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A hierarchy of length scales for solutions of the three-dimensional Navier-Stokes equations J. D. Gibbon
arXiv:1012.3604v1 [nlin.CD] 16 Dec 2010
Department of Mathematics Imperial College London, London SW7 2AZ, UK email:
[email protected]
Dedicated to David Levermore on the occasion of his 60th birthday. Abstract Moments of the vorticity are used to define and estimate a hierarchy of time-averaged inverse length scales λ−1 m for the three-dimensional Navier-Stokes equations on a periodic box [0, L]3 . The result is 3/2αm , Lλ−1 m ≤ c Re
where
αm =
2m . 4m − 3
When m = 1 the Re3/4 -bound coincides with the Kolmogorov length of statistical physics, but for higher moments these bounds rapidly increases to Re3 . The implications of these results for computational resolution are discussed.
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Introduction
Resolution issues in computations of solutions of the three-dimensional Navier-Stokes equations are not only closely associated with the problem of regularity but they also raise the question of how resolution length scales can be defined and estimated. The Kolmogorov −1 3/4 school of statistical turbulence suggests that the inverse Kolmogorov length λ−1 k ∼ L Re for a system of size L3 has a cut-off in its −5/3 energy spectrum at kc ∼ L−1 Re3/4 . The wave-numbers k > kc are considered to lie in what is called the dissipation range [1, 2]. Significant energy lying in this range can provoke intermittent events in the vorticity and strain fields characterized by violent, spiky departures away from space-time averages whose corresponding statistics are non-Gaussian in character [3, 4, 5, 6, 7]. Whether significant energy actually cascades down to the micro/nano-scales where the equation fails to be a valid model is intimately entwined not only with the open question of regularity but also with the use of the Navier-Stokes equations as a limit of kinetic theory [8]. This phenomenon continues to pose severe computational challenges [9, 10]. In Kolmogorov’s statistical theory the objects that are used to study intermittency are the ensemble-averaged velocity structure functions h |u(x + r) − u(x)|p iens.av. ∼ r ζp
(1.1)
the departure of whose exponents ζp from a linear relation1 is thought to be caused by intermittent behaviour [1, 2]. It is clear, however, that these structure functions are not best 1
Kolmogorov predicted a linear relation between ζp and p : departure from this is called ‘anomalous scaling’ and is usually manifest by ζp lying on a concave curve below linear for p > 3. The two coincide for p = 3.
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suited for Navier-Stokes analysis : the task of this paper is to discuss what could replace these in the Navier-Stokes context and what information could be gleaned from them. While higher gradients of the velocity field would undoubtedly capture intermittent behaviour, these would be unreachable computationally for all practical purposes. A better diagnostic of spikiness in Navier-Stokes solutions is a sequence of Lp -norms, or higher moments, of the vorticity ω = curl u defined through the set of frequencies (p = 2m for m > 1) Ωm (t) =
L
−3
Z
2m
|ω|
dV
V
1/2m
+ ̟0 ,
(1.2)
where the domain V = [0, L]3 is taken to be periodic while the basic frequency ̟0 = νL−2 is present simply for technical reasons. Ω21 is the enstrophy which is related to the energy dissipation rate whereas the higher moments will naturally pick up events at smaller scales. The setting is the incompressible (div u = 0), forced, three-dimensional Navier-Stokes equations for the velocity field u(x, t) ∂t u + u · ∇u = ν∆u − ∇p + f (x) .
(1.3)
Traditionally, most estimates in Navier-Stokes analysis have been found in terms of the Grashof 2 number Gr, which is expressed in terms of the root mean square (frms = L−3 kf k22 ) of the divergence-free forcing f (x) (see [11, 12, 13, 14]) but it would be more helpful to express these in terms of the Reynolds number Re to facilitate comparison with the results of statistical physics. The definitions of Gr and Re are Gr = L3 frms ν −2 ,
Re = U0 Lν −1 .
(1.4)
Doering and Foias [15] used the idea of defining U0 as U02 = L−3 hkuk22 iT
(1.5)
where the time average h · iT over an interval [0, T ] is defined by 1 hg(·)iT = lim sup T g(0)
Z
T
g(τ ) dτ .
