- Email: [email protected]
Inverse cascade and energy transfer in forced low-Reynolds number two-dimensional turbulence
Fluid Dynamics Research 34 (2004) 289 – 297
Inverse cascade and energy transfer in forced low-Reynolds number two-dimensional turbulence J. Legaa;∗ , T. Passotb a
Department of Mathematics, University of Arizona, building 89, 617 N. Santa Rita, P.O. Box 210089, Tucson, AZ 85721-0089, USA b CNRS, Observatoire de la Cˆote d’Azur, Boˆ1te Postale 4229; 06304 Nice Cedex 4, France Received 14 October 2003; received in revised form 1 February 2004; accepted 11 February 2004 Communicated by M.-E. Brachet
Abstract Using numerical simulations of the forced two-dimensional Navier–Stokes equation, it is shown that the amount of energy transferred to large scales is related to the Reynolds number in a unique fashion. It is also observed that the critical value of the initial Reynolds number for the onset of an inverse cascade is lowered as the scale of the forcing approaches the size of the system, or in the presence of anisotropy. This study is motivated by recent experiments with bacterial colonies, and their description in terms of a hydrodynamic model. c 2004 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. PACS: 47.27.Cn; 47.27.Eq Keywords: Two-dimensional turbulence; Inverse cascade; Biological ;ows
1. Introduction This work is motivated by recent experiments (Mendelson et al., 1999; Kessler, 2000) with colonies of Bacillus subtilis growing on ;at surfaces, in which collective behaviors of bacteria are observed in the form of whirls and jets. These structures are dynamic; occur at a scale much larger (50–100 times) than that of an individual bacterium; alternate in time (cycles consisting of a clockwise whirl, a jet, a counterclockwise whirl and a jet in opposite direction are observed; each cycle takes about 1 s to complete); and cease to exit if bacteria die or stop swimming (Mendelson et al., 1999). We recently introduced a hydrodynamic model (Lega and Passot, 2003) in which ∗
Corresponding author. Tel.: +1-520-621-4350; fax: +1-520-621-8522. E-mail addresses: [email protected] (J. Lega), [email protected] (T. Passot).
c 2004 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. 0169-5983/$30.00 All rights reserved. doi:10.1016/j.;uiddyn.2004.02.002
290
J. Lega, T. Passot / Fluid Dynamics Research 34 (2004) 289 – 297
the dynamics of a bacterial colony and of its boundary is described by advection–reaction– diFusion equations coupled to a two-dimensional hydrodynamic equation for the complex ;uid made of bacteria and water. In such a macroscopic model, bacterial activity is globally described by a small-scale forcing in the hydrodynamic equation. Our numerical simulations have shown that partial transfer of energy towards large scales takes place in such a system, even at relatively low-Reynolds numbers. This leads to the formation of structures which are reminiscent of the whirls and jets observed in experiments (Mendelson et al., 1999; Kessler, 2000). However, collectives of bacteria are typically very-low-Reynolds-number systems, even though recent experiments suggest that hydrodynamic eFects may be important (Mendelson et al., 1999; Kessler, 2000; Wu and Libchaber, 2000). In order to validate the use of a hydrodynamic model to describe the whirls and jets seen in the experiments, it is therefore necessary to estimate the minimum value of the Reynolds number for which energy transfer takes place in the forced low-Reynolds number two-dimensional Navier–Stokes equation. This paper addresses this question by means of a series of numerical experiments, and also describes eFects which may promote energy transfer at even lower Reynolds numbers. In the two-dimensional Navier–Stokes equation, inverse transfer of energy typically leads to the formation of large-scale vortices. The existence of such an inverse cascade in two- and three dimensions is often understood as resulting from a negative total (molecular plus eddy) viscosity for large-scale secondary ;ows, which thus grow from random perturbations of the small-scale ;ow (Kraichnan, 1976). In particular, anisotropy and asymmetry in space and time of the basic ;ow (Sivashinsky and Yakhot, 1985; Bayly and Yakhot, 1986; Yakhot and Sivashinsky, 1987; Libin et al., 1987; Yakhot and Pelz, 1987; Frisch et al., 1987; Dubrulle and Frisch, 1991), as well as long-lived small-scale ;uctuations (Hefer and Yakhot, 1989; Zhang and Frenkel, 1998) are important, although not necessary (Sivashinsky and Frenkel, 1992; Gama et al., 1994) factors for such an instability to develop. In this paper, we report on numerical experiments of forced two-dimensional turbulence in which we measure the percentage of energy shared by Fourier modes with wave number smaller than that of the forcing. We show that this quantity is a function of the Reynolds number of the system, whether or not an inverse cascade occurs. We also study the eFect of conKnement (i.e. of making the scale at which the forcing is applied comparable to the size of the system) and of anisotropy on the relationship between the Reynolds number initially imposed by the forcing and the amount of energy transferred to larger scales. Our results indicate that one of the eFects of conKnement and/or anisotropy is to decrease the critical value of the initial Reynolds number above which an inverse cascade occurs. We then discuss how these results may be relevant to the modeling of bacterial colonies. 2. Partial energy transfer and transition to turbulence We consider the two-dimensional Navier–Stokes equation for an incompressible ;uid of density = 1 and dynamic viscosity . This equation reads @v + (v · ∇)v = −∇p + ∇2 v + F; @t ∇ · v = 0;
