Towards a new friction model for shallow water equations through an interactive viscous layer

Transcription

1 Towards a new friction model for shallow water equations through an interactive viscous layer François James, Pierre-Yves Lagrée o, Minh H. Le and Mathilde Legrand o Sorbonne Université, CNRS, UMR 719, Institut Jean Le Rond d'alembert, F-755 Paris, France Laboratoire d'hydraulique Saint Venant ENPC, CEREMA, EDF R&D, Chatou, France, Mathématiques Analyse, Probabilités, Modélisation Orléans MAPMO, Université d'orléans & CNRS UMR 7349, BP 6759, F-4567 Orléans Cedex, France francois.james@univ-orleans.fr Abstract The derivation of shallow water models from Navier-Stokes equations is revisited. An improved velocity prole is proposed, based on the superposition of an ideal uid and a viscous layer inspired by the Interactive Boundary Layer interaction used in aeronautics. This leads to a new friction law which takes into account the phase-lag between friction and topography, a key-point in the instability mechanism involved in erosion. The resulting system is an extended shallow water model consisting of three depth-integrated equations: the two rst ones are mass and momentum conservation in which a slight correction on hydrostatic pressure has been made; and the third one, known as von Kármán equation, describes evolution of the viscous layer. This coupled model is shown to be hyperbolic under the thin viscous layer condition; a Godunov-type nite volume scheme is also proposed. Several numerical examples are provided which emphasize the ability of the model to recover the phase-lag behaviour and possible reverse ow. Keywords: shallow water, viscous layer, friction, phase-lag, Prandtl equation, von Kármán equation. 1 AMS subject classications: Primary: 35B4, 35D3, 35L6, 35Q9, 49K Introduction Many phenomena in uvial or maritime hydraulics involve free surface ows in shallow waters for the study e.g. of oods and tides. Shallow water equations were originally introduced by Saint-Venant in 1871 [1] in the context of channel modelling. Since then, the model has been widely extended and is used in the modelling and numerical simulation of a number of natural or man-made phenomena such as river ow [6, 5], ood forecasting [7, 33], pollutant transport [6, 48], dam-break [1, 53], tsunami [, 3, 45], overland ow [13, 16, 5], soil erosion [8, 4]. The shallow water system can be derived from the incompressible Navier-Stokes equations under several hypotheses; the main one being the long wave approximation meaning that the characteristic wavelength is much larger than the water depth. Two consequences follow then: the hydrostatic pressure law, and the viscosity vanishing in the horizontal direction. Next, to proceed from Navier-Stokes to shallow water, the equations are integrated along the vertical direction. At this point, care has to be taken of the vertical velocity prole, which on the one hand has to be approximated to deal with nonlinearities of the momentum ux, but on the other hand drives the bottom boundary condition, hence the friction phenomena. Two classical assumptions on the longitudinal velocity prole along the vertical direction lead to explicit integrations. The rst one is a viscous Poiseuille-like i.e. parabolic prole on the whole water depth which gives rise to a linear with respect to the depth-averaged velocity friction term, sometimes referred to as laminar friction. The second one is a constant prole, somehow corresponding to an ideal uid; but, by construction, there is a priori no friction term in the integrated equations. Friction has to be added afterwards using empirical 1

2 laws such as Manning, Chézy, etc see e.g. [11]. The main drawback of these classical approaches is the nonadaptability of the friction terms for large variations of velocity because the assumed proles parabolic or at do not hold. We intend here to pay a particular attention to the fact that these empirical laws are unable to describe the uid inertia or more precisely to predict the phase-lag between the bottom friction and a perturbation of the bed which is known as an essential mechanism for dune or ripple formation [3]. Indeed, it is well reported that for the case of a sub-critical ow on a bump, the maximum of the friction must be slightly shifted upstream of the crest to drag the particle from the troughs up to the crest. Consequently, a coupling of classical shallow water equations with a mass conservation equation for sediment transport e.g. Exner equation [17] for bedload case cannot predict the bed instability; see [1, 34] for more details. We look for a more exible model with a better understanding of how the no-slip boundary condition gives rise to the friction term in the depth-integrated equations. This will allow to recover this phase-lag phenomenon as well as recirculation of the ow boundary layer separation. This is done by using an asymptotic description of the uid as a superposition of an ideal uid over a viscous layer located at the bottom. The thickness of the viscous layer is quantied by a small parameter δ related to the inverse of the Reynolds number of the ow. Integrating the incompressibility equation under this consideration leads to the same mass conservation equation of the usual shallow water system. On the contrary, the integration of the momentum equation exhibits major dierences. On the one hand, it turns out that the order of magnitude of the friction term is larger as expect: precisely of order δ, while the above mentioned Poiseuille prole leads to a δ order of magnitude. In the case of an ideal uid δ =, the model degenerates, of course, into the classical shallow water one. On the other hand, we introduce a new closure for the momentum ux which involves an additional pressure law of order δ. At this stage, we obtain a system of two equations which are similar in structure with the usual shallow water system, but involving several additional unknown functions related to the viscous layer. The next step consists therefore in a careful analysis of the viscous layer. Following a classical methodology in aerodynamics, see e.g. [49], we integrate the Prandtl equation along the vertical axis to obtain the so-called von Kármán equation. It describes the evolution of the so-called displacement thickness δ 1 see Figure 1, which is involved in the denition of the afore mentioned corrective pressure, and can be interpreted as some physical thickness of the viscous layer. The system has to be complemented by the velocity equation of the ideal uid, since it is involved in the von Kármán equation. We will discuss on the eects for ows over short bumps. The acceleration induced by the bump will change a lot the basic ow so that the shape velocity prole is no longer a half-poiseuille nor a at one. This study aims to understand this kind of ows which are not taken into account by the shallow water equations themselves. The outline of the paper is as follows. In a rst section we recall the Navier-Stokes system, and state the long wave approximation which is convenient for shallow water approximation. Next we turn to the viscous layer analysis, and derive Prandtl and von Kármán equations. Velocity proles are also introduced. The third section is devoted to the derivation of various formulations of the Extended Shallow Water System. In Section 4 we derive some formal properties of the model, together with the numerical scheme. Finally, we evidence several properties of the model by numerical simulations. 1 From Navier-Stokes to shallow water equations In this section we recall how classical models for shallow waters are obtained from Navier-Stokes equations. The rst assumption is a long wave approximation, stating that indeed we deal with a thin layer of water. Next, we integrate along the vertical direction, assuming a given velocity prole on the whole water depth. 1.1 Navier-Stokes equations We consider a uid evolving in a time-dependant domain Ω t = R {f b x y ηt, x}. This thin layer is limited below by a xed bottom represented by a function y = f b x and above by the free surface described by y = ηt, x. We denote the water depth h = η f b. In this study, the properties of the air above the free surface are completely neglected see Figure 1. Our starting point is to consider the dimensionless Navier-Stokes equations expressing the mass and momentum conservations of an incompressible Newtonian uid [49]. For the sake of simplicity, we limit ourselves in this work to consider only laminar ows. Indeed, the asymptotic from the Navier-Stokes equations is clearer

