Numerical solution of unsteady Navier–Stokes equations on curvilinear meshes

The objective of the present work is to extend our FDS-based third-order upwind compact schemes by Shah et al. (2009) [8] to numerical solutions of the unsteady incompressible Navier–Stokes equations in curvilinear coordinates, which will save much computing time and memory allocation by clustering grids in regions of high velocity gradients. The dual-time stepping approach is used for obtaining a divergence-free flow field at each physical time step. We have focused on addressing the crucial issue of implementing upwind compact schemes for the convective terms and a central compact scheme for the viscous terms on curvilinear structured grids. The method is evaluated in solving several two-dimensional unsteady benchmark flow problems.     

Numerical simulation of gas flow in rough microchannels: hybrid kinetic–continuum approach versus Navier–Stokes

The flow field in a rough microchannel is numerically analyzed using a hybrid solver, dynamically coupling kinetic and Navier–Stokes solutions computed in rarefied and continuum subareas of the flow field, respectively, and a full Navier–Stokes solver. The rough surface is configured with triangular roughness elements, with a maximum relative roughness of 5 % of the channel height. The effects of Mach number, Knudsen number (or Reynolds number) and roughness height are investigated and discussed in terms of Poiseuille number and mass flow rate. Discrepancies between full Navier–Stokes and hybrid solutions are analyzed, assessing the range of validity of Navier–Stokes equations provided with first-order slip boundary conditions for modeling gas flow along a rough surface. Results indicate that the roughness increases Poiseuille number and decreases mass flux in comparison with those for the smooth microchannel. Increasing rarefaction results in further enhancement of roughness effect. At the same time, the compressibility effect is more noticeable than the roughness one, although the compressibility effect is alleviated by increase in the rarefaction. It was found that, although the Navier–Stokes solution of the flow in a smooth channel is accurate up to Kn = 0.1, when relative roughness height is higher than 1.25 % significant errors already appear at Kn = 0.02.     

Numerical solution of the incompressible Navier–Stokes equations with exponential type schemes

A new exponential type finite-difference scheme of second-order accuracy for solving the unsteady incompressible Navier–Stokes equation is presented. The driven flow in a square cavity is used as the model problem. Numerical results for various Reynolds numbers are given, and are in good agreement with those presented by Ghia et al. (Ghia, U., Ghia, K.N. and Shin, C.T., 1982, High-Re solutions for incompressible flow using the Navier–Stokes equations and a multi-grid method. Journal of Computational Physics, 48, 387–411.).     

Numerical investigation of effectivity indices of space-time error indicators for Navier–Stokes equations

In this paper we propose two error indicators aimed at estimating the space discretization error and the time discretization error for the unsteady Navier–Stokes equations. We define a space error indicator for evaluating the quality of the mesh and a time error indicator for evaluating the time discretization error. Moreover, we verify the reliability of the estimations through numerical experiments and we propose an effective space-time adaptive strategy for the unsteady Navier–Stokes equations. Such technique is based on two residual-based error indicators that suitably drive the mesh and the timestep-length modifications. Adaptive simulations show that the presented strategy allows to obtain accurate solutions in efficient way.     

Discretization of the Navier–Stokes equations on non-smooth grids using auxiliary points

The colocated scheme for the incompressible Navier–Stokes equations is improved on structured non-Cartesian grids. The method relies on a finite volume discretization and on the use of auxiliary points to locally approximate gradients following a two-point discretization. Enhanced accuracy is demonstrated for two-dimensional cases on strongly distorted meshes by computing Poiseuille flow and a flow in a differentially heated cavity. Received: 23 February 1999 / Accepted: 17 June 1999     

Isogeometric analysis of the isothermal Navier–Stokes–Korteweg equations

This paper is devoted to the numerical simulation of the Navier–Stokes–Korteweg equations, a phase-field model for water/water-vapor two-phase flows. We develop a numerical formulation based on isogeometric analysis that permits straightforward treatment of the higher-order partial–differential operator that represents capillarity. We introduce a new refinement methodology that desensitizes the numerical solution to the computational mesh and achieves mesh invariant solutions. Finally, we present several numerical examples in two and three dimensions that illustrate the effectiveness and robustness of our approach.     

