Three-dimensional shear driven turbulence with noise at the boundary
aa r X i v : . [ m a t h . A P ] S e p THREE-DIMENSIONAL SHEAR DRIVEN TURBULENCE WITH NOISE ATTHE BOUNDARY
WAI-TONG LOUIS FAN, MICHAEL JOLLY, AND ALI PAKZAD
Abstract.
We consider the incompressible 3D Navier-Stokes equations subject to a shear inducedby noisy movement of part of the boundary. The effect of the noise is quantified by upper boundson both the expected value and the variance of the dissipation rate. The expected value estimaterecovers the bound in [14] for the deterministic case. The movement of the boundary is given byan Ornstein–Uhlenbeck process; a potential for over-dissipation is noted if the OU process werereplaced by the Wiener process. INTRODUCTION
Noise is added to turbulence models for a variety of reasons, both practical and theoretical. Forexample, the onset of turbulence is often related to the randomness of background movement [34].In any turbulent flow there are unavoidably perturbations in boundary conditions and materialproperties; see [39, Chapter 3]. The addition of noise in a physical model can be interpreted asa perturbation from the model. There is considerable evidence supporting the stabilization ofsolutions by noise (see, e.g., [1, 9, 20, 28]). However, the effect of noise in turbulent flow is far fromcompletely understood.This paper concerns the Kolmogorov dissipation law associated with the incompressible Navier-Stokes equations (NSE) in a 3-dimensional box D = [0 , L ] × [0 , h ] subject to a shear induced bynoisy movement of one wall. Specifically, we consider the following differential equation, du + ( u · ∇ u − ν ∆ u + ∇ p ) dt = 0 , ∇ · u = 0 , (1.1)with random boundary conditions given by the following: u is L − periodic in the x and x directions, u ( x , x , , t ) = ( X t , , ⊤ and u ( x , x , h, t ) = (0 , , ⊤ , (1.2)where ν > X t = X t ( ω ) : Ω → R , t ∈ R + ,is a given continuous-in-time stochastic process. The stochastic processes u ( x , x , x , t ; ω ) and p ( x , x , x ; ω ) represent respectively the velocity field and the pressure.The Kolmogorov dissipation law is tied to a phenomenon in turbulence called the energy cascade,which can be explained in 3 main steps. 1 − In the absence of a body force, the kinetic energy isintroduced into the large scales of the fluid between the parallel plates by the effects of the moving plate. This energy is called energy input . 2 − The large eddies break up into smaller eddies throughvortex stretching over an intermediate range , where the energy is transferred to smaller scales andthe energy dissipation due to the viscous force is negligible. 3 − At small enough scales (expectedto be ∼ Re − / , where Re is the Reynolds number defined in (1.3)) dissipation dominates and theenergy in those smallest scales decays to zero exponentially fast.Based on the above description the dissipation is effective at the end of a sequence of processes.Therefore, the rate of dissipation, which measures the amount of energy lost by the viscous force,is determined by the first process in the sequence, which is the energy input. The persistent forcedriving the shear flow is the motion of the bottom wall { ( x , x ,
0) : ( x , x ) ∈ [0 , L ] } . The timeaveraged energy dissipation rate must balance the drag exerted by the walls on the fluid. In termsof the characteristic speed U , the large eddies have energy of order U and time scale τ = h/U , sothe rate of energy input can be scaled as U /τ = U /h . This suggests the Kolmogorov dissipationlaw for time-averaged energy dissipation rate ε (Kolmogorov 1941); ε ∼ U h . Here a ∼ b means a . b and b . a ; a . b means a ≤ c b for a nondimensional universal constant c .The energy dissipation rate has been widely studied in the literature in the deterministic case[7, 12, 16, 18, 25, 27, 30, 31, 36–38]. Doering and Constantin proved in [14] a rigorous asymptoticbound directly from the Navier-Stokes equations. Their bound is of the form ε . U h , as Re → ∞ , where Re = U h/ν . (1.3)In this paper we derive an upper bound on the expected value of the energy dissipation rateas well as its variance in terms of characteristics of an Ornstein-Uhlenbeck process that is movingpart of the boundary. Our estimate recovers (1.3) in the limit as the variance of the noise tendsto 0. The key to the analysis is the choice of a stochastic background flow and the treatment of astochastic integral (with respect to the Wiener process) as a local martingale.Since the work of Bensoussan and Temam [3] in 1973, there has been substantial advance inunderstanding the stochastic Navier-Stokes equations, see for example [2, 5, 6, 34, 35, 43] and thereferences therein. Recently in [11], the exact dissipation rate is obtained for the stochasticallyforced Navier-Stokes equations under an assumption of energy balance. In all those works theequation always contains noise as a forcing term. Other than the analysis of symmetries of apassive scalar advected by a shear flow in which a boundary moves as a stochastic process in [8],to the best of our knowledge, there is no other work concerning the equations of the motion withstochastic boundary conditions.
