How close are shell models to the 3D Navier-Stokes equations?
HHow close are shell models to the D Navier–Stokes equations?
Dario Vincenzi ‡ and John D Gibbon Universit´e Cˆote d’Azur, CNRS, LJAD, 06100 Nice, France Department of Mathematics, Imperial College London, London SW7 2AZ, UKE-mail: [email protected], [email protected]
Abstract.
Shell models have found wide application in the study of hydrodynamicturbulence because they are easily solved numerically even at very large Reynoldsnumbers. Although bereft of spatial variation, they accurately reproduce themain statistical properties of fully-developed homogeneous and isotropic turbulence.Moreover, they enjoy regularity properties which still remain open for the three-dimensional (3 D ) Navier–Stokes equations (NSEs). The goal of this study is to makea rigorous comparison between shell models and the NSEs. It turns out that onlythe estimate of the mean energy dissipation rate is the same in both systems. Theestimates of the velocity and its higher-order derivatives display a weaker Reynoldsnumber dependence for shell models than for the 3 D NSEs. Indeed, the velocity-derivative estimates for shell models are found to be equivalent to those correspondingto a velocity gradient averaged version of the 3 D Navier–Stokes equations (VGA-NSEs), while the velocity estimates are even milder. Numerical simulations over awide range of Reynolds numbers confirm the estimates for shell models.
1. Introduction
Three-dimensional (3 D ) incompressible turbulent flows are characterized by a cascadeof kinetic energy from the length scales at which the flow is generated to the scales atwhich viscous dissipation becomes predominant [1–4]. Kinetic energy is usually injectedat large scale by a body forcing or the boundary conditions, and it is transferred at aconstant rate to smaller scales by nonlinear interactions between the Fourier modes ofthe velocity. The cascade is strongly dissipated when the viscous-dissipation range isreached. The range between the forcing and viscous scales is known as inertial rangeand is characterized by a kinetic-energy spectrum of the form E ( k ) ∼ k − / , where k is the wavenumber. As viscosity is reduced, the dissipation range shrinks, while theinertial range extends to smaller and smaller length scales. The consequence of this isthat, in the limit of vanishing viscosity, the mean energy dissipation rate tends to anonzero value [5, 6], which is known as the ‘dissipative anomaly’.A mathematically rigorous description of the generation of small scales in turbulent3 D Navier–Stokes flows can be achieved by considering the L m -norms of the velocity ‡ Also Associate, International Centre for Theoretical Sciences, Tata Institute of FundamentalResearch, Bangalore 560089, India. a r X i v : . [ n li n . C D ] S e p hell models and the Navier–Stokes equations u over a periodic cube V = [0 , L ] , volume integrals and norms can bedefined as H n,m ( t ) = (cid:90) V | ∇ n u | m d V . (1)The well known scaling property of the NSEs u ( x , t ) → µ − u ( x /µ, t/µ ) suggests thedefinition of a doubly-labelled set of dimensionless, invariant quantities [10] F n,m ( t ) = ν − L /α m,m H / mn,m , (2)for 0 (cid:54) n < ∞ and 1 (cid:54) m (cid:54) ∞ , where ν is the kinematic viscosity and α n,m = 2 m m ( n + 1) − . (3)It was shown in [9, 10] that for 1 (cid:54) n < ∞ and 1 (cid:54) m (cid:54) ∞ , together with n = 0for 3 < m (cid:54) ∞ (cid:10) F α n,m n,m (cid:11) T (cid:54) c n,m Re + O (cid:0) T − (cid:1) ( Re (cid:29) , (4)where Re is the Reynolds number and the time average up to time T > (cid:104)·(cid:105) T = T − (cid:90) T · d t . (5)Moreover, the set of estimates in (4) encompasses all the known a priori bounds forweak solutions of the 3 D NSEs equations and shows how these bounds arise naturallyfrom scale invariance [9, 10]. In those references it has been shown that a hierarchy ofspatially averaged length scales (cid:96) n,m ( t ) can be constructed from the F n,m in the followingmanner : (cid:0) L(cid:96) − n,m (cid:1) n +1 := F n,m . (6)Higher values of n allow the detection of smaller scales, while higher values of m accountfor stronger deviations from the mean. Using (4) and (6), followed by a H¨older inequality,one finds that (cid:10) L(cid:96) − n,m (cid:11) T (cid:54) c n,m Re n +1) αn,m ( Re (cid:29) . (7)As noted in [9], while the estimate for the first in the hierarchy is Re / and is consistentwith the inverse Kolmogorov length, the limit as n, m → ∞ is finite and is consistentwith the fact that viscosity ultimately dissipates the cascade of energy § .The Re dependence of the moments of ∇ u has been studied within the multifractalformalism [1, 11, 12] and in numerical simulations of both the 3 D NSE [13] and theBurgers equation [14]. The calculation of H n,m for high Re and large values of n and m nonetheless requires large numerical simulations (see [15] for the n = 1 case). At high Re , indeed, the injection and viscous-dissipation scales are widely separated, and thecascade process activates a wide range of length scales : an empirical argument due to § Strictly speaking there is a limit to the value of Re beyond which kinetic scales are reached and theNSEs become invalid. hell models and the Navier–Stokes equations Re / (see also [1]). For this reason, the direct numerical simulationof high- Re flows have remained a great challenge [17–21]. In order to study the propertiesof fully developed turbulence, simplified models have thus been introduced that retainsome of the properties of the NSEs but are much more tractable both theoretically andnumerically. Among these, shell models of the energy cascade have played a majorrole [1, 22–24]. They consist of a system of ordinary differential equations for a set ofcomplex scalar variables