Derivation of an intermediate viscous Serre-Green-Naghdi equation
aa r X i v : . [ m a t h . A P ] J a n Derivation of an intermediate viscous Serre–Green–Naghdiequation
Denys Dutykh a Herv´e Le Meur b , a Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chamb´ery, France b LAMFA, CNRS, UMR 7352, Universit´e Picardie Jules Verne, 80039 Amiens, France
Received *****; accepted after revision +++++Presented by XXXXXX
Abstract
In this note we present the current status of the derivation of a viscous Serre–Green–Naghdi system. For thisgoal, the flow domain is separated into two regions. The upper region is governed by inviscid Euler equations,while the bottom region (the so-called boundary layer) is described by Navier-Stokes equations. We consider aparticular regime linking the Reynolds number and the shallowness parameter. The computations presented inthis note are performed in the fully nonlinear regime. The boundary layer flow reduces to a Prantdl-like equation.Further approximations seem to be needed to obtain a tractable model.
R´esum´eObtention des ´equations interm´ediaires de Serre–Green–Naghdi visqueuse.
Dans cette note nouspr´esentons l’´etat actuel de l’obtention du syst`eme de type Serre–Green–Naghdi visqueux dans un canal. Nouss´eparons le domaine fluide en deux couches. La couche sup´erieure est d´ecrite par les ´equations d’Euler tandis quela couche limite en-dessous, ob´eit aux ´equations de Navier-Stokes. Nous consid´erons un r´egime pleinement nonlin´eaire o`u le nombre de Reynolds est li´e au rapport de la longueur d’onde typique `a la profondeur moyenne.La dynamique de l’´ecoulement dans la couche limite se ram`ene `a une ´equation de type Prantdl. Des hypoth`esessuppl´ementaires sont n´ecessaires afin d’obtenir un mod`ele utilisable en pratique.
Version fran¸caise abr´eg´ee
Nous tentons d’obtenir un mod`ele r´eduit aux ´equations de l’´ecoulement d’un fluide visqueux dans uncanal peu profond. Nous ne supposons pas l’amplitude des vagues comme petite (r´egime non lin´eaire).Comme dans le cas du r´egime lin´eaire (cf. [5]), nous r´esolvons les ´equations dans la zone principale
Email addresses:
[email protected] (Denys Dutykh),
[email protected] (Herv´e Le Meur).
Preprint submitted to Elsevier Science 13 janvier 2020 ouvern´ee par des ´equations d’Euler pour arriver `a (13). Puis nous tentons la mˆeme r´esolution dans lacouche limite, mais ne pouvons aller plus loin que (18). Cette derni`ere ´equation est de type Prandtl. Elleest connue pour un comportement tr`es sensible `a chacun de ses termes (cf. [4]) et donc laisse peu d’espoirpour ˆetre simplifi´ee afin d’obtenir un mod`ele 1D.
1. Introduction
The water wave theory has been essentially developed in the framework of the inviscid, and very oftenalso irrotational, Euler equations. However, various viscous effects are inevitably present in laboratoryexperiments and even more in the real world. Thus, the conservative conventional models have to besupplemented with dissipative effects to improve the quality of their predictions. A straightforward energybalance asymptotic analysis shows that the main dissipation takes place at the bottom boundary layer[1, Section §
2] (or at the lateral walls if they are also present [2]). In this way, the corresponding longwave Boussinesq-type systems have been derived taking into account the boundary layer effects [3]. In [5],the author derives the viscous Boussinesq model without the irrotationality assumption. Other articlesalready took the vorticity into account, even for fully nonlinear Boussinesq equations (here called Serre-Green-Naghdi or SGN) [6]. Fully nonlinear models are becoming very popular. In the present note wereport the current status of the derivation of a viscous counterpart of the well-known SGN equations.The asymptotic regime relates the Reynolds number to the shallowness parameter.