Organization of this paper.
In section 2, we will introduce the necessary notation and prelim-inary results needed in the proceeding sections. In section 3, we will state the main result of thiswork. We will set up an almost sure bound starting form the energy equation in section 4. From
HREE-DIMENSIONAL SHEAR DRIVEN TURBULENCE WITH NOISE AT THE BOUNDARY 3 there, we will derive an upper bound on the mean value and variance of the energy dissipationrespectively in sections 5 and 6. The concluding Section 7 contains some open problems in thisdirection. 2.
Definitions and Notations
We take (Ω , A , F , P ) to be a complete, filtered probability space equipped with a filtration F = {F t ; t ∈ R + } . Let W = ( W t ) t ∈ R + be a standard Brownian motion (a.k.a. the Wiener process)adapted to F . The L norm and inner product will be denoted by k · k and ( · , · ) respectively, whileall other norms will be labeled with subscripts.As is tacitly assumed with most work on shear flow in the deterministic case (see, for instance,[22, Section 2]), we take the path-wise solution to be regular enough to satisfy the energy equality.That is, u = ( u t ) t ∈ R + is a stochastic process defined on (Ω , A , P ), adapted to the filtration F , andfor P -almost all sample points ω ∈ Ω, u is an element in L loc ( R + ; W , ( D )) ∩ L ∞ loc ( R + ; L ( D )) thatsolves (1.1)-(1.2) in the classical deterministic case where X t is replaced by a constant speed, U .Since u ∈ L loc ( R + ; W , ( D )) almost surely, we have(2.1) Z t k∇ u k ds < ∞ P − a.s. for all t ∈ R + . In experiments, it is natural to take a long but fixed time interval [0 , T ] and compute the time-average(2.2) h ǫ i T := 1 | D | T Z T ν k∇ u ( t, · , ω ) k L dt . It is shown in [22] that the effect of T in finite-time averages of physical quantities in turbulencetheory, including the energy dissipation rate, can be controlled by parameters such as Re. In oursetting, this finite-time average in (2.2) is a random variable whose mathematical expectation canbe approximated by taking average over a number of samples in the experiments. Definition 2.1.
We take the time-averaged expected energy dissipation rate for (1.1)-(1.2) is definedby ε := lim sup T →∞ E [ h ǫ i T ] . (2.3)Our main result, Theorem 3.1 below, is an upper bound for ε in terms of the characteristicsof the noise added to the movement of the boundary. Moreover, to assess the deviation from theexpectation, we also obtain an upper bound forlim sup T →∞ V ar [ h ǫ i T ] . WAI-TONG LOUIS FAN, MICHAEL JOLLY, AND ALI PAKZAD
Remark . We note that by Fatou’s lemmalim sup T →∞ E [ h ǫ i T ] ≤ E (cid:20) lim sup T →∞ h ǫ i T (cid:21) . Hence our upper bound on ε defined in (2.3) does not imply one when the order of the lim sup andexpectation are reversed. Definition 2.2.