3 Figure 1 Domain under consideration: the water layer is dened by the depth h, the bottom is a given function f b and η is the free surface. Two families of velocity proles are displayed for the ow over the topography, rst with the usual half-poiseuille description dashed, and second with the at prole with a boundary layer plain. Note that the shear slope of the velocity at the wall is completely dierent in those two descriptions. and the resulting description is quantitative. A similar study for turbulent ows can be made with a modied Reynolds tensor. Nondimensionalization has been made with the same characteristic length h, e.g. a reference water depth, for both the abscissa and the ordinate. The dimensionless Navier-Stokes equations write x u + y v = 1.1 t u + u x u + v y u = x p + 1 u Re 1. t v + u x v + v y v = y p 1 F r + 1 v Re 1.3 where u, v are the horizontal and vertical velocities respectively and p is the pressure. We have dened the Reynolds Re and Froude F r numbers given by Re := u h ν, F r := u gh in which u and ν being the reference velocity and kinematic viscosity respectively and the constant g stands for the acceleration due to gravity. The Reynolds number expresses the ratio between the inertia force and the viscosity; the Froude number represents the ratio between the kinetic and potential energies. The system is complemented with the following boundary conditions: at the bottom y = f b x: no-slip condition, i.e. u = v =, at the free surface y = ηt, x: kinematic boundary condition: v = t η + u x η x u p continuity of the stress tensor: σ n =, where σ = x v + y u is the stress tensor y u + x v y v p x η and n = is the normal to the free surface Long wave scaling Up to now, no hypothesis has been taken into account for the size order of the characteristic quantities u, h. We have in mind applications to rivers or coastal ows for which the following conditions may be observed: the Reynolds number is large, the horizontal velocity has small variation along the vertical, the vertical velocity is small compared to the horizontal velocity. We introduce the aspect ratio ε := h L 1, 3

4 where L is a characteristic wave length. Let us start by investigating the third one, which justies the following scaling for the velocities: v = εṽ, u = ũ. Then the mass conservation equation 1.1 enforces also a scaling for the space variables y x since Hence there are two options for the variable scaling: 1. Long wave scaling: = x u + y v = x ũ + ε y ṽ. x = x t, y = ỹ, t = ε ε. It means that the long wave hypothesis needs a long time study. Furthermore, since f b x = f b x we have f b x = ε f b x and so the bottom slope needs to be small enough.. Thin layer scaling: x = x, y = εỹ, t = t This scaling restricts the study of small vertical velocity only to a thin water depth which tends to zero when ε. It is the classical scaling used in the boundary layer approach. So, the long wave scaling is compatible with the hypothesis about small vertical velocity compared to horizontal velocity. Let us see the consequences for the set of equations and the boundary conditions: x ũ + ỹṽ =, ε [ tũ + ũ xũ + ṽ ỹũ] = ε x p + 1 [ ε xũ + Re ỹũ ], 1.4 ε [ tṽ + ũ xṽ + ṽ ỹṽ] = 1 F r ỹ p + ε [ ε xṽ + Re ỹṽ ], 1.5 ũ = ṽ = at ỹ = f b, t η + ũ x η ṽ = at ỹ = η, ε ε x ũ p x η ε x ṽ ỹũ ε x η ỹũ + ε x ṽ ỹṽ = at ỹ = η. p Taking an approximation at order Oε leads to: Cancellation of the viscosity in horizontal direction. Equation 1.4 reduces to: tũ + ũ xũ + ṽ ỹũ = x p + 1 εre ỹũ. Simplied version of the stress tensor continuity at the free surface: p =, ỹũ = at ỹ = η. Hydrostatic pressure with 1.5 and p = at the surface: ỹ p = 1 1 = p = η ỹ. F r F r This approximation implies that x p does not depend on ȳ. In other words, the pressure gradient is conserved over the vertical. This result for pressure at order Oε is already observed see e.g. [9]. Remark 1.1. We emphasize here that the above properties, which are classical shallow water hypotheses, are brought out solely by the long wave approximation. 4

