shear in the inertial range. The forcing, the suction and the dissipation terms are therefore taken in order that the substantial inertial subrange be maintained. We expect that the qualitative features of the inertial subrange do not depend on the forms of these terms, although the details of the flow field may. There are some difficulties to perform numerical simulations in the Cartecian coordinate system because of the inhomogeneous term in eq. (1). Then we introduce the new coordinate system (X, Y) = (x - yyt, y) which becomes skewed as time increases [6]. In terms of this new coordinate system, eq. (1) is expressed as follows: 0to'
0 '
at - - U ~ - x - J ( 0 , ~ o ' ) -
u4A'2w' + f + d ~ , (3)
oJ'=A'~o,
A r
u'-
oO OY'
L,'- aO OX '
+ -~-yt~-- R
This equation is solved as an initial-value problem in a doubly periodic square domain of width 2w in the new coordinate system ( X , Y ) by the dealiased pseudospectral method and the modified R u n g e - K u t t a scheme. We use 256 modes for the Y-direction and 128 for the X-direction. The value of v4 is 2.4 × 10 -6. To allow the simulation to progress for a substantial time, it is necessary to redistribute modes at regular intervals [7]. It should be noted that the wavenumbers in the original coordinates are related to those in the new coordinates as (kx, k y ) = (Kx, K y - ytKx). The details of the numerical scheme will be explained in the forthcoming papers. In the following sections, all the results of numerical
S. Toh et al. / Two-dimensional shear flow turbulence
2re
a
571
b 27r
Y
Y
0
0 0
X"
2re
0
X
2rr
X
27r
d
C 27r
:Tr
Y
Y
0
0 0
X
27r
0
Fig. 1. C o n t o u r s o f v o r t i c i t y in t h e d e c a y i n g c a s e f o r y = 1 at (a) t = 0, (b) t = 10, (c) t = 20 a n d (d) t = 30. T h e c o n t o u r i n t e r v a l is 2. R e g i o n s w i t h positive v o r t i c i t y a r e s h a d e d .
simulations are represented in the original coordinates (x, y). 3. Vortex selection
In this section, we examine the emergence of coherent vortices, the selection of vortices by the mean flow and the dependence of flow patterns on the relative vorticity of turbulent fluctuations to that of the mean flow. We use a random initial condition where the amplitudes of Fourier modes of vorticity obey a scalar function, {ck/[1 + (k/ko)4]} 1/2 and their phases are randomized.
First, we treat freely decaying case. In fig. 1, the contour plots of vorticity for y = 1 are shown. From the random initial condition, many small vortices emerge. Vortices with the opposite sign of vorticity to that of the mean flow are elongated by the shear. Because of the periodic boundary conditions, the elongated vortices continue to fill the domain, and are homogenized by the viscosity. On the other hand, vortices with the same sign of vorticity as the mean flow repeat merging and grow into larger ones. These large vortices are reminiscent of the Great Red Spot on Jupiter
[5].
S. Toh et a L / Two-dimensional shear flow turbulence
572
Next we study the forced cases. We use the following forcing and suction to make turbulence stationary:
f=foexp(iO)
for3
= 0
otherwise,
d~=uok -2
(4)
for k < 3,
= 0
otherwise,
(5)
where the phase ~ is a random variable for each 100
a I
50
'
~
time
0
b
i
time
"'v ;i
50
~\tIU-'\\,j,L/.t'lk.l~t.jr'tjlt\'%\.l
k/
100
Ji
,
50
1
50
Ji
100
50
.t.¢
100
^ . \JC] -'1\~ '~'\ i'~\J'~\it~l\.