J. Lega, T. Passot / Fluid Dynamics Research 34 (2004) 289 – 297
291
where F is a forcing and the pressure p is determined to ensure that the vector Keld v remains divergence free. Our simulations are performed by means of a pseudo-spectral code with periodic boundary conditions. The time step is 0.1, the size of the system is 8 × 8 and distances in Fourier space are measured in units of 2=8 = 14 inverse units of lengths. Collocation points in real space deKne a numerical lattice M of size Nx × Ny . In what follows, Nx = Ny = N , and, depending on the scale at which the system is forced, either N = 128 or N = 256. Initial conditions are v = 0. The forcing is a white noise in time whose Fourier spectrum has Gaussian-distributed random phases and is supported on the region between the two squares centered at the origin and of side lengths 2(kf −1) and 2(kf +1), where kf deKnes the scale at which the forcing is applied. More precisely, the forcing F is solenoidal and the Fourier transform Fˆ x (k) (resp. Fˆ y (k)) of its x (resp. y) component is such that (X + i Y ) F0 A(|kx; y |) if kf − 1 6 |kx |; |ky | 6 kf + 1; Fˆ x; y (k) = 0 otherwise; X = −ln(1 − a1 )sin(2a2 ); Y = −ln(1 − a3 )cos(2a4 ); (1) where k = kx2 + ky2 , F0 is the amplitude of the forcing, the ai ’s are random variables uniformly distributed between 0 and 1, the function A(q) is given by A(q) = 0
if q = kf − 1 or q = kf or q = kf + 1; 4 A(kf ) A(kf + 1) 1 4 2 = 1− = 1− ; ; A(kf − 1) kf A(kf − 1) kf + 1
and A(kf − 1) is chosen such that the amount of energy injected into the system is independent on the value of kf and therefore only proportional to F0 . At any given time, we deKne the total energy in the system as 1 2 E= (vx2 + vy2 ) = 2 vˆx + vˆ2y ; N M
Mˆ
where vx and vy are the components of the velocity Keld v, vˆx and vˆy are their discrete Fourier transforms, and Mˆ is the numerical lattice in Fourier space. We also deKne the amount of energy below a given wave number k as 1 N E(k) = 2 vˆ2x + vˆ2y ; 0 6 k 6 ; N 2 2 2 2 kx +ky 6 k
and use this quantity to calculate an energy density , deKned by E(0) (0) = ; E(N=2) (k) =
E(k) − E(k − 1) ; E(N=2)
16k 6
N : 2
292
J. Lega, T. Passot / Fluid Dynamics Research 34 (2004) 289 – 297
Note that is proportional to the power spectrum of v and that E = E(N=2). We then measure a characteristic length scale L for each realization of the velocity Keld v according to P L = 4= k;
kP =
N=2
(k)k;
k=0
and deKne the corresponding Reynolds number as 1 2 v L ; v = 2 Re = vx + vy2 N M
(recall = 1). Since the value of F0 (set to 0.02) and the boundary conditions (periodic) are Kxed, the only two parameters in this study are kf , the wave number at which the system is forced, and , the dynamic viscosity of the ;uid. For kf Kxed and for small enough, an inverse cascade occurs, in which (i) the total energy in the system, E, increases linearly as a function of time, (ii) the fraction of energy below forcing, ¡ = E(kf − 3)=E;
200
0.986
180
0.982 ε <
E
tends to 100% as t increases, and (iii) kP decreases. This is illustrated in Fig. 1, where we show E, ¡ and kP as functions of time, in the case where = 0:00013 and kf = 30, and in Fig. 3, where we show the corresponding energy ;ux (solid line). There is indeed a deKnite region of wave number space below forcing where a roughly constant, although strongly ;uctuating, negative energy ;ux is present, indicating the presence of an inverse cascade. Fig. 1(d) displays the energy spectrum,
160
0.978
140 0.974 120 12000
14000
(a)
16000 time
12000
18000
14000
(b)
4
16000
18000
time 10
0
10
-5
3.8
δ(k+1)
_ k
3.6 3.4
-5/3
- 10
10
3.2 3 2.8 12000 (c)
10 14000
16000 time
- 15
1
18000 (d)
10
100
1000
mode index (k + 1)
P as Fig. 1. Plots of (a) the total energy E, (b) the fraction of energy below forcing ¡ , and (c) the mean wave vector k, functions of time. The energy density (k) at t =18000 is shown in (d) in log–log scale. For this run, =0:00013; kf =30, and N = 256.
J. Lega, T. Passot / Fluid Dynamics Research 34 (2004) 289 – 297 12
0.7
10
0.6 0.5 0.4
ε<
E
8 6
0.3
4
0.2
2 0
0.1 0 2000
(a)
6000
10000
0
14000 10
0
10
-5
10
- 10
10
- 15
δ(k+1)
28 26 24 22 0 2000
(c)
6000 10000 time
0 2000
(b)
time 30
_ k
293
14000
(d)
1
6000 10000 time
14000
10 100 mode index (k + 1)
1000
Fig. 2. Plots of (a) the total energy E, (b) the fraction of energy below forcing ¡ , and (c) the mean wave vector kP as functions of time. The energy density (k) at t = 12000 is shown in (d) in log–log scale. For this run, = 0:0002; kf = 30, and N = 256.
