Dynamics of scaled norms of vorticity for the three-dimensional Navier-Stokes and Euler equations

Transcription

1 Available online at wwwsciencedirectco Procedia IUAM ) opological Fluid Dynaics: heory and Applications Dynaics of scaled nors of vorticity for the three-diensional Navier-Stokes and Euler equations J D Gibbon Departent of Matheatics, Iperial College London SW7 2AZ, UK Abstract A series of nuerical experients is suggested for the three-diensional Navier-Stokes and Euler equations on a periodic doain based on a set of L 2 -nors of vorticity Ω for 1 hese are scaled to for the diensionless sequence D = ϖ 1 Ω)α where ϖ is a constant frequency and α =2/4 3) A nuerically testable Navier-Stokes regularity criterion coes fro coparing the relative agnitudes of D and D +1 while another is furnished by iposing a critical lower bound on D dτ he behaviour of the D is also iportant in the Euler case in suggesting a ethod by which possible singular behaviour ight also be tested c he Published Authors by Published Elsevier by Ltd Elsevier Selection BV Open and/or access peer-review under CC BY-NC-ND under responsibility license of K Bajer, Y Kiura, & HK Moffatt Selection and/or peer-review under responsibility of the Isaac Newton Institute for Matheatical Sciences, University of Cabridge Keywords: Navier-Stokes; Euler; vorticity, weak solutions; length scales 1 Introduction he challenges that face those concerned with the nuerical integration of the three-diensional incopressible Euler and Navier-Stokes equations for a velocity field ux, t) on a 3D periodic doain V =[,L] 3 per Du Dt = νδu p div u = 1) ν =for the Euler equations) are also reflected in the challenges faced by analysts in their attepts to understand the regularity of these equations he best known and ost effective result in which analysis has guided nuerics is the Beale-Kato-Majda BKM) theore [1, 2], which says that solutions of the three-diensional incopressible Euler equations are controlled fro above by ω dτ If this is finite then no blow-up can occur at tie t Moreover, any nuerical singularity in the vorticity field of the type ω t t) p ust have p 1 for the singularity not to be a nuerical artefact he BKM criterion has becoe a standard feature in Euler coputations : see the papers in the special volue [3] his present paper is concerned with regularity criteria that for a consistent fraework for the Euler and Navier- Stokes equations and which are testable nuerically In both cases it is often not clear when a nuerically observed E-ail address: jdgibbon@icacuk he Authors Published by Elsevier BV Open access under CC BY-NC-ND license Selection and/or peer-review under responsibility of the Isaac Newton Institute for Matheatical Sciences, University of Cabridge doi: 1116/jpiuta21336