An accelerated algorithm for Navier–Stokes equations

In this paper, the numerical solution of the Navier–Stokes equations by the Characteristic-Based-Split (CBS) scheme is accelerated with the Minimum Polynomial Extrapolation (MPE) method to obtain the steady state solution for evolution incompressible and compressible problems.The CBS is essentially a fractional time-stepping algorithm based on an original finite difference velocity-projection scheme where the convective terms are treated using the idea of the Characteristic-Galerkin method. In this work, the semi-implicit version of the CBS with global time-stepping is used for incompressible problems whereas the fully-explicit version is used for compressible flows.At the other end, the MPE is a vector extrapolation method that transforms the original sequence into another sequence converging to the same limit faster then the original one without the explicit knowledge of the sequence generator.The developed algorithm, tested on two-dimensional benchmark problems, demonstrates the new computational features arising from the introduction of the extrapolation procedure to the CBS scheme. In particular, the results show a remarkable reduction of the computational cost of the simulation.     

Numerical simulations of solutions of a two-level averaged Navier–Stokes system

An averaging procedure for the Navier–Stokes equations has been proposed in an earlier article [I. Moise, R.M. Temam, Renormalization group method. Application to Navier–Stokes Equation, Discrete Contin. Dyn. Syst. 6 (1) (2000) 191–210]. This averaging procedure is based on a two-level decomposition of the solution into low and high frequencies. The aim of the present article is to investigate, with the help of numerical simulations, the behavior of the small scales of the corresponding system. Space-periodic solutions with a non-resonant period are considered. The time evolution of the averaged and standard (non-averaged) small scales are compared at different Reynolds numbers and for different values of the cut-off level used to separate large and small scales of the flow variables. The numerical results illustrate the efficiency of the proposed averaging procedure for the Navier–Stokes equations. The averaged small scales provide an accurate prediction of the time-averaged small scales of the Navier–Stokes solutions. As the computational cost is reduced for the averaged equations, long time integrations on more than 50 eddy-turnover times have been performed for cut-off levels ensuring a proper resolution of the large scales. In these cases, development of instabilities in the averaged small scale equation is observed.     

On pressure–velocity coupled time-integration of incompressible Navier–Stokes equations using direct inversion of Stokes operator or accelerated multigrid technique

Two fully pressure–velocity coupled approaches to time-integration of two- and three-dimensional Navier–Stokes equations discretized by finite volume method are proposed and verified. The first approach utilizes a direct sparse matrix solver to inverse the Stokes operator. In the second approach a multigrid iterative solver is accelerated by a modification of CLGS smoother that allows for derivation of an analytical solution for velocity and pressure corrections belonging to a whole row or column of finite volumes. Both approaches are tested by two- and three-dimensional natural convection benchmark problems. It is concluded that the analytical solution accelerated CLGS technique (ASA-CLGS) can be considered as a promising tool for solution of time-dependent three-dimensional fluid dynamics problems.     

On the Numerical Controllability of the Two-Dimensional Heat,Stokes and Navier–Stokes Equations

The aim of this work is to present some strategies to solve numerically controllability problems for the two-dimensional heat equation, the Stokes equations and the Navier–Stokes equations with Dirichlet boundary conditions. The main idea is to adapt the Fursikov–Imanuvilov formulation, see Fursikov and Imanuvilov (Controllability of Evolutions Equations, Lectures Notes Series, vol 34, Seoul National University, 1996); this approach has been followed recently for the one-dimensional heat equation by the first two authors. More precisely, we minimize over the class of admissible null controls a functional that involves weighted integrals of the state and the control, with weights that blow up near the final time. The associated optimality conditions can be viewed as a differential system in the three variables \(x_1\), \(x_2\) and t that is second-order in time and fourth-order in space, completed with appropriate boundary conditions. We present several mixed formulations of the problems and, then, associated mixed finite element Lagrangian approximations that are relatively easy to handle. Finally, we exhibit some numerical experiments.     

