A postprocessing mixed finite element method for the Navier–Stokes equations

A fully discrete postprocessing mixed finite element scheme is considered for solving the time-dependent Navier–Stokes equations. In the PP method, we only consider a non-linear equation in the coarse-level subspace and a linear problem in the fine-level subspace. The analysis shows that the PP scheme can reach the same accuracy as the standard Galerkin method with a very fine mesh size h by an appropriate choice of H. Numerical examples are provided that confirm both the theoretical analysis and the corresponding improvement in computational efficiency.     

A new immersed boundary method for compressible Navier–Stokes equations

This paper presents an immersed boundary method for compressible Navier–Stokes equations in irregular domains, based on a local radial basis function approximation. This approach allows one to define a reconstruction of the radial basis functions on each irregular interface cell to treat both the Dirichlet and Neumann boundary conditions accurately on the immersed interfaces. Several numerical examples, including problems with available analytical solutions and the well-documented flow past an airfoil, are presented to test the proposed method. The numerical results demonstrate that the proposed method provides accurate solutions for viscous compressible flows.     

A fictitious domain finite element method for simulations of fluid–structure interactions: The Navier–Stokes equations coupled with a moving solid

The paper extends a stabilized fictitious domain finite element method initially developed for the Stokes problem to the incompressible Navier–Stokes equations coupled with a moving solid. This method presents the advantage to predict an optimal approximation of the normal stress tensor at the interface. The dynamics of the solid is governed by Newton׳s laws and the interface between the fluid and the structure is materialized by a level-set which cuts the elements of the mesh. An algorithm is proposed in order to treat the time evolution of the geometry and numerical results are presented on a classical benchmark of the motion of a disk falling in a channel.     

Higher order spectral/hp finite element models of the Navier–Stokes equations

In this paper, we present higher order least-squares finite element formulations for viscous, incompressible, isothermal Navier–Stokes equations using spectral/hp basis functions. The second-order Navier–Stokes equations are recast as first-order system of equations using stresses as auxiliary variables. Both steady-state and transient problems are considered. For a better coupling of pressure and velocity, especially in transient flows, an iterative penalisation strategy is employed. The outflow-type boundary conditions are applied in a weak sense through the least-squares functional. The formulation is verified by solving various benchmark problems like the lid-driven cavity, backward-facing step and flow over cylinder problems using direct serial solver UMFPACK.     

A combination of parabolized Navier–Stokes equations and level-set method for stratified two-phase internal flow

A novel methodology is proposed for the numerical computation of pressure-driven gravity-stratified flows along channels comprising two immiscible phases. The parabolized Navier–Stokes equations are combined with the level set approach, resulting into a downstream-marching problem in which the solution is computed at each cross-section based on upstream information only. A main difficulty in the implementation of the approach for internal flows is the conservation of the mass flow rates, which is addressed by extending to two-phase flows the method proposed by Patankar and Spalding (1972) and Raythby and Schneider (1979), and by adding an explicit forcing term in the equation for the advection of the level function. The combination of high-order finite differences and sparse storage and algebra used here allows a fully-coupled integration of the parabolized equations, as opposed to the more classical segregated approaches. This enables a very efficient calculation of the complete downstream-developing flow field.     

A spectral solver for the Navier–Stokes equations in the velocity–vorticity formulation

A numerical algorithm intended for the study of flows in a cylindrical container under laminar flow conditions is proposed. High resolution of the flow field, governed by the Navier–Stokes equations in velocity–vorticity formulation relative to a cylindrical frame of reference, is achieved through spatial discretisation by means of the spectral method. This method is based on a Fourier expansion in the azimuthal direction and an expansion in Chebyshev polynomials in the (nonperiodic) radial and axial directions. Several regularity constraints are used to take care of the coordinate singularity. These constraints are implemented, together with the boundary conditions at the top, bottom and mantle of the cylinder, via the tau method. The a priori unknown boundary values of the vorticity are evaluated by means of the influence-matrix technique. The compatibility between the mathematical and numerical formulation of the Navier–Stokes equations is established through a tau-correction procedure. The resolved flow field exhibits high-precision satisfaction of the incompressibility constraints for velocity and vorticity and the definition of the vorticity. The performance of the solver is illustrated by resolution of several configurations representative of generic three-dimensional laminar flows.     

