Inviscid flow separation at the crest of an erodible dune

The question of the place where a fluid flow separates on a curved obstacle is among the most difficult in fluid mechanics. A criterion based on the cancellation, at that place, of the inviscid flow singularity was proposed by Brillouin [Annal. Chimie Phys. 23, 145 (1911)] and Villat [J. Math. Pures Appl. 10, 231 (1914)], which is at the starting point of the triple-deck theory. A similar question arises for the flow over an erodible sand dune with sharp crest, where the a priori unknown dune profile and location of the crest result from the coupling of the fluid flow with the sand motion at the dune surface. We show, by computing the potential flow with Levi-Civita's conformal transformation and using a standard closure law for the shear stress driving the sand flux, that the Brillouin-Villat condition here provides the appropriate criterion. We emphasize that within the present model where the bed shear stress is in phase with the flow velocity, so that the flat bed is linearly stable, it is the separation which allows the existence of a self-preserving dune shape. For a dune traveling without deformation on a nonerodible ground, the Brillouin-Villat condition eventually selects its velocity. A parallel is drawn with the Kutta-Joukovski condition for the flow around an airfoil.

Figure 1

(a) Potential flow over an obstacle on a flat ground, with separation point $S$, (b) corresponding points and streamlines in the $F$ plane, and (c) in the Levi-Civita $\zeta$ plane.

Figure 2

(a) Flow above the exponential obstacle (15) with $\ell =2$ and $x_s=1$ imposed (${\varphi }_s=0.882, a_0=-2/\pi {tan}^{-1}(1/\ell )=-0.295, a_1=-0.155$, and $a_3=0.0184$); dashed line: virtual profile (15) beyond the separation point; (b) velocities $u$ ($-$) and $v$ (- -) along the streamline $\psi =0$; inset: enlargement of the small rectangles around the points $(x_s,u_s)$ and $(x_s,v_s)$.

Figure 3

(a) Streamlines and (b) $u$ and $v$ along the streamline $\psi =0$ for the same exponential profile as in Fig. 2, now unbounded. The Brillouin-Villat condition $B=0$ gives $x_s=1.91$; inset: enlargement around the points $(x_s,u_s)$ and $(x_s,v_s)$, showing that the singularity has disappeared (${\varphi }_s=1.78, a_1=-0.257$, and $a_3=-0.0239$).

Figure 4

(a) Streamlines over the smooth polynomial obstacle $h/\ell ={(x/\ell )}^2-{(x/\ell )}^3$ with $\ell =3$: The Brillouin-Villat condition gives $x_s=1.53$; (b) velocity components $u$ and $v$; inset: enlargement around the points $(x_s,u_s)$ and $(x_s,v_s)$ (${\varphi }_s=1.29, a_0=0, a_1=0.332$, and $a_3=0.0688$).

Figure 5

Position $x_s$ of the separation point on the exponential profile (15) as a function of the slope ${\ell }^{-1}$ at the foot $O$, according to the Brillouin-Villat condition: (+), nonlinear solution ($B=0$); ($\circ$), linear solution (24).

Figure 6

(a) Streamlines over a traveling erodible dune for the transport law (26); (b) corresponding velocities $u$ and $v$ along the streamline $\psi =0$; inset: enlargement around the points $(x_s,u_s)$ and $(x_s,v_s)$. The Brillouin-Villat condition selects $x_s=1.12$ and the dune velocity and volume $V=17.0$ and $A=0.259$ ($\mu =0.5, u_t=0.5$).

Figure 7

Same as Figure 6, for the linearized sand transport law (41); the computed position of the separation point is $x_s=1.12$ and the dune velocity and volume are $V=25.2$ and $A=0.148$ ($\mu =0.5, u_t=0.5$).

Figure 8

Same as Fig. 6, for the Meyer-Peter and Müller law (42); the computed position of the separation point is $x_s=1.10$ and the dune velocity and volume are $V=15.9$ and $A=0.221$ ($\mu =0.5, u_t=0.5$).

Figure 9

Fluid velocity $\left|w\right|$ in the vicinity of the separation point, with the linearized sand transport law (41): (-.-), $x_s=1.10$ imposed; ($-$), $x_s=1.12$ satisfying the Brillouin-Villat condition; (- -), $x_s=1.16$ imposed.