2 4 J D Gibbon / Procedia IUAM ) spike in the vorticity or strain fields reains finite or is a anifestation of a singularity It is well known that onitoring the global enstrophy, or H 1 -nor ω 2, pointwise in tie deterines Navier-Stokes regularity [4, 5], while the onitoring of ω likewise deterines the fate of Euler solutions However, the range of L p -nors between these ay be useful he basic objects are a set of frequencies based on L 2 -nors of the vorticity field ω = curl u ) 1/2 Ω t) = L 3 ω 2 dv 1 2) V Hölder s inequality insists that Ω Ω +1 he Navier-Stokes and Euler equations are invariant under the re-scaling x = ɛx ; t = ɛ 2 t ; u = ɛu ; p = ɛ 2 p If the doain length L is also re-scaled as L = εl then Ω re-scales as Ω = ε 2 Ω as expected If, however, L is not re-scaled but kept fixed then Ω re-scales as where Ω α = ɛω α, α = ) It turns out that this strange scaling is particularly iportant and provides a otivation for the definition of the set of diensionless quantities D t) = ϖ 1 Ω α ), 1 5) where α 1 =2and α =1/2 For the Navier-Stokes equations the frequency ϖ is easily defined as ϖ = νl 2 he case of the Euler equations is ore difficult as there is no obvious aterial constant to replace ν in the definition of ϖ A circulation Γ has the sae diensions as that of ν but it ust be taken around soe chosen initial data : for instance, in [6], initial data was taken to be a pair of anti-parallel vortex tubes, in which case Γ could be chosen as the circulation around one of these For a discussion of the variety of conclusions that can be drawn fro nuerical experients see [6, 7, 8, 9, 1, 11] No proof exists, as yet, of the existence and uniqueness of solutions of either the 3D Navier-Stokes or Euler equations for arbitrarily long ties A tie-honoured approach has been to look for inial assuptions that achieve this result [4, 5] In fact, uch of what is known about solutions of both the 3D Navier-Stokes and Euler equations is encapsulated in the sequence of tie integrals D 2 dτ based on the continuu lying between D 1 = ϖ 2 L 3 ω 2 dv D = ϖ 1 ω 1/2 ) 7) V A well-known tie-integral regularity condition for the Navier-Stokes equatons is that the first in the sequence in 6) should be finite [4, 5] : that is D2 1 dτ < In contrast, the boundedness of the last in the sequence in 6) at = is exactly the Beale-Kato-Majda criterion ω dτ < for the regularity of solutions of the Euler equations [1, 2, 3] For three-diensional Navier-Stokes turbulence, it has to be aditted that arbitrarily iposed regularity assuptions, such as D2 dτ <, while atheatically interesting, have little foundation in physics : see the discussion of this point in [12] However, what is known, without any assuptions, is that weak solutions in the sense of Leray [13]) obey the tie integral [14] D dτ c tre 3 + η 1 ), where η 1 is a constant depending upon D ) his result plays two roles In 3 it is shown that it leads to a definition of a continuu of inverse length scales Lλ 1 the upper bound on the first of which is the well-known Kologorov 3) 6) 8)

3 J D Gibbon / Procedia IUAM ) scale proportional to Re 3/4 he ore general upper bound is discussed in 3 and is given by Lλ 1 cre 3/2α hus the λ for >1correspond to deeper length scales associated with the higher L 2 -nors of vorticity iplicit within the D In addition, the agnitude of the bounded tie integral in 8) is also significant Let us consider whether the saturation, or near saturation, of this tie integral plays any role in the regularity question In the forced case it has been shown in [12] that if a critical lower bound is iposed on this tie integral in ters of the Grashof nuber Gr then this leads to exponential collapse in the D t) In fact boundedness fro above of any one of the D also iplies the boundedness of D 1 which iediately leads to the existence and uniqueness of solutions While it can be argued that the iposition of this lower bound is physically artificial the result is nevertheless intriguing because it suggests that if the value of the integral 8) is sufficiently large then solutions are under control, which is surely counter-intuitive Once it dips below this critical value then regularity could potentially break down he iportance of this echanis lies in the role it ay play in understanding the phenoenon of Navier-Stokes interittency his is discussed in 5 where the critical lower bound is expressed in ters of the ore physical Reynolds nuber Re tre 3δ ) t + η 2 D dτ, δ 1 9) he range of δ is estiated and it is shown that δ 1 2 for large, thus allowing considerable slack between the critical lower bound in 9) and the upper bound in 8) he D are coparatively easy quantities to calculate fro a nuerical schee and is thus it is worth exploring whether regularity criteria can be gleaned fro the relative agnitudes of the D or their tie integrals his is the task of this paper In 2 the Euler equations are discussed in these ters where two versions of a nuerical experient are suggested for testing singular or non-singular behaviour 2 he incopressible 3D Euler equations Whether the three-diensional Euler equations develop a singularity in a finite tie still reains an open proble but a variety of super-weak solutions have recently been shown to exist [15, 16, 17, 18, 19, 2] Let Γ be the circulation around soe chosen initial data such that ϖ =ΓL 2, as discussed in 1 his defines ϖ within the definition of D Proposition 1 Provided solutions of the three-diensional Euler equations exist, for 1 < the D forally satisfy the following differential inequality ) ξ D+1 Ḋ c ϖ D 3, D ξ = ) 1) Proof : he tie derivative of the Ω obeys 2L 3 Ω 2 1 Ω = d dt ω 2 dv 2 ω 2 u dv ) 1 ) ) 1 2 ω 2 2 dv ω 2+1) 2+1) dv u 2+1) 2+1) dv 2L 3 c 1, Ω Ω 11) where we have used u p c p ω p, for 1 p< hus it transpires that Ω c 1, Ω+1 Ω ) +1 Ω 2, 1 < 12) Note that the case = is excluded : it was shown in [1] that a logarithic H 3 = V 3 u 2 dv factor is needed such that u c ω 1 + ln H 3 )