in which a k −5=3 slope is clearly visible below forcing. For larger values of , an inverse cascade does not occur but transfer of energy towards larger scales may still take place. In such a case, the quantities E, ¡ and kP asymptotically ;uctuate about a constant value for large times, 1 as illustrated in Fig. 2. If dissipation (i.e. if ) is further increased, the amount ¡ of energy transferred towards scales larger than that of the forcing becomes negligible. Fig. 3 shows time averages of the kinetic energy ;ux R(k)=N 3 as a function of the wave number k for three diFerent values of : = 0:00013 (as in Fig. 1), for which an inverse cascade occurs; =0:0002 (as in Fig. 2), for which there is only partial transfer of energy; and = 0:00033, for which there is almost no transfer of energy towards larger scales. The kinetic energy ;ux is deKned as (see for instance Lesieur (1997), Chapter VI) T (j); R(k) = − 06j 6k
where the kinetic energy transfer T (k) is obtained by averaging over an annulus of radius k in Fourier plane, S(k); T (k) = k −1¡k6k
1
In “borderline” cases, E appears to ;uctuate about a constant value, even though the corresponding energy ;ux, which also ;uctuates strongly, is roughly constant and negative at small k’s. We do not use the phrase “inverse cascade” to describe such situations, since dissipation at small k’s is still suUcient to balance inverse transfer of energy.
294
J. Lega, T. Passot / Fluid Dynamics Research 34 (2004) 289 – 297 40 20
Π(k)/ N3
0 −20 −40 −60 −80
0
20
40
60
80
100
120
140
k
Fig. 3. Plot of the time-averaged kinetic energy ;ux R(k)=N 3 for kf = 30, N = 256, and three diFerent values of . Solid line: = 0:00013 and the ;ux is averaged over 20 outputs; dashed line: = 0:0002, ;ux averaged over 6 outputs; dashed –dotted line: = 0:00033, ;ux averaged over 6 outputs.
and S(k) is given by (Rose and Sulem, 1978) S(k) = Jm ((k · uˆp )(uˆ∗k · uˆq ) + (k · uˆq )(uˆ∗k · uˆp )) ; p+q=k
where uˆk is the Fourier transform of the vector u = (ux ; uy ) at k = (kx ; ky ) and ∗ denotes complex conjugate. We see from Fig. 3 and the preceding ones that the amount of energy transferred towards larger scales is correlated with the presence of large negative ;uxes at small wave numbers. The above results can be summarized independently of the values of and kf . Fig. 4 shows ¡ as a function of Re for kf = 5 (diamonds and down-triangles), kf = 10 (squares), kf = 20 (up-triangles) and kf = 30 (circles), and for diFerent values of . We use temporal averages in the absence of an inverse cascade (Klled symbols), and instantaneous values when an inverse cascade occurs (open symbols). All the points, including those obtained from instantaneous values, appear to lie in the vicinity of the graph of a universal function E(Re) which has a steep gradient near Re = Rec 9. We can thus deKne Rec as the critical Reynolds number above which signiKcant energy transfer towards large scales takes place. We also note that as the amount of energy transferred to larger scales increases, the time-average of the velocity Keld solving the forced Navier–Stokes equation gets more and more structured. 3. Energy transfer, con nement and anisotropy We now turn to the question of relating ¡ to the initial Reynolds number deKned by ReI =
v f Lf ;
Lf = 4=kf ;
ε<
J. Lega, T. Passot / Fluid Dynamics Research 34 (2004) 289 – 297 100 90 80 70 60 50 40 30 20 10 0 0
295
60 50 40 30 20 10 0
80
160
0
2
240 Re
4
6
8
320
10
12
400
480
Fig. 4. Plot of the fraction of energy below forcing, ¡ , as a function of the Reynolds number, Re, for kf = 5 (diamonds for forcing (1) and down-triangles for forcing (2)), kf = 10 (squares), kf = 20 (up-triangles) and kf = 30 (circles). Open symbols indicate that an inverse cascade occurs. The dashed line is here to guide the eyes and represents a sketch of the graph of the function E (see text). The inset shows an enlargement of the graph for small Reynolds numbers. For these runs, N = 256 if kf = 30 and N = 128, otherwise.
100 90 80 70
ε<
60 50 40 30 20 10 0
0
1
2
3
4
5 6 Re I
7
8
9
10
Fig. 5. Plots of the fraction of energy below forcing, ¡ , as a function of the initial Reynolds number, ReI , for kf = 5 (diamonds and down-triangles), kf = 10 (squares), kf = 20 (triangles) and kf = 30 (circles). Open symbols indicate that an inverse cascade occurs. Solid curves correspond to the forcing given by (1) and the dashed curve to the anisotropic forcing (2). For these runs, N = 256 if kf = 30 and N = 128 otherwise.
where v f is the averaged magnitude of the velocity Keld vf induced by the forcing. We approximate vf by the velocity Keld solving the Navier–Stokes equation at a short time, speciKcally t = 100 (recall that our initial condition is v = 0). Our simulations indicate that for each value of kf ; ¡ is a function of ReI whose graph has a shape similar to that of E. However, the threshold value ReIc above which ¡ becomes signiKcant now depends on the properties of the forcing. As shown in Fig. 5, this threshold is displaced towards small values of the Reynolds number as kf is decreased. In other words, the critical value of the initial Reynolds number above which signi@cant energy transfer towards large scales takes place is reduced in con@ned systems. Moreover, for an anisotropic forcing
296
J. Lega, T. Passot / Fluid Dynamics Research 34 (2004) 289 – 297
Fa deKned by Fˆ ax (k) = %Fˆ x (k);
Fˆ ay (k) = (1 − %)Fˆ y (k)
(2)
with % = 0:75 and Fˆ x; y as in (1), this threshold is even lower, as shown by the dashed curve in Fig. 5.
