A Residual-Based Compact Scheme for the Compressible Navier–Stokes Equations

A simple and efficient time-dependent method is presented for solving the steady compressible Euler and Navier–Stokes equations with third-order accuracy. Owing to its residual-based structure, the numerical scheme is compact without requiring any linear algebra, and it uses a simple numerical dissipation built on the residual. The method contains no tuning parameter. Accuracy and efficiency are demonstrated for 2-D inviscid and viscous model problems. Navier–Stokes calculations are presented for a shock/boundary layer interaction, a separated laminar flow, and a transonic turbulent flow over an airfoil.     

A particle–particle hybrid method for kinetic and continuum equations

We present a coupling procedure for two different types of particle methods for the Boltzmann and the Navier–Stokes equations. A variant of the DSMC method is applied to simulate the Boltzmann equation, whereas a meshfree Lagrangian particle method, similar to the SPH method, is used for simulations of the Navier–Stokes equations. An automatic domain decomposition approach is used with the help of a continuum breakdown criterion. We apply adaptive spatial and time meshes. The classical Sod’s 1D shock tube problem is solved for a large range of Knudsen numbers. Results from Boltzmann, Navier–Stokes and hybrid solvers are compared. The CPU time for the hybrid solver is 3–4 times faster than for the Boltzmann solver.     

Simulating 2D Flows with Viscous Vortex Dynamics

Approximate solutions of the two-dimensional Navier–Stokes equation can be constructed as a superposition of viscous Lamb vortices. Requiring minimum deviation from the Navier–Stokes equation, one gets a set of ordinary differential equations for the positions, strength and width of the vortices. We calculate the deviation of the solution from the Navier–Stokes equation in the square norm. The time dependence of this error is determined and discussed.     

The Small Scales of the Stochastic Navier–Stokes Equations Under Rough Forcing

We prove that the small scale structures of the stochastically forced Navier–Stokes equations approach those of the naturally associated Ornstein–Uhlenbeck process as the scales get smaller. Precisely, we prove that the rescaled kth spatial Fourier mode converges weakly on path space to an associated Ornstein–Uhlenbeck process as |k| . In addition, we prove that the Navier–Stokes equations and the naturally associated Ornstein–Uhlenbeck process induce equivalent transition densities if the viscosity is replaced with hyper-viscosity. This gives a simple proof of unique ergodicity for the hyperviscous Navier–Stokes system. We show how different strengthened hyperviscosity produce varying levels of equivalence.     

Odd Viscosity

When time reversal is broken, the viscosity tensor can have a nonvanishing odd part. In two dimensions, and only then, such odd viscosity is compatible with isotropy. Elementary and basic features of odd viscosity are examined by considering solutions of the wave and Navier–Stokes equations for hypothetical fluids where the stress is dominated by odd viscosity.     

Uniform Finite Element Error Estimates with Power-Type Asymptotic Constants for Unsteady Navier–Stokes Equations

Uniform error estimates with power-type asymptotic constants of the finite element method for the unsteady Navier–Stokes equations are deduced in this paper. By introducing an iterative scheme and studying its convergence, we firstly derive that the solution of the Navier–Stokes equations is bounded by power-type constants, where we avoid applying the Gronwall lemma, which generates exponential-type factors. Then, the technique is extended to the error estimate of the long-time finite element approximation. The analyses show that, under some assumptions on the given data, the asymptotic constants in the finite element error estimates for the unsteady Navier–Stokes equations are uniformly power functions with respect to the initial data, the viscosity, and the body force for all time t>0. Finally, some numerical examples are shown to verify the theoretical predictions.     

