Leopold Škerget

COMPUTATIONAL FLUID DYNAMICS BY BOUNDARY-DOMAIN INTEGRAL METHOD

Very fast development of computing enabled also the development of numerical fluid dynamics. It is numerical modelling and simulation of flow circumstances, including numerical experiments by the computer. Such procedure may have several important advantages over physical measurements on a laboratory model. It is of great importance that fluid properties (density, viscosity, compressibility, etc.) may be simply and arbitrarily changed, numerical experiment does not disturb the flow, plane flows can simply be simulated what may not be the case with laboratory experiments. The numerical experiment also has its own drawbacks and disadvantages, known to all numerical procedures, since the numerical solution represents a result of a discrete equation systems, which are not completely identical to basic physical laws of mechanics of continua. Discretisation often changes quantitatively and qualitatively the behavior of equations and thus also the solutions. Numerical simulation has also similar limitations like a laboratory experiments, since the solutions are individual discrete values only, not the functions of the flow fields.

ITERATIVE METHODS IN SOLVING NAVIER–STOKES EQUATIONS BY THE BOUNDARY ELEMENT METHOD

The solution of Navier–Stokes equations of time-dependent incompressible viscous fluid flow in planar geometry by the Boundary Domain Integral Method (BDIM) is discussed. The introduction of a subdomain technique to fluid flow problems is considered and improved in order to maintain the stability of BDIM. To avoid problems with flow kinematics computation in the sudomain mesh, a segmentation technique is proposed which combines the original BDIM with its subdomain variant and preserves its numerical stability. In order to reduce the computational cost of BDIM, which greatly depends on the solution of systems of linear equations, iterative methods are used. Conjugate gradient methods, conjugate gradients squared and an improved version of the biconjugate gradient method BiCGSTAB, together with the generalized minimal residual method, are used as iterative solvers. Different types of preconditioning, from simple Jacobi to incomplete LU factorization, are carried out and the performance of chosen iterative methods and preconditioners are reported. Test examples include backward facing step flow and flow through tubular heat exchangers. Test computation results show that BDIM is an accurate approximation technique which, together with the subdomain technique and powerful iterative solvers, can exhibit some significant savings in storage and CPU time requirements.

Boundary-domain integral method using a velocity-vorticity formulation

Abstract The paper deals with the numerical solution of fluid dynamics (transport phenomena in incompressible fluid flow) using a boundary-domain integral method. A velocity-vorticity formulation of the Navier-Stokes equations is adopted, where the kinematic equation is written in its parabolic version.

Natural convection flows in complex cavities by BEM

A numerical method for the solution of the Navier‐Stokes equations is developed using an integral representation of the conservation equations. The velocity‐vorticity formulation is employed, where the kinematics is given with the Poisson equation for a velocity vector, while the kinetics is represented with the vorticity transport equation. The corresponding boundary‐domain integral equations are presented along with discussions of the kinetics and kinematics of the fluid flow problem. The boundary‐domain integral formulation is developed and tested for natural convection flows in closed cavities with complex geometries.

Mixed boundary elements for laminar flows

We present a mixed boundary element formulation of the boundary domain integral method (BDIM) for solving diffusion-convective transport problems. The basic idea of mixed elements is the use of a continuous interpolation polynomial for conservative field function approximation and a discontinuous interpolation polynomial for its normal derivative along the boundary element. In this way, the advantages of continuous field function approximation are retained and its conservation is preserved while the normal flux values are approximated by interpolation nodal points with a uniquely defined normal direction. Due to the use of mixed boundary elements, the final discretized matrix system is overdetermined and a special solver based on the least squares method is applied. Driven cavity, natural and forced convection in a closed cavity are studied. Driven cavity results at Re = 100, 400 and 1000 agree better with the benchmark solution than Finite Element Method or Finite Volume Method results for the same grid density with 21 × 21 degrees of freedom

Boundary element method for natural convection in non-Newtonian fluid saturated square porous cavity