which can be regarded loosely as the amplitudes of the Fouriercomponents of the velocity field. The structure of the equations mimics that of theNSEs in Fourier space. The nonlinear part has a form that recalls the vortex-stretchingterm, but the interactions between the velocity variables are local. A linear small-scaledissipation and generally a forcing are also included. Because of their scalar nature,shell models are unable to provide information on the spatial structure of the velocityfield. However, they successfully reproduce the statistical properties of space-averagedquantities, such as the kinetic-energy spectrum, the velocity structure functions or theviscous-dissipation rate in isotropic turbulence. From a mathematical point of view,stronger results have been proved for shell models than for the 3 D NSEs : for instance,the global regularity of strong solutions and the existence of a finite-dimensional inertialmanifold [25, 26] (see [27] for analogous results on stochastic shell models).Questions still remain, however, over how close the mathematical results of shell-models are to those for the NSEs. Shell-models are bereft of spatial variation whilethe behaviour of solutions of the NSEs equations differ widely depending upon theirdimension. Indeed, although the notion of velocity gradient as a spatio-temporal fieldin shell models is not available, it is easy to define the analogue of the volume integral ofpowers of the velocity derivatives. For instance, the shell-model analogues of enstrophyand helicity have been studied extensively [22–24]. Nevertheless, it is not clear wherethe exact correspondence lies. The goal of this paper is to investigate the analogue of H n,m for shell models, both mathematically and numerically, in relation to the NSEsequations to see if there is a consistent correspondence between the two.The paper is organized in the following steps. Section 2 introduces the shell modeland the mathematical framework. We consider the ‘Sabra’ model [28], but the resultsare general and, in particular, also hold for the GOY model [29, 30], the only differencebeing in the constants that appear in the estimates.The starting point of our study is a bound for the mean energy-dissipation rate,which corresponds to the m = n = 1 case. This is obtained in section 3 by adapting toshell models the methods used by Doering and Foias [31] for the NSEs (see also [32,33] forthe application of the same methods to magnetohydrodynamics and binary mixtures).The estimate for the dissipation rate coincide, as expected, with that for the 3 D NSEs.In sections 4.1 and 4.2, we keep m = 1 but move to general n , i.e. we study H n ≡ H n, , the analogue of the L -norm of the n -th order derivative of the velocity. Wefirst prove two differential relations connecting H n and H n +1 , in the spirit of the “ladder” hell models and the Navier–Stokes equations H n and to estimate the time average (cid:28) H n +1 n (cid:29) T interms of Re . This latter result is the counterpart of a bound proved by Foias, Guillop´eand Temam [38] for weak solutions of the 3 D NSEs. It is further extended to general n and m in section 4.3, where it is shown that, in terms of the α n,m defined above in (3)and (4), the shell-model equivalent is α n,m = 4 n + 1 , (8)which is independent of m .The form of these bounds and, in particular, their insensitivity to m , raise thequestion of how close these results are to the NSEs in any dimension. Comparing (8)with (3), we find that α n,m is greater for shell models than for the 3 D NSEs for all n > (cid:54) m (cid:54)
0. Therefore, the Re dependence of the high-order velocityderivatives differs in the two systems and, in shell models, is significantly weaker. Itis indeed discovered in section 5 that, as far as the velocity-derivative estimates areconcerned, the real PDE-equivalent of the shell models considered here is not the full3 D NSEs themselves but a version of these that we have called the ‘velocity gradientaveraged Navier–Stokes equations’ (VGA-NSEs). While less specific in its definition asintermittency in multi-fractal theories [1], intermittent events in solutions of the NSEshave the property that excursions in ∇ u depart strongly from its average (cid:107) ∇ u (cid:107) ,thereby implying that for very short periods of time L / (cid:107) ∇ u (cid:107) ∞ (cid:107) ∇ u (cid:107) (cid:29) , (9)whereas making the approximation L / (cid:107) ∇ u (cid:107) ∞ (cid:107) ∇ u (cid:107) = 1 , (10)has the effect of suppressing strong events in ∇ u . The VGA-NSEs are obtained byusing (10) in the differential inequalities for the NSEs. In fact, it can be thought of inthe following way : a Gagliardo–Nirenberg inequality shows that (cid:107) ∇ u (cid:107) ∞ (cid:107) ∇ u (cid:107) (cid:54) c n κ / n , κ n = (cid:18) (cid:107) ∇ n +1 u (cid:107) (cid:107) ∇ u (cid:107) (cid:19) /n . (11)The wave-number κ n ( t ) behaves as a higher moment of the enstrophy spectrum and hasa lower bound expressed as L − ≤ κ n ( t ). (10) occurs when one uses only the minimumof the right-hand side of the inequality in (11). One of the main results of this paperis that for all n (cid:62) (cid:54) m (cid:54) ∞ , the bounds and the exponents in the variousbounded time-averages of VGA-NSEs and the shell models are equivalent.The velocity estimates for the shell models are even milder than those for theVGA-NSEs. Indeed, it is shown that for the VGA-NSEs (cid:10) (cid:107) u (cid:107) m (cid:11) T (cid:54) c m ( νL − ) Re ( m (cid:62) , (12) hell models and the Navier–Stokes equations (cid:104)(cid:107) u (cid:107) m (cid:105) T scale as Re .Finally, section 6 concludes the paper by summarizing the estimates for shell modelsand comparing them with numerical simulations of the ‘Sabra’ model over a wide rangeof values of Re .