4 42 J D Gibbon / Procedia IUAM ) We wish to convert inequality 12) to one in ters of D Ω+1 Ω ) +1 Ω = ϖ D+1 D ) +1 ) α α +1 D +1 α + 1 α ) 1 D ) = ϖ D 2 D having used the fact that 1 1 ) β =2, α +1 α β = ) 13) Substitution into 12) copletes the proof here are now at least two interesting routes for the integration of 1) 1 Firstly if a finite-tie singularity is suspected then divide 1) by D 3 ε where [D t)] 2 ε F 1,ε t) =c 2 ε)ϖ For instance, for ε =we have where D 2 t) 1 [D t )] 2 ε) F 1,ε t) t D+1 1 [D t )] 2 F 1, t) F 1, t) =2c ϖ t D+1 D D with ε<2 to obtain 14) ) ξ D ε dτ, ξ = ) 15) 16) ) ξ dτ 17) A singularity in the upper bound of inequality 16) is not necessarily significant What is ore significant is whether the solution tracks this singular upper bound his suggests the following nuerical test : a) Is F 1, linear in t? b) If so, then test whether D 2 c, t) C with c, c 18) uniforly in If such behaviour occurs it suggests, but does not prove, that the D ay be blowing up close to the upper bound 2 If exponential or super-exponential growth is suspected then divide 1) only by D the case ε =2) and integrate ) ξ D+1 D D t )expϖ D 2 dτ 19) t D he rate of growth of the integral with respect to t is of interest Does it reain finite for as long as the integrated solution reains reliable? he copanion paper is this volue by Kerr addresses soe of these questions [21]

5 J D Gibbon / Procedia IUAM ) Weak solutions of Navier-Stokes and a range of scales Weak solutions are natural for the global enstrophy ω 2 2 because of the properties of projection operators he original arguent used by Leray [13] gives us the textbook result fro his energy inequality [4, 5] In ters of D 1 this is D 1 cre 3 + O 1) 2) where the tie average up tie given by is defined by F ) = 1 li sup F τ) dτ F 21) o obtain siilar results for ω 2 for >1 looks difficult not only because the properties of projection operators do not naturally extend to the higher spaces but also because ω 2 does not appear naturally in an energy inequality However, these probles have been circuvented in [14], the ain result fro which will be stated below and its very short proof repeated for the benefit of the reader : heore 1 For 1, weak solutions obey D cre 3 + O 1), 22) where c is a unifor constant Proof : he proof is based on a result of Foias, Guillopé & ea 1981) [22] their heore 31) for weak solutions Doering and Foias [23] used the square of the averaged velocity U 2 = L 3 u 2 2 to define the Reynolds nuber Re = U Lν 1 which enables us to convert estiates in Gr to estiates in Re hus the result of Foias, Guillopé & ea [22] in ters of Re becoes c N L 1 ν 2 2N 1 Re 3 + O 1), 23) H 1 2N 1 N where H N = V N u 2 dv = N 1 ω 2 dv, 24) V and where H 1 = V u 2 dv = V ω 2 dv hen an interpolation between ω 2 and ω 2 is written as ω 2 c N, N 1 ω a 2 ω 1 a 3 1) 2, a = 2N 1), 25) for N 3 ω 2 is raised to the power A, which is to be deterined ω A 2 c A N, = c A N, c A N, N 1 ω aa ) 1 H 1 2N 1 N 2 ω 1 a)a 2 1 H 1 2 aa2n 1) 2N 1 N 2 aa2n 1) H a)a 1 H 1 a)a aa2n 1) 2 aa2n 1) 1 An explicit upper bound in ters of Re is available only if the exponent of H 1 within the average is unity ; that is 1 a)a 2 aa 2N 1) =1 A = = α 27) 26)