4. Conclusions These numerical investigations indicate that in the forced two-dimensional Navier–Stokes equation, the amount ¡ of energy eventually transferred to scales larger than that of the forcing is a function of the asymptotic Reynolds number of the system only. The quantity ¡ can, however, also be thought of as a function of the initial Reynolds number, calculated from the velocity Keld initially imposed by the forcing. In this case, this function is parametrized by kf , whose reciprocal measures the spatial scale Lf at which the forcing is imposed. The eFect of conKnement, as well as that of anisotropy, is to increase the amount of energy transferred to larger scales, for a given value of ReI . This study is limited to the case of periodic boundary conditions, but we expect the main result, i.e. the existence of a universal curve giving the amount of energy transferred to larger scales as a function of the Reynolds number, to be valid for other types of boundary conditions, at least for systems of large enough size. The eFects of conKnement and anisotropy should also be similar to those described here, even for diFerent types of boundary conditions. This study may be relevant to the understanding of complex bacterial systems in the following sense. For bacterial colonies growing on agar plates, whirls and jets are only observed in regions wet enough to allow the bacteria to swim. Typically, this occurs in a relatively narrow band close to the colony boundary (Mendelson et al., 1999), and Knite-size eFects are therefore important. In the hydrodynamic model of (Lega and Passot (2003)), ;agellar activity at the bacterial level is modelled as a small-scale forcing. The isotropic or anisotropic nature of this forcing, as well as the scale at which it is imposed are currently diUcult to quantify. However, the above results indicate that if one can show that the microscopic bacterial dynamics leads to a forcing whose Reynolds number is of order of a few units, then partial transfer of energy is expected to take place. This would then provide a deKnite justiKcation for the whirls and jets observed in a variety of experiments involving bacteria (Mendelson et al., 1999; Kessler, 2000; Wu and Libchaber, 2000). A possible approach toward quantifying the microscopic forcing imposed by a dense collective of bacteria is to estimate the correlation length of the velocity Keld imposed by a collection of randomly distributed doublets. Such considerations are, however, beyond the scope of this paper and will be the topic of further investigation.
Acknowledgements This material is based upon work supported by the National Science Foundation under Grants No. 9909866 and 0075827 to J.L. and by CNRS (Centre National de la Recherche ScientiKque)/NSF Grant No. 9166 to T.P.
J. Lega, T. Passot / Fluid Dynamics Research 34 (2004) 289 – 297
297
References Bayly, B.J., Yakhot, V., 1986. Positive- and negative-eFective-viscosity phenomena in isotropic and anisotropic Beltrami ;ows. Phys. Rev. A 34, 381–391. Dubrulle, B., Frisch, U., 1991. Eddy viscosity of parity-invariant ;ows. Phys. Rev. A 43, 5355–5364. Frisch, U., She, Z.S., Sulem, P.L., 1987. Large-scale ;ow driven by the anisotropic kinetic alpha eFect. Physica D 28, 382–392. Gama, S., Vergassola, M., Frisch, U., 1994. Negative eddy viscosity in isotropically forced two-dimensional ;ow: linear and nonlinear dynamics. J. Fluid Mech. 260, 95–126. Hefer, D., Yakhot, V., 1989. Inverse energy cascade in a time-dependent ;ow. Phys. Fluids A 1, 1383–1386. Kessler, J.O., 2000. Dynamics of swimming bacteria at low and high volume fractions. in: Fiedler, B., GrVoger K., Sprekels, J. (Eds.), International Conference on DiFerential Equations, Vol. 2. World ScientiKc, Singapore. pp. 1284–1287. Kraichnan, R.H., 1976. Eddy viscosity in two and three dimensions. J. Atmos. Sci. 33, 1521–1536. Lega, J., Passot, T., 2003. Hydrodynamics of bacterial colonies: a model. Phys. Rev. E 67, 031906 1–18 Lesieur, M., 1997. Turbulence in Fluids. Kluwer Academic Publishers, Boston. Libin, A., Sivashinsky, G., Levich, E., 1987. Long-wave instability of periodic ;ows at large Reynolds numbers. Phys. Fluids 30, 2984–2986. Mendelson, N.H., Bourque, A., Wilkening, K., Anderson, K.R., Watkins, J.C., 1999. Organized cell swimming motions in Bacillus subtilis colonies: patterns of short-lived whirls and jets. J. Bacteriol. 181, 600–609. Rose, H.A., Sulem, P.L., 1978. Fully developed turbulence and statistical mechanics. J. Phys. (Paris) 39, 441–484. Sivashinsky, G., Yakhot, V., 1985. Negative viscosity eFect in large-scale ;ows. Phys. Fluids 28, 1040–1042. Sivashinsky, G.I., Frenkel, A.L., 1992. On negative eddy viscosity under conditions of isotropy. Phys. Fluids A 4, 1608–1610. Wu, X.-L., Libchaber, A., 2000. Particle diFusion in a quasi-two-dimensional bacterial bath. Phys. Rev. Lett. 84, 3017–3020. Yakhot, V., Pelz, R., 1987. Large-scale structure generation by anisotropic small-scale ;ows. Phys. Fluids 30, 1272–1277. Yakhot, V., Sivashinsky, G., 1987. Negative-viscosity phenomena in three-dimensional ;ows. Phys. Rev. A 35, 815–820. Zhang, X., Frenkel, A.L., 1998. Large-scale instability of generalized oscillating Kolmogorov ;ows. SIAM J. Appl. Math. 58, 540–564.