0 0
time
0
time
25
50
50
10
20
25
Fig. 2. Time development of the total energy (solid line), the enstrophy (dashed line) and the palinstrophy (dash-dotted line), for (a) 3' = 0, (b) 3" = 1, (c) 3' = 2 and (d) ~ = 4. T h e energy, the enstrophy and the palinstrophy are respectively multiplied by a factor of 10, 5, 1/50.
m o d e and f0 = 4.0 and u 0 = 1.5. Because wavenumbers are time-dependent in our numerical scheme, we force the modes in the shell so that the forcing is kept almost isotropic. In fig. 2, the time series of the palinstrophy, the enstrophy and the energy after transient periods are shown for 3' = 0, 1,2,4. It is noted that the fast variations of the energy and the enstrophy are due to the random forcing. For 7 = 1, all three quantities have their maximum average values, and their variations are most intense. This suggests the strength of turbulence is enhanced by the shear as far as the shear rate 7 is not so large. F o r shearless t w o - d i m e n s i o n a l t u r b u l e n c e , McWilliams inferred that turbulence fluctuations are produced in the stretched regions surrounding the coherent vortices. It is then natural to expect that the elongation of vortices by the shear may also contribute to the production of turbulence. In fig. 3, the contour plot of vorticity is shown for 7 = 1. This figure is a snapshot and shows a typical pattern. As in the decaying case, most localized vortices have the same sign of the vorticity as the mean flow. Contrary to the decaying case, their sizes do not grow without limit and seem to be of the scales in inertial range like vortices seen in shearless forced two-dimensional turbulence. Vortices with the positive vorticity are elongated, while those with positive but relatively large vorticity survive. The larger the values of 3' become, the more vortices are elongated by the mean flow (see fig. 3). This asymmetry in vortex parity is qualitatively explained by the following ideas for the inviscid cases. Kida found a class of solutions of an elliptic vortex with uniform vorticity in a uniform straining and vorticity [8]. Particularly for the linear shear (our case), it is pointed out that vortices with the value of vorticity between 0 and (3 + 2v/2-)3' ( = w c) are elongated. Otherwise stationary elliptic vortices exist and their axes are parallel to the x or y axis. Though these results are exactly true only for uniform vortices and for inviscid cases, some of the dynamics of vortices
S. Toh et al. / Two-dimensional shear flow turbulence
573
to separate coherent structures from turbulent regions in shearless two-dimensional turbulence. In Weiss' method [9], the relative magnitudes of strain and vorticity, Q = S 2 + S 2 - w 2 is defined where S 1 = Ol,l / O x - Ov/Oy, S 2 = Ov/Ox + Ou/Oy and ~o = Ov/Ox - Ou/Oy. The region with Q > 0 is regarded as turbulent, and one with Q < 0 is neutral, which then corresponds roughly to coherent vortices. As a matter of fact, f Q is related to the eigenvalue of the time variation of the vorticity gradient on a fluid percel. That is, a region with Q > 0 is elongated and that with
are qualitatively explained by means of Kida's vortices. Actually, the typical positive values of vorticity of vortices are larger than toc for y = 1, of the same order as toc for 3' = 2 and smaller than toc for 3' = 4. These are consistent with the observation that the larger the shear rate becomes, the less the vortices with positive vorticity can be observed for the same forcing. It is noted that the value of vorticity itself decreases with the shear rate. We can also explain the asymmetry by extending Weiss' method, which was used in refs. [2, 4]
a 27r
Y
0
X
27r
b
c
27r
27r
Y
Y
0 0
X
27r
0
X
21r
Fig. 3. Contours of vorticity in the forced case for (a) y = 1, (b) y = 2 and (c) 3' = 4. The c o n t o u r intervals are (a) 3, (b) 2, and (c) 1. Regions with positive vorticity are shaded.
574
S. Toh et a L / Two-dimensional shear flow turbulence
two kinds of cascade processes are due to nonlinear interaction between different wavenumber components. In our case, however, the linear mean flow makes the turbulent flow anisotropic and the cascade process is somewhat deformed. The enstrophy flux in the turbulence with linear mean flow is given by the sum of two kinds of terms,
27r
eg Y •
r
D
FNL(k) = fflkl>k R e [ w ( - k )
q
FL(k ) 0
X
=
J(g~, to)] d k ,
- fflkl=k3"kxkylto(k)12 dk,
(6)
(7)
27r
Fig. 4. The black regions have large negative Q~ - Q ( _< - 20) and the shaded ones have positive Q s - Q (>-20). In region A, Qs < 0 while Q < 0, because of large positive value of the additional term (see fig. 3a).