5 To summarize, the long wave approximation of the Navier-Stokes equations consists of the following system, sometimes called the RNSP equations Reduced Navier-Stokes/Prandtl [38]. This will be our reference system for the remaining of this article. In these equations, it is more convenient to dene an eective Reynolds Re h number which takes into account the aspect ratio of the model Re h := εre. Dropping all the tildes on to the variables and unknowns, the system writes: x u + y v =, 1.6 t u + u x u + v y u = x p + 1 Re yu, h 1.7 y p = 1 F r, 1.8 t η + u x η v = at y = η, 1.9 p =, y u = at y = η, 1.1 u = v = at y = f b In the limit Re h, the case of an incompressible ideal uid, the RNSP equations degenerate to the wellknown hydrostatic Euler system. Indeed, the equations of mass conservation 1.6, hydrostatic pressure 1.8 and boundary condition at free surface are unchanged, but the momentum conservation equation is replaced by t u + u x u + v y u = x p. 1.1 Therefore we conclude this paragraph by giving the behaviour of horizontal velocity. Proposition 1.. A regular solution u of the hydrostatic Euler system is needed constant over the vertical, so we write u := u e t, x, and veries the following equation: t u e + u e x u e = x p Proof. Applying a partial derivative in y on equation 1.1 and taking into account the incompressibility 1.6 together with the hydrostatic pressure law 1.8 leads to d dt yu =. It means that y u remains constant along the characteristic curve x t = u, y t = v, hence y u = y u x,η = by condition 1.1 where x, η is the foot of characteristic curve starting at the free surface y = η. Consequently the horizontal velocity u of hydrostatic Euler system is independent on y. As it is well-known in Ideal Fluid theory, the depth-independent horizontal velocity component leads to a slip velocity at the bed. Thus, the no-slip boundary condition 1.11 is not relevant for ideal uid and has to be replaced by a weaker one called the non-penetration condition v e y=fb = u e f b Recovering some connection between the ideal uid equations and the no-slip condition is precisely the aim of the viscous layer theory, which we will present in section. 1.3 Classical shallow water equations Let us recall briey the classical way to obtain the shallow water model by vertical integration the RNSP equations over the whole water depth. Denition 1.3. The water depth h and the depth-averaged horizontal velocity U are given by h := η f b, hu := η f b u dy

6 Integrating the mass conservation equation 1.6, from f b to η, yields η = x u dy + v y=η v y=fb f b η = x u dy u y=η x η + u y=f b f b + v y=η v y=fb f b η = x u dy + t η f b in which we have applied the kinematic condition 1.9 at the surface and the no-slip condition 1.11or the weaker one 1.14at the bottom. As f b is time-independent, we can write t η = t η f b ; and by using denition 1.15, the mass conservation in its integrated form reads t h + x hu = In the same way, we now derive the depth-integrated momentum balance equation. The main point to notice at this stage is the appearance of the nonlinear momentum ux and the friction term. Indeed, by integrating the momentum equation 1.7 over the depth and using the fact that x p does not depend on y, and also the condition y u y=η = from 1.1, we get 1 η η y u y=fb h y p = t u dy + u x u + v y u dy Re h f b η f b η = t u dy + u x u dy + uv y=η uv y=f b f b f b η η = t u dy + x f b f b u dy u t η + u x η v y=η + uuf b v y=f b. Using denition 1.15 and applying the free surface condition 1.9 and the bottom condition 1.11 or 1.14, we can rewrite the integrated momentum conservation equation under the form t hu + x βhu = h x p τ b, 1.17 in which we have introduced the so-called Boussinesq coecient β and the bottom shear stress τ b, also called friction, which are dened by η f b u dy := βhu, τ b := 1 Re h y u y=fb Therefore, the knowledge of β and τ b requires the knowledge of the ow. It can be checked that β 1 since, by denition 1.18, we can write β = η 1 u dy h U f b Without complementary equations, a closure relation on the velocity prole is needed in order to compute the Boussinesq coecient and to express the friction term in function of the conservative variables h, hu. Let us recall two constitutive proles which are often adopted in the context of shallow water ows. Flat prole. This is the most adopted approach in hydraulic river modelling where the ow is rather turbulent. Scaling analysis reveals that the velocity prole is quasi-at except within a very thin-layer close to the river bed. Based on this consideration, the velocity prole can be assumed to be at over the whole water depth, so that β equals one. The friction term τ b is not properly dened in this context. The ow can be described using the Euler system resulting in an inviscid shallow water modelequation 1.17 without friction. A large family of empirical friction laws exist, which express the friction as a quadratic function of U with a friction coecient C f = ORe 1/4 for a smooth bottom, see [49]. This coecient depends on h and U as well, for instance with Chézy, Manning laws, see [11] for a bibliographical study. In summary, this kind of model consists in writing β = 1, τ b = 1 C f U. 1. 6

7 We emphasize again that the friction term derives here from empirical considerations. Poiseuille prole. This type of prole is inspired from a analytic solution of the RNSP equations in the case of an uniform ow on a negative constant slope. The balance between the friction and the driving force of the slope gives a self-similar parabolic solution, also known as the half-poiseuille or Nusselt solution: This choice of prole leads to u U = 3 ζ 1 ζ, ζ := y f b 1. h β = 6 5, τ b = 3 Re h U h. 1.1 It means that the friction is indeed linear with respect to the mean velocity, and is referred to as laminar friction. Using such a prescribed prole in shallow water equations leads to some important restriction of the model, in particular when dealing with large variation of the velocity. For example it has been reported in [8] that a constant value for the Boussinesq coecient is less adapted to describe the dynamics of the uid layer close to a dry-wet transition. Remark. We conclude this section by evidencing that both these models do not present a phase-lag for the ow over a bump. Let us study a steady linearized solution of the usual shallow water equations. Consider such a small perturbation of the bed f b that we can write in the form f b = εfb 1,where ε is just a small parameter and not necessarily the aspect ratio dened before. We look for the solution in the form h = h + εh 1, U = U + εu For high Reynolds numbers, we see from relations 1. and 1.1 that the friction in negligeable. We can consider therefore h, U solution of the following frictionless and steady state shallow water equations x hu =, x βhu + 1 F r h = 1 F r hf b. 1.3 Inserting 1. in 1.3 and identifying powers of ε leads to a cascade of equations for each terms h i, U i, i =, 1. Then, it should be checked that the zero-order terms h and U are needed constant. For rst-order terms, a straightforward calculation leads to h h x U 1 + U x h 1 =, F r βu x h 1 = h F r f b 1. By introducing F r the local Froude number, we can express the linearised solution under the form F r := βu h F r, h = h 1 + F r 1f b, U = U + U 1 h 1 F r f b. 1.4 As we can see, U are exactly in-phase with f b. As a consequence, local maxima of the friction estimated by empirical formulas 1. or 1.1 are always reached at f b =, that is at the crest of the bump. Indeed, because x h = x U = if f b = by 1.4, it follows that xτ b = h τ b x h + U τ b x U =. Viscous layer analysis We turn now to the main step towards the model we look for. It mainly consists in dividing the uid in two layers: an ideal uid layer dealing with the free surface; a thin viscous layer with the no-slip condition at the bottom. In the rst layer addressing to ideal uid, we take advantage of the explicit integration along the vertical. In the second one describing viscous layer, we take into account the viscosity in the vertical direction and recover some friction in the integrated equations. This section is devoted to the study of the viscous layer, and to the analysis of the interactions between the two layers. 7