2. The ‘Sabra’ shell model
Shell models of turbulence describe the velocity field by means of a sequence of complexvariables u j , j = 1 , , , . . . , which represent its Fourier components. In the ‘Sabra’model [28], the variables u j satisfy the following equations :˙ u j = i( ak j +1 u ∗ j +1 u j +2 + bk j u j +1 u ∗ j − − ck j − u j − u j − ) − νk j u j + f j , (13)where k j = k λ j ( k > λ >
1) are logarithmically-spaced wave numbers, ν is thekinematic viscosity, and the f j are complex and represent the Fourier amplitudes of theforcing. The ‘boundary conditions’ for the velocity variables are u = u − = 0, while k − plays the role of the largest spatial scale in the system. The coefficients a , b , c arereal and satisfy a + b + c = 0 . (14)This condition ensures that the kinetic energy, E = ∞ (cid:88) j =1 | u j | , (15)is conserved when ν = 0 and f j = 0 for all j . In the inviscid, unforced case andunder condition (14), the shell model also possesses a second quadratic invariant, whichfor suitable values of a , b , c can be interpreted as either a generalized helicity or ageneralized enstrophy [22]. The parameters of the shell model can also be tuned so as togenerate an inverse cascade of energy from small to large scales, as in two-dimensionalturbulence [39]. In the following, however, we shall not impose any additional constrainton a , b , c other than (14).Various forcings have been considered in the literature, such as those that act onlyon few low- j shells and mimic the injection of energy at large scales [28, 30], those thatimpose a constant energy input [40], or those with a power-law ‘spectrum’ [41, 42]. Weconsider a constant-in-time deterministic forcing, but the results are easily generalizedto time-dependent f j . Following [31], we define the forcing in a way as to isolate itsmagnitude from its shape. We take f j = F φ j − j f , (16)where F is a complex constant, j f (cid:62)
1, and the shape function φ p is such that φ p = 0if p <
0. Thus, k f = k λ j f is the characteristic wavenumber of the forcing. We alsoassume ∞ (cid:88) p =0 | φ p | < ∞ (17) hell models and the Navier–Stokes equations ∞ (cid:88) p =0 λ − p | φ p | = 1 . (18)The assumption in (17) means that the ‘energy’ of the forcing is finite, while (18) isa normalization condition on the shape function. In sections 4 and 5, we shall furtherrequire that the forcing has a maximum wavenumber k max .Under assumption (17), it was shown in [25] that if the energy E is bounded attime t = 0, then it stays bounded at any later time. This allows us to define the rootmean square velocity U = (cid:104) E (cid:105) / T , (19)where the time average up to time T has been introduced in (5). In addition, thetime-averaged dissipation rate (cid:15) = ν (cid:42) ∞ (cid:88) j =1 k j | u j | (cid:43) T (20)is also bounded for all T > U , k f , and | F | , we can then define theReynolds and Grashof numbers as Re = Uνk f and Gr = | F | ν k f , (21)respectively. The latter is a dimensionless measure of the forcing, while the formerquantifies the response of the system.Finally, we note that it is possible to introduce a suitable functional setting forthe study of (13), in which a solution u = ( u , u . . . ) is regarded as an elementof the sequences space (cid:96) over the field of complex numbers, with scalar product( u , v ) = (cid:80) ∞ j =1 u j v ∗ j for any u , v ∈ (cid:96) [25]. Here, however, we follow the physicalnotation and work with the variables u j directly.
3. The time-averaged energy dissipation rate
An estimate of the time-averaged energy dissipation rate (cid:15) , defined in (20), is an essentialelement of the present study as results on the high-order derivatives of the velocity arebased on this. It was once conventional to write estimates for (cid:15) in terms of the Grashofnumber Gr until Doering and Foias [31] introduced a method that converted these intoestimates in terms of the Reynolds number Re , which is much more useful for comparisonwith other theories of turbulence. The methods used here are adapted from Doeringand Foias [31].Let us first introduce the constants that will appear in the estimate for (cid:15) : A = | a | λ + | b | + | a + b | λ − , B γ = sup p (cid:62) λ − (2 γ − p | φ p | , C γ = ∞ (cid:88) p =0 λ γp | φ p | . (22) hell models and the Navier–Stokes equations A is a function of the parameters of the shell model, while B γ and C γ are fixed bythe shape of the forcing. The exponent γ is a real number and must be such that B γ and C γ are finite; in particular, the normalization condition in (18) implies C − = 1.It is important to stress that B γ and C γ depend neither on the amplitude nor on thecharacteristic wavenumber of the forcing. We shall also make use of the following result : Lemma 1.
For any γ ∈ R such that C γ is finite, ∞ (cid:88) j =1 k γj | f j | = C γ k γf | F | . (23) Proof.
By using the definition of the forcing in (16) and rearranging the terms in thesum, we obtain : ∞ (cid:88) j =1 k γj | f j | = | F | ∞ (cid:88) j = j f k γj | φ j − j f | = | F | ∞ (cid:88) p =0 k γp + j f | φ p | (24)= | F | ∞ (cid:88) p =0 k γ λ γ ( p + j f ) | φ p | = | F | ( k λ j f ) γ ∞ (cid:88) p =0 λ pγ | φ p | . (25)Replacing the definitions of k f and C γ in (25) yields the result.As discussed above at the beginning of this section we now use the method ofDoering and Foias [31] to estimate the time-averaged dissipation rate (cid:15) . Theorem 1.
Let the forcing ( f , f , . . . ) be as in Sect. 2 and the initial energy E (0) bebounded. Then the time-averaged energy dissipation rate satisfies (cid:15) (cid:54) ν k f (cid:0) c Re + c Re (cid:1) + O( T − ) , (26) where the constants c = (cid:112) C C − γ ) C − γ , c = A √ C B γ C − γ , (27) depend on the parameters a , b , λ of the shell model and on the shape of the forcing ( φ , φ , . . . ) , but are uniform in ν , k , k f , | F | . The value of γ ∈ R may be chosen in away as to minimize c and c , but it must nonetheless be such that C − γ , C − γ ) , B γ arefinite. Remark 1.