(1.6)
0
Clearly, Gr is fixed provided f is L2 -bounded, while Re is the system response to this forcing. A brief look at Leray’s energy inequality shows why this definition of U0 is of value Z Z d 2 1 |ω|2 dV + kf k2 kuk2 , (1.7) |u| dV ≤ −ν 2 dt V V leading to Z
2
2
Ω21
(1.8)
≤ ̟02 Gr Re + O(̟02 ) + O T −1 .
(1.9)
T
which in turn gives2
T
≤ ν 2 L−1 Gr Re + O T −1
|ω| dV
V
The additive ̟02 -term in (1.9) is present because of the ̟0 -term in the definition of Ωm which provides a lower bound for Ωm : from now on it will be dropped.
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Doering & Foias [15] then showed that for Navier-Stokes solutions, Gr ≤ c Re2 , which gives
2 Ω1 T ≤ c ̟02 Re3 + O T −1 . (1.10)
In fact ν Ω21 T is the time-averaged energy dissipation rate per unit volume over [0, T ] and allows us to form and bound from above the inverse Kolmogorov length scale λ−1 k
2 ν Ω1 T 3/4 λ−4 ⇒ Lλ−1 + O(T −1/4 ) . (1.11) k = k ≤ c Re ν3 An estimate for the inverse Taylor micro-scale λT ms can also be found from (1.8) !1/2
kωk22 T −1 ≤ c Re1/2 + O(T −1/2 ) . λT ms :=
kuk22 T
(1.12)
Both these upper bounds gratifyingly coincide with the results of statistical turbulence theory [1, 2] although the fact that they are bounds allows for structures to occur in a flow whose natural scales are larger [16]. The question is now clear : can we construct and bound from above a sequence of inverse length scales associated with the higher moments Ωm ?
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A scaling property & length scale estimates
Leray’s energy inequality (1.7) is valid for weak solutions and thus the estimate (1.10) is rigorous. An equivalent result for the Ωm for m > 1 is problematic because the evolution equation for vorticity ω = curl u expressed as ∂t ω + u · ∇ω = ν∆ω + ω · ∇u + curlf
(2.1)
needs to be used despite the fact that existence and uniqueness remain an open problem. While it is possible to subscribe to the view that difficulties in flow resolution could be a symptom of the lack of uniqueness of weak solutions, in tandem it ought also to be acknowledged that these difficulties may simply be caused by the practical challenges of working on a system where even the naturally largest scale (other than L) lies close to the edge of computational resolution. The spirit of this paper is such that (strong) solutions are assumed to exist on an interval [0, T ] where T is taken sufficiently large3 , thus allowing the differentiation of Ωm and the use of (2.1). This assumption allows us to estimate an infinite hierarchy of time averages on [0, T ] without appealing to point-wise estimates that the solution of the regularity problem would require. In turn, these time averages allow us to define and explore the natural length scales inherent in the system. In [17] it has been shown, making minor modifications, that the Ωm satisfy : Lemma 1 Provided strong solutions exist on the interval [0, T ] and with 1 ≤ m < ∞ and n = 21 (m + 1), the Ωm (t) satisfy 2(m+1) ˙m − 3 ̟0 Ωm+1 4m(m+1)/3 Ω + c2,m ̟0 2m−1 Ωn2m−1 + c3 ̟0 Re2 . ≤− (2.2) Ωm c1,m Ωm 3
While existence and uniqueness of solutions is easily proved for small values of T [11, 12, 13], larger values than this are necessary to make sense of long-time averages.
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Remarks : (i) The differential inequality (2.2) has been included here for illustrative purposes. To gain control over solutions and thus prove regularity, the neighbours Ωm+1 and Ωm would need to stretch away from each other thereby increasing the natural lower bound of unity Ωm+1 /Ωm ≥ 1. While this can be shown to be true on some intervals within [0, T ] it is not clear how to prove this for the whole interval [17]. This is consistent with the phenomenon of intermittency where turbulent bursts intersperse flatter and more controlled behaviour. (ii) The central term on the right hand side of (2.2) corresponds to the vortex stretching. The most straightforward way of estimating it would be to obtain a term proportional to kωk∞ which is not helpful. The alternative, and the one pursued here, is to estimate the contribution this term makes to d (Ω2m m )/dt in the following way Z m Z Z 1 m+1 m+1 2(m+1) m+1 |ω|2(m−1) ω · (ω · ∇u) dV ≤ |ω| dV |∇u| dV V V V 2m Ωm+1 ≤ cm Ω2m (2.3) m Ωn Ωm where n = 12 (m + 1) and where a Riesz transform has been used to obtain the Ωn -term. Use of a H¨older inequality allows this to be split and combined with the negative term on the right hand side of (2.2). The case m = ∞ must be excluded as it needs a ln H3 term [18]. The next step is to see that (2.2) has a scaling property such that the transformation Dm = (̟0−1 Ωm )αm ;
αm =
2m 4m − 3
(2.4)
turns it into −1
(̟0 αm )
D˙ m Dm+1 2m(4m+1)/3 2 1 Dm + c2,m Dn2 + c3,m Re2 . ≤− Dm c1,m Dm
(2.5)
The time average of Dm obey : Theorem 1 ([17]) For 1 ≤ m ≤ ∞ the time average of Dm on [0, T ] satisfy hDm iT ≤ c Re3 + Re2 + O T −1 .
