Unsteady cavitating flow often contains vapor structures with a wide range of different length scales,from micro-bubbles to large cavities,which issues a big challenge to precisely investigate its evolution mechanism ...Unsteady cavitating flow often contains vapor structures with a wide range of different length scales,from micro-bubbles to large cavities,which issues a big challenge to precisely investigate its evolution mechanism by computational fluid dynamics(CFD)method.The present work reviews the development of simulation methods for cavitation,especially the emerging Euler-Lagrange approach.Additionally,the progress of the numerical investigation of hot and vital issues is discussed,including cavitation inception,cloud cavitation inner structure and its formation mechanism,cavitation erosion,and cavitation noise.It is indicated that the Euler-Lagrange method can determine cavitation inception point better.For cloud cavitation,the Euler-Lagrange method can reveal the source of microbubbles and their distribution law inside the shedding cloud.This method also has advantages and great potential in assessing cloud cavitation-induced erosion and noise.With the ever-growing demands of cavitation simulation accuracy in basic research and engineering applications,how to improve the Euler-Lagrange method’s stability and applicability is still an open problem.To further promote the application of this advanced CFD simulation technology in cavitation research,some key issues are to be solved and feasible suggestions are put forward for further work.展开更多
Collision between particles plays an important role in determining the hydrodynamic characteristics of gas-solid flow in a fluidized bed.In the present work,earlier work(Loha,Chattopadhyay.& Chatterjee,2013) was e...Collision between particles plays an important role in determining the hydrodynamic characteristics of gas-solid flow in a fluidized bed.In the present work,earlier work(Loha,Chattopadhyay.& Chatterjee,2013) was extended to study the effect of the elasticity of particle collision on the hydrodynamic behavior of a bubbling fluidized bed filled with 530-μm particles.The Eulerian-Eulerian two-fluid model was used to simulate the hydrodynamics of the bubbling fluidized bed.where the solid-phase properties were calculated by applying the kinetic theory of granular flow.To investigate the effect of the elasticity of particle collision,different values of the coefficient of restitution were applied in the simulation and their effects were studied in detail.Simulations were performed for two different solid-phase wall boundary conditions.No bubble formation was observed for perfectly elastic collision.The bubble formation started as soon as the coefficient of restitution was set below 1.0,and the space occupied by bubbles in the bed increased with a decrease in the coefficient of restitution.Simulation results were also compared with experimental data available in the literature,and good agreement was found for coefficients of restitution of 0.95 and 0.99.展开更多
Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynom...Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynomials, Bernoulli numbers;its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained;the formulae for obtaining all π<sup>m</sup> as series on k<sup>-m</sup> and for expanding functions into series of Euler polynomials.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.52176041,12102308).
文摘Unsteady cavitating flow often contains vapor structures with a wide range of different length scales,from micro-bubbles to large cavities,which issues a big challenge to precisely investigate its evolution mechanism by computational fluid dynamics(CFD)method.The present work reviews the development of simulation methods for cavitation,especially the emerging Euler-Lagrange approach.Additionally,the progress of the numerical investigation of hot and vital issues is discussed,including cavitation inception,cloud cavitation inner structure and its formation mechanism,cavitation erosion,and cavitation noise.It is indicated that the Euler-Lagrange method can determine cavitation inception point better.For cloud cavitation,the Euler-Lagrange method can reveal the source of microbubbles and their distribution law inside the shedding cloud.This method also has advantages and great potential in assessing cloud cavitation-induced erosion and noise.With the ever-growing demands of cavitation simulation accuracy in basic research and engineering applications,how to improve the Euler-Lagrange method’s stability and applicability is still an open problem.To further promote the application of this advanced CFD simulation technology in cavitation research,some key issues are to be solved and feasible suggestions are put forward for further work.
文摘Collision between particles plays an important role in determining the hydrodynamic characteristics of gas-solid flow in a fluidized bed.In the present work,earlier work(Loha,Chattopadhyay.& Chatterjee,2013) was extended to study the effect of the elasticity of particle collision on the hydrodynamic behavior of a bubbling fluidized bed filled with 530-μm particles.The Eulerian-Eulerian two-fluid model was used to simulate the hydrodynamics of the bubbling fluidized bed.where the solid-phase properties were calculated by applying the kinetic theory of granular flow.To investigate the effect of the elasticity of particle collision,different values of the coefficient of restitution were applied in the simulation and their effects were studied in detail.Simulations were performed for two different solid-phase wall boundary conditions.No bubble formation was observed for perfectly elastic collision.The bubble formation started as soon as the coefficient of restitution was set below 1.0,and the space occupied by bubbles in the bed increased with a decrease in the coefficient of restitution.Simulation results were also compared with experimental data available in the literature,and good agreement was found for coefficients of restitution of 0.95 and 0.99.
文摘Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynomials, Bernoulli numbers;its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained;the formulae for obtaining all π<sup>m</sup> as series on k<sup>-m</sup> and for expanding functions into series of Euler polynomials.