8 We introduce a small parameter δ, whose magnitude will be specied below. It is related to the thickness of the viscous layer, but does not correspond to its actual physical value. We follow the classical strategy used in the boundary layer theory [46, 49] except that in that case δ, whereas we keep a nite value here. The rst step is to rescale again the RNSP equations with the thin layer scaling to obtain a set of the well-known Prandtl equations. The next step consists in vertical integrating these equations over the viscous layer. This leads to the so-called von Kármán equation, where extra unknowns are introduced. Finally, some suitable assumptions have to be made on the velocity prole in order to obtain a closed model..1 Prandtl equations We introduce the following change of variables, referred to as the Prandtl shift: x = x, y = δȳ + f b, t = t, p = p, ū = u, v = v f b u..1 δ By this, the RNSP equations 1.6, 1.7, 1.8 and 1.11 are transformed into a set of boundary layer equations on a at bottom x ū + ȳ v =, tū + ū x ū + v ȳū = x p + f b δ ȳ p + 1 Re ȳū, h δ 1 δ ȳ p = 1 F r, ū = v = at ȳ =. There are several possible scalings for δ in terms of Re h : if δ veries Re h δ 1, we recover the ideal uid equations; if δ satises Re h δ 1, we obtain ȳū = that leads to ū = due to the continuity of the stress tensor and the no-slip condition. So we do not consider this trivial case; the last possibility is Re h δ 1 which balances the convective terms and the diusive one called dominant balance. This scaling, which respects to the least degeneracy principle [54], allows to preserve as many as possible terms in the equations; so most of the physical phenomena are described. This is why in what follows, we consider the scaling δ = 1 Reh 1.. With this choice of δ, the viscous term appears with the same order as the other terms in the momentum equation. We obtain the Prandtl equations written in viscous layer variables: x ū + ȳ v =.3 tū + ū x ū + v ȳū = x p f b F r + ȳū.4 ȳ p = δ.5 F r ū = v = when ȳ =.6 We notice that this system of equations are in the same form as the Prandtl equations obtained directly from Navier-Stokes equations with classical boundary layer scaling [49], ch. VII except for the topography term in momentum equation.4. Up to now, we do not have enough boundary conditions for the viscous layer. The natural connection consists in assuming that the velocity at the top of the viscous layer coincides with the velocity of inviscid layer. Precisely, we impose the following matching boundary condition ū t, x, η = u e t, x, where η := η f b,.7 δ which is obviously compatible with the Prandtl shift.1 since x = x and t = t. It is worth noticing that in classical boundary layer theory the limit is ū t, x, ȳ u e t, x and called asymptotic matching. 8

9 . Von Kármán equation The von Kármán equation expresses the defect of velocity between the ideal uid and the viscous layer. A classical way to obtain such an equation consists in writing the Prandtl equation, then introducing the velocity defect u e ū, and nally integrating it on the viscous layer see Schlichting [49]. We introduce the following two integrated quantities: Denition.1. Let U be the depth-averaged velocity. We dene the displacement thickness δ 1 given by the momentum thickness δ given by η hu = h δδ 1 u e,.8 f b u dy = h δδ 1 + δ u e..9 Physically, the displacement thickness expresses the distance by which the ground should be displaced to obtain an ideal uid with velocity u e with the ow rate hu see Figure. In the same way, the momentum thickness accounts for the loss of momentum in the viscous layer. h u u e δδ 1 Figure The ux of mass is the same in the viscous layer and in a equivalent layer of ideal uid shifted by an amount of δδ 1. A simple computation from.8 and.9 leads to the following expressions for these quantities: η δ 1 = 1 ū η ū dȳ, δ = 1 ū dȳ. u e u e u e In the limit δ, we recover the classical formulæ of δ 1, δ in boundary layer scaling [49]: + δ 1 = 1 ū + ū dȳ, δ = 1 ū dȳ. u e u e u e Proposition.. The displacement and momentum thicknesses are ruled by the so-called von Kármán equation: t u e δ 1 + u e δ 1 x u e + x u eδ = τ b,.1 where τ b denotes the parietal constraints: τ b := ȳū ȳ= = τ b δ..11 Proof. First we notice that the Prandtl shift leads to following relations x = x f b δ ȳ, y = 1 δ ȳ. Momentum equation 1.13 for inviscid ow can be rewritten as tū e + ū e x ū e = x p + f b δ ȳ p = x p f b F r, in which we have used.5 to rewrite the right-hand side. The dierence between this equation and.4 gives tū e ū + ū e x ū e ū x ū v ȳū = ȳū. 9

