a r X i v :n l i n /0506049v 1 [n l i n .C D ] 23 J u n 2005Enstrophy dissipation in freely evolving two-dimensional
turbulence
Chuong V.Tran ∗
Mathematics Institute,University of Warwick,Coventry CV47AL,UK
(Dated:February 8,2008)Abstract Freely decaying two-dimensional Navier–Stokes turbulence is studied.The conrvation of vor-ticity by advective nonlinearities renders a class of Casimirs that decays under viscous effects.A rigorous constraint on the palinstrophy production by nonlinear transfer is derived,and an up-per bound for the enstrophy dissipation is obtained.This bound depends only on the decaying Casimirs,thus allowing the enstrophy dissipation to be bounded from above in terms of initial data of the flows.An upper bound for the enstrophy dissipation wavenumber is derived and the new result is compared with the classical dissipation wavenumber.PACS numbers:47.27.Gs,47.27.Eq
1
In1969,Batchelor1adapted Kolmogorov’s equilibrium theory for three-dimensional(3D) turbulence to tw
o-dimensional(2D)turbulence,on the basis of a phenomenologically anal-ogous property between the two systems.For a3Dfluid,decrea of the viscosityνis accompanied by increa of the mean-square vorticity,a conquence of the magnification of the vorticity by stretching of vortex lines,so that in the inviscid limit the energy dissipation is nonzero.For a2Dfluid,decrea of the viscosity enhances convective mixing,in which isovorticity lines get extended and brought clor to one another,1giving ri to increa of the mean-square vorticity gradients(twice the palinstrophy),so that in the inviscid limit the rate of enstrophy(half the mean-square vorticity)dissipation can approach afinite valueχ. On the basis of this analogy,Batchelor1applies the familiar arguments of the Kolmogorov equilibrium theory for the small-scale components of3D turbulence to the2D ca,where the roles of the energy and energy dissipation in the original theory are played by the enstro-phy and enstrophy dissipation.This means that the statistical properties of the small-scale components of the turbulence depend only on the two dimensional parametersχandν.The enstrophy dissipationχis thus an important dynamical quantity in Batchelor’s theory.One of its prominent role is in the expression of the enstrophy spectrum Z(k)of the so-called enstrophy inertial range,which is presumably formed when an initial enstrophy rervoir spreads out in a virtually inviscid region of wavenumber space:
Z(k)=Cχ2/3k−1,(1)
where C is a universal constant and k is the wavenumber.Another important role ofχis in the determination of the dissipation wavenumber kν:
χ1/6
kν=
palinstrophy by convective mixing is fully understood.This letter takes a direct approach to this problem by deriving a rigorous upper bound for the nonlinear term reprenting the palinstrophy production rate.Equating this bound to the viscous dissipation term yields a constraint,from which upper bounds forχand for the enstrophy dissipation wavenumber k d (to be defined later in this letter)can be derived.The bounds are found to be completely described in terms of initial data of theflows.The derived enstrophy dissipation wavenumber is consistent with the classical prediction of kνgiven by(2),in the n that they both have the same functional dependence onν.A novel result of this study is thatχand k d can be estimated in terms of initial data of the turbulence,while they(more precilyχand kν) are esntially undetermined in the Batchelor theory.
In the vorticity formulation,the freely evolving2D Navier–Stokes equations governing the motion of an incompressiblefluid confined to a doubly periodic domain are
∂tξ+J(ψ,ξ)=ν∆ξ,(3)
猪肝汤whereξ(x,t)is the vorticity,J(θ,ϑ)=θxϑy−θyϑx,νthe kinematic viscosity,andψ(x,t)the stream function.The vorticity is defined in terms of the stream function and of the velocity v byξ=∆ψ=ˆn·∇×v,whereˆn is the normal vector to thefluid domain.Equivalently, v can be recovered fromψandξby v=(−ψ
,ψx)=(−∆−1ξy,∆−1ξx).
y
An importance property of the advective nonlinear term in(3)is that it conrves the kinetic energy and an infinite class of integrated quantities,known as Casimirs,including the enstrophy.The latter conrvation law is attributed to the fact that vorticity is conr-vatively redistributed in physical space by the advective transfer.While the conrvation of energy and enstrophy impos strict constraints on turbulentflows and has been explored in the literature to a great extent,2−19the conrvation of Casimirs,other than the enstro-phy,ems to render little additional knowledge of theflows and has received much less attention.20−22In the prent ca of unforced dynamics,it is well known that a wide class of the Casimirs decays under the action of viscosity.Here our main int
erest is in the following Casimirs: |ξ|p ,where · denotes a spatial average.By taking the time derivative of |ξ|p and using(3)one obtains
d
where the cond equation is obtained by integration by parts,upon which the nonlinear term identically vanishes.For p=2,Eq.(4)governs the decay of |ξ|2 (twice the enstrophy),for which the dissipation term becomes2χ,which is the subject of this study.The right-hand side of(4)is negative for p>1.Hence |ξ|p decays in time,for p>1.It follows that
|ξ(t)|p ≤ |ξ(0)|p ,(5)
for p>1and t≥0.In particular,in the limiting ca p→∞,one has
||ξ(t)||
∞≤||ξ(0)||
西非海牛
∞
,(6)
for t≥0,where||ξ||
∞
denotes the L∞norm ofξ.