The
Ornstein–Uhlenbeck process is the strong solution to the Itˆo stochastic differ-ential equation d X t = θ ( U − X t ) dt + σdW t , where W t denotes the Wiener process, and θ > σ > X t is explicitlygiven by X t = X e − θt + U (1 − e − θt ) + σ Z t e − θ ( t − s ) dW s (2.4)and has stationary distribution given by the normal distribution with mean U and variance σ θ . Ifthe initial distribution satisfies X ∼ N ( U, σ θ ), then X t ∼ N ( U, σ θ ) for all t ≥ X is a stationary OU process.Intuitively, the OU process is a Wiener process plus a tendency to move towards a location U ,where the tendency is greater when the process is further away from that location. In (2.4), θ is the decay-rate which measures how strongly the system reacts to perturbations, and σ is thevariation or the size of the noise. If the Ornstein–Uhlenbeck process (2.4) is stationary, then it canbe represented in terms of a time-dependent Wiener process X t = U + σ √ θ e − θt W e θt , where W t isthe standard Wiener process and the equality is in distribution.3. Statement of the Results
Theorem 3.1.
Suppose the stochastic process u satisfies (1.1) and boundary conditions (1.2), where X t is a stationary Ornstein–Uhlenbeck process (2.4) . Assume that the initial condition u is suchthat E ( k u k ) < ∞ . Then the energy dissipation rate (2.3) satisfies ε := lim sup T →∞ E [ h ǫ i T ] . (cid:18) Re + Uθ h + 1 Re h θU (cid:19) σ + (cid:16) U + U (cid:16) σ θ (cid:17) + (cid:16) σ θ (cid:17)(cid:17) / h . (3.1) Moreover, the variance of h ǫ i T satisfies (3.2) lim sup T →∞ V ar [ h ǫ i T ] ≤ lim sup T →∞ E [ h ǫ i T ] . h (cid:26) ν σ U + " U + U σ θ + U (cid:18) σ θ (cid:19) + (cid:18) σ θ (cid:19) + U (cid:20) U + U (cid:16) σ θ (cid:17) + (cid:16) σ θ (cid:17) (cid:21) + ν θ U (cid:16) σ θ (cid:17) (cid:27) . HREE-DIMENSIONAL SHEAR DRIVEN TURBULENCE WITH NOISE AT THE BOUNDARY 5
In the above estimate on the mean of dissipation rate (3.1), as the variance σ of the disturbancefrom U tends to 0, we recover the upper bound in Kolmogorov’s dissipation law,lim σ → ε ≤ U h , which is also consistent with the rate proven for the Navier-Stokes equations in [14]. The constantssuppressed by the use of . in (3.1) and (3.2) are explicitly given in (5.5) and (6.10). Remark . Since U is the mean velocity of the bottom wall, X t has the dimension of velocity.Therefore, θ scales as , and σ has dimension velocity √ time . Therefore, one can check that the resultsin Theorem 3.1 are also dimensionally consistent.4. An almost sure bound for the energy dissipation
The difficulty in the analysis of the shear flow (1.2) is due to the effect of the random nonhomo-geneous boundary condition on the flow. We overcome this difficulty by constructing a carefullychosen stochastic background flow. This construction is based on the Hopf extension [24].
Stochastic Background Flow.