10 Through.3 and.6, we can rearrange the term v = ȳ xū. Furthermore we get tū e ū + ū e ū x ū e + ū x ū e ū + ȳū ȳ x ū dȳ = ȳū. Using integration by parts, the last term in the left-hand side can be rewritten as ȳ ȳ ȳū x ū dȳ = ū x ū + ȳ ū x ū dȳ. Now we integrate the resulting equation over ȳ, between and η, together with the matching boundary condition.7 to obtain the momentum integral equation t η η ū e ū dȳ + x ū e ū e ū dȳ + η ū x ū e ū dȳ η = ȳū ȳ=. ū x ū dȳ + u e η x ū dȳ η The last three terms of the left-hand side are now rewritten as x ūū e ū dȳ. Moreover, since u e is independent of ȳ, we have the relations u e δ 1 = η fb /δ u e ū dȳ, u eδ = η fb /δ ūu e ū dȳ. By the fact that t = t and x = x in Prandtl shift, all unknowns of the equation are independent on ȳ. So we can drop as well all bar symbols in the derivatives and by using denition.11 we obtain the nal form.1 of von Kármán equation. Finally, relation between the friction τ b and the rescaled one τ b in.11 comes from Prandtl shift since by denition 1.18 we have τ b = δ y u y=fb = δ ȳū ȳ= = δ τ b. The von Kármán equation.1 is independent on the exact shape of velocity. Together with equation 1.13 on the velocity u e of ideal uid, these governing equations give only a partial representation of the boundary layer, since they involve four unknowns, namely u e, δ 1, δ, and τ b. To proceed further towards an integrated model, we turn now to propose specic shapes of velocity to close the equations..3 Velocity prole in the viscous layer Through the viscous layer, the velocity ū varies from at the bottom to external velocity u e in order to match the ideal uid. Therefore we introduce a prole function ϕ as well as a scaling factor t, x, chosen in such a way that quanties the physical thickness of the viscous layer. Following [49], we wish to have ū t, x, ȳ u e Therefore we choose < η, and a prole function ϕξ such that ϕ =, ϕξ 1 = 1, η 1 ȳ ȳ = ϕ = ϕξ, ξ := t, x..1 1 ϕ dξ := α 1 < +, Hence, by denition of δ 1 and δ, we can write η δ 1 = 1 ū dȳ = δ = ū u e u e 1 ū u e 1 dȳ = 1 1 ϕ dξ = α 1, 1 ϕ1 ϕ dξ = α. ϕ1 ϕ dξ := α <

11 To link these variables, we introduce the shape factor H which only depends on the prole function ϕ H := δ 1 δ = The parietal constraints can also be expressed in terms of ϕ and u e as 1 1 ϕ dξ ϕ1 ϕ dξ 1 τ b = ȳū ȳ= = ϕ u e = α 1ϕ u e = f H u e,.15 δ 1 where the parameter f is known as the friction factor see [38, 49] and f H := α 1 ϕ. Using denitions.14 and.15, we can rewrite the Von Kármán equation.1 in following form u t u e δ 1 + u e δ 1 x u e + e δ 1 x = f H u e..16 H δ 1 At this stage, choosing a prole of velocity in viscous layer consists in imposing a closure formulae on the shape factor H and the friction factor f. In such a way, the Von Kármán consists in expressing only the global dependence between u e and the displacement thickness δ 1. Several shapes can be used for the prole, including turbulent ones. As far as laminar proles are concerned, we refer to [49, ch.x] for elements of comparisons between dierent proles. We shall assume that ϕ depends solely on the variable ξ according to the similarity principle [49] on velocity prole over a at plane at zeroincidence. For the sake of clarity, let us briey present in what follows several classical prole functions in viscous layer from which we establish some instructive laws on H and f for our stu dy. Polynomial prole. This kind of approach is known as the Pohlhausen solution which consists in considering a polynomial approximation of velocity prole. The polynomial coecients are chosen such that ϕξ veries boundary condition.13. The rst and also the simplest case consists in a linear prole δ 1 ϕξ = ξ, H = 3, f =.167. Higher order polynomial can be derived by imposing additional conditions = ϕ 1 = ϕ 1 = which mean that the transition between the viscous layer and the inviscid layer must be smooth. Thus, the second and third-order proles write ϕξ = ξ ξ, H =.5, f =.67, ϕξ = 3 ξ 1 ξ3, H =.7, f =.8. We can notice an interesting feature is that although these proles are quite dierent, both lead to very close values of H, f. Nevertheless, they do not allow the observation of separation for decelerated ows. This diculty can be overcome only by using a fourth-order polynomial with a free parameter Λ. The resulting prole function, namely Pohlhausen4, takes the form ϕξ = ξ ξ 3 + ξ 4 + Λ 6 ξ1 ξ3. The parameter Λ is related to the pressure gradient, and therefore so the variation of velocity of the inviscid ow velocity. Indeed, taking the second-order derivative ϕ and by evaluating Prandtl momentum equation.4 at ȳ = we deduce Λ = ϕ = u e ȳū ȳ= = u e x p = x u e, where the last equality comes from a stationary consideration, for the sake of simplicity, of momentum equation 1.1 of inviscid layer. As ϕξ 1 by denition, it is concluded that Λ 1. In particular, for values of Λ 1, the velocity proles show negative regions that correspond to reverse ow see Figure 3, left side. We turn now to study the shape and friction factors based on the prole under consideration. First, substituting the fourth-order polynomial into denitions.14 and.15 allows to express H, f as explicit functions of Λ, but they are omitted here for the sake of compactness. Nevertheless we notice that these relations are just formal since the "physical thickness" t, x remains unknown once the velocity ū in viscous layer has not yet 11

12 H f ϕ = ū/ue.4 Parabolic profile Pohlhausen4 profile HΛ1 4.1 fλ ξ Λ 1 Figure 3 Polynomial approximation of the velocity prole: left parabolic vs Pohlhausen of order 4 pro- les with Λ = 1,, 1, 4; right closure on the shape factor H and and the friction f factors based on Pohlhausen of order 4. Note that reverse ows f < are possible. been solved. In practical, it is more convenient to replace by the displacement thickness δ 1. A possible way, according to [37, 38], is to introduce a new parameter Λ 1, inspired from the denition of Λ, given by 36 Λ 36 Λ Λ 1 := δ1 x u e = x u e = Λ in which we have used the relation δ 1 / = 36 Λ/1 obtained by substituting the fourth-order polynomial into denition of δ 1. Moreover, equation.17 leads to Λ 1 being monotone on the physical range Λ 1. Consequently, the factors H, f can also be expressed as functions of Λ 1. We represent on Figure 3, on the right side, the functions HΛ 1 and f Λ 1 with Λ ranging from 4 to 1 that corresponds to 6 Λ Finally, we present in Tab. 1 values of H and f corresponding to special cases: Λ = 1 limit of physical range, Λ = no pressure gradient namely Blasius solution and Λ = 1 incipient separation. Case Λ Λ 1 H f Limit case Blasius case Incipient separation Table 1 Specic solutions of Pohlhausen of order four prole. Numerical prole. Polynomial proles, despite their simplicity, are rather articial. Their construction is based only on some suitable boundary conditions. An alternative approach, that might be more interesting, is to use exact solutions of boundary layer equations in order to establish more physical closures. Such an approach can be done by employing the solution to Falkner-Skan equation [18]. It plays an important role to illustrate the main physical features of boundary layer phenomena. This solution describes the form of an external laminar boundary layer of a ow over a wedge. The Blasius solution for a at plate is a particular case of this solution. Falkner-Skan equation consists a third-order boundary value problem whose resolution is still complicate see e.g. [9, 31, 56]. We do not present here details on the resolution of Falkner-Skan equation but focus the construction of closure formulae and compare against the obtained results with that given by Pohlhausen4. First, we solve Falkner-Skan equation on whole physical range of pressure gradient, corresponding to the case of accelerated, decelerated and reverse ows, to obtain all value of the triplet Λ 1, H, f. Next, we nd out a numerical relation between these parameters by inspiring from the approach presented for Pohlhausen4. On Figure 4, it is found that the Pohlhausen4 closure, although its purely algebraic derivation, presents a good agreement with 1

