Submarine sand ripples formation in a viscous fluid: 2D and 3D linear stability analysis

Transcription

1 Marine Sandwave and River Dune Dnamics 1 & April 4 - Enschede, the Netherlands Submarine sand ripples formation in a viscous fluid: D and 3D linear stabilit analsis V. Langlois (1) and A. Valance (1) Groupe Matière Condensée et Matériaux, UMR 666, Université Rennes I Rennes cedex - FRANCE Abstract We investigate analticall the linear stabilit of a planar sand bed sheared b a laminar boundar laer flow in a D and 3D configuration. The sand transport is described b a continuum phenomenological model taking into account the local bed shear stress (calculated from the resolution of the flow over a deformed sand bed), the grain inertia and the local bed slope. We find that the competition between the destabilizing effect of fluid inertia and the stabilizing ones of grain inertia and gravit leads to the selection of a single mode, that is longitudinal (i.e., parallel to the flow direction). The 3D calculation shows that there exists a band of unstable modes in the oblique direction, which can couple to the main longitudinal mode in the non-linear regime to give birth to a two-dimensional pattern no longer invariant in the transverse direction. 1. Introduction River dunes are structures appearing on river beds made of sand or gravel, i.e. under a unidirectional water flow. It means that a granular surface sheared b a continuous flow is unstable and can develop regular patterns. In this paper we investigate the theoretical mechanisms of this instabilit and derive analtical uations for ripples growth. The first theoretical works on this instabilit considered potential flows but could not account for the instabilit without adding artificial phase lags between the flow and the bed profile. Other approaches consisted in solving the turbulent flow over the sand bed. These models are able to predict the ripple instabilit but the wavelength evaluation of the resulting unstable modes depends greatl on the wa how the turbulence is parameterised. Recentl, Charru (Charru, ) showed that the turbulence is not necessar to get ripple instabilit. We present here a D model, inspired from that of Charru, where the sand transport takes into account both the bed slope effect and the grain inertia one. The later is shown to be the preponderant stabilizing mechanism at high particle Renolds numbers and to determine the most dangerous wavelength. We also extend the model to the 3D configuration, and show that oblique modes are unstable and can couple to the most unstable longitudinal mode in the non-linear regime. As a result, we expect that at the first stage of the instabilit, a pattern invariant in the transverse direction first emerges and then evolves towards a brick pattern.. D model uations We consider a Newtonian and viscous fluid flowing over a deformed sand bed. Outside the boundar laer this flow is assumed to be unidirectional and uniform. The dimensions of the bed deformation are supposed to be small compared to those of the laminar boundar laer. The main assumption is the quasi-stationarit: the flow is calculated as if the bed were fixed. This means that the flow adapts itself instantaneousl to a change of the bed profile. This allows us to solve the stationar hdrodnamic uations. Within this approximation the flow uations are given b: ρ ( u grad) u = grad p + η u (1) 168

2 Marine Sandwave and River Dune Dnamics 1 & April 4 - Enschede, the Netherlands div u = () and are associated to the following boundar conditions : u (h) = on the sand bed surface (3) u( L) = U e x at the top of the boundar laer (4) where h is the height of the bed profile, L the thickness of the boundar laer and outside the boundar laer. U is the flow velocit The basic state of the flow corresponds to the situation where the sand bed surface remains flat and horizontal. In that case, the flow velocit profile is given b a simple linear profile: U z U z = ( ) = γ z (5) L where γ is the shear rate. The sand bed is described as a continuum. The uilibrium flux of transported grains in a full developed state is given b the Meer-Peter law (Fredsøe, 199): q ( Θ Θ ) 3/ c (6) where Θ is the Shields parameter defined as Θ = τ / ρ ( s 1) gd ( τ is the shear stress at the sand bed, s the relative densit of the sand in comparison with that of the fluid, d the sand grain diameter and ρ the densit of the fluid). Θc is the critical Shields number above which grains start to move. The critical Shields parameter is found to depend on the local slope of the bed as follows: 1 h Θ = Θ + c c 1 (7) tanφs x where φ s is the internal angle of friction and Θc is a numerical constant corresponding to the critical Shields number for an horizontal flat sand bed. A model based on the division of the grains into two separate populations (respectivel static and moving grains), inspired from the BCRE model of granular avalanches (Bouchaud, 1995), is used to account for the inertia of grains (Valance, 3). This analsis leads to the expression of the actual flux of grains: q q q = x l l stands for the characteristic length necessar for a grain to reach the uilibrium with the flow, starting from an out-of-uilibrium situation. It depends on the drag force exerted b the fluid on the grain and in the linear approximation l = f (Re p ) d s where f is a function of the particle Renolds number Re p wich is constant for high values of Re p and scales as Re p for low ones. (8) 169