Our key idea here is to choose the boundary layer thickness δ = δ t ( ω ) in the background flow to be random and time-dependent, namely,(4.1) δ = δ t ( ω ) = ν | X t ( ω ) | + U ) . We then let(4.2) φ ( ω ) = φ t ( x ; ω ) = (cid:18) − x δ t ( ω ) (cid:19) X t ( ω ) if 0 ≤ x ≤ δ t ( ω )0 otherwise , and define the stochastic background flow Φ = Φ t ( x , x , x ; ω ) as:(4.3) Φ( ω ) := ( φ ( ω ) , , ⊤ . The boundary layer is denoted by D δ = (0 , L ) × (0 , δ ) . Note that δ ∈ (0 , h ) if ν U < h , and that for all U ≥ ≤ − δ | X t | ν ≤ . Remark . It is worth mentioning that Φ is a divergence free stochastic vector field which alsosatisfies the non-homogeneous boundary conditions (1.2), by our construction in (4.2). In addition, δ in (4.1) is determined so as to absorb a term in (4.13).The key idea to estimate the mean value of the dissipation rate is to decompose the velocity, u = v + Φ , WAI-TONG LOUIS FAN, MICHAEL JOLLY, AND ALI PAKZAD where Φ is a stochastic, incompressible background field (4.3), carrying the inhomogeneities ofthe problem and v is a fluctuating incompressible field which is unforced and hence of arbitraryamplitude. Therefore using (1.1) and (4.2) we have dv = du − d Φ = − ( u · ∇ u − ν ∆ u + ∇ p ) dt + ( (cid:0) ( x δ − d X t , , (cid:1) ⊤ if 0 ≤ x ≤ δ . Now use the Itˆo’s product rule to obtain(4.4) v · dv = 12 d ( v · v ) −
12 ( x δ − d h X i t , for 0 ≤ x ≤ δ. Inserting u = v + Φ in (1.1), we find the stochastic process v satisfies, dv + d Φ = − ( v · ∇ v + v · ∇ Φ+Φ · ∇ v + Φ · ∇ Φ − ν ∆ v − ν ∆Φ + ∇ p ) dt, ∇ · v = 0 . The boundary conditions for v are periodic in the x and x directions while in the x direction, v ( x , x , , t ) = v ( x , x , h, t ) = 0 . The energy equation for v , obtained by taking the dot product of v with the above stochasticequation, integrating over D , and integrating by parts, is Z D v · dv dx | {z } I + Z D v · d Φ dx | {z } II = (cid:0) − ( v · ∇ v, v ) | {z } III − ( v · ∇ Φ , v ) | {z } IV − (Φ · ∇ v, v ) | {z } V − (Φ · ∇ Φ , v ) | {z } VI − ν k∇ v k − ν ( ∇ v, ∇ Φ) | {z } VII (cid:1) dt. (4.5)We shall estimate each numbered term in (4.5).
Term I.
Using Proposition 5.1 (ii) and (4.4) together with a direct calculation, we have Z D v · dv dx = 12 d k v k − Z D h ( x δ − d h X i t i dx = 12 d k v k − Z D δ h ( x δ − dx i d h X i t = 12 d k v k − δL σ dt. (4.6) Term II.
From (2.4) it follows that Z D v · d Φ dx = Z D δ v · dφ dx = (cid:20)Z D δ v (1 − x δ ) dx (cid:21) d X t = (cid:20)Z D δ v (1 − x δ ) dx (cid:21) θ ( U − X t ) dt + σ (cid:20)Z D δ v (1 − x δ ) dx (cid:21) dW t . (4.7) HREE-DIMENSIONAL SHEAR DRIVEN TURBULENCE WITH NOISE AT THE BOUNDARY 7
Term III.
Using the incompressibility of v , along with integration by parts, we get( v · ∇ v, v ) = 0 . Term IV.
Applying the Cauchy-Schwarz inequality (twice), we first estimate as (cid:12)(cid:12)(cid:12)(cid:12)Z L Z L v v dx dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z L Z L Z x ∂v ∂x ( x , x , ξ ) dξ Z x ∂v ∂x ( x , x , η ) dη dx dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ x k ∂v ∂x k k ∂v ∂x k . Using this together with Young’s inequality, we have | ( v · ∇ Φ , v ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z D δ v v ∂φ∂x dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) X t δ (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)Z L Z L Z δ v v dx dx dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) X t δ (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)Z δ (cid:20)Z L Z L v v dx dx (cid:21) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) X t δ (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)Z δ (cid:20) x k ∂v ∂x k k ∂v ∂x k (cid:21) dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) X t δ (cid:12)(cid:12)(cid:12)(cid:12) δ k ∂v ∂x k k ∂v ∂x k≤ δ | X t | (cid:20) k ∂v ∂x k + 12 k ∂v ∂x k (cid:21) ≤ δ | X t | k∇ v k . (4.8) Term V.
Using integration by parts, one can show that,(Φ · ∇ v, v ) = 0 . Term VI.
A pointwise calculation leads to Φ · ∇
Φ = 0, hence,(Φ · ∇ Φ , v ) = 0 . Term VII.