The switch from Re to Re behaviour in (26) is observed in the numericalcomputations displayed in Fig. 1.Proof. We begin by writing the evolution equation for the energy. Towards this end,we multiply (13) by u ∗ j and the complex conjugate of (13) by u j . We then add the tworesulting equations and sum over j :d E d t = − ν ∞ (cid:88) j =1 k j | u j | + ∞ (cid:88) j =1 ( f j u ∗ j + f ∗ j u j ) . (28) hell models and the Navier–Stokes equations E ( T ) + 2 ν (cid:90) T (cid:32) ∞ (cid:88) j =1 k j | u j ( t ) | (cid:33) d t (cid:54) E (0) + 2 (cid:112) C U | F | T , (29)whence (cid:15) (cid:54) (cid:112) C U | F | + E (0)2 T . (30)To express this bound in terms of Re , we need to estimate | F | in terms of U and k f .We multiply (13) by k − γj f ∗ j , sum over j , and average over time : (cid:42) ∞ (cid:88) j =1 k − γj f ∗ j ˙ u j (cid:43) T = ∞ (cid:88) j =1 k − γj | f j | − (cid:42) ν ∞ (cid:88) j =1 k − γj u j f ∗ j (cid:43) T + (cid:42) i ∞ (cid:88) j =1 k − γj f ∗ j (cid:0) ak j +1 u ∗ j +1 u j +2 + bk j u j +1 u ∗ j − − ck j − u j − u j − (cid:1)(cid:43) T . (31)From (28), it is easy to see that E ( t ) is bounded by a time-independent constant [25].This follows from using k j < k for all j > T − ).The first term on the right-hand side is calculated from Lemma 1 as : ∞ (cid:88) j =1 k − γj | f j | = C − γ k − γf | F | . (32)The second term is estimated by using the Cauchy–Schwarz inequality : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:42) ν ∞ (cid:88) j =1 ( k − γj f ∗ j ) u j (cid:43) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) (cid:113) C − γ νU k − γf | F | . (33)We estimate the third term by moving the forcing out of the sum and using again theCauchy–Schwarz inequality : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:42) i ∞ (cid:88) j =1 k − γj f ∗ j (cid:0) ak j +1 u ∗ j +1 u j +2 + bk j u j +1 u ∗ j − − ck j − u j − u j − (cid:1)(cid:43) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) k − γ +1 f | F | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:42) ∞ (cid:88) j =1 λ − ( j − j f )(2 γ − φ j − j f ( aλu ∗ j +1 u j +2 + bu j +1 u ∗ j − + ( a + b ) λ − u j − u j − ) (cid:43) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) AB γ U k − γ +1 f | F | . (34) hell models and the Navier–Stokes equations | F | (cid:54) (cid:112) C − γ C − γ νU k f + A B γ C − γ U k f + O (cid:0) T − (cid:1) . (35)Inserting (35) into (30) and rearranging finally yields the estimate for (cid:15) .The implications of (26) for turbulent flows have been discussed thoroughly in [31]within the context of the 3 D NSEs (see also [43]). Here we briefly mention the shell-model counterpart of the main points :(i) The bound on (cid:15) can be rewritten as (cid:15)U k f (cid:54) c Re + c + O( T − ) . (36)Thus, in the high- Re limit the saturation of the bound recovers the empiricalprediction (cid:15) ∼ U k f [1].(ii) The estimate of (cid:15) can be converted into bounds for the Kolmogorov dissipationwavenumber k η = ( (cid:15)/ν ) / , the Taylor microscale k T = ( (cid:15)/νU ) / , and the Taylor-microscale Reynolds number, Re λ = U/νk T . The saturation of these bounds for Re → ∞ is consistent with the empirical predictions k η ∼ Re / , k T ∼ Re / , Re λ ∼ Re / for 3 D homogeneous and isotropic turbulence [1].(iii) A lower bound for the time-averaged dissipation rate can also be derived by usingthe shell-model version of the Poincar´e inequality : (cid:15) (cid:62) νk (cid:42) ∞ (cid:88) j =1 | u j | (cid:43) T = νk U , (37)where we have used k > k j for all j >
1. The latter bound can be rewritten as (cid:15)U k f (cid:62) (cid:18) k k f (cid:19) Re − . (38)Therefore, the small- Re scaling in (36) is sharp. Moreover, if we take j f = 1 and φ p = δ p, , then k f = k and c = 1. As a consequence, the upper and lower boundson (cid:15) coincide for Re →
0, i.e. (cid:15) behaves as (cid:15)/U k f = Re − . This means that thelower bound on (cid:15) is also optimal.(iv) Dividing (35) by ν k f yields : Gr (cid:54) c (cid:48) Re + c (cid:48) Re (39)with c (cid:48) = c / √ C and c (cid:48) = c / √ C . This bound establishes a relation between theforcing (represented by Gr ) and the response of the system (represented by Re ).As mentioned earlier, the proof of Theorem 1 parallels that of Doering and Foias [31]for the NSEs. By using the same approach, it is possible to obtain estimates of (cid:15) interms of Gr analogous to those available for the NSEs. It can indeed be shown thatfor Gr → (cid:15) coincide, and hence the time-averageddissipation rate behaves as (cid:15) = ν k f Gr , while for Gr → ∞ it satisfies the lower bound (cid:15) (cid:62) c ν k k f Gr , where the constant c is uniform in ν , k , | F | , and k f . hell models and the Navier–Stokes equations
4. High-order velocity derivatives
To investigate higher-order derivatives of the velocity, we now consider the sequence ofinfinite sums H n = ∞ (cid:88) j =1 k nj | u j | , n (cid:62) , (40)which represent the shell-model analogues of the L -norms (cid:107) ∇ n u (cid:107) L . Note that H isthe energy E , while the time average of H is proportional to (cid:15) : (cid:15) = ν (cid:104) H (cid:105) T . (41)We also denote the equivalent sums for the forcing variables asΦ n = ∞ (cid:88) j =1 k nj | f j | , n (cid:62) . (42)Recall from Lemma 1 that Φ n = C n k nf | F | . H n The following theorem shows that there exist two ladders of differential inequalitiesthat connect H n and H n +1 and reproduce the analogous ladder inequalities for theNSEs [34, 35]. Theorem 2.
Let n (cid:62) and assume that the forcing ( f , f , . . . ) is such that Φ n < ∞ .Then H n satisfies
12 ˙ H n (cid:54) − νH n +1 + c n H n sup j (cid:62) k j | u j | + H n Φ n (43 a ) and
12 ˙ H n (cid:54) − ν H n +1 + d n ν H n sup j (cid:62) | u j | + H n Φ n (43 b ) with c n = λ − n +1 (cid:0) | a | λ − n + | b | + | a + b | λ n (cid:1) , d n = c n λ . (44) Proof.