2. Primary equations
Consider the flow of an incompressible liquid in a physical two-dimensional space over a flat bottom andwith a free surface. We assume additionally that the fluid is homogeneous (i.e. the density ρ = const)and the gravity acceleration g is constant. For the sake of simplicity, in this study we neglect all otherforces (such as the Coriolis force and friction). Hence, we deal with pure gravity waves. We introduce aCartesian coordinate system O ˜ x ˜ y . The horizontal line O ˜ x coincides with the still water level ˜ y = 0 andthe axis O ˜ y points vertically upwards. The fluid layer is bounded below by the horizontal solid bottom˜ y = − d and above by the free surface ˜ y = ˜ η (˜ x, ˜ t ) .In order to make the equations dimensionless, we choose a characteristic horizontal length ℓ , verticalheight of the free surface A and mean depth d . All this enables us to define a characteristic velocity c = √ gd . Then one may define dimensionless independent variables:˜ x = ℓx, ˜ y = dy, ˜ t = tℓ/c . This enables us to define the dimensionless fields:˜ u = c u, ˜ v = dc ℓ v, ˜ p = ˜ p atm − ρgd y + ρgd p, ˜ η (˜ x, ˜ y, ˜ t ) = Aη ( x, y, t ) . We also define some dimensionless numbers, characteristic of the flow: ε = Ad , µ = d ℓ , Re = ρc dν . The system of Navier-Stokes equations can then be written in 2D and in dimensionless variables:2 u t + uu x + vu y − / Re ( µu xx + u yy /µ ) + p x = 0 µ ( v t + uv x + vv y ) − µ / Re ( µv xx + v yy /µ ) + p y = 0 u x + v y = 0 − ( p − εη ) I + 2Re µu x ( u y + µ v x ) / u y + µ v x ) / µv y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) εη n = 0 on y = εηη t + u ( y = εη ) η x − v ( y = εη ) /ε = 0 on y = εηu ( y = −
1) = v ( y = −
1) = 0 , (1)where we denote u | εη = u ( y = εη ) = u ( x, y = εη ( x, t ) , t ).One could assume the fields to be small around the hydrostatic flow (which is lifted by the changeof field from ˜ p to p ), so around ( u, v, p, η ) ≃
0. Such an assumption is contradictory with our nonlinearassumption where ε is assumed not to be small. Yet, should we make this assumption, we would be led toa linear system identical (up to changes of variables) to System (7) of [5]. The study of the linear regimesuggests to assume, not only in the Boussinesq regime:Re ≃ µ − . (2)Below, we solve the problem in the bulk part where Euler’s equations are justified to apply (Section2.1), then try to solve the velocity in the boundary layer (Section 2.2). In this last section, we are led toPrandtl’s equation that prohibits any further advance to the best of our knowledge.[??? `a nettoyer] What is the size of the boundary layer where the no-slip condition forces the fluid tohave a large gradient of velocity ? In the same way as in [5], one may assume it is of size µ : y = − µ γ. (3)One might be surprised that the gravity-viscosity layer be of size O ( µ ) (or a little larger) while oneusually assumes the size of the viscous layer to be of size O (Re − / ) = O ( µ / ). Indeed the classical termstems from the 1 / Re u yy term which is replaced here by 1 / ( µ Re) u yy . So µ Re us = Re classical and the sizeof the boundary layer is Re − / classical = ( µ Re us ) − / = ( µ − ) − / = µ !2.1. Resolution in the upper part (Euler)