Direct calculation shows that k ∂φ∂x k = (cid:18) L δ (cid:19) | X t | . WAI-TONG LOUIS FAN, MICHAEL JOLLY, AND ALI PAKZAD
Therefore using the Cauchy-Schwarz inequality and Young’s inequality, we find | ν ( ∇ v, ∇ Φ) | ≤ ν Z Ω (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂x (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ∂v ∂x (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ ν k ∂φ∂x k k ∂v ∂x k≤ ν (cid:18) L δ (cid:19) | X t | k∇ v k≤ νδ L | X t | + ν k∇ v k . (4.9)Using the estimates for all the seven terms above in (4.5) yields,12 d k v k + 3 ν k∇ v k dt + σ (cid:20)Z D δ v (1 − x δ ) dx (cid:21) dW t ≤ δL σ dt + (cid:12)(cid:12)(cid:12)(cid:12)Z D δ v (1 − x δ ) dx (cid:12)(cid:12)(cid:12)(cid:12) θ | U − X t | dt + (cid:20) δ | X t | k∇ v k + νL | X t | δ (cid:21) dt. (4.10)The second term on the right hand side of inequality (4.10) can be bounded above as follows.Since v vanishes on the bottom wall, we express v ( x , x , x ) as R x ∂v ∂ζ ( x , x , ζ ) dζ , and applythe Cauchy-Schwarz inequality twice,(4.11) (cid:12)(cid:12)(cid:12)(cid:12)Z D δ v (cid:16) − x δ (cid:17) dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z L Z L Z δ (cid:16) − x δ (cid:17) Z x ∂v ∂x ( x , x , ξ ) dξ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z L Z L Z δ (cid:16) − x δ (cid:17) x / Z x (cid:12)(cid:12)(cid:12)(cid:12) ∂v ∂x (cid:12)(cid:12)(cid:12)(cid:12) dξ ! / dx ≤ δ / Z L Z L Z δ (cid:12)(cid:12)(cid:12)(cid:12) ∂v ∂x (cid:12)(cid:12)(cid:12)(cid:12) dξ ! / dx dx ≤ δ / L k∇ v k . Now applying this and then Young’s inequality,(4.12) (cid:12)(cid:12)(cid:12)(cid:12)Z D δ v (1 − x δ ) dx (cid:12)(cid:12)(cid:12)(cid:12) θ | U − X t | ≤ θ δ L k∇ v k | U − X t | ≤ ν k∇ v k + 19 ν δ L ( U − X t ) θ . Hence inserting estimate (4.12) in (4.10), and collecting terms, we have the following stochasticequation. 12 d k v k + (cid:18) − δ | X t | ν (cid:19) ν k∇ v k dt + σ (cid:20)Z D δ v (cid:16) − x δ (cid:17) dx (cid:21) dW t ≤ (cid:20) L δ σ + νL | X t | δ + 19 ν δ L ( U − X t ) θ (cid:21) dt. (4.13)The stochastic differential inequality (4.13) is interpreted in its integral form. HREE-DIMENSIONAL SHEAR DRIVEN TURBULENCE WITH NOISE AT THE BOUNDARY 9
With our choice of δ in (4.1), the stochastic integral inequality (4.13) gives, for all T ≥ k v ( T ) k − k v k + 14 Z T ν k∇ v k dt + Z T σ (cid:20)Z D δ v (cid:16) − x δ (cid:17) dx (cid:21) dW t ≤ ν L Z T σ | X t | + U dt + 2 L Z T | X t | ( | X t | + U ) dt + 172 L ν θ Z T ( U − X t ) ( | X t | + U ) dt. (4.14)Note that we can ignore the term k v ( T ) k on the left of (4.14) to obtain the following almostsure upper bound for the energy dissipation. Lemma 4.2 is the main result of this section. Lemma 4.2.