Now the main result of this letter can be readily derived.By multiplying(3)by∆ξand taking the spatial average of the resulting equation,one obtains the equation governing the evolution of the palinstrophy |∇ξ|2 /2:
1
dt
家园共育图片细胞融合实验|∇ξ|2 = ∆ξJ(ψ,ξ) −ν |∆ξ|2
= ξJ(ψx,ξx) + ξJ(ψy,ξy) −ν |∆ξ|2
≤ |ξ|(|∇ψx||∇ξx|+|∇ψy||∇ξy|) −ν |∆ξ|2
≤||ξ||
∞
|∇ψx|2+|∇ψy|2 1/2 |∇ξx|2+|∇ξy|2 1/2−ν |∆ξ|2
草莓清洗方法≤||ξ||
∞
|ξ|2 1/2 |∆ξ|2 1/2−ν |∆ξ|2 .(7) In(7),the cond equation is obtained via the two elementary identities
∆J(ψ,ξ)=J(ψ,∆ξ)+2J(ψx,ξx)+2J(ψy,ξy)(8) and
∆ξJ(ψ,ξ) =− ξJ(ψ,∆ξ) .(9) The H¨o lder inequality is ud in the fourth step,and the last step can be en by expressingψ
(andξ)in terms of Fourier ries.The triple-product term||ξ||
∞
|ξ|2 1/2 |∆ξ|2 1/2reprents an upper bound for the palinstrophy production rate.It can be en that the palinstrophy necessarily ceas to grow as its dissipationν |∆ξ|2 reaches this bound.It follows tha
t as the palinstrophy grows to and reaches a maximum(d |∇ξ|2 /dt≥0),the following inequality necessarily holds
|∆ξ|2 1/2≤||ξ||
∞
|ξ|2 1/2
Since |∇ξ|2 can be bounded from above in terms of |∆ξ|2 ,ineq.(10)can be ud to derive an explicit upper bound for the palinstrophy.By H¨o lder inequality one has18
|∇ξ|2 2
|∆ξ|2 ≥
.(12)
ν
It follows that the enstrophy dissipationχis bounded from above by
|ξ|2 .(13)
χ≤||ξ||
∞
It is notable that the upper bound forχin(13)is expressible in terms of two decaying
and |ξ|2 ,so that it can be bounded from above in terms of initial Casimirs,namely||ξ||
∞
data of theflows.More accurately,χcan be bounded from above in terms of the initial
末日使者and |ξ|2 are intensive vorticityfield only.In passing,it is worth mentioning that since||ξ||
∞
independent of the domain size,the constraint(13)is size-independent.
Let us denote by k d and k D the wavenumbers defined by |∇ξ|2 1/2/ |ξ|2 1/2and |∆ξ|2 1/2/ |∇ξ|2 1/2,respectively.For“regular”spectra,k d(k D)specifies where,in wavenumber space, |∇ξ|2 ( |∆ξ|2 )is mainly distributed.In other words,k d(k D)spec-ifies where,in wavenumber space,the enstrophy(palinstrophy)dissipation mainly occurs. By“regular”it is meant that the enstrophy is not highly concentrated in any particular regions of wavenumber space that would result in vere steps in the enstrophy spectrum. By(11),the dissipation wavenumbers satisfy k d≤k D,and the inequality sign“≤”can become“≪”.This can be realized if the palinstrophy spectrum around k d does not fall offso steeply.More quantitatively,by the definition of k d,one can expect the palinstrophy spectrum around k d to be shallower than k−1.(Becau,otherwi,most of the contribution to |∇ξ|2 would come from k<k d,making the ratio |∇ξ|2 1/2/ |ξ|2 1/2significantly lower than k d,a contradiction to the very definition of k d.)This means that the spectrum of |∆ξ|2 around k d is shallower than k1.Hence,most of the contribution to |∆ξ|2 can come from k≫k d if the palinstrophy spectrum beyond k d becomes steeper than k−1and falls offto k−3gradually.This allows for the possibility k D≫k d to be realized.In any ca,k d should be well beyond the end of the enstrophy k>k d Z(k)dk/ k<k d Z(k)dk≈0,and k D
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should be well beyond the palinstrophy k>k D P(k)dk/ k<k D P(k)dk≈0,where Z(k)and P(k)are the enstrophy and palinstrophy spectra,respectively.In other words,the enstrophy spectrum around k d should be steeper than k−1,and the palinstrophy(enstrophy) spectrum around k D should be steeper than k−1(k−3).阳光体育运动
Our primary concern is an estimate of k d when |∇ξ|2 achieves a maximum.It is likely that k d achieves a global maximum then.Equation(7)and the subquent equations(10) and(12)imply
k d k D≤||ξ||
∞
ν 1/2≤
||ξ(0)||∞ν1/2
= k d |ξ|2 1/2
一岁宝宝奶粉用量k d =
|ξ|2 1/2