A Newton iterative mixed finite element method for stationary conduction–convection problems

In this article, a Newton iterative mixed finite element method is presented for solving the stationary conduction–convection problems in two dimensions. The stability and the errors generated by both partitioning the space and solving nonlinear equations are analysed, which show that our method is stable and has good precision. Finally, some numerical experiments are given to confirm its effect.     

Viscous boundary layers for the Navier–Stokes equations with the Navier slip conditions

We tackle the issue of the inviscid limit of the incompressible Navier–Stokes equations when the Navier slip-with-friction conditions are prescribed on impermeable boundaries. We justify an asymptotic expansion which involves a weak amplitude boundary layer, with the same thickness as in Prandtl’s theory and a linear behavior. This analysis holds for general regular domains, in both dimensions two and three.     

High-order ALE method for the Navier–Stokes equations on a moving hybrid unstructured mesh using flux reconstruction method

The flux reconstruction (FR) formulation can unify several popular discontinuous basis high-order methods for fluid dynamics, including the discontinuous Galerkin method, in a simple, efficient form. An arbitrary Lagrangian–Eulerian (ALE) extension to the high-order FR scheme is developed here for moving mesh fluid flow problems. The ALE Navier–Stokes equations are derived by introducing a grid velocity. The conservation law are spatially discretised on hybrid unstructured meshes using Huynh’s scheme (Huynh 2007) on anisotropic elements (quadrilaterals) and using Correction Procedure via Reconstruction scheme on isotropic elements (triangles). The temporal discretisation uses both explicit and implicit treatments. The mesh movement is described by node positions given as a time series, instead of an analytical formula. The geometric conservation law is tested using free stream preservation problem. An isentropic vortex propagation test case is performed to show the high-order accuracy of the developed method on both moving and fixed hybrid meshes. Flow around an oscillating cylinder shows the capability of the method to solve moving boundary viscous flow problems, with the numeric method further verified by comparison of the result on a smoothly deforming mesh and a rigid moving mesh.     

A Variational Approach for the Navier–Stokes System

We introduce a variational approach to treat the regularity of the Navier–Stokes equations both in dimensions 2 and 3. Though the method allows the full treatment in dimension 2, we seek to precisely stress where it breaks down for dimension 3. The basic feature of the procedure is to look directly for strong solutions, by minimizing a suitable error functional that measures the departure of feasible fields from being a solution of the problem. By considering the divergence-free property as part of feasibility, we are able to avoid the explicit analysis of the pressure. Two main points in our analysis are:

A gas-kinetic theory based multidimensional high-order method for the compressible Navier–Stokes solutions

This paper presents a gas-kinetic theory based multidimensional high-order method for the compressible Naiver–Stokes solutions. In our previous study, a spatially and temporally dependent third-order flux scheme with the use of a third-order gas distribution function is employed.However, the third-order flux scheme is quite complicated and less robust than the second-order scheme. In order to reduce its complexity and improve its robustness, the secondorder flux scheme is adopted instead in this paper, while the temporal order of method is maintained by using a two stage temporal discretization. In addition, its CPU cost is relatively lower than the previous scheme. Several test cases in two and three dimensions, containing high Mach number compressible flows and low speed high Reynolds number laminar flows, are presented to demonstrate the method capacity.     