A comparison of parallel solvers for the incompressible Navier–Stokes equations

The paper compares coupled multigrid methods and pressure Schur complement schemes (operator splitting schemes) for the solution of the steady state and time dependent incompressible Navier–Stokes equations. We consider pressure Schur complement schemes with multigrid as well as single grid methods for the solution of the Schur complement problem for the pressure. The numerical tests have been carried out on benchmark problems using a MIMD parallel computer. They show the superiority of the coupled multigrid methods for the considered class of problems. Received: 14 October 1997 / Accepted: 11 February 1998     

Fast and reliable solution of the Navier–Stokes equations including chemistry

In this paper, we describe recent developments in the design and implementation of Navier-Stokes solvers based on finite element discretization. The most important ingredients are residual driven a posteriori mesh refinement, fully coupled defect-correction iteration for linearization, and optimal multigrid preconditioning. These techniques were systematically developed for computing incompressible viscous flows in general domains. Recently they have been extended to compressible low-Mach flows involving chemical reactions. The potential of automatic mesh adaptation together with multilevel techniques is illustrated by several examples, (1) the accurate prediction of drag and lift coefficients, (2) the determination of CARS-signals of species concentration in flow reactors, (3) the computation of laminar flames. Received: 30 April 1999 / Accepted: 17 June 1999     

Fully iterative ILU preconditioning of the unsteady Navier–Stokes equations for GPGPU

In this work we investigate the numerical difficulties that arise in optimizing the efficiency of Newtonian fluids simulations on a massively parallel computing hardware like a GPU. In particular, we will concentrate on the resulting algebraic problem. We will present an approximate, fully-iterative, ILU preconditioner and we will show that this solution is very efficient on a GPU if compared with an intrinsic massively parallel preconditioner like the diagonal preconditioner, which indeed goes faster than more robust techniques, like ILU, despite their strong decrease in the number of iterations. We refer to GMRES as the iterative scheme used to solve the linear system. In particular, we will deal with the solution of incompressible flows with variable density and we will investigate the interplay between Reynolds and Atwood numbers. We will show that the numerical simulation at medium–high Reynolds numbers produces linear systems whose matrices can be reasonably preconditioned with the diagonal preconditioner, while at low Reynolds numbers the higher viscosity of the fluid makes the diagonal preconditioner ineffective in the solution time requested from GMRES and, decreasing the Reynolds number, unable to let GMRES converge at all. In this situation, we will show how an adequate iterative approach to the parallel solution of the triangular systems that result from the ILU preconditioning, turns out to be robust and efficient. We will show numerical results for variable-density fluids, discretized with the scheme described in Calgaro et al. (2008), in classical benchmarks and, in particular, in the well-known Rayleigh–Taylor instability.     

Topology optimization of steady and unsteady incompressible Navier–Stokes flows driven by body forces

This paper presents the topology optimization method for the steady and unsteady incompressible Navier–Stokes flows driven by body forces, which typically include the constant force (e.g. the gravity) and the centrifugal and Coriolis forces. In the topology optimization problem, the artificial friction force with design variable interpolated porosity is added into the Navier–Stokes equations as the conventional method, and the physical body forces in the Navier–Stokes equations are penalized using the power-law approach. The topology optimization problem is analyzed by the continuous adjoint method, and solved by the finite element method in conjunction with the gradient based approach. In the numerical examples, the topology optimization of the fluidic channel, mass distribution of the flow and local velocity control are presented for the flows driven by body forces. The numerical results demonstrate that the presented method achieves the topology optimization of the flows driven by body forces robustly.