With probability one, the following inequality holds for all
T > . (4.15) Z T ν k∇ v k dt + 4 M T ≤ k v k + Y T , where (4.16) Y T := Z T ν L σ | X t | + U + 8 L ( | X t | + U | X t | ) + 118 L ν θ ( U − X t ) ( | X t | + U ) dt and (4.17) M T := Z T σ (cid:20)Z D δ v (1 − x δ ) dx (cid:21) dW t . Remark . The term hR D δ v (1 − x δ ) dx i dW t in (4.7) is the first time that we need to make senseof an Itˆo integral. Assumption (2.1) and (4.11) ensure that M T is a local martingale with quadraticvariation, h M i T = σ Z T (cid:20)Z D δ v (1 − x δ ) dx (cid:21) dt (4.18) ≤ σ Z T δ L k∇ v k dt (4.19) = ν L σ Z T k∇ v k ( | X t | + U ) dt, (4.20)where the last equality follows from the definition of δ in (4.1). Remark . Consider the stochastic integral. For each n ≥ τ n := inf (cid:26) t ≥ Z t k∇ v k ( | X t | + U ) ds > n (cid:27) . The stochastic integral R T ∧ τ n k∇ v k ( | X t | + νh ) dW t (as a process indexed by T ) is a martingale and hence(4.21) E [ M T ∧ τ n ] = E "Z T ∧ τ n k∇ v k ( | X t | + U ) dW t = 0 for all T ∈ [0 , ∞ ) , n ∈ N . From (4.15), for all n ≥
1, we have P -a.s.,(4.22) Z T ∧ τ n ν k∇ v k dt + M T ∧ τ n ≤ k v k + Y T ∧ τ n . Taking expectation E on both sides, applying (4.21), and observe that Y t is nondecreasing in t , weobtain(4.23) E Z T ∧ τ n ν k∇ v k dt ≤ E [2 k v k + Y T ] . By Assumption (2.1), the process h M i t does not blow up in the sense that lim n →∞ τ n = + ∞ almostsurely. Hence E R T ∧ τ n k∇ v k dt → E R T k∇ v k dt as n → ∞ and therefore(4.24) E Z T ν k∇ v k dt ≤ E [2 k v k + Y T ] . Estimation of the Mean Value
With Remark 4.4 in mind, the rest of proof is as follows. To construct the estimate on E [ h ǫ i T ],take the expected value of (4.15) over (Ω , A , F , P ), then average it over [0 , T ], and finally take thelimit superior as T → ∞ .We shall estimate the four terms in Y t (4.16) separately in (5.1:5.4). To this end we need thefollowing standard properties for the stationary OU process and Gaussian random variables (for aproof and additional properties see [17]). Proposition 5.1.
Let X t be a stationary Ornstein–Uhlenbeck process (2.4) . The following hold forall t ≥ .(i) X t ∼ N ( U, σ θ ) ,(ii) h X i t = σ t , where h X i t is the quadratic variation of X ,(iii) E (cid:2) | X t | (cid:3) = σ θ ,(iv) E (cid:2) | X t | (cid:3) = U + 6 U (cid:16) σ θ (cid:17) + 3 (cid:16) σ θ (cid:17) and therefore E (cid:2) | U − X t | (cid:3) = 3 (cid:16) σ θ (cid:17) ,(v) E (cid:2) | X t | (cid:3) = U + 15 U σ θ + 45 U (cid:16) σ θ (cid:17) + 15 (cid:16) σ θ (cid:17) . For the first term on the right of (4.16), simply from | X t | ≥
0, we have(5.1) 1 T E (cid:20)Z T σ | X t | + U dt (cid:21) ≤ T E (cid:20)Z T σ U dt (cid:21) = 1
U σ . Recall that, X t has normal distribution with mean U and variance σ θ for all t ∈ R + under P . ByFubini’s theorem and Jensen’s inequality, the second term is estimated as(5.2) 1 T E (cid:20)Z T | X t | dt (cid:21) = 1 T Z T Z Ω | X | dP dt ≤ T Z T (cid:18)Z Ω | X | dP (cid:19) / dt ≤ (cid:0) E | X t | (cid:1) / = (cid:18) U + 6 U (cid:16) σ θ (cid:17) + 3 (cid:16) σ θ (cid:17) (cid:19) / . HREE-DIMENSIONAL SHEAR DRIVEN TURBULENCE WITH NOISE AT THE BOUNDARY 11