HLLC scheme for the preconditioned pseudo-compressibility Navier–Stokes equations for incompressible viscous flows

A new HLLC (Harten-Lax-van leer contact) approximate Riemann solver with the preconditioning technique based on the pseudo-compressibility formulation for numerical simulation of the incompressible viscous flows has been proposed, which follows the HLLC Riemann solver (Harten, Lax and van Leer solver with contact resolution modified by Toro) for the compressible flow system. In the authors' previous work, the preconditioned Roe's Riemann solver is applied to the finite difference discretisation of the inviscid flux for incompressible flows. Although the Roe's Riemann solver is found to be an accurate and robust scheme in various numerical computations, the HLLC Riemann solver is more suitable for the pseudo-compressible Navier--Stokes equations, in which the inviscid flux vector is a non-homogeneous function of degree one of the flow field vector, and however the Roe's solver is restricted to the homogeneous systems. Numerical investigations have been performed in order to demonstrate the efficiency and accuracy of the present procedure in both two- and three-dimensional cases. The present results are found to be in good agreement with the exact solutions, existing numerical results and experimental data.     

Functional and generalized separable solutions to unsteady Navier–Stokes equations

The paper studies unsteady Navier–Stokes equations with two space variables. It shows that the non-linear fourth-order equation for the stream function with three independent variables admits functional separable solutions described by a system of three partial differential equations with two independent variables. The system is found to have a number of exact solutions, which generate new classes of exact solutions to the Navier–Stokes equations. All these solutions involve two or more arbitrary functions of a single argument as well as a few free parameters. Many of the solutions are expressed in terms of elementary functions, provided that the arbitrary functions are also elementary; such solutions, having relatively simple form and presenting significant arbitrariness, can be especially useful for solving certain model problems and testing numerical and approximate analytical hydrodynamic methods. The paper uses the obtained results to describe some model unsteady flows of viscous incompressible fluids, including flows through a strip with permeable walls, flows through a strip with extrusion at the boundaries, flows onto a shrinking plane, and others. Some blow-up modes, which correspond to singular solutions, are discussed.     

Global convergence strategies for a spectral-element space-time discontinuous-Galerkin discretization of the Navier Stokes–equations

ABSTRACT

This paper presents two global convergence strategies for a spectral-element, space-time discretisation of the Navier–Stokes equations. The first employs a hierarchical temporal mesh subdivision and polynomial order reduction to approximate the high-order solution. The second generalises Pseudo-Transient Continuation for steady problems to a space-time system.     

An efficient parallel high-order compact scheme for the 3D incompressible Navier–Stokes equations

This article provides a strategy for solving incompressible turbulent flows, which combines compact finite difference schemes and parallel computing. The numerical features of this solver are the semi-implicit time advancement, the staggered arrangement of the variables and the fourth-order compact scheme discretisation. This is the usual way for solving accurately turbulent incompressible flows. We propose a new strategy for solving the Helmholtz/Poisson equations based on a parallel 2d-pencil decomposition of the diagonalisation method. The compact scheme derivatives are computed with the parallel diagonal dominant (PDD) algorithm, which achieves good parallel performances by introducing a bounded numerical error. We provide a new analysis of its effect on the numerical accuracy and conservation features. Several numerical experiments, including two simulations of turbulent flows, demonstrate that the PDD algorithm maintains the accuracy and conservation features, while conserving a good parallel performance, up to 4096 cores.     

Navier–Stokes Equations with Navier Boundary Conditions for an Oceanic Model

We consider the Navier–Stokes equations in a thin domain of which the top and bottom surfaces are not flat. The velocity fields are subject to the Navier conditions on those boundaries and the periodicity condition on the other sides of the domain. This toy model arises from studies of climate and oceanic flows. We show that the strong solutions exist for all time provided the initial data belong to a “large” set in the Sobolev space H ¹. Furthermore we show, for both the autonomous and the nonautonomous problems, the existence of a global attractor for the class of all strong solutions. This attractor is proved to be also the global attractor for the Leray–Hopf weak solutions of the Navier–Stokes equations. One issue that arises here is a nontrivial contribution due to the boundary terms. We show how the boundary conditions imposed on the velocity fields affect the estimates of the Stokes operator and the (nonlinear) inertial term in the Navier–Stokes equations. This results in a new estimate of the trilinear term, which in turn permits a short and simple proof of the existence of strong solutions for all time.