Fuel droplet vaporization and spray combustion theory

Three-dimensional flow over two identical (solid or liquid) spheres which are held fixed relative to each other with the line connecting their centres normal to a uniform I stream is investigated numerically at Reynolds numbers 50, 100, and 150. We consider the lift, moment, and drag coefficients on the spheres and investigate their dependence on the distance between the two spheres. The computations show that, for a given Reynolds number, the two spheres are repelled when the spacing is of the order of the diameter but are weakly attracted at intermediate separation distances. For small spacing, the vortical structure of the near wake is significantly different from that of the axisymmetric wake that establishes at large separations. The partially confined flow passing between the two spheres entrains the flows coming around their other sides. Our results agree with available experimental and numerical data.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112093000229

Three-dimensional flow over two spheres placed side by side

The three-dimensional flow past two identical spherical bubbles moving side by side in a viscous fluid is studied numerically by solving the full Navier–Stokes equations. The bubble surface is assumed to be clean so that the outer flow obeys a zero-shear-stress condition. The present study describes the interaction between the two bubbles over a wide range of Reynolds number (, S being the distance between the bubble centres normalized by the bubble radius). The flow structure, the vorticity field, the sign of the interaction force and the magnitude of the drag and lift forces are analysed; in particular the latter are compared with analytical expressions available in the potential flow limit and in the limit of low-but-finite Reynolds number. This study sheds light on the role of the vorticity generated at the bubble surface in the interaction process. When vorticity remains confined in a boundary layer whose thickness is small compared to the distance between the two bubbles, the interaction is dominated by the irrotational mechanism that results in an attractive transverse force. In contrast, when viscous effects are sufficiently large, the vorticity field about each bubble interacts with that existing about the other bubble, resulting in a repulsive transverse force. Computational results combined with available high-Reynolds-number theory provide empirical expressions for the drag and lift forces in the moderate-to-large Reynolds number regime. They show that the transverse force changes sign for a critical Reynolds number whose value depends on the separation. Using these computational results it is shown that, depending on their initial separation, freely moving bubbles may either reach a stable equilibrium separation or move apart from each other up to infinity.

Hydrodynamic interactions between two spherical bubbles rising side by side in a viscous liquid

Interactive processes in gasification and combustion. Part I: Liquid drop arrays and clouds

The motion of two equal spherical bubbles moving along their line of centres in a viscous liquid is studied numerically in bispherical coordinates. The unsteady Navier-Stokes equations are solved using a mixed spectral/finite-difference scheme for Reynolds numbers up to 200. Free-slip conditions at the bubble surfaces are imposed, while the normal stress condition is replaced by the sphericity constraint under the assumption of small Weber number. The vorticity shed by the upstream bubble affects the drag on the trailing bubble in a very complex fashion that appears to be quite beyond the power of existing asymptotic analyses. The separation between two equal bubbles rising in line under the action of buoyancy is predicted to reach an equilibrium value dependent on the Reynolds number. This result is at variance with experiment. The explanation offered of this difference casts further doubt on the feasibility of a simplified simulation of bubbly liquid dynamics.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112094003733

On the in-line motion of two spherical bubbles in a viscous fluid

Heat and momentum transfer around a pair of spheres in viscous flow

Hydrodynamics and heat transfer in sphere assemblages—cylindrical cell models

The spherical cell model for transport in assemblages of spheres, which stems from the low Reynolds number hydrodynamics, has a serious physical limitation when extended to the range of intermediate Reynolds numbers. Spherical symmetry cannot persist around a sphere, given a definite direction of convection. The boundary conditions of the spherical cell model are inappropriate and a cubic cell model would best represent the system. However, in order to avoid undue mathematical difficulties peculiar to three-dimensional problems, a cylindrical cell model is proposed.

A numerical solution of the Navier-Stokes equations expressed in vorticity-stream function variables has been performed in conjunction with the cylindrical cell model for various values of porosity and Reynolds number. A much better agreement of the drag coefficient with experimental results is obtained by the cylindrical cell model than by the spherical cell model.

Reuven Tal

Papers

Heat and momentum transfer around a pair of spheres in viscous flow

Hydrodynamics and heat transfer in sphere assemblages—cylindrical cell models

Cylindrical cell model for the hydrodynamics of particle assemblages at intermediate reynolds numbers

Periodic solutions of heat transfer for flow through a periodic assemblage of spheres

Reassessment of solutions of finite difference equations in fluid mechanics