6 44 J D Gibbon / Procedia IUAM ) as desired Using the estiate in 23), and 2) for H 1, the result follows c N, can be iniized by choosing N =3 c 3, does not blow up even when = ; thus we take the largest value of c α 3, and call this c Following the stateent of heore 1 and otivated by the definition of the Kologorov length for =1,a continuu of length scales λ can be defined thus : D := Lλ 1 ) 2α, 28) in which case 28) becoes Lλ 1 cre 3/2α 29) When =1, α 1 =2, and thus Lλ 1 1 cre 3/4, which is consistent with Kologorov s statistical theory [22, 23] However, the bounds on λ 1 becoe increasingly large with increasing reflecting how the L 2 -nors can detect finer scale otions 4 A regularity criterion based on the relative sizes of D and D +1 Consider two -dependent constants c 1, and c 2, and two frequencies ϖ 1, and ϖ 2, defined by ϖ 1, = ϖ α c 1 1, ϖ 2, = ϖ α c 2, 3) In [12], using a standard contradiction strategy on a finite interval of existence and uniquness [, ), it was shown that for the decaying Navier-Stokes equations the D obey the following theore in which the dot represents differentiation with respect to tie : heore 2 For 1 < on [, ) the D t) satisfy the set of inequalities { ) ρ Ḋ D 3 D+1 ϖ 1, + ϖ 2,}, 31) D where ρ = ) In the forced case there is an additive ter ϖ 3,Re 2 D he obvious conclusion is that solutions coe under control pointwise in t provided where D +1 t) c ρ D t) c ρ =[c 1, c 2, ] 1/ρ his is a nuerically testable criterion : if 32) holds then the D ust decay in tie It was shown in [12] that this can be weakened to a tie integral result : heore 3 For any value of 1 <, if the integral condition is satisfied ) 1+Z ln dτ, Z = D +1 /D 34) c 4, with c 4, = [ 2 ρ c 1, c 2, ) ] ρ 1, then D t) D ) on the interval [, t] Given the nature of c 4, 2 it is clear that there ust be enough regions of the tie axis where D +1 > c 4, 1)D to ake the integral positive he D are easily coputable fro Navier-Stokes data herefore, an interesting nuerical experient would be to test : 1 Whether the D are ordered as tie evolves such that D D +1 or D D +1? 2 Do the D cross over fro one regie to the other? 3 How significant are the initial conditions and the Reynolds nuber in this behaviour? 32) 33)

7 J D Gibbon / Procedia IUAM ) Body-forced Navier-Stokes equations 51 A critical lower bound on D dτ in ters of Re In [12, 26] the body-forced Navier-Stokes equations were considered in ters of the Grashof nuber Gr Itis ore useful to to consider this in ters of the Reynolds nuber Re he inclusion of the forcing in 31) odifies 1 this but requires the introduction of a third frequency ϖ 3, = ϖ α c 3, Ḋ D 3 { ϖ 1, D+1 D ) ρ + ϖ 2,} + ϖ 3, Re 2 D, 35) where ρ = )and γ = 1 2 α ) 1 Let Δ be defined by 2 Δ 6) Δ =3{δ 2 + ρ γ ) ρ γ } 36) he following result shows that if a critical lower bound is set on D dτ then D will decay exponentially Note that the case =1is excluded : heore 4 If there exists a value of lying in the range 1 <<, with initial data [D )] 2 <C Re Δ, for which the integral lies on or above the critical value c tre 3δ ) t + η 2 D dτ and δ and η 2 lie in the ranges 37) 2/3+ρ γ <δ < 1, and η 2 η 1 Re 3δ 1), 38) 2+ρ γ then D t) decays exponentially on [, t] Reark: δ 1/2 for large so enough slack lies between the upper and lower bounds on D dτ Proof : o proceed, divide by D 3 to write 35) as 1 d 2 dt D 2 ) X D 2 A lower bound for X dτ can be estiated thus : and so D +1 dτ = ) ϖ2, X = ϖ 1, D+1 D [ D+1 D D+1 D ) ρ ] 1 D 2 ρ ρ 2 D ρ dτ ) ρ D 2 ϖ 3, Re 2 39) ) ρ ) 1 D 2 ρ t ) ρ 2 ρ dτ D dτ t 1/ρ 4) ) ρ X dτ ϖ 1, t 1 D +1 dτ ) ϖ t ρ 2 3,tRe 2 41) D dτ 1 For the forced case the definition of Ω requires an additive ϖ ter to act as a lower bound [12, 26]

