We consider the inviscid limit for the two-dimensional Navier–Stokes equations in the class of integrable and bounded vorticity fields. It is expected that the difference between the Navier–Stokes and Euler velocity fields vanishes in $L^2$ with an order proportional to the square root of the viscosity constant $\nu $. Here, we provide an order $ (\nu /|\log \nu | )^{\frac 12\exp (-Ct)}$ bound, which slightly improves upon earlier results by Chemin.

The Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore, the convergence of the expansion and the range of its coefficients, as the force varies are investigated.

In this article, we study theoretically and numerically the interaction of a vortex induced by a rotating cylinder with a perpendicular plane. We show the existence of weak solutions to the swirling vortex models by using the Hopf extension method, and by an elegant contradiction argument, respectively. We demonstrate numerically that the model could produce phenomena of swirling vortex including boundary layer pumping and two-celled vortex that are observed in potential line vortex interacting with a plane and in a tornado.

In this paper, we present some efficient numerical schemes to solve a two-phase hydrodynamics coupled phase field model with moving contact line boundary conditions. The model is a nonlinear coupling system, which consists the Navier-Stokes equations with the general Navier Boundary conditions or degenerated Navier Boundary conditions, and the Allen-Cahn type phase field equations with dynamical contact line boundary condition or static contact line boundary condition. The proposed schemes are linear and unconditionally energy stable, where the energy stabilities are proved rigorously. Various numerical tests are performed to show the accuracy and efficiency thereafter.

A comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flows to DNS of turbulent flows, are presented to assess the performance of these schemes. Numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.

This paper deals with an analytical solution of an oscillatory flow in a channel filled with a porous medium saturated with a viscous fluid. The consideration of porosity in the channel is the basic idea of the paper. The oscillatory waves in the channel with porous medium are produced due to self-excited pressure disturbances caused by inevitable fluctuation in a suction rate at the porous walls. The ensuing steady axial velocity and the time dependent oscillatory axial velocity are found analytically using perturbation method and WKB approximation. The important physical quantities like the velocity profile, amplitude of the oscillation and penetration depth of the oscillatory velocity have been given special emphasis in this analysis. The effects of porosity of the medium on these quantities are calculated analytically and examined graphically. We find that the amplitude of oscillatory velocity and the penetration depth of the oscillatory axial velocity decrease with increasing values of inverse Darcy parameter. The oscillations in the fluid can be minimized by decreasing the permeability of the medium.

A general analysis of the hydrodynamic limit of multi-relaxation time lattice Boltzmann models is presented. We examine multi-relaxation time BGK collision operators that are constructed similarly to those for the MRT case, however, without explicitly moving into a moment space representation. The corresponding ‘moments’ are derived as left eigenvectors of said collision operator in velocity space. Consequently we can, in a representation independent of the chosen base velocity set, generate the conservation equations. We find a significant degree of freedom in the choice of the collision matrix and the associated basis which leaves the collision operator invariant. We explain why MRT implementations in the literature reproduce identical hydrodynamics despite being based on different orthogonalization relations. More importantly, however, we outline a minimal set of requirements on the moment base necessary to maintain the validity of the hydrodynamic equations. This is particularly useful in the context of position and time-dependent moments such as those used in the context of peculiar velocities and some implementations of fluctuations in a lattice-Boltzmann simulation.

The Onera elsA CFD software is both a software package capitalizing theinnovative results of research over time and a multi-purpose tool for applied CFD andmulti-physics. The research input from Onera and other laboratories and the feedback fromaeronautical industry users allow enhancement of its capabilities and continuousimprovement. The paper presents recent accomplishments of varying complexity from researchand industry for a wide range of aerospace applications: aircraft, helicopters,turbomachinery...

In this paper, we review the recent development of phase-field models and their numerical methods for multi-component fluid flows with interfacial phenomena. The models consist of a Navier-Stokes system coupled with a multi-component Cahn-Hilliard system through a phase-field dependent surface tension force, variable density and viscosity, and the advection term. The classical infinitely thin boundary of separation between two immiscible fluids is replaced by a transition region of a small but finite width, across which the composition of the mixture changes continuously. A constant level set of the phase-field is used to capture the interface between two immiscible fluids. Phase-field methods are capable of computing topological changes such as splitting and merging, and thus have been applied successfully to multi-component fluid flows involving large interface deformations. Practical applications are provided to illustrate the usefulness of using a phase-field method. Computational results of various experiments show the accuracy and effectiveness of phase-field models.

We couple different flow models, i.e. a finite element solver for the Navier-Stokes equations and a Lattice Boltzmann automaton, using the framework Peano as a common base. The new coupling strategy between the meso- and macroscopic solver is presented and validated in a 2D channel flow scenario. The results are in good agreement with theory and results obtained in similar works by Latt et al. In addition, the test scenarios show an improved stability of the coupled method compared to pure Lattice Boltzmann simulations.

For robust discretizations of the Navier-Stokes equations with small viscosity, standardGalerkin schemes have to be augmented by stabilization terms due to the indefiniteconvective terms and due to a possible lost of a discrete inf-sup condition. For optimalcontrol problems for fluids such stabilization have in general an undesired effect in thesense that optimization and discretization do not commute. This is the case for thecombination of streamline upwind Petrov-Galerkin (SUPG) and pressure stabilizedPetrov-Galerkin (PSPG). In this work we study the effect of different stabilized finiteelement methods to distributed control problems governed by singular perturbed Oseenequations. In particular, we address the question whether a possible commutation error inoptimal control problems lead to a decline of convergence order. Therefore, we givea priori estimates for SUPG/PSPG. In a numerical study for a flow withboundary layers, we illustrate to which extend the commutation error affects theaccuracy.

An exploratory study is performed to investigate the use of a time-dependent discreteadjoint methodology for design optimization of a high-lift wing configuration augmentedwith an active flow control system. The location and blowing parameters associated with aseries of jet actuation orifices are used as design variables. In addition, a geometricparameterization scheme is developed to provide a compact set of design variablesdescribing the wing shape. The scaling of the implementation is studied using severalthousand processors and it is found that asynchronous file operations can greatly improvethe overall performance of the approach in such massively parallel environments. Threedesign examples are presented which seek to maximize the mean value of the liftcoefficient for the coupled system, and results demonstrate improvements as high as 27%relative to the lift obtained with non-optimized actuation. This lift gain is more thanthree times the incremental lift provided by the non-optimized actuation.