13 Falkner-Skan when Λ 1 corresponding to accelerated ows. In particular, exact value of Blasius solution Λ 1 =, H =.59, f =. is very close to that given by Pohlhausen4, see again Tab. 1. However, these closures diverge in regions of decelerated and reverse ows Λ 1 <. Incipient separation f = is reached at Λ 1 = 1.9, H = 4 while it is Λ 1 = 1.9, H = 3.5 for Pohlhausen of order 4. Finally, a value Λ 1 =.6, H =.74 is found as limit of physical range of Falkner-Skan solution. 7 6 Fanlkner-Skan Pohlhausen4.59 exp.37λ Fanlkner-Skan Pohlhausen4 1.54/H 1/H 5 Separation. H 4 Blasius f.1 Blasius Separation Λ H Figure 4 Falkner-Skan vs Pohlhausen order 4 closures: shape factor left and friction factor right. Lagrée and Lorthois [38] proposed the following ad-hoc closure based on Falkner-Skan solution {.59e.37Λ 1 if Λ H = 1 <.6, 4 and f.74 otherwise, = 1.5 H H As we can see on gure 4, this numerical law presents a good agreement near Blasius solution both accelerated and decelerated ows. These regions are also the cases the most concerned in river hydraulic application i.e. with small bed perturbation. Especially, when plotting f as function of H, an excellent agreement is found. This is why we adopt.18 for our numerical study of the present model. Finally, it is important to emphasize that both polynomial and numerical closure for H and f are based on steady solutions of boundary-layer equations. 3 Extended shallow water model We are now in position to obtain the extended model we are looking for. Depth-integrating the mass and momentum conservation equations of RNSP system results shallow water equations 1.16 and 1.17, as presented in Sec Compared to that of classical shallow water model [1], we have noticed, on one hand, the dependence δ 1, δ of momentum equation, and on other hand, the parietal constraints arise in the right-hand side at order 1 in δ. This motivates a new closure for the momentum ux and so the system has to be coupled with the von Kármán equation presented above. 3.1 Towards the extended model This section is devoted precisely to the coupling between depth-integrated shallow water equations and the von Kármán equation. It merely emphasizes that relation.9 does not give any closure for the momentum ux; it can be entirely completed by a choice of the momentum thickness δ. Complete closure can be achieved by the study of the viscous layer, as we did above in Section.3. In what follows, we show that expression.9 for the momentum ux is in fact the most convenient, since it takes into account the eect of the viscous layer, and we clarify clarify the role of the von Kármán equation. To this end, let us start from the system of depth-integrated equations 1.16, 1.17 and ideal uid equation 1.13; and for which we rewrite momentum equation 1.17 with a generic form of the ux together with relation.11 on parietal constraints : t hu + x J = h x p δ τ b,

14 where J is the momentum ux for which we seek a closure. We evidence now the fact that a convenient denition of J allows to recover the von Kármán equation from this system of integrated equations. Proposition 3.1. Let h, U, u e, J be solution to 1.13, 1.16 and 3.1. Assume δ 1 is dened by.8. Then δ 1 and δ solve the von Kármán equation.1 if and only if there holds J = h δδ 1 + δ u e. 3. Proof. We start from the von Kármán equation and introduce.8 to obtain t hu e t hu + hu e U x u e + δ x u eδ = δ τ b. To this equation we add 3.1, the parietal term disappears, leading to t hu e + x J + hu e x u e hu x u e + δ x u eδ = h x p. Developing the time derivative and simplifying with 1.13 we obtain u e t h + x J hu x u e + x δu e δ =. Finally, we use 1.16 to eliminate the time derivative, regroup terms and get x J hue U + δδ u e =, so that, up to a constant which can be taken equal to zero by considering that the ux is zero when the velocity is zero, we have J = hu e U δδ u e, 3.3 which together with.8 gives precisely 3.. Conversely, we consider.8 and use successfully the mass and momentum balance equations t u e δδ1 = t hu e t hu = u e t h + h t u e + x h δδ1 + δ u e + h x p + δ τ b = u e x hu hu e x u e h x p + x hu e δ x δ1 + δ u e + h x p + δ τ b = u e x hu e + u e x δδ 1 u e hu e x u e + x hu e δ x δ1 + δ u e + δ τb = δ u e x δ 1 u e + x δ 1 + δ u e τ b. Noting that x δ 1 u e = u e x δ 1 u e + δ 1 u e x u e we recover as required the von Kármán equation. In the above proposition, we only use the three equations 1.13, 1.16, 3.1 and denition.8 of the displacement thickness. Replacing the physical denition.8 by von Kármán equation.1 will give back the physical denition in the sense of characteristics as explained in the next proposition. Proposition 3.. Let J be dened by 3., and h, U, δ 1, u e be a smooth solution to the system of equations 1.13, 1.16 and 3.1 together with von Kármán equation.1. Denoting by δ 1 the thickness obtained using.8, we have t u e δ 1 δ 1 u e x u e δ 1 δ 1 =. In other words the error between these displacement thickness is constant along the characteristics of the ideal uid. Hence, if initially δ 1 = δ 1 then it is true for all times. Proof. From.8 t u e δδ 1 = t hu e t hu = u e t h + h t u e + x h δδ1 + δ u e + h x p + δ τ b = u e x hu hu e x u e h x p + x hu e δ x δ1 + δ u e + h x p + δ τ b = u e x hu e + u e x δδ 1u e hu e x u e + x hu e δ x δ1 + δ u e + δ τb = δ u e x δ 1u e + x δ 1 + δ u e τ b. Now using the von Kármán equation to eliminate δ, we obtain δ t u e δ 1 = δ u e x δ 1u e + x δ 1 u e t u e δ 1 u e δ 1 x u e which is the desired result. = δ u e x ue δ 1 δ 1 t u e δ 1, 14

