A stencil adaptive algorithm for finite difference solution of incompressible viscous flows

The numerical simulation of incompressible viscous flows using primitive variable formulation requires that the pressure be obtained by solving the pressure Poisson equation (PPE). It is mentioned in the literature on incompressible flows [1,2] that when all the pressure boundary conditions are of Neumann type, then the pressure compatibility condition (PCC) has to be satisfied for guaranteeing the existence of solution. The main aim of this paper is to show that the PCC is not an important issue as has been claimed in the literature [1–3]. On the other hand, it will be shown that the crucial issue, in numerical simulation of incompressible viscous flows, is to ensure solenoidality of the velocity field accurately. We will further show that the accuracy of solenoidality depends on the choice of pressure boundary conditions. The demonstration is based on the computation for 2-D lid-driven cavity problem for Re=100 on 1292 uniform grid.

A stabilized finite element method for incompressible flow and heat transfer: A natural derivation based on the use of local time-steps

Comparison investigations of numerical simulations of incompressible viscous flows by the SPH (smoothed particle hydrodynamics) and the MPS (moving particle semi-implicit) are presented. A dam-break problem is chosen as the test case. In the calculation with the SPH method, weakly compressible model is used, i.e. WCSPH, which describes water as a nearly incompressible fluid, while in MPS method, the pressure Poisson equation is introduced to keep the density of fluid to be constant. The numerical results show that the two particle methods are robust and flexible, numerical results qualitatively agree with the experimental data. It can be seen that both the SPH method and the MPS method can be easily applied to the complex free surface flow problems.

Simulation of incompressible viscous flows past a circular cylinder by hybrid FD scheme and meshless least square-based finite difference method

A new code and methodology are introduced for solving the general relativistic magnetohydrodynamic (GRMHD) equations in fixed background spacetimes using time-explicit, finite-volume discretization. The code has options for solving the GRMHD equations using traditional artificial-viscosity (AV) or non-oscillatory central difference (NOCD) methods, or a new extended AV (eAV) scheme using artificial-viscosity together with a dual energy-flux-conserving formulation. The dual energy approach allows for accurate modeling of highly relativistic flows at boost factors well beyond what has been achieved to date by standard artificial viscosity methods. It provides the benefit of Godunov methods in capturing high Lorentz boosted flows but without complicated Riemann solvers, and the advantages of traditional artificial viscosity methods in their speed and flexibility. Additionally, the GRMHD equations are solved on an unstructured grid that supports local adaptive mesh refinement using a fully threaded oct-tree (in three dimensions) network to traverse the grid hierarchy across levels and immediate neighbors. A number of tests are presented to demonstrate robustness of the numerical algorithms and adaptive mesh framework over a wide spectrum of problems, boosts, and astrophysical applications, including relativistic shock tubes, shock collisions, magnetosonic shocks, Alfven wave propagation, blast waves, magnetized Bondi flow, and the magneto-rotational instability in Kerr black hole spacetimes.

This study is aimed at developing a novel computational framework that seamlessly incorporates the feedback forcing model and adaptive mesh refinement mesh refinement (AMR) techniques in the immersed-boundary (IB) lattice Boltzmann method (LBM) approach, so that challenging problems, including the interactions between flowing fluids and moving objects, can be numerically investigated. Owing to the feedback forcing based IB model, the advantages, such as simple mechanics principle, explicit interpolations, and inherent satisfaction of no-slip boundary condition for solid surfaces are fully exhibited. Additionally, the "bubble' function is employed in the local mesh refinement process, so that the solution of second order accuracy at newly generated nodes can be obtained only by the spatial interpolation but no temporal interpolation. Focusing on both steady and unsteady flow around a single cylinder and bi-cylinders, a number of test cases performed in this study have demonstrated the usefulness and effectiveness of the present AMR IB-LBM approach.