On Backtracking Failure in Newton–GMRES Methods with a Demonstration for the Navier–Stokes Equations

In an earlier study of inexact Newton methods, we pointed out that certain counterintuitive behavior may occur when applying residual backtracking to the Navier–Stokes equations with heat and mass transport. Specifically, it was observed that a Newton–GMRES method globalized by backtracking (linesearch, damping) may be less robust when high accuracy is required of each linear solve in the Newton sequence than when less accuracy is required. In this brief discussion, we offer a possible explanation for this phenomenon, together with an illustrative numerical experiment involving the Navier–Stokes equations.     

h-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Being implicit in time, the space-time discontinuous Galerkin discretization of the compressible Navier–Stokes equations requires the solution of a non-linear system of algebraic equations at each time-step. The overall performance, therefore, highly depends on the efficiency of the solver. In this article, we solve the system of algebraic equations with a h-multigrid method using explicit Runge–Kutta relaxation. Two-level Fourier analysis of this method for the scalar advection–diffusion equation shows convergence factors between 0.5 and 0.75. This motivates its application to the 3D compressible Navier–Stokes equations where numerical experiments show that the computational effort is significantly reduced, up to a factor 10 w.r.t. single-grid iterations.     

Spectral/hp least-squares finite element formulation for the Navier–Stokes equations

We consider the application of least-squares finite element models combined with spectral/hp methods for the numerical solution of viscous flow problems. The paper presents the formulation, validation, and application of a spectral/hp algorithm to the numerical solution of the Navier–Stokes equations governing two- and three-dimensional stationary incompressible and low-speed compressible flows. The Navier–Stokes equations are expressed as an equivalent set of first-order equations by introducing vorticity or velocity gradients as additional independent variables and the least-squares method is used to develop the finite element model. High-order element expansions are used to construct the discrete model. The discrete model thus obtained is linearized by Newton’s method, resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method. Spectral convergence of the L₂ least-squares functional and L₂ error norms is verified using smooth solutions to the two-dimensional stationary Poisson and incompressible Navier–Stokes equations. Numerical results for flow over a backward-facing step, steady flow past a circular cylinder, three-dimensional lid-driven cavity flow, and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation.     

Superdiffusivity of Two Dimensional Lattice Gas Models

It was proved [Navier–Stokes Equations for Stochastic Particle System on the Lattice. Comm. Math. Phys. (1996) 182, 395; Lattice gases, large deviations, and the incompressible Navier–Stokes equations. Ann. Math. (1998) 148, 51] that stochastic lattice gas dynamics converge to the Navier–Stokes equations in dimension d=3 in the incompressible limits. In particular, the viscosity is finite. We proved that, on the other hand, the viscosity for a two dimensional lattice gas model diverges faster than (log t)^1/2. Our argument indicates that the correct divergence rate is (log t)^1/2. This problem is closely related to the logarithmic correction of the time decay rate for the velocity auto-correlation function of a tagged particle.     

Topology optimization of unsteady incompressible Navier–Stokes flows

This paper discusses the topology optimization of unsteady incompressible Navier–Stokes flows. An optimization problem is formulated by adding the artificial Darcy frictional force into the incompressible Navier–Stokes equations. The optimization procedure is implemented using the continuous adjoint method and the finite element method. The effects of dynamic inflow, Reynolds number and target flux on specified boundaries for the optimal topology of unsteady Navier–Stokes flows are presented. Numerical examples demonstrate the feasibility and necessity of this topology optimization method for unsteady Navier–Stokes flows.     

Comparison of Kinetic Theory and Hydrodynamics for Poiseuille Flow

Comparison of particle (DSMC) simulation with the numerical solution of the Navier–Stokes (NS) equations for pressure-driven plane Poiseuille flow is presented and contrasted with that of the acceleration-driven Poiseuille flow. Although for the acceleration-driven case DSMC measurements are qualitatively different from the NS solution at relatively low Knudsen number, the two are in somewhat better agreement for pressure-driven flow.     

The incompressible Navier–Stokes equations from black hole membrane dynamics

We consider the dynamics of a d+1 space–time dimensional membrane defined by the event horizon of a black brane in (d+2)-dimensional asymptotically Anti-de Sitter space–time and show that it is described by the d-dimensional incompressible Navier–Stokes equations of non-relativistic fluids. The fluid velocity corresponds to the normal to the horizon while the rate of change in the fluid energy is equal to minus the rate of change in the horizon cross-sectional area. The analysis is performed in the Membrane Paradigm approach to black holes and it holds for a general non-singular null hypersurface, provided a large scale hydrodynamic limit exists. Thus we find, for instance, that the dynamics of the Rindler acceleration horizon is also described by the incompressible Navier–Stokes equations. The result resembles the relation between the Burgers and KPZ equations and we discuss its implications.     