In the upper part, y ≫ − µ and µ is small. So one may drop the Laplacian and keep from (1): u t + uu x + vu y + p x = O ( u yy / ( µ Re)) + O ( µ ) µ ( v t + uv x + vv y ) + p y = O ( µ ) u x + v y = 0 − p + εη = O ( u y | εη / Re) + O ( µ/ Re) on y = εη ( p − εη ) εη x + 2( − µu x εη x + ( u y + µ v x )) / Re = 0 on y = εηε ( η t + u | εη η x ) = v | εη on y = εη. (4)First, one may notice that the viscosity terms are no more present inside the domain. It is argued in [5]that one may (and even must) then drop the fifth equation from this system, due to the fact that thefluid is indeed no more viscous in this part of the domain.It is classical to use (4) to get v = v | εη − Z yεη u x d y ′ , (5)3here v | y = εη is given by (4) . One may use this vertical velocity in (4) to compute p y . Thanks to (4) ,one has: p = εη − µ " ( y − εη ) (( v | εη ) t + u x | εη εη t ) + (cid:18)Z yεη u (cid:19) (cid:18) ( v | εη ) x + u x | εη εη x − (cid:18)Z yεη u x (cid:19) v | εη (cid:19) − Z yεη Z y ′ εη u xt − Z yεη u Z y ′ εη u xx ! + Z yεη u x Z y ′ εη u x ! + O (cid:18) u y | εη Re (cid:19) + O (cid:16) µ Re (cid:17) + O ( µ ) . (6)So we have both v (thanks to (5)) and p (thanks to (6)) and may rewrite (4) with the only fields u and η : u t + uu x + u y (cid:18) v | εη − Z yεη u x (cid:19) + εη x − µ " ( y − εη ) (( v | εη ) t + u x | εη εη t )+ (cid:18)Z yεη u (cid:19) (( v | εη ) x + u x | εη εη x ) − (cid:18)Z yεη u x (cid:19) v | εη − Z yεη Z y ′ εη u xt − Z yεη u Z y ′ εη u xx ! + Z yεη u x Z y ′ εη u x ! x = O (cid:18) ( u y | εη ) x Re (cid:19) + O (cid:16) µ Re (cid:17) + O (cid:18) u yy µ Re (cid:19) . (7)In order to take off the dependence on y of this equation, we integrate between y = − µ γ ∞ and y = εη ( x, t ) and we define: H µ,γ ∞ = 1 + εη − µ γ ∞ , and ¯ u ( x, t ) = 1 H µ,γ ∞ Z εη − µ γ ∞ u ( x, y ) d y. (8)We also need a lemma that will enable to commute the integration and the x differentiation under anassumption: Lemma 2.1
Let F a C function defined in Ω = { ( x, y ) /x ∈ R , − µ γ ∞ < y < εη ( x ) } , such that if ∀ x, F ( x, y = εη ) = 0 , then Z εη − µ γ ∞ ∂F∂x ( x, y )d y = ∂∂x Z εη − µ γ ∞ F ( x, y )d y. (9)The proof is very simple and left to the interested reader.Thanks to Lemma 2.1, one may commute the x differentiation of the square bracket in Equation (7)with the integral since the terms in the square brackets vanish at y = εη . An integration by parts of the R u y (cid:16) v | εη − R yεη u x (cid:17) d y term, and the treatment of R ( u ) x leads to (below, we write H = H µ,γ ∞ ): H ¯ u t + (cid:18)Z εη − µ γ ∞ u (cid:19) x + H εη x + (¯ u − u | − µ γ ∞ ) ( εη t + ( H ¯ u ) x ) − ¯ u ( H ¯ u ) x − µ (cid:20) − H ∂ t + ¯ u∂ x ) ( v | εη ) + εη t ( u x | εη − ¯ u x ) + εη x (¯ uu x | εη − ¯ u x u | εη ))+ Z εη − µ γ ∞ Z yεη ( u − ¯ u )d y ′ d y × (( v | εη ) x + u x | εη εη x ) − Z εη − µ γ ∞ Z yεη ( u − ¯ u ) x d y ′ d y v | εη − Z εη − µ γ ∞ Z yεη "Z y ′ εη u xt + u Z y ′ εη u xx − u x Z y ′ εη u x d y ′ d y x = O (cid:18) ( u y | εη ) x Re (cid:19) + O (cid:18) u yy µ Re (cid:19) . (10)We need now the following (double) assumption: u ( x, y, t ) = ¯ u ( x, t ) + µ ˜ u ( x, y, t ) , (11)4ith Z εη − µ γ ∞ ˜ u = 0 and ¯ u ( x, t ) = 1 H µ,γ ∞ Z εη − µ γ ∞ u ( x, y ) d y. (12)The mean ¯ u is the same as before. Notice that the expansion of a function around its mean value ¯ u is notan assumption. The real assumption is that the discrepancy with the mean is small ( O ( µ )). An other wayto formulate this assumption is to look at an expansion in µ , in which one assumes that the zeroth orderterm does not depend on y and that the next order term is zero-mean value. This gives two different waysto see its consequences. Last but not least, this assumption is proved to be true in Lemma 11 (Eq. (77))of [5] in case of a Boussinesq flow (where ε is small) without the assumption of irrotationality in the Eulerpart of the flow. We remind the reader that we still assume we solve the Euler equations and not yet theNavier-Stokes ones. So we are coherent.Upon this assumption, (10) simplifies to: H ¯ u