We multiply (13) by k nj u ∗ j and the complex conjugate of (13) by k nj u j . We thensum to obtain˙ H n = − νH n +1 + ∞ (cid:88) j =1 k nj (cid:2) i u ∗ j ( ak j +1 u ∗ j +1 u j +2 + bk j u j +1 u ∗ j − − ck j − u j − u j − + f j ) + c . c . (cid:3) , (45)where ‘c.c.’ stands for ‘complex conjugate’. The forcing term is estimated by using theCauchy–Schwarz inequality : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:88) j =1 k nj u ∗ j f j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:88) j =1 ( k nj u ∗ j )( k nj f j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) H n Φ n . (46) hell models and the Navier–Stokes equations a . We have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ∞ (cid:88) n =1 k nj k j +1 u ∗ j u ∗ j +1 u j +2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) | a | λ − n +1 sup j (cid:62) ( k j | u j | ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:88) j =1 ( k nj +1 u ∗ j +1 )( k nj +2 u j +2 ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (47) (cid:54) | a | λ − n +1 H n sup j (cid:62) ( k j | u j | ) , (48)where we have used k nj = λ − np k nj + p and the Cauchy–Schwarz inequality. The termswith coefficients b and c = − ( a + b ) are treated in a similar manner. The first ladderinequality is thus proved by using (46) and the estimates for the nonlinear terms in (45).To prove (43 b ), we start again from (45). The forcing term is estimated as above.The term with coefficient a is now manipulated as follows : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ∞ (cid:88) j =1 k nj k j +1 u ∗ j u ∗ j +1 u j +2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) | a | λ sup j (cid:62) | u j | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:88) j =1 (cid:0) k nj u ∗ j +1 (cid:1) (cid:0) k n +1 j u j +2 (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (49) (cid:54) | a | λ − n − H n H n +1 sup j (cid:62) | u j | , (50)where we have used the Cauchy–Schwarz inequality. We then estimate the terms withcoefficient b and c in a similar way and use Young’s inequality to find (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:88) j =1 k nj u ∗ j (cid:2) ( ak j +1 u ∗ j +1 u j +2 + bk j u j +1 u ∗ j − − ck j − u j − u j − ) + c . c . (cid:3)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (51) (cid:54) (cid:112) d n H n H n +1 sup j (cid:62) | u j | (cid:54) νH n +1 + 2 d n ν H n sup j (cid:62) | u j | , (52)where d n is defined in (44). Finally, we combine the first term on the right-hand sideof (52) with the viscous term in (45) and add the estimate of the forcing term to get(43 b ).The structure of the ladder inequalities makes it evident that control over a low- n rung of the ladder automatically yields control over all the higher-order rungs [34, 35].Since sup j (cid:62) | u j | (cid:54) H and H is bounded [25], inequality (61 b ) can be used to provethat there are absorbing balls for all the H n . The existence of absorbing balls was provedin [25] by using different methods. Here we show how this result follows immediatelyfrom the ladder inequalities and, in addition, we estimate the radius of the absorbingball for H n under the assumption that Φ n is finite. Corollary 1.
Let n (cid:62) and assume the forcing is such that Φ n < ∞ , then lim sup t →∞ H n (cid:54) ν k n +1) f (cid:104) ˜ d n ρ n +1) Gr n +1) + (cid:101) C n ρ n +2 Gr (cid:105) , (53) where ρ = k f /k and ˜ d n = 2 n d nn , (cid:101) C n = 2 nn +2 C nn +2 n . (54) Proof.
By using sup j (cid:62) | u j | (cid:54) H and the inequality (see the Appendix for the proof) H n (cid:54) H n +1 H nn +1 n +1 , (55) hell models and the Navier–Stokes equations b ) as˙ H n (cid:54) − H n (cid:34) ν H n n H n − d n ν H − n H n (cid:35) . (56)It follows thatlim sup t →∞ H n (cid:54) n d nn ν − n lim sup t →∞ H n +10 + 2 nn +2 ν − nn +2 Φ nn +2 n lim sup t →∞ H n +2 . (57)From Lemma 1, we haveΦ n = C n k nf | F | = C n ν k n +6 f Gr . (58)In addition, it was shown in [25] thatlim sup t →∞ H (cid:54) ν (cid:18) k f k (cid:19) k f Gr . (59)Inserting (58) and (59) into (57) yields the advertised result.It is also useful to reformulate the ladder inequalities in terms of the quantities K n = H n + τ Φ n with τ = ν − k − , (60)which incorporate the contribution of the forcing. This can be achieved under theadditional assumption that the forcing has a cutoff in the spectrum, i.e. there exists amaximum wavenumber k max = k λ j max such that f j = 0 for j > j max . Corollary 2. If n (cid:62) and the forcing has a maximum wavenumber k max and is suchthat Φ n < ∞ , then K n satisfies
12 ˙ K n (cid:54) − νK n +1 + c n K n sup j (cid:62) k j | u j | + ν (cid:0) k + k (cid:1) K n (61 a ) and
12 ˙ K n (cid:54) − ν K n +1 + d n ν K n sup j (cid:62) | u j | + ν (cid:0) k + k (cid:1) K n . (61 b ) Proof.
The strategy for deriving (43 a ) from (61 a ) is the same as for the NSEs [7, 35].Note first that ˙ H n = ˙ K n . Then, add and substract ντ Φ n +1 to the right-hand sideof (43 a ) to obtain the negative definite term in (61 a ). The remaining two terms ofthe H n inequality are expressed in terms of K n via the obvious bounds H n (cid:54) K n andΦ n (cid:54) τ − K n . Finally, we are left with the term ντ Φ n +1 , which is estimated by using τ Φ n +1 (cid:54) Φ n +1 K n / Φ n (cid:54) k K n .Inequality (61 b ) is proved in exactly the same manner. hell models and the Navier–Stokes equations (cid:28) H n +1 n (cid:29) T We now make use of the first ladder inequality and the estimate for (cid:15) to prove the shell-model analogue of a Navier–Stokes result of Foias, Guillop´e and Temam [38]. It oughtto be noted that the exponent of H n in the bound below is greater than that found forthe 3 D NSEs. The reason for this difference between the shell model and the 3 D NSEsis discussed in Sect. 5.
Theorem 3.