Also by Fubin’s theorm, the third term can be written as(5.3) 1 T E (cid:20)Z T | X t | dt (cid:21) = E (cid:2) | X t | (cid:3) = σ θ . Since U − X t is a centered normal variable with variance σ θ , we again interchange the order ofintegration to obtain the following bound on the fourth term of Y t in (4.16)(5.4) 1 T E (cid:20)Z T ( U − X t ) ( | X t | + U ) dt (cid:21) ≤ U T E (cid:20)Z T | U − X t | dt (cid:21) = 1 U E (cid:2) | U − X t | (cid:3) = 1 U σ θ . Now take the expectation of (4.15), divide by T and | D | = L h , and use the above estimates(5.1), (5.2), (5.3), (5.4), and (4.21) to obtain(5.5) ε ≤ (cid:18) νU h + 4 Uθ h + 136 ν θh U (cid:19) σ + 8 (cid:16) U + 6 U (cid:16) σ θ (cid:17) + (cid:16) σ θ (cid:17)(cid:17) / h , which also can be represented as(5.6) ε ≤ (cid:18)
13 1Re + 4
Uθ h + 136 1Re h θU (cid:19) σ + 8 (cid:16) U + 6 U (cid:16) σ θ (cid:17) + (cid:16) σ θ (cid:17)(cid:17) / h . Remark . One could replace U in (4.1) with the dimensionally consistent term νh . Proceeding asbefore, one arrives at(5.7) ε . (cid:18) νh θ + h θν (cid:19) σ + 8 (cid:16) U + U (cid:16) σ θ (cid:17) + Big ( σ θ (cid:17)(cid:17) / h . This avoids the singularity in (5.5) as U →
0, but replaces it with one as ν → . Remark . Based on our current analysis, if we were to instead take X t to be Brownian motion,i.e., X t = W t , this would result in a potential over-dissipation of the model, since,1 T E (cid:20)Z T | X t | dt (cid:21) = 1 T Z T E (cid:2) W t (cid:3) dt = 12 T → ∞ , as T → ∞ . Remark . If θ →
0, the estimate (5.5) also allows for over-dissipation of the model E [ h ε i ] → ∞ .This observation is also consistent with Remark 5.3 because while θ →
0, the Ornstein–Uhlenbeckprocess → the Wiener process in (2.4).6. Estimation of the Variance
Define, for
T > E T := Z T ν k∇ v k dt. Lemma 4.2 tells us that(6.2) E T ≤ k v k + Y T + | M T | . The variance of E T is bounded above by the second moment, and(6.3) E [ |E T | ] ≤ E (cid:2) k v k + | Y T | + | M T | (cid:3) . Bound E [ | M T | ] . From (4.18) we have E [ | M T | ] = E [ h M i T ] ≤ ν L σ Z T k∇ v k ( | X t | + U ) dt ≤ ν L σ U E [ E T ] . (6.4)6.2. Bound E [ | Y T | ] . We apply the Cauchy-Schwarz inequality to (4.16) to obtain | Y T | ≤ T Z T (cid:20) ν L σ | X t | + U + 8 L ( | X t | + U | X t | ) + 118 L ν θ ( U − X t ) ( | X t | + U ) (cid:21) dt. (6.5)Hence E [ | Y T | ] ≤ T Z T E (cid:20) ν L σ | X t | + U + 8 L ( | X t | + U | X t | ) + 118 L ν θ ( U − X t ) ( | X t | + U ) (cid:21) dt (6.6) = T E (cid:20) ν L σ | X t | + U + 8 L ( | X t | + U | X t | ) + 118 L ν θ ( U − X t ) ( | X t | + U ) (cid:21) (6.7)because X t has normal distribution with mean U and variance σ θ for all t . By the Cauchy-Schwarzinequality ( P ni =1 a i ) ≤ n P ni =1 a i with n = 4, which applied to the above expectation gives(6.8) E (cid:20) ν L σ | X t | + U + 8 L ( | X t | + U | X t | ) + 118 L ν θ ( U − X t ) ( | X t | + U ) (cid:21) ≤ (cid:26) ν L σ U + 64 L E [ X t ] + 64 L U E [ X t ] + L ν θ U E [( U − X t ) ] (cid:27) = 4 (cid:26) ν L σ U + 64 L " U + 15 U σ θ + 45 U (cid:18) σ θ (cid:19) + 15 (cid:18) σ θ (cid:19) +64 L U (cid:20) U + 6 U (cid:16) σ θ (cid:17) + 3 (cid:16) σ θ (cid:17) (cid:21) + L ν θ U (cid:16) σ θ (cid:17) (cid:27) := Θ , where in the last equality we used Proposition 5.1 to estimate the moments of normal randomvariable.6.3. Summarizing.
