Hybrid LES/URANS Simulation of Rayleigh-Bénard Convection Using BEM

In this paper,we develop and test a unified hybrid LES/URANS turbulence model with two different Large Eddy Simulation(LES)turbulence models.The numerical algorithm is based on the Boundary Element Method.In the existing hybrid LES/URANS turbulence model we implemented a new Smagorinsky LES turbulence model.The hybrid LES/URANS turbulence model is unified,which means that the LES/URANS interface is changed dynamically during simulation using a physical quantity.In order to define the interface between LES and unsteady Reynolds Averaged Navier Stokes(URANS)zones during the simulation,we use the Reynolds number based on turbulent kinetic energy as a switching criterion.This means that the flow characteristics define where the sub-grid scale or URANS effective viscosity and thermal conductivity are used in the governing equations in the next time step.In unified hybrid turbulence models,only one set of governing equations is used for LES and URANS regions.The developed hybrid LES/URANS model was tested on non-isothermal,unsteady and turbulent Rayleigh-Bénard Convection and compared with an existing model,where LES is based on turbulent kinetic energy.The hybrid turbulence model was implemented within a numerical algorithm based on the Boundary-Domain Integral Method,where a single domain and sub-domain approaches were used.The numerical algorithm uses governing equations written in a velocity-vorticity form.The false transient time scheme is used for the kinematics equation.

Two efficient methods for solving Schlömilch’s integral equation

Purpose–In this paper,the exact solutions of the Schlömilch’s integral equation and its linear and non-linear generalized formulas with application are solved by using two efficient iterative methods.The Schlömilch’s integral equations have many applications in atmospheric,terrestrial physics and ionospheric problems.They describe the density profile of electrons from the ionospheric for awry occurrence of the quasi-transverse approximations.The paper aims to discuss these issues.Design/methodology/approach–First,the authors apply a regularization method combined with the standard homotopy analysis method to find the exact solutions for all forms of the Schlömilch’s integral equation.Second,the authors implement the regularization method with the variational iteration method for the same purpose.The effectiveness of the regularization-Homotopy method and the regularizationvariational method is shown by using them for several illustrative examples,which have been solved by other authors using the so-called regularization-Adomian method.Findings–The implementation of the two methods demonstrates the usefulness in finding exact solutions.Practical implications–The authors have applied the developed methodology to the solution of the Rayleigh equation,which is an important equation in fluid dynamics and has a variety of applications in different fields of science and engineering.These include the analysis of batch distillation in chemistry,scattering of electromagnetic waves in physics,isotopic data in contaminant hydrogeology and others.Originality/value–In this paper,two reliable methods have been implemented to solve several examples,where those examples represent the main types of the Schlömilch’s integral models.Each method has been accompanied with the use of the regularization method.This process constructs an efficient dealing to get the exact solutions of the linear and non-linear Schlömilch’s integral equation which is easy to implement.In addition to that,the accompanied regularization method with each of the two used methods proved its efficiency in handling many problems especially ill-posed problems,such as the Fredholm integral equation of the first kind.