t + H ¯ u ¯ u x + HH x + (¯ u − u | − µ γ ∞ ) ( H t + ( H ¯ u ) x ) − µ (cid:20) − H ∂ t + ¯ u∂ x ) ( v | εη ) − H u xt + ¯ u ¯ u xx − ¯ u ¯ u x ) (cid:21) x = O (cid:18) ( u y | εη ) x Re (cid:19) + O (cid:18) u yy µ Re (cid:19) + O ( µ ) . (13) Remark 1 The attention may be drawn to the fact that H t + ( H ¯ u ) x = εη t + H x ¯ u + H ¯ u x = v | εη + H ¯ u x + O ( µ ) = v | − µ γ ∞ + O ( µ ) . In the Euler case, v | − µ γ ∞ = 0 since the flow does not cross the boundary. So we would not need tocompute u | − µ γ ∞ . Resolution in the boundary layer
We write the system that applies in the layer, extracted from (1): u t + uu x + vu y − µu xx / Re − u yy / ( µ Re) + p x = 0 ,µ ( v t + u v x + v v y ) − µ v xx / Re − µv yy / Re + p y = 0 ,u x + v y = 0 ,u ( y = −
1) = v ( y = −
1) = 0 . (14)This system may be rewritten with the change of variables justified in (3) y = − µ γ , where γ > γ ∞ . We also use the assumption (2) on Re such thatRe = R µ − where R is a constant. We should have tilded the fields but would have dropped the tildesoon after. So we omit them. When precision is needed, we denote u BL = u ( x, γ, t ) the horizontal velocityin the boundary layer. The system writes: u t + u u x + v u γ /µ − µ /R u xx − u γγ /R + p x = 0 ,µ (cid:0) v t + u v x + v v γ /µ (cid:1) − ( µ /R ) v xx − ( µ /R ) v γγ + p γ /µ = 0 ,u x + v γ /µ = 0 ,u ( x, γ = 0 , t ) = v ( x, γ = 0 , t ) = 0 . (15)As is classical, we first compute v (owing to (15 ) and (15 ): v ( x, γ, t ) = 0 − µ Z γγ =0 u x d γ ′ . (16)Then we can compute the differentiated pressure from (15) that proves p γ = O ( µ ). As a consequence, p BL ( x, γ, t ) = p BL ( γ → γ ∞ ) + O ( µ ) , p BL ( γ → γ ∞ ) is determined thanks to a matching condition with the bottom of the upper part(Euler part). From (6), and owing to the already stated assumption (11, 12) the pressure in the boundarylayer is, up to O ( µ ): p BL ( x, γ, t ) = εη ( x, t ) + µ (cid:2) H ( ∂ t + ¯ u∂ x )( v | εη ) + H / u xt + ¯ u ¯ u xx − ¯ u x ¯ u x ) (cid:3) + O ( µ ) . (17)Last, we may gather v BL (from (16)), p BL (from (17)) and rewrite (15) : u BLt + u BL u BLx − u BLγ Z γ u BLx ( γ ′ )d γ ′ − u BLγ γ R + εη x + µ (cid:2) H ( ∂ t + ¯ u∂ x )( v | εη ) + H / u xt + ¯ u ¯ u xx − ¯ u x ¯ u x ) (cid:3) x = O ( µ ) . (18)At that stage of the derivation, we recognize a Prandtl’s equation. We do not know how to derive a simplermodel. It is well-known that Prandtl’s equation still resists to the best of physicists and mathematicians.It was proved to be ill-posed in [4] and partially well-posed later. Moreover, so as to couple this equationwith (13), one should assume a link between u BL and u | − µ γ ∞ like identity by continuity.
3. Conclusions
We stopped our derivation, in the fully nonlinear regime, at equation (13), (18), which is only anintermediate state because we still have functions of x, γ, t . We would like to stress it out that Equations(13), (18) are still Galilean invariant despite the presence of the boundary layer. The proposed modelenjoys this property because we did not introduce any drastic simplifications yet at this level. To makefurther progress, the Prantdl-type equation should be further simplified but it seems highly speculative.One strategy could consist in assuming a particular profile of the velocity u BL in the coordinate γ similarto the one computed in [5] in the Boussinesq regime, but it is incoherent. Further research is needed toreach an effective 1D model. Acknowledgements
This research did not receive any specific grant from funding agencies in the public, commercial, ornot-for-profit sectors.
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