Q < 0 rotates. It is easy to include the m e a n flow effect in this method. The extended Q is Qs = $2 + $22 _ to2 + 2 3 ' ( $ 2 + to) and this indicates that a region with to > 0 tends to be elongated and vise versa when 3' > 0 (the vorticity of the mean flow is - 3 ' ) . This result suggests that vortices with the same sign of vorticity as the mean flow tend to survive. Comparison of Qs with Q in the case of fig. 3a shows that Qs is smaller (larger) than Q in the regions of negative (positive) vorticity. In some regions, Qs and Q have opposite signs due to the shear effect (fig. 4), which indicates that the linear shear has a stabilizing or destabilizing effect depending on the sign of the vorticity. It is also noteworthy that the maximum value of vorticity of the region A is over 15, much larger than toc estimated from Kida's model.
4. Enstrophy flux It is well known that in shearless two-dimensional turbulence, the enstrophy is transferred to a higher wavenumber region while the energy is transferred to a lower wavenumber region. These
the first of which denotes the nonlinear enstrophy transfer which has the same form as that in the shearless case, and the second is the enstrophy transfer arising from the interaction between the linear shear flow and the turbulent fluctuations. Throughout the following, we call the latter the linear enstrophy transfer term, because it is linear with respect to the turbulent fluctuations. For stationary state, the total enstrophy flux, i.e. the sum of FNL(k) and Fe(k), is equal to the enstrophy dissipation rate ~ in the inertial subrange. The two-dimensional spectrum of the enstrophy is given in fig. 5 for 3' = 0, 1, 2, 4. An isotropic distribution of the enstrophy is observed in the shearless case 3' = 0. This isotropy reflects the fact that the nonlinear transfer term is isotropic. In contrast with this, the linear transfer term is anisotropic. An inspection of the expression of FL(k) shows that the linear transfer term carries the enstrophy in wavenumber space along lines parallel to the ky-aXis from ky = ~ to ky = - o , . The stationary enstrophy distribution in a sheared case, therefore, results from a balance between this anisotropic transfer by the linear term and the nonlinear returning to isotropy. Fig. 5 for nonzero values of 3' shows this balance, with the spectrum being more anisotropic for stronger shear rate. In the case of 3' = 4, the spectrum is much elongated along the ky-aXis, indicating that
S. Toh et al. / Two-dimensional shear flow turbulence
the anisotropic linear transfer dominates over the nonlinear effect. The enstrophy fluxes by the linear and the nonlinear terms, FNL(k)and FL(k), are shown in fig. 6 together with the enstrophy dissipation term,
(8)
F,(k) = fgk
For the check of the numerical integration, the sum of FNL(k), FL(k) and Fn(k) is also shown,
b
a 40 30
3O
20
20
\
10
ky
%
0
0
-10
I0
-20
20
-30
40
-40
OkxlO
20
30
40
~ I kxlO
,
/ I I/'I 20 30
40
d
C 30
3O
20
20
I0
I0
k~
k~
0
0
-10
I0
-20
20
-30
3O
575
which should coincide with ~/ everywhere except at forced wavenumbers. In the shearless case (y = 0 ) a short inertial subrange is observed, where the nonlinear flux FNL(k) is constant and the enstrophy spectrum takes k - l - f o r m (see section 5). In the weakly sheared case (y = 1), the enstrophy is transferred by both the linear and the nonlinear mechanisms, but the nonlinear flux is larger than the linear one at k > 5. We note that the total enstrophy flux is larger than that in the shearless case. This is due not only to the participation of the linear enstrophy flux, but also to the enhancement of the nonlinear enstrophy transfer. It is interesting that the nonlinear flux still takes a constant value definitely less than -q in a wavenumber range slightly higher than the shearless case. For 7 = 2, the linear flux becomes larger than the nonlinear one. The nonlinear flux is strongly suppressed in comparison with the case of 7 = 1, while the linear flux takes a value slightly larger than that for y = 1. Thus the reversal of the order of the fluxes comes mainly from the suppression of the nonlinear enstrophy flux. This also brings about the decrease of the total enstrophy flux. Therefore the total enstrophy flux takes the maximum value for 3' = 1. For more strongly sheared case (3' = 4), the nonlinear enstrophy transfer is almost perfectly suppressed, and the linear flux dominates in the enstrophy transfer to higher wavenumbers. Note that the total enstrophy transfer is smaller than that in the case of y = 2. We also remark that as the linear flux dominates over the nonlinear one, the turbulent fluctuations of the velocity become weak and the turbulent field becomes strongly anisotropic (see section 5).