Abstract The main purpose of this work is to present the use of the Boundary Element Method (BEM) in the analysis of the natural convection in the square porous cavity saturated by the non-Newtonian fluid. The results of hydrodynamic and heat transfer evaluations are reported for the configuration in which the enclosure is heated from a side wall while the horizontal walls are insulated. The flow in the porous medium is modelled using the modified Brinkman extended Darcy model taking into account the non-Darcy viscous effects. The governing equations are transformed by the velocity–vorticity variables formulation enabling the computation scheme to be partitioned into kinematic and kinetic parts. To analyse the effects of the available non-Newtonian viscosity and to evaluate the presented approach, the power law model for shear thinning fluids ( n n >1) and in the limit for the Newtonian fluids ( n =1) is considered. Numerical model is tested also for the Carreau model adequate for many non-Newtonian fluids. Solutions for the flow and temperature fields and Nusselt numbers are obtained in terms of a modified Rayleigh number Ra ∗ , Darcy number Da , and the non-Newtonian model parameters. The agreement between the results obtained with finite difference method is very good indicating that BEM can be efficiently used for solving transport phenomena in saturated porous medium.

Boundary domain integral method for transport phenomena in porous media

A boundary domain integral method (BDIM) for the solution of transport phenomena in porous media is presented. The complete, so-called modified Navier-Stokes equations (Brinkman-extended Darcy formulation with inertial term included) have been used to describe the fluid motion in porous media. Velocity-vorticity formulation (VVF) of the conservative equations is employed. The proposed numerical scheme is tested on a particular case of natural convection and the results of flow and heat transfer characteristics of a fluid in a vertical porous cavity heated from the side and saturated with Newtonian fluid are presented in detail

Fast boundary-domain integral algorithm for the computation of incompressible fluid flow problems

We deal with the numerical solution of fluid dynamics using the boundary-domain integral method (BDIM). A velocity-vorticity formulation of the Navier-Stokes equations is adopted, where the kinematic equation is written in its parabolic form. Computational aspects of the numerical simulation of two-dimensional flows is described in detail. In order to lower the computational cost, the subdomain technique is applied. A preconditioned Krylov subspace method (PKSM) is used for the solution of systems of linear equations. Level-based fill-in incomplete lower upper decomposition (ILU) preconditioners are developed and their performance is examined. Scaling of stopping criteria is applied to minimize the number of iterations for the PKSM. The effectiveness of the proposed method is tested on several benchmark test problems

Predicting Free-Surface Vortices with Single-Phase Simulations

Abstract In this article, single-phase, computational-fluid-dynamics simulations of free-surface vortices are presented. The purpose of the simulations is to determine the appropriate turbulence model for free-surface vortices, which could later be applied to simulations of flow in various engineering systems. The water flow in the laboratory model of a free-surface vortex was numerically simulated by unsteady single-phase computations. The vortex circumferential velocity, the downward velocity inside the vortex core and the predicted length of the free-surface vortex gas core were compared with available measurements. For the two-equation turbulence models, the results indicated the importance of the curvature correction (CC). The effect of the time-step size and the choice of the advection scheme were analyzed. For the tested case, it was determined that the unsteadiness of the flow was insufficient for the correct behavior of the scale-adaptive simulation (SAS) turbulence model. With the CC option, the shear-stress-transport (SST-CC) turbulence model and the SAS-CC turbulence model can both be used for such predictions; however, the SAS-CC model was found to be more reliable. Single-phase simulations successfully predicted the gas-core length for vortices with a short gas core. However, for long cores, the length was underpredicted.

BEM for non-Newtonian fluid flow

Abstract The main purpose of this work is to present the use of boundary-domain integral method (BDIM) to analyse the flow behaviour of non-Newtonian fluids. A few available parametric viscosity models are applied representing a non-linear dependence on shear strain rate and shear stress. To evaluate the presented approach the Rayleigh–Benard natural convection was solved at different Rayleigh number values.