Inverse cascade and energy transfer in forced low-Reynolds number two-dimensional turbulence J. Legaa;∗ , T. Passotb a
Department of Mathematics, University of Arizona, building 89, 617 N. Santa Rita, P.O. Box 210089, Tucson, AZ 85721-0089, USA b CNRS, Observatoire de la Cˆote d’Azur, Boˆ1te Postale 4229; 06304 Nice Cedex 4, France Received 14 October 2003; received in revised form 1 February 2004; accepted 11 February 2004 Communicated by M.-E. Brachet
Abstract Using numerical simulations of the forced two-dimensional Navier–Stokes equation, it is shown that the amount of energy transferred to large scales is related to the Reynolds number in a unique fashion. It is also observed that the critical value of the initial Reynolds number for the onset of an inverse cascade is lowered as the scale of the forcing approaches the size of the system, or in the presence of anisotropy. This study is motivated by recent experiments with bacterial colonies, and their description in terms of a hydrodynamic model. c 2004 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. PACS: 47.27.Cn; 47.27.Eq Keywords: Two-dimensional turbulence; Inverse cascade; Biological ;ows
1. Introduction This work is motivated by recent experiments (Mendelson et al., 1999; Kessler, 2000) with colonies of Bacillus subtilis growing on ;at surfaces, in which collective behaviors of bacteria are observed in the form of whirls and jets. These structures are dynamic; occur at a scale much larger (50–100 times) than that of an individual bacterium; alternate in time (cycles consisting of a clockwise whirl, a jet, a counterclockwise whirl and a jet in opposite direction are observed; each cycle takes about 1 s to complete); and cease to exit if bacteria die or stop swimming (Mendelson et al., 1999). We recently introduced a hydrodynamic model (Lega and Passot, 2003) in which ∗
Corresponding author. Tel.: +1-520-621-4350; fax: +1-520-621-8522. E-mail addresses: [email protected] (J. Lega), [email protected] (T. Passot).
c 2004 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. 0169-5983/$30.00 All rights reserved. doi:10.1016/j.;uiddyn.2004.02.002
290
J. Lega, T. Passot / Fluid Dynamics Research 34 (2004) 289 – 297
the dynamics of a bacterial colony and of its boundary is described by advection–reaction– diFusion equations coupled to a two-dimensional hydrodynamic equation for the complex ;uid made of bacteria and water. In such a macroscopic model, bacterial activity is globally described by a small-scale forcing in the hydrodynamic equation. Our numerical simulations have shown that partial transfer of energy towards large scales takes place in such a system, even at relatively low-Reynolds numbers. This leads to the formation of structures which are reminiscent of the whirls and jets observed in experiments (Mendelson et al., 1999; Kessler, 2000). However, collectives of bacteria are typically very-low-Reynolds-number systems, even though recent experiments suggest that hydrodynamic eFects may be important (Mendelson et al., 1999; Kessler, 2000; Wu and Libchaber, 2000). In order to validate the use of a hydrodynamic model to describe the whirls and jets seen in the experiments, it is therefore necessary to estimate the minimum value of the Reynolds number for which energy transfer takes place in the forced low-Reynolds number two-dimensional Navier–Stokes equation. This paper addresses this question by means of a series of numerical experiments, and also describes eFects which may promote energy transfer at even lower Reynolds numbers. In the two-dimensional Navier–Stokes equation, inverse transfer of energy typically leads to the formation of large-scale vortices. The existence of such an inverse cascade in two- and three dimensions is often understood as resulting from a negative total (molecular plus eddy) viscosity for large-scale secondary ;ows, which thus grow from random perturbations of the small-scale ;ow (Kraichnan, 1976). In particular, anisotropy and asymmetry in space and time of the basic ;ow (Sivashinsky and Yakhot, 1985; Bayly and Yakhot, 1986; Yakhot and Sivashinsky, 1987; Libin et al., 1987; Yakhot and Pelz, 1987; Frisch et al., 1987; Dubrulle and Frisch, 1991), as well as long-lived small-scale ;uctuations (Hefer and Yakhot, 1989; Zhang and Frenkel, 1998) are important, although not necessary (Sivashinsky and Frenkel, 1992; Gama et al., 1994) factors for such an instability to develop. In this paper, we report on numerical experiments of forced two-dimensional turbulence in which we measure the percentage of energy shared by Fourier modes with wave number smaller than that of the forcing. We show that this quantity is a function of the Reynolds number of the system, whether or not an inverse cascade occurs. We also study the eFect of conKnement (i.e. of making the scale at which the forcing is applied comparable to the size of the system) and of anisotropy on the relationship between the Reynolds number initially imposed by the forcing and the amount of energy transferred to larger scales. Our results indicate that one of the eFects of conKnement and/or anisotropy is to decrease the critical value of the initial Reynolds number above which an inverse cascade occurs. We then discuss how these results may be relevant to the modeling of bacterial colonies. 2. Partial energy transfer and transition to turbulence We consider the two-dimensional Navier–Stokes equation for an incompressible ;uid of density = 1 and dynamic viscosity . This equation reads @v + (v · ∇)v = −∇p + ∇2 v + F; @t ∇ · v = 0;