Numerical simulation of incompressible flows using adaptive unstructured meshes and the pseudo-compressibility hypothesis

A distributed Lagrange multiplier/fictitious domain method for particulate flows

Numerical simulation of incompressible viscous flow continues to remain a challenging task, in particular if three space dimensions are involved. Space‐time finite element methods feature the natural construction of higher order discretization schemes. They offer the potential to achieve accurate results on computationally feasible grids. Using a temporal test basis supported on the subintervals and linearizing the resulting algebraic problems by Newton's method yield linear systems of equations with block matrices built of (k + 1) × (k + 1) saddle point systems, where k denotes the polynomial order of the variational time discretization. We demonstrate numerically the efficiency of preconditioning GMRES iterations for solving these linear systems by a V‐cycle geometric multigrid approach based on a local Vanka smoother. The studies are done for the two‐ and three‐dimensional benchmark problem of flow around a cylinder. The robustness of the solver with respect to the piecewise polynomial orderkin time is analyzed and confirmed numerically.

A directional error estimator for adaptive limit analysis

This paper presents a simulation of incompressible viscous flow within a two-dimensional square cavity. The objective is to develop a method originated from Lattice Gas (cellular) Automata (LGA), which utilises discrete lattice as well as discrete time and can be parallelised easily. Lattice Boltzmann Method (LBM), known as discrete Lattice kinetics which provide an alternative for solving the Navier–Stokes equations and are generally used for fluid simulation, is chosen for the study. A specific two-dimensional nine-velocity square Lattice model (D2Q9 Model) is used in the simulation with the velocity at the top of the cavity kept fixed. LBM is an efficient method for reproducing the dynamics of cavity flow and the results which are comparable to those of previous work.

High fidelity modeling and simulations methods need to be anchored to data collected from selected flight tests or experiments to develop robust, accurate and validated supersonic flow-simulation methods to predict the behavior of flowfield throughout the wide range flight regimes including highly rarefied supersonic gas flows.We have developed a unified flow model for compressible flows (called RNS model), based on the Generalized Hydrodynamic Equations (GHE) by Alexeev (1994), derived from generalized Boltzmann transport equation. The model is supposed to account for kinetic effects (intermediate Knudsen number, fluctuations) in the continuum approximation. This model has been explored for simulations of incompressible viscous flows for a wide range of problems and flow parameters, including high Reynolds numbers flows with thin boundary layers, demonstrating good agreement with experimental data, Fedoseyev (2012).Simulations of compressible supersonic flows is a very challenging problem as such flows can exhibit both continuum and non-continuum flow regimes. Typically, the flow can be continuous to transitional in the near field flow structure, and free molecular flow in the far field. The shock wave (bow shock) is detached from the vehicle at high altitude, and near boundary slip-flow is typical for such regimes. First results for this model has been reported in Fedoseyev (2020).In this paper we provide a comparison of simulation results of our model (called RNS, the Regularized Navier-Stokes) with the experimental data for rarefied hypersonic flows by Allegre et al. (1997). Simulations by DSMC method are also provided for comparison, the DSMC results obtained using the open source SPARTA DSMC code.The comparisons show good agreements of both RNS and DSMC with the experiment, and proove that SPARTA DSMC results can be used as a good approximation for experimental data, in cases where those are not available.

We describe in this Note a method for the numerical simulation of incompressible viscous flow around moving rigid bodies; we suppose the rigid body motions a priori known. The computational technique takes advantage of a time discretization by operator splitting à la Marchuk-Yanenko and of a finite element space discretization on a fixed mesh, to combine a Lagrange multiplier/fictitious domain treatment of the rigid body motions with an L2-projection technique, to force the incompressibility condition. The results of numerical experiments concerning flow around moving disks at Reynolds number of the order of 100 are presented. Dans cette Note on décrit une méthode pour la simulation numérique d'écoulements de fluides visqueux incompressibles autour de corps rigides dont le mouvement est connu a priori. La méthode de résolution tire avantage d'une discrétisation en temps par décomposition d'opérateurs, de type Marchuk-Yanenko, et d'une discrétisation en espace par éléments finis sur un maillage fixe, pour combiner une technique de domaines fictifs avec multiplicateurs de Lagrange avec une méthode de projection L2 pour le traitement de la condition d'incompressibilité. On présente les résultats d'essais numériques pour des écoulements bi-dimensionnels autour de disques mobiles, à des nombres de Reynolds de l'ordre de 100.