3 Marine Sandwave and River Dune Dnamics 1 & April 4 - Enschede, the Netherlands The closure of this formulation is given b the mass conservation of sand grains: h t q = x (9) The analsis consists in calculating the evolution of a perturbation of the bed of the form: ikxx+ ωt h( x, z, t) = h1 ( z) e (1) where h 1 is a small quantit. Note that the wave number k x characterises the spacing of the crests, the imaginar part of ω/k x represents the phase velocit of the perturbation whereas the real part of ω corresponds to its growth rate. It is important to note that uation (9) implies that a perturbation grows onl if the imaginar part of the transport rate q is negative, that is, in particular, if there is a phase lag between the grain flux and the bed profile. 3. Two-dimensional analsis We first calculate the flow perturbation over a one-dimensional bed. The calculation strateg emploed is the same as that exposed in (Vittor, 199) and (Charru, ). We will focus here on the situation where the height L of the boundar laer is much larger than the characteristic length scales of the sand bed deformation. This calculation leads to an analtical expression of the perturbed Shields number calculated at the sand bed to first order in h 1 : Θ 1 Ai( ie e -? iπ / 6 β 4/ 3 Ai( ξ ' )dξ where Ai is the first Air function, ξ ' π / 6 / 3 = e i ( ξ iβ ) / β. ) (11) β = k lν = k ν / γ is the dimensionless viscous length, and Then linearizing the uations governing the sediment transport and the bed height, we get a close uation for the growth rate of the bed perturbation: 3/ 1/ Θ1 k k 1 ikl ω mθc µ i (1) Θc h1 tanφ 1 k l s + where µ = ( Θ Θ c ) / Θ c. The parameter µ measures the distance from the threshold of grain motion and will be referred to as the relative shear stress excess. If we neglect the effects of the gravit (i.e., φ s = π / ) and of grain inertia (i.e., l = ), the real part of the growth rate ω of the perturbation is given on (Fig. 1). We conclude that the bed is unstable against an perturbation, and the shortest wavelengths are the most unstable. However, according to the experimental observations, one particular mode should be selected. That is wh we need to introduce stabilizing mechanisms like gravit and grain inertia. We will first include the gravit effect in the sediment transport. Second, we will take into account grain inertia effect, neglecting bed slope effect. At last, we will treat the general situation where both effects are included. 17

4 Marine Sandwave and River Dune Dnamics 1 & April 4 - Enschede, the Netherlands Fig. 1 Growth rate for small values of β 3.1 Gravit regime Setting l in uation (1), the growth rate becomes, in the long wavelength limit ( β << 1 ) : = ω.53(1 + µ ) β 4/ 3 β tan φ s (13) The selection of a particular mode results from the competition between the destabilizing mechanism of the 4/ 3 fluid flow, proportional to k, and the stabilizing one due to bed slope effect, proportional to k. The order of magnitude of the fastest growing mode is then given b the balance of these two terms. One gets : λ max = (tanφ s ) 3l ν 3/ (1 + µ ) 3/ (14) λmax The evolution of as a function of µ is shown on (Fig. ). In the case of sand grains in water flow, we obtain λ =.5 max cm for grains of diameter d = 5 µ m and shear stress excess µ = 1, which seems to be underestimated in comparison to the available experimental data. Fig. Most unstable wavelength vs. relative shear stress excess (grain inertia neglected). 171

5 Marine Sandwave and River Dune Dnamics 1 & April 4 - Enschede, the Netherlands 3. Inertia regime We neglect here the effect of the bed slope on the sediment transport but take the grain inertia into account, which is expected to be pertinent when Re p is larger than one. In this case, the growth rate reads in the long wavelength limit: ω 4 / 3.53 β.9 β l ν l 7/ 3 The grain inertia also plas a stabilizing role for short waves and scales as then given b: (15) 7/ 3 k. The most unstable mode is ρ g λmax = 19 f ( Re p ) d (16) ρ and its evolution as a function of the relative shear stress excess is shown on Fig.3. In the case of sand grains in water flow and shear rate corresponding to high particle Renolds number, we obtain λ = 9.5cm max for grains of diameter d = 5 µ m, which seems to be in better agreement with experiments. Fig. 3 Most unstable wavelength vs. relative shear stress excess (gravit neglected). 3.3 General case At last, we analze the general situation where both stabilizing effects are taken into account. Fig. 3 Most unstable wavelength vs. particle Renolds number 17