J. Lega, T. Passot / Fluid Dynamics Research 34 (2004) 289 – 297
291
where F is a forcing and the pressure p is determined to ensure that the vector Keld v remains divergence free. Our simulations are performed by means of a pseudo-spectral code with periodic boundary conditions. The time step is 0.1, the size of the system is 8 × 8 and distances in Fourier space are measured in units of 2=8 = 14 inverse units of lengths. Collocation points in real space deKne a numerical lattice M of size Nx × Ny . In what follows, Nx = Ny = N , and, depending on the scale at which the system is forced, either N = 128 or N = 256. Initial conditions are v = 0. The forcing is a white noise in time whose Fourier spectrum has Gaussian-distributed random phases and is supported on the region between the two squares centered at the origin and of side lengths 2(kf −1) and 2(kf +1), where kf deKnes the scale at which the forcing is applied. More precisely, the forcing F is solenoidal and the Fourier transform Fˆ x (k) (resp. Fˆ y (k)) of its x (resp. y) component is such that (X + i Y ) F0 A(|kx; y |) if kf − 1 6 |kx |; |ky | 6 kf + 1; Fˆ x; y (k) = 0 otherwise; X = −ln(1 − a1 )sin(2a2 ); Y = −ln(1 − a3 )cos(2a4 ); (1) where k = kx2 + ky2 , F0 is the amplitude of the forcing, the ai ’s are random variables uniformly distributed between 0 and 1, the function A(q) is given by A(q) = 0
if q = kf − 1 or q = kf or q = kf + 1; 4 A(kf ) A(kf + 1) 1 4 2 = 1− = 1− ; ; A(kf − 1) kf A(kf − 1) kf + 1
and A(kf − 1) is chosen such that the amount of energy injected into the system is independent on the value of kf and therefore only proportional to F0 . At any given time, we deKne the total energy in the system as 1 2 E= (vx2 + vy2 ) = 2 vˆx + vˆ2y ; N M
Mˆ
where vx and vy are the components of the velocity Keld v, vˆx and vˆy are their discrete Fourier transforms, and Mˆ is the numerical lattice in Fourier space. We also deKne the amount of energy below a given wave number k as 1 N E(k) = 2 vˆ2x + vˆ2y ; 0 6 k 6 ; N 2 2 2 2 kx +ky 6 k
and use this quantity to calculate an energy density , deKned by E(0) (0) = ; E(N=2) (k) =
E(k) − E(k − 1) ; E(N=2)
16k 6
N : 2
292
J. Lega, T. Passot / Fluid Dynamics Research 34 (2004) 289 – 297
Note that is proportional to the power spectrum of v and that E = E(N=2). We then measure a characteristic length scale L for each realization of the velocity Keld v according to P L = 4= k;
kP =
N=2
(k)k;
k=0
and deKne the corresponding Reynolds number as 1 2 v L ; v = 2 Re = vx + vy2 N M
(recall = 1). Since the value of F0 (set to 0.02) and the boundary conditions (periodic) are Kxed, the only two parameters in this study are kf , the wave number at which the system is forced, and , the dynamic viscosity of the ;uid. For kf Kxed and for small enough, an inverse cascade occurs, in which (i) the total energy in the system, E, increases linearly as a function of time, (ii) the fraction of energy below forcing, ¡ = E(kf − 3)=E;
200
0.986
180
0.982 ε <
E
tends to 100% as t increases, and (iii) kP decreases. This is illustrated in Fig. 1, where we show E, ¡ and kP as functions of time, in the case where = 0:00013 and kf = 30, and in Fig. 3, where we show the corresponding energy ;ux (solid line). There is indeed a deKnite region of wave number space below forcing where a roughly constant, although strongly ;uctuating, negative energy ;ux is present, indicating the presence of an inverse cascade. Fig. 1(d) displays the energy spectrum,
160
0.978
140 0.974 120 12000
14000
(a)
16000 time
12000
18000
14000
(b)
4
16000
18000
time 10
0
10
-5
3.8
δ(k+1)
_ k
3.6 3.4
-5/3
- 10
10
3.2 3 2.8 12000 (c)
10 14000
16000 time
- 15
1
18000 (d)
10
100
1000
mode index (k + 1)
P as Fig. 1. Plots of (a) the total energy E, (b) the fraction of energy below forcing ¡ , and (c) the mean wave vector k, functions of time. The energy density (k) at t =18000 is shown in (d) in log–log scale. For this run, =0:00013; kf =30, and N = 256.
J. Lega, T. Passot / Fluid Dynamics Research 34 (2004) 289 – 297 12
0.7
10
0.6 0.5 0.4
ε<
E
8 6
0.3
4
0.2
2 0
0.1 0 2000
(a)
6000
10000
0
14000 10
0
10
-5
10
- 10
10
- 15
δ(k+1)
28 26 24 22 0 2000
(c)
6000 10000 time
0 2000
(b)
time 30
_ k
293
14000
(d)
1
6000 10000 time
14000
10 100 mode index (k + 1)
1000
Fig. 2. Plots of (a) the total energy E, (b) the fraction of energy below forcing ¡ , and (c) the mean wave vector kP as functions of time. The energy density (k) at t = 12000 is shown in (d) in log–log scale. For this run, = 0:0002; kf = 30, and N = 256.