5. Momentum flux and energy spectrum -40
.
0~:
. I0
. . 20
40
30
40
0 ,u /am I 0
20
30
40
Fig. 5. C o n t o u r s of t h e t i m e - a v e r a g e d t w o - d i m e n s i o n a l ens t r o p h y s p e c t r a ~(kx, ky) for (a) 3' = 0, (b) 3" = 1, (c) 3" = 2 a n d (d) 3' = 4. T h e levels a r e t a k e n l o g a r i t h m i c a l l y with e q u a l intervals.
In this section we study the spectral characteristics of the Reynolds stress (the momentum flux in the physical space) together with the energy spectrum. The Fourier spectrum of the Reynolds
S. Toh et al. / Two-dimensional shear flow turbulence
576
a
C ~++
liil\ i
-1
i
i
1
i
i
Lirl
I
k
ii
I
I
10
I
-1
r I tl
50
i
i
~--.,
+ ++ + + ~ , + ~ I
i
i
i i~11
k
100
/
/
J~
t
lO
i
50
ii
100
b J
,\
........
i i
i+
~.
0:7
i
. . . . . .
"
"
/
'\ "" .'\ I
i/
1 I
d
/1
+
•
tI
+ +++++"~.9~.
ji
j,_l__----
k
.,,,,,,~
,~
~,,,=,--
-
"% ,/ "x,~ L
[ r
--1
i
I
1
I
I
I I Ill
k
I
I
10
i
I
-1
I t II
5O
1
100
k
10
50
lOO
Fig. 6. Time average of enstrophy spectral fluxes for (a) 3' = 0, (b) 3' = 1, (c) 3' = 2 and (d) 3' = 4. Circles denote the nonlinear flux FNL(k) and plusses the linear flux FL(k). The dash-dotted curve represents FNL(k)+FI(k), the dashed curve F,7(k) and the triangles denote F N L ( k ) + F L ( k ) + F,(k). The solid line denotes the time average of rt(t). I0 °
t0°
~
10 -I
~-
0-2
lO -~ i 0 -2
0-2
10-3
0-4
10 -4
0-5
i 0 -s
0-6
i0-5
0-z 0-8
i 0 -r
10 ° 10-t
0-g 0-~o
10-2 .
+
+÷+
-2
+++
,0, ,0 , t ,05 t
i 0 -8
a i
i i ltlJJl
3
k
10
i
I
30
I
r
I
I I I
100
lO-g lo-bO
10 -8
i b ....... i I
3
k
io
C
10 -g F
....... 30
r
k-3
i0-1o ~ oo
I
,
j I , ,,,,i 3 k IO
.... 30
,,,, ioo
Fig. 7. Time average of the energy spectrum (solid curve) and ur-spectrum (circles denoting positive values and plusses negative ones) for (a) 3' = 1, (b) 3" = 2 and (c) 3' = 4.
S. Toh et al. / Two-dimensional shear flow turbulence
stress (uv-spectrum) is given by
ff (Ikl-k)TFk~:ky-I.,(k) 12dk, and the energy spectrum
E(k) = ff~(Ikl-
(9)
E(k) by
k ) ]u__u_((~)]2d k ,
-yk3uv(k).
negative values at k > 5, consistent with an intuition that the m o m e n t u m should be transferred downward in physical space.