Let n (cid:62) and E (0) < ∞ and assume that the forcing ( f , f , . . . ) has amaximum wavenumber k max and is such that Φ n < ∞ . Then, for Re (cid:29) , (cid:28) H n +1 n (cid:29) T (cid:54) ˆ c n ν n +1 k f Re + O (cid:0) T − (cid:1) , (62) where the dimensionless positive constant ˆ c n depends on a , b , λ , n but is uniform in ν , k , k f , k max , | F | .Proof. By noting thatsup j (cid:62) k j | u j | = (cid:18) sup j (cid:62) k j | u j | (cid:19) / (cid:54) H / , (63)we turn (61 a ) into12 ˙ K n (cid:54) − νK n +1 + c n (cid:98) H / K n , (64)where we have denoted (cid:98) H / = H / + 2 νk and have used k < k max . We shall seethat the additive constant in (cid:98) H / gives a negligible contribution at large Re .We then divide (64) by K nn +1 n and time average. The time-derivative term can besimplified as follows : (cid:68) K − nn +1 n ˙ K n (cid:69) T = ( n + 1) (cid:28) dd t K n +1 n (cid:29) T = n + 1 T (cid:20) K n +1 n ( T ) − K n +1 n (0) (cid:21) . (65)The first term on the right-hand side is bounded below by ( n + 1) ( τ Φ n ) n +1 /T > T − ). We are therefore left with (cid:42) K n +1 K nn +1 n (cid:43) T (cid:54) c n ν (cid:28) K n +1 n (cid:98) H (cid:29) T + O (cid:0) T − (cid:1) (cid:54) c n ν (cid:28) K n +1 n (cid:29) T (cid:68) (cid:98) H (cid:69) T + O (cid:0) T − (cid:1) . (66)We now estimate the time average of K n +2 n +1 by using (66) and H¨older’s inequality : (cid:28) K n +2 n +1 (cid:29) T = (cid:42)(cid:32) K n +1 K nn +1 n (cid:33) n +2 K n ( n +1)( n +2) n (cid:43) T (cid:54) (cid:42) K n +1 K nn +1 n (cid:43) n +2 T (cid:28) K n +1) n (cid:29) nn +2 T (67) (cid:54) c (cid:48) n ν − n +2 (cid:28) K n +1) n (cid:29) n +1 n +2 T (cid:68) (cid:98) H (cid:69) n +2 T + O (cid:0) T − (cid:1) (68) hell models and the Navier–Stokes equations c (cid:48) n = c n +2 n . Define now the dimensionless quantities A = ν − k − f (cid:68) (cid:98) H (cid:69) T and A n = ν − n +1 k − f (cid:28) K n +1) n (cid:29) T (69)for n (cid:62)
2. The bound in (68) then takes the form (cid:104) A n +1 (cid:105) T (cid:54) c (cid:48) n (cid:104) A n (cid:105) n +1 n +2 T (cid:104) A (cid:105) n +2 T + O (cid:0) T − (cid:1) (70)and, after the use of Young’s inequality, (cid:104) A n +1 (cid:105) T (cid:54) c (cid:48) n ( n + 1) n + 2 (cid:104) A n (cid:105) T + c (cid:48) n n + 2 (cid:104) A (cid:105) T + O (cid:0) T − (cid:1) . (71)To estimate A , we invoke Jensen’s inequality, (41), and Theorem 1 for Re (cid:29) (cid:104) A (cid:105) T (cid:54) ν − k − f (cid:104) H (cid:105) T + 4 ν − k − f k (cid:104) H (cid:105) / T + 4 k − f k (cid:54) ˆ c Re + O (cid:0) T − (cid:1) . (72)Here ˆ c is a dimensionless constant that depends on a , b , λ and is uniform in ν , k , k f , k max , | F | . We now use (72) in (71) for n = 1 to estimate (cid:104) A (cid:105) T and then proceediteratively to find (cid:104) A n (cid:105) T (cid:54) ˆ c n Re + O (cid:0) T − (cid:1) . (73)We obtain the final result by writing the latter bound in dimensional form and recallingthat H n (cid:54) K n . For Navier–Stokes flows, the deviations of the velocity and its derivatives from theirmean values are captured by the norms (cid:107) ∇ n u (cid:107) L m , where 0 (cid:54) n and 1 (cid:54) m (cid:54) ∞ [10].For m < ∞ , the shell-model analogues of (cid:107) ∇ n u (cid:107) mL m are H n,m = ∞ (cid:88) j =1 k nmj | u j | m , (74)which reduce to H n when m = 1. Instead, the analogue of (cid:107) ∇ n u (cid:107) L ∞ is sup j (cid:62) k nj | u j | .By building on the results of the previous sections, we can generalize Theorem 3 to H n,m . Note once again that the exponent of H n,m in the time average differs from thatfound for weak solutions of the 3 D NSEs [10].
Theorem 4.
Under the same assumptions as in Theorem 3 and for Re (cid:29) , H n,m satisfies (cid:28) H m ( n +1) n,m (cid:29) T (cid:54) ˆ c n ν n +1) k f Re + O (cid:0) T − (cid:1) (75) if (cid:54) n , (cid:54) m < ∞ , and (cid:68) H m ,m (cid:69) T (cid:54) ν k f Re (76) if n = 0 and (cid:54) m < ∞ . In addition, for n (cid:62) (cid:28)(cid:16) sup j (cid:62) k nj | u j | (cid:17) n +1 (cid:29) T (cid:54) ˆ c n ν n +1) k f Re + O (cid:0) T − (cid:1) , (77) hell models and the Navier–Stokes equations while for n = 0 (cid:28)(cid:16) sup j (cid:62) | u j | (cid:17) (cid:29) T (cid:54) ν k f Re . (78) The constants ˆ c n depend on a , b , λ , n , but are uniform in ν , k , k f , k max , | F | .Proof. The case m = 1 was proved in Theorem 3. For 1 < m < ∞ , we use the inequality ∞ (cid:88) j =1 X j (cid:54) (cid:32) ∞ (cid:88) j =1 X /pj (cid:33) p , (79)where p (cid:62) X j (cid:62) j . When applied to H n,m , this inequality yields H n,m (cid:54) H mn . (80)If n (cid:62)
1, the result follows from raising both sides of (80) to the power 2 /m ( n + 1) andinvoking Theorem 3. For n = 0, it is proved by raising both sides of (80) to the power1 /m and using H = ν k f Re .Finally, (77) is proved by noting that (cid:18) sup j (cid:62) k nj | u j | (cid:19) n +1 = (cid:18) sup j (cid:62) k nj | u j | (cid:19) n +1 (cid:54) H n +1 n (81)and using Theorem 3, while (78) follows from sup j (cid:62) | u j | (cid:54) H .