Scaling Exponents for Active Scalars

We provide bounds for Dirichlet quotients and for generalized structure functions for 3D active scalars and Navier–Stokes equations. These bounds put constraints on the possible extent of anomalous scaling.     

Transition to Longitudinal Instability of Detonation Waves is Generically Associated with Hopf Bifurcation to Time-Periodic Galloping Solutions

We show that transition to longitudinal instability of strong detonation solutions of reactive compressible Navier–Stokes equations is generically associated with Hopf bifurcation to nearby time-periodic “galloping”, or “pulsating”, solutions, in agreement with physical and numerical observation. In the process, we determine readily numerically verifiable stability and bifurcation conditions in terms of an associated Evans function, and obtain the first complete nonlinear stability result for strong detonations of the reacting Navier–Stokes equations, in the limit as amplitude (hence also heat release) goes to zero. The analysis is by pointwise semigroup techniques introduced by the authors and collaborators in previous works.     

Coupling Biot and Navier–Stokes equations for modelling fluid–poroelastic media interaction

The interaction between a fluid and a poroelastic structure is a complex problem that couples the Navier–Stokes equations with the Biot system. The finite element approximation of this problem is involved due to the fact that both subproblems are indefinite. In this work, we first design residual-based stabilization techniques for the Biot system, motivated by the variational multiscale approach. Then, we state the monolithic Navier–Stokes/Biot system with the appropriate transmission conditions at the interface. For the solution of the coupled system, we adopt both monolithic solvers and heterogeneous domain decomposition strategies. Different domain decomposition methods are considered and their convergence is analyzed for a simplified problem. We compare the efficiency of all the methods on a test problem that exhibits a large added-mass effect, as it happens in hemodynamics applications.     

Comparing the first- and second-order theories of relativistic dissipative fluid dynamics using the 1+1 dimensional relativistic flux corrected transport algorithm

Focusing on the numerical aspects and accuracy we study a class of bulk viscosity driven expansion scenarios using the relativistic Navier–Stokes and truncated Israel–Stewart form of the equations of relativistic dissipative fluids in 1+1 dimensions. The numerical calculations of conservation and transport equations are performed using the numerical framework of flux corrected transport. We show that the results of the Israel–Stewart causal fluid dynamics are numerically much more stable and smoother than the results of the standard relativistic Navier–Stokes equations.     

The Hydrodynamic Limit of a Deterministic Particle System with Conservation of Mass and Momentum

We study the hydrodynamic limit of a deterministic one-dimensional particle system with nearest neighbour interaction and an additional regularizing force. Under its evolution mass and momentum are conserved. In the limit with Euler scaling their macroscopic distributions are shown to be governed by the compressible Navier–Stokes equations with a density dependent viscosity.     

Ergodicity for the Dissipative Boussinesq Equations with Random Forcing

We study the stationary measure for the two-dimensional Boussinesq equation with random forcing. We prove the ergodicity for the two-dimensional stochastically forced Boussinesq equation. We also study the Galerkin truncations of the three-dimensional Boussinesq equations under degenerate stochastic forcing. We follow closely the previous results on the stochastically forced Navier–Stokes equations.     

Absorbing boundary conditions for the Euler and Navier–Stokes equations with the spectral difference method

Two absorbing boundary conditions, the absorbing sponge zone and the perfectly matched layer, are developed and implemented for the spectral difference method discretizing the Euler and Navier–Stokes equations on unstructured grids. The performance of both boundary conditions is evaluated and compared with the characteristic boundary condition for a variety of benchmark problems including vortex and acoustic wave propagations. The applications of the perfectly matched layer technique in the numerical simulations of unsteady problems with complex geometries are also presented to demonstrate its capability.