8 46 J D Gibbon / Procedia IUAM ) Recall that ρ = )and γ = 1 2 α ) 1 It is not difficult to prove that Ω 2 Ω Ω 1 for >1, fro which, after anipulation into the D becoes D D α/2γ2 +1 D α/22 1 and therefore a Hölder inequality gives α ) 1+γ =1 42) D +1 dτ D dτ D dτ D 1 dτ ) γ 43) ) ργ +2 X dτ ϖ 1, t 1 D dτ ) t ργ D ϖ 3, tre 2 44) 1 dτ Inequality 39) integrates to { exp 2 } t [D t)] 2 X dτ [D )] 2 2ϖ 2, exp { 2 τ X dτ } dτ 45) 41) can be re-written as ) t ργ +2 t X dτ ϖ 1, t 1 D dτ ) t ργ D ϖ 3, tre 2 1 dτ c t ϖ 1, Re Δ ϖ 3, Re 2) 46) having used the assued lower bound in the theore and the upper bound of D 1 dτ Moreover, to have the dissipation greater than forcing requires Δ > 2 so δ ust lie in the range as in 38) because 2 < Δ 6 For large Re the negative Re 2 -ter in 45) is dropped so the integral in the denoinator of 45) is estiated as τ ) exp 2 X dτ dτ [2 c ϖ 1, ] 1 Re Δ 1 exp [ 2ϖ 1, c tre Δ]), 47) and so the denoinator of 45) satisfies Denoinator [D )] 2 c 2, c 1, 2 c ) 1 Re Δ 1 exp [ 2ϖ 1, c tre Δ]) 48) his can never go negative if [D )] 2 >c 1, c 2, 2 c ) 1 Re Δ, which eans D ) <C Re 1 2 Δ

9 J D Gibbon / Procedia IUAM ) D t) Re 3 upper bound of D t critical lower-bound of D Re 3δ t potential singularities t t 1 t 2 t Figure 1 : A cartoon of D t) versus t illustrating the phases of interittency he range of δ lies in heore 4 he vertical arrows depict the region where there is the potential for needle-like singular behaviour & thus a break-down of regularity 52 A relaxation oscillator echanis for interittency Experientally, signals go through cycles of growth and collapse [27, 28, 29, 3] so it is not realistic to expect that the critical lower bound iposed in heore 4 should hold for all tie Using the average notation t, inequality 45) shows that if D t lies above critical then D t) collapses exponentially In Figure 1 the horizontal line at Re 3δ is drawn as the critical lower bound on D t Above this critical range, D t) will decay exponentially fast However, because integrals take account of history, there will be a delay before D t decreases below the value above which a zero in the denoinator of 45) can be prevented at t 1 ) : at this point all constraints are reoved and D t) is free to grow rapidly again in t 1 t t 2 If the integral drops below critical then it is in this interval that the occurrence of singular events depicted by vertical arrows) ust still forally be considered if one occurs the solutions fails Provided a solution still exists, growth in D will be such that, after another delay, it will force D t above critical and the syste, with a re-set of initial conditions at t 2, is free to ove through another cycle hus it behaves like a relaxation oscillator he vertical arrows in Figure 1 label the region, below critical, where D dτ < c tre 3δ + η 2 ) It is in this regie where where potentially singular point-wise growth of D t) could occur which contributes little to the growth of the integral D dτ and which does not drive it past critical No control echanis for this type of growth is known and so the regularity proble reains open 49)