Putting the above and (6.4) into (6.3), we obtain E [ |E T | ] ≤ E (cid:2) k v k (cid:3) + 3 ν L σ U E [ E T ] + 3 T Θ . (6.9) HREE-DIMENSIONAL SHEAR DRIVEN TURBULENCE WITH NOISE AT THE BOUNDARY 13
Applying Jensen’s inequality and then Young’s inequality, we find α E [ E T ] ≤ α p E [ |E T | ] ≤ α E [ |E T | ]2 , where α = 3 ν L σ U has the same dimension as that of E T .Using this in (6.9), we have E [ |E T | ] ≤ E (cid:2) k v k (cid:3) + α E [ |E T | ]2 + 3 T Θso that E [ |E T | ] ≤ E (cid:2) k v k (cid:3) + α + 6 T Θ , and hence lim sup T →∞ E [ |E T | ] T ≤ . Recalling | D | = L h , we have(6.10) lim sup T →∞ E [ h ǫ i T ] = lim sup T →∞ E [ |E T | ] | D | T ≤ L h = 24 h (cid:26) ν σ K + 64 " U + 15 U σ θ + 45 U (cid:18) σ θ (cid:19) + 15 (cid:18) σ θ (cid:19) +64 U (cid:20) U + 6 U (cid:16) σ θ (cid:17) + 3 (cid:16) σ θ (cid:17) (cid:21) + ν θ U (cid:16) σ θ (cid:17) (cid:27) . Conclusion and Commentary
In this paper we have derived uniform (in T ) bounds for both the mean and the variance of theenergy dissipation rate for solutions of the incompressible Navier–Stokes equations with a boundarywall moving as a stationary Ornstein–Uhlenbeck process. We recover the bound for the deterministiccase in [14] as the variance of the OU process tends to 0. A similar argument can be used to findhigher moment bounds. A novelty of our method is the construction of a carefully chosen stochasticbackground flow Φ that depends on the stochastic forcing, as indicated in (4.1). Our technique canbe readily generalized to the case where the OU process is replaced by a general gradient system ofthe form(7.1) dX t = −∇ h ( X t ) dt + σ dW t , where σ >
0. The OU process (2.4) is the case when h ( x ) = − θ ( x − U ) /
2. It is well-known that if Z ( σ ) := Z R exp (cid:18) − σ h ( x ) (cid:19) dx < ∞ , then 1-dimensional gradient system (7.1) has a unique invariant distribution given by the Gibbsmeasure(7.2) 1 Z ( σ ) exp (cid:18) − σ h ( x ) (cid:19) . This technique can also be generalized to jump processes. The analysis herein would allow forover-dissipation of the model if the noise at the boundary were taken to be the Wiener process, asnoted in Remarks 5.3 and 5.4.Finally, it was crucial to take the limit superior in time after the expectation. Our estimate doesnot provide a bound when the operations are taken in the reverse order. It remains to find a boundin the latter case, or quantify the difference in the two expressions describing the rate of dissipation.8.
Acknowledgments
The work of M. Jolly was supported in part by NSF grant DMS-1818754. W.T. Fan is partiallysupported by NSF grant DMS-1804492 and ONR grant TCRI N00014-19-S-B001.
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E-mail address : [email protected] Department of Mathematics, Indiana University Bloomington, IN 47405, USA
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