6 Marine Sandwave and River Dune Dnamics 1 & April 4 - Enschede, the Netherlands The transition between the gravitational and inertial regimes occurs at a critical particle Renolds number Re pc which is found to depend on the grain diameter (see on Fig. 3), or more precisel on the Galileo number (Valance, 3). As an example, we obtain, in a water flow, Re p c for grain diameter of 5µ m and Re p c 8 for grain diameter of 5µ m. As a conclusion, the most dangerous mode is driven b gravit for small particle Renolds numbers and b inertia for larges ones. There exists a crossover regime at intermediate values of Re. p 4. 3D Analsis 4.1 Extension of the model in 3D We now extend the previous analsis to the three dimensional configuration, i.e., a unidirectional flow over a two-dimensional deformed bed of the form: i ( k xx+ k ) + ω t h( x,, z, t) = h1 ( z) e (17) Equations (1-) with boundar conditions (3-4) can also be analticall solved in the 3D case and the shear stress at the sand surface can be again expressed in terms of integrals of Air functions (Langlois, 3). The expression for the uilibrium flux of transported grains can be extended to the 3D situation. On the base of the analsis of Fredsoe et al (Fredsoe, 199), one can derive the following 3D law for the transport rate: ( Θ + + Θ + 3/ x nznx ) e x ( nz n ) e q = ( Θ Θ c ) (18) ( Θ x + nxnz ) + ( Θ + nnz) where n = ( nx, n, nz ) is the unit vector normal to the sand surface. We recall that Q = ( Θx, Θ, Θz ) is the dimensionless shear stress evaluated at the sand surface. The critical Shields number is found to depend on the local slope of the bed surface as follows: 1 Q Θ = Θ + c c 1 ( e ). z nzn tanφs Θ (19) Finall, the mass conservation in the 3D configuration reads h t = div q () For simplicit, we will neglect here the inertial effects so that q = included in a 3D model, but this will not be exposed in this paper. q. The inertial effects can also be 4. Results Using the same strateg as that exposed in D, we can derive an analtical expression for the growth rate of a perturbation of mode k : 173

7 Marine Sandwave and River Dune Dnamics 1 & April 4 - Enschede, the Netherlands 1/ 3/ Θ Θ Θ x k x k c µ k k 3/ 1/ 1 x 1 ω Θc µ i + i (1) Θc h1 tanφ s 1+ µ Θc h1 tanφ s where Θ 1x and Θ 1 are the x and component of the perturbed dimensionless shear stress and can be expressed in terms of integrals of Air functions (Langlois, 3). From the dispersion relation, one can plot the stabilit diagram (see Fig. 5). The left panel in Fig. 5 corresponds to the case where the stabilizing effect of gravit is neglected. In that case, arbitraril infinite large longitudinal wave number k x are unstable, as in the D situation. The novelt in comparison to the D case is the presence of unstable oblique modes (with k ). Taking into account gravit effect (right panel in Fig. 5), one gets a band of unstable modes of finite width in the longitudinal and transverse direction. The fastest growing mode is found to be on the horizontal axis: it means that the most dangerous mode is a longitudinal mode (see Fig. 6 where the growth rate is plotted as a function of the wave numbers k x and k ). As a consuence, in the first stage of the instabilit the sand bed will develop a ripple pattern invariant along the direction. However, in the nonlinear regime one can expect that the unstable oblique modes will couple with the main longitudinal mode and gives birth to a D pattern where the invariance along the direction disappears. Fig. 5 Stabilit diagram. (left: without gravitational effect; right: with the gravitational effect). Fig. 6 Growth rate for long wavelengths 174

8 Marine Sandwave and River Dune Dnamics 1 & April 4 - Enschede, the Netherlands 5. Conclusion and Perspectives We presented in this paper a linear stabilit analsis of a sand surface submitted to a viscous fluid shear in D and 3D configuration. The sand transport was described b a continuum model taking into account the local bed shear stress (calculated from the resolution of the flow over a deformed bed), the grain inertia and the bed slope. We found that the selected mode (or uivalentl the most unstable mode) results from the balance of the destabilizing effect of the fluid inertia and the stabilizing ones of gravit and grain inertia. In D as well as in 3D, the most unstable mode is longitudinal, resulting in the development of a sand pattern invariant along the transverse direction. The specificit in 3D is the presence of unstable oblique modes whose dnamics are dominated b the main longitudinal mode in the linear regime. However the are expected to couple with the main longitudinal mode in the non-linear regime and give birth to D brick pattern. This last result should be checked b a non-linear analsis. References J.-P. Bouchaud et al., Phs. Rev. Lett., Vol. 74, F. Charru and H. Mouilleron-Arnould, Instabilit of a bed of particles sheared b a viscous flow, J. Fluid Mech., Vol. 45, pp ,. F. Charru and J. Hinch, J. Fluid Mech., Vol. 414, pp ,. J. Fredsøe and R. Deigaard, Mechanics of coastal sediment transport, World Scientific, 199. V. Langlois and A. Valance, preprint, 3. A. Valance and V. Langlois, Formation of ripples over a sand bed. Part I, submitted to Phs. Rev. E. A. Valance, Formation of ripples over a sand bed. Part II, submitted to Phs. Rev. E, 3. G. Vittori and P. Blondeaux, Sand ripples under sea waves. Part 3, J. Fluid Mech., Vol. 39, pp. 3-45,