Global stability analysis of a 90°-bend pipe flow

The present work investigates the stability properties of the flow in a 90°-bend pipe with curvature $\delta =R/R_c=1/3$, with R being the radius of the cross-section of the pipe and $R_c$ the radius of curvature at the pipe centreline. Direct numerical simulations (DNS) for values of the bulk Reynolds number ${\mathit{Re}}_b=U_{b}D/\nu$ between 2000 and 3000 are performed. The bulk Reynolds number is based on the bulk velocity $U_b$, the pipe diameter D, and the kinematic viscosity $\nu$. The flow is found to be steady for ${\mathit{Re}}_b⩽2500$, with two main pairs of symmetric, counter-rotating vortices in the section of the pipe downstream of the bend. The presence of two recirculation regions is detected inside the bend: one on the outer wall and the other on the inner side. For ${\mathit{Re}}_b⩾2550$, the flow exhibits a periodic behaviour, oscillating with a fundamental non-dimensional frequency $\mathit{St}=\mathit{fD}/U_b=0.23$. A global stability analysis is performed in order to determine the cause of the transition from the steady to the periodic regime. The spectrum of the linearised Navier-Stokes operator reveals a pair of complex conjugate eigenvalues with positive real part, hence the transition is ascribed to a Hopf bifurcation occurring at ${\mathit{Re}}_{b,\mathit{cr}}\approx 2531$, a value much lower than the critical Reynolds number for the flow in a torus with the same curvature. The velocity components of the unstable direct and adjoint eigenmodes are investigated, and they display a large spatial separation, most likely due to the non-normality of the linearised Navier-Stokes operator. Thus, the core of the instability, also known in the literature as the wavemaker, is sought performing an analysis of the structural sensitivity of the unstable eigenmode to spatially localised feedbacks. The region located 15° downstream of the bend inlet, on the outer wall, is the most receptive to this kind of perturbations, and thus corresponds to where the instability originates. Since this region coincides with the outer-wall separation bubble, it is concluded that the instability is linked to the strong shear by the backflow phenomena. The present results are relevant for technical applications where bent pipes are frequently used, and their stability properties have hitherto not been studied.     

2D numerical contributions for the study of non-cohesive sediment transport beneath tidal bores

2D numerical simulations of tidal bores were obtained using the OpenFOAM CFD software to solve the Navier–Stokes equations by means of the Finite Volume Method by applying a LES turbulence model. The trajectories of non-cohesive sediment particles beneath tidal bores were estimated using a tracker method. Using the fourth order Runge–Kutta scheme, the tracker method solves the Maxey and Riley equations, which requires the knowledge of the velocity field at time t. From 2D numerical simulations of tidal bores, we proposed a classification of tidal bores with respect to the Froude number Fr (or r the ratio of water depths). For a Froude number $1<Fr<1.43$ ($1<r<1.57$), the tidal bore is undular. For a Froude number $1.43<Fr<1.57$ ($1.57<r<1.75$), the tidal bore is partially breaking, which is similar to the transitional tidal bore defined by Furgerot (2014). And for a Froude number $Fr>1.57$ ($r>1.75$), the tidal bore is totally breaking. The numerical results of trajectories of non-cohesive sediment particles are similar to the type of trajectories given by the analytical model proposed by Chen et al. (2012) with some modifications to take into account the effects of gravity, elevation, and attenuation. The parameters of modified Chen's model, ${\beta }_1$, ${\beta }_2$ and ${\beta }_3$, are linearly proportional to the Froude number Fr. This is because the level of turbulence for undular tidal bores is low. The flow induced by an undular tidal bore is not complex. This physical phenomenon is quasi linear. The parameter ${\beta }_1$, related to the front celerity of the undular tidal bore, decreases when the Froude number Fr increases. The parameter ${\beta }_2$, related to the elevation, increases when the Froude number Fr increases. And the parameter ${\beta }_3$, related to the attenuation of the secondary waves, increases when the Froude number Fr increases.