Jure Ravnik

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.

Boundary element formulations for the numerical solution of two-dimensional diffusion problems with variable coefficients

This paper presents new formulations of the radial integration boundary integral equation (RIBIE) and the radial integration boundary integro-differential equation (RIBIDE) methods for the numerical solution of two-dimensional diffusion problems with variable coefficients. The methods use either a specially constructed parametrix (Levi function) or the standard fundamental solution for the Laplace equation to reduce the boundary-value problem (BVP) to a boundary-domain integral equation (BDIE) or boundary-domain integro-differential equation (BDIDE). The radial integration method (RIM) is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Furthermore, a subdomain decomposition technique (SDBDIE) is proposed, which leads to a sparse system of linear equations, thus avoiding the need to calculate a large number of domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed approaches.

Integral equation formulation of an unsteady diffusionconvection equation with variable coefficient and velocity

In this paper we present an integral equation formulation for the time dependent diffusionconvection equation with variable coefficient and velocity with sources. The formulation is based on usage of the steady fundamental solution of the convectiondiffusion equation. For a known velocity and coefficient fields, which may change with location and time, the formulation avoids the usage of the gradient of the unknown field function and thus avoids making the problem nonlinear. Two discretization approaches are proposed and compared: a standard single domain boundarydomain element technique and a domain decomposition approach. The validity of the formulation and comparison of discretization approaches is preformed on several challenging test cases. Mesh convergence is reported and the advantages and disadvantages of both approaches are examined.

Choice of a turbulence model for pump intakes

This article is focused on the choice of a suitable turbulence model for simulations of an industrial pumps intake, from the perspective of accuracy and, partially, also the CPU time. Twelve steady-state and transient simulations were made on a fine computational mesh, using turbulence models such as: the shear stress transport (SST), the scale-adaptive simulation (SAS), the Reynolds stress model, the explicit algebraic Reynolds-stress model, the detached eddy simulation and the large eddy simulation (LES). The curvature-correction (CC) option was assessed for the SST and SAS turbulence models. The results were compared with the LES and with published experimental results. Although all the models could predict the main floor vortex, there were still some substantial differences. It can be able to conclude that it is better to use either the SST-CC turbulence model, due to its low-computational resources and far better results than the SST model, or the SAS-CC turbulence model, since its predictions are quite similar to the LES results. In the final step, good agreement with experimental results was shown for a longer simulation with the SAS-CC turbulence model.

Turbulence model comparison for a surface vortex simulation

A pump intake can have an important impact on a pump operation due to production of strong unsteady vortices which may cause air intake problems. Constructing a pump sump model and experimental testing is expensive, therefore numerical simulations are expected to help or even replace the experimental testing in the future. In order to understand and eventually be able to predict such surface vortices numerically, a vortex in a small chamber was simulated. A benefit of such isolated vortex test case is small number of elements in computational mesh, compared to the whole pump intake, and a controlled testing environment. For a small chamber vortex simulation, various turbulence model simulations as well as laminar and Euler simulations were evaluated. The results indicate that the SAS-CC turbulence model might be a good choice for a simulation of a pump intake. Time step increase had a moderate influence on SAS-CC results.

THREE-DIMENSIONAL DOUBLE-DIFFUSIVE NATURAL CONVECTION WITH OPPOSING BUOYANCY EFFECTS IN POROUS ENCLOSURE BY BOUNDARY ELEMENT METHOD

A three-dimensional double-diffusive natural convection with opposing buoyancy effects in a cubic enclosure fi lled with fl uid saturated porous media is studied numerically using the boundary element method (BEM). The mathematical model is based on the space-averaged Navier–Stokes equations, which are coupled with the energy and species equations. The simulation of coupled laminar viscous fl ow, heat and solute transfer is performed using a combination of single-domain BEM and subdomain BEM, which solves the velocity-vorticity formulation of governing equations. The numerical simulations for a case of negative values of buoyancy coeffi cient are presented, focusing on the situations where the fl ow fi eld becomes three-dimensional. The results are analyzed in terms of the average heat and mass transfer at the walls of the enclosure. When possible, the results are compared with previous existing numerical data published in literature.

Daftardar-Jafari Method for Solving Nonlinear Thin Film Flow Problem

Abstract The aim of this paper is to develop the Daftardar-Jafari iterative method (DJM) for a mathematical model that represents the nonlinear thin film flow of a non-Newtonian third-grade fluid on a moving belt with the aim to obtain an approximate solution of high accuracy. When applying the DJM there is no need to resort to any additional techniques such as evaluating Adomian’s polynomials as in the Adomian decomposition method (ADM) or such as using Lagrange multipliers in the variational iteration method (VIM). The accuracy of our results is numerically verified by evaluating the functions of the error remainder and the maximal error remainders. In addition, these results are analyzed by comparing the accuracy of the DJM solutions with those of the fourth order Runge-Kutta method (RKM), ADM and VIM at the same parameter values. All the evaluations have been successfully performed in an iterative way by using the symbolic manipulator Mathematica®.

On Constitutive Models for the Momentum Transfer to Particles in Fluid-Dominated Two-Phase Flows

This contribution deals with fluid flow-particle interactions in fluid dominated two phase flows. Spherical as well as non-spherical particles in the form of are considered. In the case of ellipsoids, the hydrodynamic drag force model based on the Brenner-type resistance tensor is applied. As high shear flow regions are frequently encountered in complex flow patterns, special attention is devoted to the extension of established shear lift models, that are only valid for special cases of shear flows, to a general shear lift model based on permutations of the lift tensor, originally derived by Harper and Chang. A generalized lift vector, valid for ellipsoidal particles, is derived and implemented for the computation of the lift force in general shear flows. The derived generalized shear lift force model is validated against other numerical models for ellipsoids in Couette flow, and its influence on the translational motion of ellipsoidal particles in a three-dimensional lid-driven cavity flow is studied. The computational results confirm the correctness of the proposed generalized shear lift model.

Implementation of the Rosseland and the P1 radiation models in the system of Navier-stokes equations with the boundary element method

The objective of this article is to develop a boundary element numerical model to solve coupled problems involving heat energy diffusion, convection and radiation in a participating medium. In this study, the contributions from radiant energy transfer are presented using two approaches for optical thick fluids: the Rosseland diffusion approximation and the P1 approximation. The governing Navier– Stokes equations are written in the velocity–vorticity formulation for the kinematics and kinetics of the fluid motion. The approximate numerical solution algorithm is based on a boundary element numerical model in its macro-element formulation. Validity of the proposed implementation is tested on a one-dimensional test case using a grey participating medium at radiative equilibrium between two isothermal black surfaces.

Two efficient methods for solving Schlömilch’s integral equation

In this paper, the exact solutions of the Schlomilch’s integral equation and its linear and non-linear generalized formulas with application are solved by using two efficient iterative methods. The Schlomilch’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.,First, the authors apply a regularization method combined with the standard homotopy analysis method to find the exact solutions for all forms of the Schlomilch’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 regularization-variational method is shown by using them for several illustrative examples, which have been solved by other authors using the so-called regularization-Adomian method.,The implementation of the two methods demonstrates the usefulness in finding exact solutions.,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.,In this paper, two reliable methods have been implemented to solve several examples, where those examples represent the main types of the Schlomilch’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 Schlomilch’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.