5. Comparison with the velocity gradient averaged Navier–Stokes equations
The issue in this section concerns how the velocity derivative estimates displayed inTheorem 4 compare with those for the NSEs. It is not clear that there necessarilyshould be a positive comparison, given that the 3 D NSEs are not known to be regularand their corresponding scaling exponents defined in (2) and (4) are different, namely : α n,m = 2 m m ( n + 1) − α n,m = 4 n + 1 (Shell) . (82)As we will now show, the real comparison lies with what we have called the “velocitygradient averaged Navier–Stokes equations” (VGA-NSEs). To explain the origin of thisname, let us return to the first ladder inequality for H n displayed in (43 a ), which forthe NSEs is written in the form (cid:107)
12 ˙ H n (cid:54) − νH n +1 + c n (cid:107) ∇ u (cid:107) ∞ H n , (83)where for simplicity, we have ignored the forcing term [34, 35]. As explained in (9) in §
1, the approximation where the L ∞ -norm is replaced by its spatial average (cid:107) ∇ u (cid:107) ∞ (cid:54) c n L − / (cid:107) ∇ u (cid:107) . (84) (cid:107) In this section, c n is a generic positive constant dependent on n . hell models and the Navier–Stokes equations ∇ u . Thus we are not dealing witha modified PDE but with an averaging of its solutions reflected in the behaviour of (cid:107) ∇ u (cid:107) ∞ . In terms of the H n -ladder we are dealing with12 ˙ H n (cid:54) − νH n +1 + c n L − / (cid:107) ∇ u (cid:107) H n (85)which yields the exact equivalent of Theorem 3 : Theorem 5.
For n (cid:62) , the H n for the D VGA-NSEs obey the bounds (cid:28) H n +1 n (cid:29) T (cid:54) c n L − ν n +1 Re . (86) Remark 2.
Bounds for H n,m follow in the same manner as in Theorem 4, as can beeasily seen by using approximation (10) in the proof of Theorem 1 of [9]. The relaxationof the L ∞ to the L -norm in (10) accounts for the insensitivity of the exponents to thevalue of m .Proof. To mimic the FGT-analysis of Theorem 3, and suppressing the multiplicativefactors of L and ν , we divide (85) by H nn +1 n to obtain (cid:42) H n +1 H nn +1 n (cid:43) T (cid:54) (cid:28) H / H n +1 n (cid:29) T (cid:54) (cid:104) H (cid:105) / T (cid:28) H n +1 n (cid:29) / T (87)Moreover, (cid:28) H n +2 n +1 (cid:29) T = (cid:42)(cid:32) H n +1 H nn +1 n (cid:33) n +2 H n ( n +1)( n +2) n (cid:43) T (cid:54) (cid:42) H n +1 H nn +1 n (cid:43) n +2 T (cid:28) H n +1 n (cid:29) nn +2 T (88)Let X n = (cid:28) H n +1 n (cid:29) T (89)then from (88) and (87) we have X n +1 (cid:54) n + 2 (cid:42) H n +1 H nn +1 n (cid:43) T + nn + 2 X n (cid:54) n + 2 ( (cid:104) H (cid:105) T + X n ) + nn + 2 X n = 1 n + 2 (cid:104) H (cid:105) T + n + 1 n + 2 X n (90)Since X = (cid:104) H (cid:105) T (cid:54) Re we have estimates for every n (cid:62) n (cid:62)
1. What of the velocity field represented by n = 0? hell models and the Navier–Stokes equations Lemma 2.
For < m (cid:54) ∞ , the velocity field for the D VGA-NSEs obey the bounds (cid:10) (cid:107) u (cid:107) m (cid:11) T (cid:54) c m Re . (91) Remark 3.
The problem lies in understanding what is happening in the range (cid:54) m (cid:54) , which remains an open problem.Proof. Firstly we note that (cid:107) u (cid:107) m (cid:54) c (cid:107) ∇ u (cid:107) a ∞ (cid:107) u (cid:107) − ap (92)for p > m and a = p − m ) p (3+2 m ) . Moreover, (cid:107) u (cid:107) p (cid:54) c (cid:107) ∇ u (cid:107) A ∞ (cid:107) u (cid:107) − Ap (93)where A = p − m ) p − , p > m >
3. Then the L ∞ → L replacement as in (10) gives (cid:107) u (cid:107) m (cid:54) c L − m − m (cid:107) ∇ u (cid:107) for m > . (94)This is exactly a ‘less intermittent’ form of Sobolev’s inequality which allows somevariation in the L m -norm on the left-hand side instead of L alone, as in its standardform.Comparing (76) with (91) shows that the equivalence between the shell model andthe VGA-NSEs only holds at the level of the velocity derivatives. In shell models, thedependence of the velocity field on Re is even weaker than in the VGA-NSEs.