(2.6)
Remark : Two versions of the proof of this result were shown in Appendix B of reference [17] : the first is based on the division by Dm H¨older inequality manipulations of (2.5) to produce a generating inequality with hDm+1 iT on the left hand side and hDm iT on the right : the result in (1.10) for m = 1 starts off the process. The second is based on the time-average of fractions of Sobolev norms proved in [19] and allows m = ∞ and with a better constant in (2.6). The same scaling with the exponent αm appears in both proofs but there are some problems where the use of the moments Ωm is more physically realistic [20]. Based on the definition of the inverse Kolmogorov length scale λ−1 k in (1.11) it can easily be seen that a generalization of this to a hierarchy of inverse lengths λ−1 m suggests the definition 2αm := hDm iT , (Lλ−1 m )
(2.7)
which are interpreted as resolution lengths in the space-time averaged sense for 1 ≤ m ≤ ∞ :
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3 2 Lλ−1 m ≤ c Re + Re
1/2αm
+ O T −1/2αm .
(2.8)
How is this result to be interpreted? Many turbulent structures such as tubes and sheets have natural inverse gross length scales lying in the range between Re1/2 and Re3/4 – see [1, 21, 22, 23] – but high gradient crinkles forming at finer scales may ultimately grow to be dominant and then become the cause of resolution difficulties. For m > 1 the λm are interpreted here as the length scales corresponding to these deeper intermittent events. The upper bounds displayed in (2.8), as the Table shows, range from the Kolmogorov Re3/4 at m = 1 to Re3 for m = ∞. Computationally it may be hard to get far beyond m = 1 : for example as little as m = 9/8 corresponds to a bound Re1 which is close to modern resolutions even for modest values of Re. Thereafter the rise in the exponent is steep. Indeed, in the very high m limit the Re3 bound has an exponent four times greater than the Kolmogorov length, which lies well below molecular scales where the equations are invalid. m
1
9/8
3/2
2
3
...
∞
3/2αm
3/4
1
3/2
15/8
9/4
...
3
Table 1: Some values of the Re-exponent 3/2αm = 3 1 −
3 4m
.
The exponent αm = 2m/(4m−3) within the time average in (2.6) appears as a natural scaling of the partial differential equations, consistent with H¨older and Sobolev inequalities, but it would need to be significantly larger to give enough regularity for existence and uniqueness. (2.5) illustrates this point : ignoring the negative term on the right hand side allows us to integrate it immediately to obtain Z t 2 Dm (t) ≤ Dm (0) exp ̟0 αm Dn (τ ) dτ . (2.9) 0
The square inside the time integral is too strong for the bounds that we possess. So far there is no evidence that other scalings exist nor are there new methods on the horizon that would suggest any. Acknowledgements: Thanks are due to Charles Doering, Darryl Holm & Joerg Schumacher for discussions.
References [1] Frisch, U., 1995, Turbulence: The legacy of A. N. Kolmogorov, Cambridge University Press (Cambridge). [2] Davidson, P. A., 2004, Turbulence, Oxford University Press (Oxford) [3] Batchelor, G. K. & Townsend, A. A., 1949, The nature of turbulent flow at large wave-numbers. Proc R. Soc. Lond. A. 199, 238–255. [4] Kuo, A. Y.-S. & Corrsin, S., 1971, Experiments on internal intermittency and fine-structure distribution functions in fully turbulent fluid. J. Fluid Mech. 50, 285–320. [5] Meneveau, C. & Sreenivasan, K., 1991, The multi-fractal nature of turbulent energy dissipation. J. Fluid Mech. 224, 429–484.
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