in which a k −5=3 slope is clearly visible below forcing. For larger values of , an inverse cascade does not occur but transfer of energy towards larger scales may still take place. In such a case, the quantities E, ¡ and kP asymptotically ;uctuate about a constant value for large times, 1 as illustrated in Fig. 2. If dissipation (i.e. if ) is further increased, the amount ¡ of energy transferred towards scales larger than that of the forcing becomes negligible. Fig. 3 shows time averages of the kinetic energy ;ux R(k)=N 3 as a function of the wave number k for three diFerent values of : = 0:00013 (as in Fig. 1), for which an inverse cascade occurs; =0:0002 (as in Fig. 2), for which there is only partial transfer of energy; and = 0:00033, for which there is almost no transfer of energy towards larger scales. The kinetic energy ;ux is deKned as (see for instance Lesieur (1997), Chapter VI) T (j); R(k) = − 06j 6k
where the kinetic energy transfer T (k) is obtained by averaging over an annulus of radius k in Fourier plane, S(k); T (k) = k −1¡k6k
1
In “borderline” cases, E appears to ;uctuate about a constant value, even though the corresponding energy ;ux, which also ;uctuates strongly, is roughly constant and negative at small k’s. We do not use the phrase “inverse cascade” to describe such situations, since dissipation at small k’s is still suUcient to balance inverse transfer of energy.
294
J. Lega, T. Passot / Fluid Dynamics Research 34 (2004) 289 – 297 40 20
Π(k)/ N3
0 −20 −40 −60 −80
0
20
40
60
80
100
120
140
k
Fig. 3. Plot of the time-averaged kinetic energy ;ux R(k)=N 3 for kf = 30, N = 256, and three diFerent values of . Solid line: = 0:00013 and the ;ux is averaged over 20 outputs; dashed line: = 0:0002, ;ux averaged over 6 outputs; dashed –dotted line: = 0:00033, ;ux averaged over 6 outputs.
and S(k) is given by (Rose and Sulem, 1978) S(k) = Jm ((k · uˆp )(uˆ∗k · uˆq ) + (k · uˆq )(uˆ∗k · uˆp )) ; p+q=k
where uˆk is the Fourier transform of the vector u = (ux ; uy ) at k = (kx ; ky ) and ∗ denotes complex conjugate. We see from Fig. 3 and the preceding ones that the amount of energy transferred towards larger scales is correlated with the presence of large negative ;uxes at small wave numbers. The above results can be summarized independently of the values of and kf . Fig. 4 shows ¡ as a function of Re for kf = 5 (diamonds and down-triangles), kf = 10 (squares), kf = 20 (up-triangles) and kf = 30 (circles), and for diFerent values of . We use temporal averages in the absence of an inverse cascade (Klled symbols), and instantaneous values when an inverse cascade occurs (open symbols). All the points, including those obtained from instantaneous values, appear to lie in the vicinity of the graph of a universal function E(Re) which has a steep gradient near Re = Rec 9. We can thus deKne Rec as the critical Reynolds number above which signiKcant energy transfer towards large scales takes place. We also note that as the amount of energy transferred to larger scales increases, the time-average of the velocity Keld solving the forced Navier–Stokes equation gets more and more structured. 3. Energy transfer, con nement and anisotropy We now turn to the question of relating ¡ to the initial Reynolds number deKned by ReI =
v f Lf ;
Lf = 4=kf ;
ε<
J. Lega, T. Passot / Fluid Dynamics Research 34 (2004) 289 – 297 100 90 80 70 60 50 40 30 20 10 0 0
295
60 50 40 30 20 10 0
80
160
0
2
240 Re
4
6
8
320
10
12
400
480
Fig. 4. Plot of the fraction of energy below forcing, ¡ , as a function of the Reynolds number, Re, for kf = 5 (diamonds for forcing (1) and down-triangles for forcing (2)), kf = 10 (squares), kf = 20 (up-triangles) and kf = 30 (circles). Open symbols indicate that an inverse cascade occurs. The dashed line is here to guide the eyes and represents a sketch of the graph of the function E (see text). The inset shows an enlargement of the graph for small Reynolds numbers. For these runs, N = 256 if kf = 30 and N = 128, otherwise.
100 90 80 70
ε<
60 50 40 30 20 10 0
0
1
2
3
4
5 6 Re I
7
8
9
10
Fig. 5. Plots of the fraction of energy below forcing, ¡ , as a function of the initial Reynolds number, ReI , for kf = 5 (diamonds and down-triangles), kf = 10 (squares), kf = 20 (triangles) and kf = 30 (circles). Open symbols indicate that an inverse cascade occurs. Solid curves correspond to the forcing given by (1) and the dashed curve to the anisotropic forcing (2). For these runs, N = 256 if kf = 30 and N = 128 otherwise.
where v f is the averaged magnitude of the velocity Keld vf induced by the forcing. We approximate vf by the velocity Keld solving the Navier–Stokes equation at a short time, speciKcally t = 100 (recall that our initial condition is v = 0). Our simulations indicate that for each value of kf ; ¡ is a function of ReI whose graph has a shape similar to that of E. However, the threshold value ReIc above which ¡ becomes signiKcant now depends on the properties of the forcing. As shown in Fig. 5, this threshold is displaced towards small values of the Reynolds number as kf is decreased. In other words, the critical value of the initial Reynolds number above which signi@cant energy transfer towards large scales takes place is reduced in con@ned systems. Moreover, for an anisotropic forcing
296
J. Lega, T. Passot / Fluid Dynamics Research 34 (2004) 289 – 297
Fa deKned by Fˆ ax (k) = %Fˆ x (k);
Fˆ ay (k) = (1 − %)Fˆ y (k)
(2)
with % = 0:75 and Fˆ x; y as in (1), this threshold is even lower, as shown by the dashed curve in Fig. 5.