6. Simulations and concluding remarks
To test the mathematical estimates, we have performed numerical simulations of theSabra model. The parameters are the typical ones used in studies of 3 D turbulence : a = 1, b = c = − / k = 2 − , λ = 2 [30]. The forcing has the form f j = F δ j, with F = 5 × − (1 + i), and the viscosity is varied between ν = 10 − and ν ≈ × .We truncate the system to N shells by imposing the additional boundary conditions u N +1 = u N +2 = 0, where N is varied between 8 and 27 depending on the value of ν . The numerical integration uses a second-order slaved Adams–Bashforth scheme [45]with time step d t = 10 − .Figures 1 to 3 show (cid:15) , Gr , and (cid:68) H m ( n +1) n,m (cid:69) T for different values of n and m as afunction of Re . To facilitate the reading of the figures, the relevant definitions andestimates are summarized in Table 1. The values of Re vary from the ‘laminar’ regime,in which the shell model relaxes to a fixed point, to the fully turbulent regime, which ischaracterized by a k − / j spectrum over several decades of wavenumbers.The simulations clearly show that the mathematical estimates in Table 1 accuratelydescribe the behaviour of the shell model as a function of Re . Figure 2(b) also indicatethat, for Re (cid:28)
1, the scaling of (cid:68) H n +1) n (cid:69) T depends on n , as may be inferred from theproof of Theorem 3 (see (70) to (72)). Related to this, in Fig. 3(a) the small- Re scaling hell models and the Navier–Stokes equations Definition Estimate Reference (cid:15) = ν (cid:68)(cid:80) ∞ j =1 k j | u j | (cid:69) T (cid:15) (cid:54) ν k f (cid:0) c Re + c Re (cid:1) (cid:15) U − k − f (cid:54) c Re − + c (26)(36) Gr = | F | /ν k f Gr (cid:54) c (cid:48) Re + c (cid:48) Re (39) H n = (cid:80) ∞ j =1 k nj | u j | (cid:68) H n +1 n (cid:69) T (cid:54) ˆ c n ν n +1 k f Re ( n (cid:62) , Re (cid:29)
1) (62) H n,m = (cid:80) ∞ j =1 k nmj | u j | m (cid:68) H m ( n +1) n,m (cid:69) T (cid:54) ˆ c n ν n +1) k f Re ( n (cid:62) , Re (cid:29)
1) (75) H ,m = (cid:80) ∞ j =1 | u j | m (cid:68) H /m ,m (cid:69) T (cid:54) ν k f Re (76) (cid:28)(cid:16) sup j (cid:62) k nj | u j | (cid:17) n +1 (cid:29) T (cid:54) ˆ c n ν n +1) k f Re ( n (cid:62)
1) (77) (cid:28)(cid:16) sup j (cid:62) | u j | (cid:17) (cid:29) T (cid:54) ν k f Re (78) Table 1.
Summary of the main estimates and definitions. The O (cid:0) T − (cid:1) correctionshave not been included for simplicity. of (cid:68) H m ( n +1) n,m (cid:69) T depends on n but not on m , as a consequence of H n,m being controlledby H mn (see (80)).Our conclusion is that shell models behave more closely to the 3 D VGA-NSEs thanthe NSEs themselves. They both have identical scaling exponents in their time averagesof their velocity derivatives which are reflected in the suppression of strong events of ∇ u , as proposed in equation (10). The actual properties of shell models for the velocityfield itself are even milder than the estimates for the VGA-NSEs : compare (76) in Table1 with (91).Finally, we ask how much more regularity do solutions of shell models possessthan those for the NSEs? This is shown up by comparing the estimates for velocityderivatives. Consider the scaling exponents α n,m defined in (3) which appear in (4). Itis not difficult to replicate this result in D = 3 , , α n,m is replacedby α n,m,D α n,m,D = 2 m m ( n + 1) − D (95)and the relation involving F n,m in (4) is replaced by (cid:68) F (4 − D ) α n,m,D n,m,D (cid:69) T (cid:54) c n,m,D Re . (96) hell models and the Navier–Stokes equations − − − − − ν − k f − ε Re Re Re − − ε / U k f ReRe − Re Figure 1.
Rescaled time-averaged energy dissipation rate as a function of the Reynoldsnumber.
In all these estimates, the larger the exponent the more regularity we have. Under whatconditions is the 4 / ( n + 1) of shell models greater than (4 − D ) α n,m,D ?4 n + 1 ≥ (4 − D ) α n,m,D ? (97)The answer turns out to be2 D { m ( n + 1) − } (cid:62) , (98)and is thus always true when n (cid:62) m (cid:62) D . Equality holds onlyat the level of the energy dissipation rate when n = m = 1. The same result impliesthat, exception made for the time-averaged dissipation rate, the Re -dependence of thevelocity derivatives is weaker for shell models than for the D -dimensional NSEs for anyinteger D . Curiously, in a formal manner, equality also holds in the limit D →
0, whichcorresponds to the “Navier–Stokes equations on a point”, which has zero dimension.Given that shell models have no spatial variation the physical correspondence betweenthe two is intriguing.
Acknowledgments
The authors are grateful to Samriddhi Sankar Ray for several useful suggestions. JohnGibbon acknowledges the award of a Visiting Professorship at the Universit´e Cˆote dAzurduring the months of November 2018 and April 2019 and the kind hospitality of theLaboratoire J. A. Dieudonn´e. hell models and the Navier–Stokes equations − − − − G r Re Re Re − − − − − ν − / ( n + ) k f − 〈 H n / ( n + ) 〉 T Re Re n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 Figure 2. (a) Grashof number as a function of the Reynolds number; (b) Time averageof H n +1 n rescaled by ν n +1 k f as a function of the Reynolds number. − − − − − ν − / ( n + ) k f − 〈 H n , m / m ( n + ) 〉 T Re Re n = 1, m = 2 n = 2, m = 2 n = 3, m = 2 n = 1, m = 3 n = 2, m = 3 n = 3, m = 3 〈 (sup k j | u j |) 〉 − − − − − ν − k f − 〈 H , m / m 〉 T Re m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 Re Figure 3.
Time averages of (a) H m ( n +1) n,m rescaled by ν n +1 k f and (b) H m ,m rescaledby ν k f as a function of the Reynolds number. Appendix
Inequality (55) is proved by induction on n [35]. By using the Cauchy–Schwarzinequality, we find H = ∞ (cid:88) j =1 | u j | (cid:0) k j | u j | (cid:1) (cid:54) H H . (A.1)We then assume H n (cid:54) H n +1 H nn +1 n +1 (A.2) hell models and the Navier–Stokes equations H n +1 as H n +1 (cid:54) H n H n +2 (cid:54) H n +1) H n n +1) n +1 H n +2 , (A.3)which yields H n +1 (cid:54) H n +2 H n +1 n +2 n +2 . (A.4)This completes the proof by induction. References [1] Frisch U 1995
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