10 48 J D Gibbon / Procedia IUAM ) Conclusion he variables D, as defined in 5), have proved useful in expressing the Navier-Stokes and Euler regularity probles in a natural anner heir use also poses soe interesting questions For instance, while the Ω ust be ordered because of Hölder s inequality, this is not the case with the D because the α decrease with heore 2 suggests that the regie D +1 /D c, where c is a constant only just above unity, guarantees the decay of D and hence control over Navier-Stokes solutions In ters of nuerical experients, it would be interesting to see, fro a variety of initial conditions, which of the two regies D +1 D or D +1 D 5) are predoinant and whether there is a cross-over fro one to the other If so, does this depend heavily on the initial conditions, such as the contrasting rando or anti-parallel vortex initial conditions? Does it also depend on the size of Re? Likewise, do solutions of the Euler equations, when in their interediate and late growth phases, track a singular upper bound as in 18)? Acknowledgeent : I would like to thank Darryl Hol of Iperial College London and Bob Kerr of the University of Warwick for discussions on this topic References [1] Beale J, Kato, Majda AJ Rearks on the breakdown of sooth solutions for the 3D Euler equations Coun Math Phys 1984;94:61 66 [2] Majda AJ, Bertozzi AL Vorticity and incopressible flow Cabridge University Press; 21 [3] Euler equations 25 years on Eyink G, Frisch U, Moreau R, Sobolevskii A, editors Physica D 28;237: [4] Constantin P, Foias C he Navier-Stokes equations Chicago University Press; 1988 [5] Foias C, Manley O, Rosa R, ea R Navier-Stokes equations and turbulence Cabridge University Press; 21 [6] Kerr RM Evidence for a singularity of the three-diensional incopressible Euler equations Phys Fluids A 1993;5: [7] Bustaente MD, Kerr RM 3D Euler in a 2D syetry plane Physica D 28; ): [8] Hou Y, Li R Dynaic depletion of vortex stretching and non blow-up of the 3-D incopressible Euler equations J Nonlinear Sci 26;16: [9] Hou Y Blow-up or no blow-up? he interplay between theory & nuerics Physica D 28; ): [1] Grafke, Hoann H, Dreher J, Grauer R Nuerical siulations of possible finite tie singularities in the incopressible Euler equations Coparison of nuerical ethods Physica D 28; ): [11] Gibbon JD he three-diensional Euler equations: Where do we stand? Physica D 28;237: [12] Gibbon JD Conditional regularity of solutions of the three diensional Navier-Stokes equations & iplications for interittency J Math Phys 212;53:11568 [13] Leray J Sur le ouveent d un liquide visqueux eplissant l espace Acta Math 1934;63: [14] Gibbon JD A hierarchy of length scales for weak solutions of the three-diensional Navier-Stokes equations Co Math Sci 211;1: [15] Shnirelan A On the non-uniqueness of weak solution of the Euler equation Coun Pure Appl Math 1997;5: [16] De Lellis C, Székelyhidi L he Euler equations as a differential inclusion Ann Math 29;2)173): [17] De Lellis C, Székelyhidi L On adissibility criteria for weak solutions of the Euler equations Arch Ration Mech Anal 21;195: [18] Wiedeann E Existence of weak solutions for the incopressible Euler equations Annales de l Institut Henri Poincaré C) Nonlinear Analysis 211;285): [19] Bardos C, iti ES Euler equations of incopressible ideal fluids Russ Math Surv 27;62: [2] Bardos C, iti ES Loss of soothness and energy conserving rough weak solutions for the 3D Euler equations Discrete and continuous dynaical systes 21;32): [21] Kerr RM he growth of vorticity oents in the Euler equations his volue [22] Foias C, Guillopé C, ea R New a priori estiates for Navier-Stokes equations in Diension 3 Co PDEs 1981;6: [23] Doering CR, Foias C Energy dissipation in body-forced turbulence J Fluid Mech 22;467: [24] Frisch U urbulence : the legacy of A N Kologorov Cabridge University Press; 1995 [25] Doering CR he 3D Navier-Stokes Proble Annu Rev Fluid Mech 29;41: [26] Gibbon JD Regularity and singularity in solutions of the three-diensional Navier-Stokes equations Proc Royal Soc A 21;466: [27] Batchelor GK, ownsend AA he nature of turbulent flow at large wave nubers Proc R Soc Lond A 1949;199: [28] Kuo AY-S, Corrsin S Experients on internal interittency and fine structure distribution functions in fully turbulent fluid, J Fluid Mech 1971;5: [29] Sreenivasan K, On the fine scale interittency of turbulence J Fluid Mech 1985;151:81 13 [3] Meneveau C, Sreenivasan K he ultifractal nature of turbulent energy dissipation J Fluid Mech 1991;224: