Peter D. Minev

Error Analysis of Pressure-Correction Schemes for the Time-Dependent Stokes Equations with Open Boundary Conditions

The incompressible Stokes equations with prescribed normal stress (open) boundary conditions on part of the boundary are considered. It is shown that the standard pressure-correction method is not suitable for approximating the Stokes equations with open boundary conditions, whereas the rotational pressure-correction method yields reasonably good error estimates. These results appear to be the first ever published for splitting schemes with open boundary conditions. Numerical results in agreement with the error estimates are presented.

A fictitious domain formulation for flows with rigid particles

In this paper, we present a development of the fictitious domain method proposed in Ref. C. Diaz-Goano, P. Minev, K. Nandakumar, A fictitious domain/finite element method for particulate flows, J. Comput. Phys. 192 (2003) 105]. The main new feature of the modified method is that after a proper splitting, it avoids the need to use Lagrange multipliers for imposition of the rigid body motion and instead, it resolves the interaction force between the two phases explicitly. Then, the end-of-step fluid velocity is a solution of an integral equation. The most straightforward way to resolve it is via an iteration but a direct extrapolation is also possible. If the latter approach is applied then the fictitious domain formulation becomes fully explicit with respect to the rigid body constraint and therefore, the corresponding numerical procedure is much cheaper. Most of the numerical results presented in this article are obtained with such an explicit formulation.

A finite element technique for multifluid incompressible flow using Eulerian grids

The paper presents a finite element method for 3D incompressible fluid flows with capillary free boundaries. It uses a fixed Eulerian grid of 10 nodes (P2 - P1) tetrahedra and tracks the free boundary using a six nodes (P2) triangular surface grid. In order to improve the mass conservation properties of the method, a local enrichment of the finite element basis in the elements intersected by the free boundaries is employed. In addition to the surface tracking, it also advects a smooth indicator function for an easy identification of the fluid properties in the different parts of the domain. The advective part of the Navier-Stokes equations is split and integrated with a characteristic method. The remaining generalized Stokes problem is resolved by means of an inexact outer-inner (Uzawa) iteration with a properly chosen preconditioner. The performance of this technique is evaluated on several problems involving droplets in viscous liquids.

Computational fluid dynamics modelling and experimental study of erosion in slurry jet flows

The main objective of the present work is to show that one can use computational fluid dynamics (CFD) to predict the flow patterns and particle impact profiles on a specimen and use such data together with phenomenological erosion models to predict both the total erosion rates and the erosion patterns on surfaces. The surface wear-out pattern from profilometry measurements and the total weight loss data of the eroded material were obtained from experiments to validate the model predictions. In the experiments, four different parameters, viz. the impingement velocity, impingement angle, sand concentration and properties of target materials were varied. Their effect on the total weight loss and the erosion patterns are investigated. Two different experimental conditions were used – viz. one in which the specimen and the jet were submerged and the other in which both were exposed (non-submerged). Experimental studies show that total weight loss has a power-law relation with respect to impingement velocity or sand concentration. The weight loss decreases as the hardness of material increases, but not in a linear fashion. Two corrosion resistance materials, 304 stainless steel and chrome white alloy, showed the same corrosion and synergism weight loss although their hardness was much different, 190 HV (Vickers hardness) for 304 stainless steel and 763 HV for chrome white alloy. This implies that the corrosion and synergism weight loss is mainly dominated by the chemical composition of the material and not by its mechanical properties. The results also differ substantially for the submerged and non-submerged cases. When the specimen and the jet were submerged, the erosion scar was typical ‘W’ style. But when sample and jet were not submerged, the scar was much flatter. A phenomenological erosion model that results in a simple algebraic equation has been developed, and the predicted results are in reasonable agreement with the experimental measurements. When they are used locally together with the CFD predictions of the flow field, the agreement with the erosion patterns and the total weight loss is improved significantly.

A characteristic/finite element algorithm for time-dependent 3-D advection-dominated transport using unstructured grids

Abstract An algorithm based on operator splitting has been successfully implemented for solving unsteady, advection-dominated transport problems in 3-D. Specifically, the general operator-integration-factor splitting method of Maday et al. is applied to the unsteady advection–diffusion equation with source/sink terms. The algorithm incorporates a 3-D characteristic Galerkin scheme to treat advection, and a standard Galerkin treatment of the diffusion and source/sink terms. Up to third-order operator splitting was implemented and validated against several analytical solutions. The algorithm showed the expected error behaviour and good performance in modeling advection-dominated transport problems. The practical utility and effectiveness of the proposed numerical scheme was further demonstrated by solving the Graetz–Nusselt problem, i.e. high Peclet number mass/heat transport in a fully developed pipe flow.

Direct numerical simulation of free falling sphere in creeping flow

In the present study, direct numerical simulations (DNS) are performed on single and a swarm of particles settling under the action of gravity. The simulations have been carried out in the creeping flow range of Reynolds number from 0.01 to 1 for understanding the hindrance effect, of the other particles, on the settling velocity and drag coefficient. The DNS code is a non-Lagrange multiplier-based fictitious-domain method, which has been developed and validated by Jin et al. (2008; A parallel algorithm for the direct numerical simulation of 3D inertial particle sedimentation. In: Conference proceedings of the 16th annual conference of the CFD Society of Canada). It has been observed that the time averaged settling velocity of the particle in the presence of other particles, decreases with an increase in the number of particles surrounding it (from 9 particles to 245 particles). The effect of the particle volume fraction on the drag coefficient has also been studied and it has been observed that the computed values of drag coefficients are in good agreement with the correlations proposed by Richardson and Zaki (1954; Sedimentation and fluidization: part I. Transactions of the Institution of Chemical Engineers, 32, 35–53) and Pandit and Joshi (1998; Pressure drop in packed, expanded and fluidised beds, packed columns and static mixers – a unified approach. Reviews in Chemical Engineering, 14, 321–371). The suspension viscosity-based model of Frankel and Acrivos (1967; On the viscosity of a concentrated suspension of solid spheres. Chemical Engineering Science, 22, 847–853) shows good agreement with the DNS results.

High-Order Time Stepping for the Incompressible Navier--Stokes Equations

This paper introduces a high-order time stepping technique for solving the incompressible Navier--Stokes equations which, unlike coupled techniques, does not require solving a saddle point problem at each time step and, unlike projection methods, does not produce splitting errors and spurious boundary layers. The technique is a generalization of the artificial compressibility method; it is unconditionally stable (for the unsteady Stokes equations), can reach any order in time, and uncouples the velocity and the pressure. The condition number of the linear systems associated with the fully discrete vector-valued problems to be solved at each time step scales like

A scalable parallel algorithm for the direct numerical simulation of three-dimensional incompressible particulate flow

O(\tau h^{-2})

High-order time stepping for the Navier-Stokes equations with minimal computational complexity

, where

Convergence analysis of a class of massively parallel direction splitting algorithms for the Navier-Stokes equations in simple domains

\tau