15 In summary, propositions 3.1 and 3. show that there are two equivalent possibilities to compute δ 1 in order to close the system. The rst one consists in adding the algebraic relation.8, which gives somewhat an equation of state. The second one makes use of the von Kármán equation that we can also rewrite in a conservative form t δ 1 u e + x H δ 1u e = τ b + u e x δ 1 u e, 3.4 in order to see that δ 1 u e appears as a conserved quantity. Putting together the results of Proposition 3.1 and the closure formulas.14 and.15 on the velocity prole in the viscous layer, we can now rewrite momentum equation 1.17or 3.1of the viscous shallow water model under the form h t hu + x δδ H u e + h F r = hf b F r δ τ b. 3.5 Moreover, the relations between h, U, u e and δ 1 through the displacement thickness.8 and.9 directly imply the following expressions for the momentum ux η f b u dy = h δδ 1 + δ u e = hu + δδ 1 δ δδ 1/hu e. The last expression clearly emphasizes that we are able to compute a non constant Boussinesq coecient, indeed from 1.19 we have β = 1 + δ δ 1 δ δδ 1/hu e hu = H δδ 1 h + O δ. 3.6 For δ = we recover the classical shallow water system with Boussinesq coecient equal to 1. As soon as viscosity eects arise, that is δ >, we have not only the friction term on the right-hand side of 3.5 but also a correction of the same order one in δ to the hydrostatic pressure. Notice that in [47] a similar correction in the ux of the shallow water system is proposed to improve the study of roll-waves. In the context of a thin viscous layer δδ 1 /h 1 we consider here, one can observe from 3.6 that the Boussinesq coecient is very close to unity, so that its impact on the the velocity correction is negligible. Therefore we focus on the study of the friction term. 3. Final equivalent ideal uid formulations The aim of this section is to propose extended shallow water ESW models involving an ideal uid with velocity u e and to couple them with the von Kármán equation describing evolution of the displacement thickness δ 1. The rst step towards these models consists in providing two systems where we keep equation 1.16 for mass conservation and constitutive relation.8 on the displacement thickness. With these relations, momentum equation 3.5 can easily be reformulated as h t u e + u e x u e + 1 F r xh + f b δ t δ 1 u e + x H δ 1u e u e x u e δ 1 τ b =. It is clear on this formulation that among the three following equations: inviscid momentum 1.13, viscous momentum 3.5 and von Kármán 3.4, once two equations are satised then so is the third one. From this consideration, the ESW can be expressed with two equivalent formulations: the rst one describes an ideal uid living on a viscous layer, which becomes some apparent topography; the second one represents an equivalent ideal uid over the whole depth h. It turns out that this last formulation is the most suitable for numerical studies. Here these systems are obtained by straightforward manipulations of the equations, but it is noteworthy that they can be derived as well from the Euler system with appropriate boundary conditions, as we shall see below. Apparent topography formulation. Let us derive the rst system which describes an ideal uid lying above the viscous layer of thickness δδ 1, see again gure. First, we dene the eective depth H := h δδ 1 of ideal uid. The constitutive relation.8 allows to write the ux of mass hu = Hu e. Next, multiplying the ideal uid 15

16 equation 1.13 by H and substituting the mass balance equation 1.16 into it, the resulting equation together with the mass balance equations and the von Kármán equation form the following coupled system t H + x Hu e + t δδ 1 =, t Hu e + x Hu e + H F r + u e t δδ 1 = H f F r b + x δδ 1, 3.7 t δ 1 u e + x H δ 1u e = u e x δ 1 u e + τ b. One can see that the two rst equations of the system represent a shallow ow, of thickness H, over a modied topography, namely f b + δδ 1. This has to be related to the so-called apparent topography formulation, where for numerical purposes the friction term is rewritten as the derivative of some function, see [4, 5]. Here this derivative arises in a natural way, together with additional time derivatives in the mass and momentum equations. As mentioned, the apparent topography formulation can be obtained as well from integration of the Euler system over the vertical, but from the modied topography f b + δδ 1 to the free surface η. The key point is that the boundary condition on the interface between the viscous layer and the ideal uid is a non-penetration condition v e y=fb + δδ 1 = u e f b + x δδ As for the classical shallow water model, it is straightforward to obtain from the rst two equations of system 3.7 the following energy balance equation: Hu e t + H + f b + δδ 1 Hu F r + x u e e + HH + f b + δδ 1 F r = u e t δδ In contrast with the classical shallow water system, the dissipation of energy here is driven by the dynamical behaviour in time of the displacement thickness δ 1. If t δδ 1, as for the Blasius-Stokes solution presented below in Section 5., we have actually dissipation of energy by the bottom friction, thus some stability of the solutions. If this not the case, the problem of energy dissipation is open. Interactive Boundary Layer formulation. This model describes an ideal uid on the whole water depth h. Multiplying 3.4 by δ and substituting it into 3.5 leads to a model being very similar to the usual shallow water one. The resulting system reads t h + x hu e δδ 1 u e =, t hu e + x hu e + h F r = hf b F r + u e x δδ 1 u e, 3.1 t δ 1 u e + x H δ 1u e = u e x δ 1 u e + τ b. Once the conservative variables h, hu e and δ 1 u e are solved, the averaged velocity U, if needed, can be recovered using relation.8. An interesting feature of this formulation is that the friction term disappears from momentum equation and has to be incoporated in von Kármán equation that governs the viscous layer. Comparing with the former formulation, system 3.1 is represented under a more conservative form which would be appropriate for numerical discretization. This formulation of the model can also be obtained by integrating the Euler system over the whole water depth, but the non-penetration condition 1.14 has to be replaced by a slightly modied one, called transpiration condition, that writes v e y=fb = u e f b + δ x δ 1 u e which is in fact a formulation of the Interactive Boundary Layer IBL or Viscous Inviscid Interaction in aerodynamics, see [36]. Comparing with 1.14, one can see that the transpiration condition is a correction of order one in δ of the non-penetration condition due to the development of viscous layer; why system 3.1 is called IBL formulation. For the sake of completeness, let us briey recall how the condition 3.11 is derived. We wish to estimate the vertical velocity v e of the ideal uid at the bottom y = f b. We rst notice that, since y u e =, integrating incompressibility equation 1.6 of ideal uid over the whole water depth yields v e y=fb = v e y=η + η f b x u e. 16