6. Conclusion
(10)
where 6 denotes Dirac's delta function. The uvspectrum vanishes in the statistical sense for shearless turbulence. It should be noted that the uv spectrum is directly related to the linear enstrophy flux F L ( k ) b y the following relation: FL(k ) =
577
(11)
Therefore if the linear enstrophy flux is constant over some wavenumber range, then the uv spectrum shows a k-3-1aw in the same range. We show in fig. 7 the uv-spectra and the energy spectra. For the weakly sheared case (y = 1), both the uv-spectrum and the energy spectrum shows a power law with the exponent close to - 4 . At present we have no explanation for this power law of the uv-spectrum. As for the energy spectrum, we note that there are some coherent vortices in the vorticity field owing to the weakness of the shear rate (fig. 3a). The energy spectrum in this case, whose exponent is stepper than - 3 , can be explained by the effect of these vortices just as in the case of shearless two-dimensional turbulence [10]. As the shear rate becomes larger, both spectra show less steep power laws. For the strongly sheared case (y = 4) the uv-spectrum shows a power law k -3. This is consistent with eq. (11), which indicates that when the linear transfer term is dominant in the enstrophy transfer process, i.e. Fe(k) ~ ~7, the Fourier spectrum of the Reynolds stress has a power laws, uv(k) ~ k -3. The energy spectrum also behaves nearly as k -3, and the coherent vortices are no longer observed (fig. 3c). We note that in each case the uv-spectrum takes
The nonlinear development of vortices and turbulence characteristics in a uniform shear flow are studied numerically. The vortices with the same sign of vorticity as that of the background shear flow survives against the shearing effect, while the vortices with the opposite sign are stretched and finally disappear. These results are consistent with Marcus's result on vortex selection [5], and also with the Kida's result [8] for an elliptic vortex in a strained flow. In the turbulence with the background uniform shear flow, there are two kinds of enstrophy fluxes in wavenumber space; one is due to the nonlinear interaction among turbulent fluctuations, and the other to the interaction between turbulent fluctuations and the background shear flow. The latter interaction is a linear one with respect to the turbulent fluctuation and may be called the linear flux. This linear flux has a direct relation to the spectrum of the Reynolds stress. As the shear rate increases, the nonlinear flux gets suppressed, which results in the dominance of the linear flux in the enstrophy transfer. The magnitude of the turbulent fluctuations get also weak with the increase of the shear rate. The Fourier spectrum of the Reynolds stress appears to have a power-law form (uv(k) ~ k -a) for weakly sheared case. As the shear rate increases, the dominance of the linear flux leads to a power form of k - 3 of the Reynolds stress.
Acknowledgements One of the authors (S.T.) would like to express his sincere thanks to the staffs of CNLS for their hospitality and kind help given to him at an unexpected accident. This work was supported in
578
s. Toh et al. / Two-dimensional shear flow turbulence
part by a Grant-in-Aid
(no. 02740192) from the
Ministry of Education,
Science, and Culture
of
Japan.
References [1] A. Babiano, C. Basdevant, B. Legras and R. Sadourny, J. Fluid Mech. 194 (1987) 379. [2] M.E. Brachet, M. Meneguzzi, H. Politano and P.L. Sulem, J. Fluid Mech. 194 (1988) 333.
[3] S. Kida, M. Yamada and K. Ohkitani, Fluid Dyn. Res. 4 (1988) 271. [4] J.C. McWilliams, J. Fluid Mech. 146 (1984) 21. [5] P.S. Marcus, Nature 331 (1988) 693; J. Fluid Mech. 215 (1990) 393. [6] Y. Kaneda, T. Gotoh and N. Bekki, Phys. Fluids A 1 (1989) 1225. [7] R.S. Rogallo, NASA Tech. Mere. 81315 (1981). [8] S. Kida, J. Phys. Soc. Japan 50 (1981) 3517. [9] J. Weiss, Physica D 48 (1991) 273. [10] R. Benzi, P. Patarnello and P. Santangelo, J. Phys. A. 21 (1988) 1221.