A new and efficient direct numerical method with second-order convergence accuracy was developed for fully resolved simulations of incompressible viscous flows laden with rigid particles. The method combines the state-of-the-art immersed boundary method (IBM), the multi-direct forcing method, and the lattice Boltzmann method (LBM). First, the multi-direct forcing method is adopted in the improved IBM to better approximate the no-slip/no-penetration (ns/np) condition on the surface of particles. Second, a slight retraction of the Lagrangian grid from the surface towards the interior of particles with a fraction of the Eulerian grid spacing helps increase the convergence accuracy of the method. An over-relaxation technique in the procedure of multi-direct forcing method and the classical fourth order Runge-Kutta scheme in the coupled fluid-particle interaction were applied. The use of the classical fourth order Runge-Kutta scheme helps the overall IB-LBM achieve the second order accuracy and provides more accurate predictions of the translational and rotational motion of particles. The preexistent code with the first-order convergence rate is updated so that the updated new code can resolve the translational and rotational motion of particles with the second-order convergence rate. The updated code has been validated with several benchmark applications. The efficiency of IBM and thus the efficiency of IB-LBM were improved by reducing the number of the Lagragian markers on particles by using a new formula for the number of Lagrangian markers on particle surfaces. The immersed boundary-lattice Boltzmann method (IBLBM) has been shown to predict correctly the angular velocity of a particle. Prior to examining drag force exerted on a cluster of particles, the updated IB-LBM code along with the new formula for the number of Lagrangian markers has been further validated by solving several theoretical problems. Moreover, the unsteadiness of the drag force is examined when a fluid is accelerated from rest by a constant average pressure gradient toward a steady Stokes flow. The simulation results agree well with the theories for the short- and long-time behavior of the drag force. Flows through non-rotational and rotational spheres in simple cubic arrays and random arrays are simulated over the entire range of packing fractions, and both low and moderate particle Reynolds numbers to compare the simulated results with the literature results and develop a new drag force formula, a new lift force formula, and a new torque formula. Random arrays of solid particles in fluids are generated with Monte Carlo procedure and Zinchenko's method to avoid crystallization of solid particles over high solid volume fractions. A new drag force formula was developed with extensive simulated results to be closely applicable to real processes over the entire range of packing fractions and both low and moderate particle Reynolds numbers. The simulation results indicate that the drag force is barely affected by rotational Reynolds numbers. Drag force is basically unchanged as the angle of the rotating axis varies.

The unsteady preconditioned characteristic boundary conditions (UPCBCs) based on the artificial compressibility (AC) method are formulated and applied at artificial boundaries for the direct numerical simulation (DNS) of incompressible flows. The compatibility equations including the unsteady terms are mathematically derived in the generalized curvilinear coordinates and then incorporated as boundary conditions (BCs) in a high-order accurate incompressible flow solver. The spatial derivative terms of the system of equations are discretized using the fourth-order compact finite difference (FD) scheme, consistent with the high-order accuracy required for the DNS. The time integration is carried out using an implicit dual time-stepping method, and the robustness and performance of the numerical algorithm are enhanced by applying the preconditioning technique and calculating the automated AC parameter based on the local flow conditions. At first, the steady viscous incompressible flow for a two-dimensional plane jet is computed to assess the accuracy and performance of the solution method by applying the UPCBCs in comparison with the simplified BCs. Then, the simulation of the unsteady incompressible viscous flow around a two-dimensional NACA0012 airfoil at the Reynolds number 800 and the angle of attack 20° is performed, and it is indicated that the computational cost of the solution is considerably decreased by implementing the UPCBCs at the artificial far-field boundary compared with the simplified BCs, which is remarkably valuable for the DNSs. Finally, the computation of the transition to unsteady for the incompressible flow around a three-dimensional NACA0012 wing at the same conditions of the two-dimensional case is successfully performed, and the results obtained are compared with those of previous studies that exhibit good agreement. Indications are that the present solution method by applying the UPCBCs at the artificial far-field boundary can be effectively used for the DNSs of the incompressible flows.