17 Next, we have on one hand δ v ȳ= η = v e y=η f bu e from Prandtl shift.1; on the other hand the vertical velocity v is recovered through the incompressibility relation.3 in the viscous layeras we can write ȳ v = x u e ū x u e. Integrating this later equation over the depth to obtain δ v ȳ= η = δ x u e δ 1 η f b x u e. Putting things together leads to the required transpiration boundary condition Numerical method It was decided that numerical analysis and the design of a nite volume solver was made for the IBL formulation. Numerical resolution of system 3.1 was made into two steps: in the rst stage, we solve the following conservation law expressing the evolution by convection t W + x F W = SW, 4.1 where the conservative variable W, the ux F W and the non-conservative source term SW are dened by h hu e δδ 1 u e W = hu e, F W = hu e + h, SW = F r δ 1 u e H δ 1u e F r + u e x δδ 1 u e ; u e x δ 1 u e in the second stage, we take the friction term τ b and modify the displacement thickness δ 1 This later equation will be discretized by a semi-implicit scheme. 4.1 Eigenvalue analysis f b t δ 1 u e = f H δ 1 u e. 4. We are interested in studying the hyperbolicity behaviour of the present model that can be considered as a stability criteria. To this end, we rst rewrite 4.1 under a quasi-linear form t V + AV x V = in which V := W, f b t. The system is said to be hyperbolic if the convective matrix AV is R-diagonalizable or strictly hyperbolic when the eigenvalues are distinct. Estimating these eigenvalues is also important to design an explicit nite volume scheme for the system; the time step t for a mesh size x has to be satised the well-known CFL condition t x λ max, 4.3 where λ max is the largest eigenvalue expressing the fastest wave speed of the system. This condition ensures that information of each Riemann problem at a cell's interface does not cross more than one cell. Regarding on the proposed Pohlhausen or FalknerSkan closures, the shape factor H is a function of Λ 1, so H depends only on u e and δ 1 u e. As eigenvalues are invariant by changing variables [3], it is therefore more convenient to make the variable change V Y V := h, u e, δ 1 u e, f b t and study eigenvalues of the associated convective matrix u e h δ ÃY = F r u e F r a b u e, where we have denoted the partial derivatives a := u e H δ 1u e, b := δ 1 u e H δ 1u e. Their explicit expressions can be computed as well once a closure formula for the shape factor is provided. 17

18 The characteristic polynomial of ÃY writes P λ := detã λ Id = λ [ b u e λ u e λ F r h δf r a ]. Therefore, the system is hyperbolic if P λ has three distinct and non-zero real roots. In other words, there are 3 intersection points between the third-order polynomial curve P SW λ and the straight horizontal line d δ: P SW λ := b u e λ u e λ F r h, d δ := δf r a. These solutions can be seen as perturbation order one in δ of the roots of P SW ; and the behaviour of the system is close to an uncoupled shallow water equations and the Von Kármán one. The two rst roots λ 1, express the propagation velocities of ideal uid while the last one λ 3 approximate that of the viscous layer h λ 1, = u e ± F r, λ 3 = b u e. 4.4 It is straightforward to verify that for the case of closure.18 based on Falkner-Skan solution, the third wave speed reads λ 3 = u e H Λ 1, for Λ 1 <.6, 4.5 and in particular when Λ 1 =, i.e. the Blasius solution, the viscous layer propagates downstream at velocity λ 3 = u e /H.39u e see gure 6-left. Denoting λ < λ + the roots of the second-order derivative polynomial P SW λ, the system is hyperbolic when d δ lies between P SW λ ±, see gure 5, that is P SW λ < δa F r < P SW λ +. Because P SW λ has always 3 roots, it therefore follows that P SW λ < < P SW λ +. For δ small enough, as considered in this study δ 1, the characteristic polynomial admits thus 3 real eigenvalues and the system turns out to be hyperbolic. d δ λ L λ 1 λ 3 λ λ R λ P SW λ Figure 5 Roots of characteristic polynomial of the coupled system. Even within the hyperbolic regime, the eigenstructure is given only implicitly, due to the form of nonlinear coupling between the ideal uid and viscous layer. As a result, when dealing with numerical methods requiring characteristic eld decomposition, e.g. Roe type method, the eigenvalues need to be computed by numerical root nding. In the absence of analytic expressions for the eigenvectors, building desirable properties for such a scheme may be more dicult. We propose hereafter a HLL type scheme [7] having advantage as the numerical ux arises directly from the governing equations 4.1 and only an estimation of lowest and the largest wave speeds λ L,R is required. Such a wave speeds estimation can be done by using accurate Nickalls's bounds [43], which writes λ L,R := 1 3 u e + b u e b + 3h F r in which we have used 4.5 provided from the Falkner-Skan closure., b = u e Λ 1, 4.6 H 18