Accelerated flows past a rigid sphere or a spherical bubble. Part 1. Steady straining flow

Abstract

This work reports the first part of a series of numerical simulations carried out in order to improve knowledge of the forces acting on a sphere embedded in accelerated flows at finite Reynolds number, Re. Among these forces added mass and history effects are particularly important in order to determine accurately particle and bubble trajectories in real flows. To compute these hydrodynamic forces and more generally to study spatially or temporally accelerated flows around a sphere, the full Navier–Stokes equations expressed in velocity–pressure variables are solved by using a finite-volume approach. Computations are carried out over the range 0.1 ≤ Re ≤ 300 for flows around both a rigid sphere and an inviscid spherical bubble, and a systematic comparison of the flows around these two kinds of bodies is presented.Steady uniform flow is first considered in order to test the accuracy of the simulations and to serve as a reference case for comparing with accelerated situations. Axisymmetric straining flow which constitutes the simplest spatially accelerated flow in which a sphere can be embedded is then studied. It is shown that owing to the viscous boundary condition on the body as well as to vorticity transport properties, the presence of the strain modifies deeply the distribution of vorticity around the sphere. This modification has spectacular consequences in the case of a rigid sphere because it influences strongly the conditions under which separation occurs as well as the characteristics of the separated region. Another very original feature of the axisymmetric straining flow lies in the vortex-stretching mechanism existing in this situation. In a converging flow this mechanism acts to reduce vorticity in the wake of the sphere. In contrast when the flow is divergent, vorticity produced at the surface of the sphere tends to grow indefinitely as it is transported downstream. It is shown that in the case where such a diverging flow extends to infinity a Kelvin–Helmholtz instability may occur in the wake.Computations of the hydrodynamic force show that the effects of the strain increase rapidly with the Reynolds number. At high Reynolds numbers the total drag is dramatically modified and the evaluation of the pressure contribution shows that the sphere undergoes an added mass force whose coefficient remains the same as in inviscid flow or in creeping flow, i.e. CM = ½, whatever the Reynolds number. Changes found in vorticity distribution around the rigid sphere also affect the viscous drag, which is markedly increased (resp. decreased) in converging (resp. diverging) flows at high Reynolds numbers.

Effect of turbulence on the drag and lift of a particle

“…20,28 -30 Extensive tests on the accuracy of the simulation technique have been performed and documented. 20,31 Results on spectral decay at various points within the computational domain, sensitivity to grid resolution, and detailed comparison with prior simulations and ex- 18 Good agreement is also observed with the numerical results obtained by Mittal 32 and Magnaudet et al 33 The effect of domain size was investigated by doubling the size of the computational domain to 60 times the particle radius. The mean drag coefficient obtained for the case of a linear shear flow varied by less than 0.02% and the corresponding change in mean lift was even smaller.…”

Preferential accumulation of bubbles in Couette-Taylor flow patterns

“…For spherical bubbles, the added-mass coefficient C M is known to be constant and equal to 1 2 whatever the Reynolds number. [21][22][23] The drag coefficient C D depends on the instantaneous bubble Reynolds number Re p =2͉v − u͉R / . As we are mostly concerned with bubble Reynolds numbers in the range 0.1-10, we select a C D correlation based on results obtained in direct numerical simulations with Re p Ͻ 50, namely 23…”

Dynamics of a two-dimensional upflowing mixing layer seeded with bubbles: Bubble dispersion and effect of two-way coupling

“…In the above expression, the drag coefficient C D depends on the instantaneous bubble Reynolds number Re p =2͉v − u͉R / ͑ being the kinematic viscosity of the carrying fluid͒. Based on direct numerical simulations around a single bubble, 30 32 showed that the history force may be neglected for moderate accelerations of the bubble at such high Re p . The added-mass coefficient C M is now known to be constant and equal to 1 / 2 whatever the Reynolds number.…”