4. Conclusions These numerical investigations indicate that in the forced two-dimensional Navier–Stokes equation, the amount ¡ of energy eventually transferred to scales larger than that of the forcing is a function of the asymptotic Reynolds number of the system only. The quantity ¡ can, however, also be thought of as a function of the initial Reynolds number, calculated from the velocity Keld initially imposed by the forcing. In this case, this function is parametrized by kf , whose reciprocal measures the spatial scale Lf at which the forcing is imposed. The eFect of conKnement, as well as that of anisotropy, is to increase the amount of energy transferred to larger scales, for a given value of ReI . This study is limited to the case of periodic boundary conditions, but we expect the main result, i.e. the existence of a universal curve giving the amount of energy transferred to larger scales as a function of the Reynolds number, to be valid for other types of boundary conditions, at least for systems of large enough size. The eFects of conKnement and anisotropy should also be similar to those described here, even for diFerent types of boundary conditions. This study may be relevant to the understanding of complex bacterial systems in the following sense. For bacterial colonies growing on agar plates, whirls and jets are only observed in regions wet enough to allow the bacteria to swim. Typically, this occurs in a relatively narrow band close to the colony boundary (Mendelson et al., 1999), and Knite-size eFects are therefore important. In the hydrodynamic model of (Lega and Passot (2003)), ;agellar activity at the bacterial level is modelled as a small-scale forcing. The isotropic or anisotropic nature of this forcing, as well as the scale at which it is imposed are currently diUcult to quantify. However, the above results indicate that if one can show that the microscopic bacterial dynamics leads to a forcing whose Reynolds number is of order of a few units, then partial transfer of energy is expected to take place. This would then provide a deKnite justiKcation for the whirls and jets observed in a variety of experiments involving bacteria (Mendelson et al., 1999; Kessler, 2000; Wu and Libchaber, 2000). A possible approach toward quantifying the microscopic forcing imposed by a dense collective of bacteria is to estimate the correlation length of the velocity Keld imposed by a collection of randomly distributed doublets. Such considerations are, however, beyond the scope of this paper and will be the topic of further investigation.
Acknowledgements This material is based upon work supported by the National Science Foundation under Grants No. 9909866 and 0075827 to J.L. and by CNRS (Centre National de la Recherche ScientiKque)/NSF Grant No. 9166 to T.P.
J. Lega, T. Passot / Fluid Dynamics Research 34 (2004) 289 – 297
297
References Bayly, B.J., Yakhot, V., 1986. Positive- and negative-eFective-viscosity phenomena in isotropic and anisotropic Beltrami ;ows. Phys. Rev. A 34, 381–391. Dubrulle, B., Frisch, U., 1991. Eddy viscosity of parity-invariant ;ows. Phys. Rev. A 43, 5355–5364. Frisch, U., She, Z.S., Sulem, P.L., 1987. Large-scale ;ow driven by the anisotropic kinetic alpha eFect. Physica D 28, 382–392. Gama, S., Vergassola, M., Frisch, U., 1994. Negative eddy viscosity in isotropically forced two-dimensional ;ow: linear and nonlinear dynamics. J. Fluid Mech. 260, 95–126. Hefer, D., Yakhot, V., 1989. Inverse energy cascade in a time-dependent ;ow. Phys. Fluids A 1, 1383–1386. Kessler, J.O., 2000. Dynamics of swimming bacteria at low and high volume fractions. in: Fiedler, B., GrVoger K., Sprekels, J. (Eds.), International Conference on DiFerential Equations, Vol. 2. World ScientiKc, Singapore. pp. 1284–1287. Kraichnan, R.H., 1976. Eddy viscosity in two and three dimensions. J. Atmos. Sci. 33, 1521–1536. Lega, J., Passot, T., 2003. Hydrodynamics of bacterial colonies: a model. Phys. Rev. E 67, 031906 1–18 Lesieur, M., 1997. Turbulence in Fluids. Kluwer Academic Publishers, Boston. Libin, A., Sivashinsky, G., Levich, E., 1987. Long-wave instability of periodic ;ows at large Reynolds numbers. Phys. Fluids 30, 2984–2986. Mendelson, N.H., Bourque, A., Wilkening, K., Anderson, K.R., Watkins, J.C., 1999. Organized cell swimming motions in Bacillus subtilis colonies: patterns of short-lived whirls and jets. J. Bacteriol. 181, 600–609. Rose, H.A., Sulem, P.L., 1978. Fully developed turbulence and statistical mechanics. J. Phys. (Paris) 39, 441–484. Sivashinsky, G., Yakhot, V., 1985. Negative viscosity eFect in large-scale ;ows. Phys. Fluids 28, 1040–1042. Sivashinsky, G.I., Frenkel, A.L., 1992. On negative eddy viscosity under conditions of isotropy. Phys. Fluids A 4, 1608–1610. Wu, X.-L., Libchaber, A., 2000. Particle diFusion in a quasi-two-dimensional bacterial bath. Phys. Rev. Lett. 84, 3017–3020. Yakhot, V., Pelz, R., 1987. Large-scale structure generation by anisotropic small-scale ;ows. Phys. Fluids 30, 1272–1277. Yakhot, V., Sivashinsky, G., 1987. Negative-viscosity phenomena in three-dimensional ;ows. Phys. Rev. A 35, 815–820. Zhang, X., Frenkel, A.L., 1998. Large-scale instability of generalized oscillating Kolmogorov ;ows. SIAM J. Appl. Math. 58, 540–564.