Stabilization and scalable block preconditioning for the Navier–Stokes equations

This study compares several block-oriented preconditioners for the stabilized finite element discretization of the incompressible Navier–Stokes equations. This includes standard additive Schwarz domain decomposition methods, aggressive coarsening multigrid, and three preconditioners based on an approximate block LU factorization, specifically SIMPLEC, LSC, and PCD. Robustness is considered with a particular focus on the impact that different stabilization methods have on preconditioner performance. Additionally, parallel scaling studies are undertaken. The numerical results indicate that aggressive coarsening multigrid, LSC and PCD all have good algorithmic scalability. Coupling this with the fact that block methods can be applied to systems arising from stable mixed discretizations implies that these techniques are a promising direction for developing scalable methods for Navier–Stokes.     

Immersed-boundary methods for general finite-difference and finite-volume Navier–Stokes solvers

We present an immersed-boundary algorithm for incompressible flows with complex boundaries, suitable for Cartesian or curvilinear grid system. The key stages of any immersed-boundary technique are the interpolation of a velocity field given on a mesh onto a general boundary (a line in 2D, a surface in 3D), and the spreading of a force field from the immersed boundary to the neighboring mesh points, to enforce the desired boundary conditions on the immersed-boundary points. We propose a technique that uses the Reproducing Kernel Particle Method [W.K. Liu, S. Jun, Y.F. Zhang, Reproducing kernel particle methods, Int. J. Numer. Methods Fluids 20(8) (1995) 1081–1106] for the interpolation and spreading. Unlike other methods presented in the literature, the one proposed here has the property that the integrals of the force field and of its moment on the grid are conserved, independent of the grid topology (uniform or non-uniform, Cartesian or curvilinear). The technique is easy to implement, and is able to maintain the order of the original underlying spatial discretization. Applications to two- and three-dimensional flows in Cartesian and non-Cartesian grid system, with uniform and non-uniform meshes are presented.     

Stable and accurate schemes for the compressible Navier–Stokes equations

Minimal stencil width discretizations of combined mixed and non-mixed second-order derivatives are analyzed with respect to accuracy and stability. We show that these discretizations lead to stability for Cauchy problems. With a careful boundary treatment, we also show that the stability holds for initial-boundary value problems. The analysis is verified by numerical simulations of Burgers’ and Navier–Stokes equations in two and three space dimensions.     

Algebraic Splitting for Incompressible Navier–Stokes Equations

Fully discretized incompressible Navier–Stokes equations are solved by splitting the algebraic system with an approximate factorization. This splitting affects the temporal convergence order of velocity and pressure. The splitting error is proportional to the pressure variable, and a simple analysis shows that the original convergence order of the time-integration scheme can be retained by solving for incremental pressure. The combination of splitting and incremental pressure is shown to be equivalent to an error-correcting method using the full pressure. In numerical experiments employing a third-order time-integration scheme and various orders for the pressure increment, the splitting error is shown to control the convergence order, and the full order of the scheme is recaptured for both velocity and pressure. The difference between perturbing the momentum or the continuity equation is also explored.     

An Asymptotic-Preserving all-speed scheme for the Euler and Navier–Stokes equations

We present an Asymptotic-Preserving ‘all-speed’ scheme for the simulation of compressible flows valid at all Mach-numbers ranging from very small to order unity. The scheme is based on a semi-implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly. This discretization, which is the key to the Asymptotic-Preserving property, provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incompressible regime. The divergence-free condition on the velocity in the incompressible regime is respected, and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations. The implicit treatment of the acoustic part allows the time-step to be independent of the Mach number. The scheme is conservative and applies to steady or unsteady flows and to general equations of state. One and two-dimensional numerical results provide a validation of the Asymptotic-Preserving ‘all-speed’ properties.     

Resistance law for a turbulent Taylor–Couette flow at very large Taylor numbers

Based on the semi-empirical model of the transport of the specific rate of turbulence energy dissipation, it has been concluded that the resistance laws are observed for a turbulent Taylor–Couette flow between independently rotating coaxial cylinders for very large Taylor numbers.     

Velocity–vorticity–helicity formulation and a solver for the Navier–Stokes equations

For the three-dimensional incompressible Navier–Stokes equations, we present a formulation featuring velocity, vorticity and helical density as independent variables. We find the helical density can be observed as a Lagrange multiplier corresponding to the divergence-free constraint on the vorticity variable, similar to the pressure in the case of the incompressibility condition for velocity. As one possible practical application of this new formulation, we consider a time-splitting numerical scheme based on an alternating procedure between vorticity–helical density and velocity–Bernoulli pressure systems of equations. Results of numerical experiments include a comparison with some well-known schemes based on pressure–velocity formulation and illustrate the competitiveness on the new scheme as well as the soundness of the new formulation.     

Gibbsian Dynamics and Ergodicity¶for the Stochastically Forced Navier–Stokes Equation

We study stationary measures for the two-dimensional Navier–Stokes equation with periodic boundary condition and random forcing. We prove uniqueness of the stationary measure under the condition that all “determining modes” are forced. The main idea behind the proof is to study the Gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time-history of the low modes. Received: 21 November 2000 / Accepted: 9 December 2000     

Improvements on open and traction boundary conditions for Navier–Stokes time-splitting methods

We present in this paper a numerical scheme for incompressible Navier–Stokes equations with open and traction boundary conditions, in the framework of pressure-correction methods. A new way to enforce this type of boundary condition is proposed and provides higher pressure and velocity convergence rates in space and time than found in the present state of the art. We illustrate this result by computing some numerical and physical tests. In particular, we establish reference solutions of a laminar flow in a geometry where a bifurcation takes place and of the unsteady flow around a square cylinder.     

Liouville-Type Theorems for the Forced Euler Equations and the Navier–Stokes Equations

In this paper we study the Liouville-type properties for solutions to the steady incompressible Euler equations with forces in ${\mathbb {R}^N}$ . If we assume “single signedness condition” on the force, then we can show that a ${C^1 (\mathbb {R}^N)}$ solution (v, p) with ${|v|^2+ |p| \in L^{\frac{q}{2}}(\mathbb {R}^N),\,q \in (\frac{3N}{N-1}, \infty)}$ is trivial, v = 0. For the solution of the steady Navier–Stokes equations, satisfying ${v(x) \to 0}$ as ${|x| \to \infty}$ , the condition ${\int_{\mathbb {R}^3} |\Delta v|^{\frac{6}{5}} dx < \infty}$ , which is stronger than the important D-condition, ${\int_{\mathbb {R}^3} |\nabla v|^2 dx < \infty}$ , but both having the same scaling property, implies that v = 0. In the appendix we reprove Theorem 1.1 (Chae, Commun Math Phys 273:203–215, 2007), using the self-similar Euler equations directly.     

Navier–Stokes Limit for a Thermal Stochastic Lattice Gas

We study a stochastic particle system on the lattice whose particles move freely according to a simple exclusion process and change velocities during collisions preserving energy and momentum. In the hydrodynamic limit, under diffusive space-time scaling, the local velocity field u satisfies the incompressible Navier–Stokes equation, while the temperature field solves the heat equation with drift u. The results are also extended to include a suitably resealed external force.     

Slip Boundary Conditions for the Compressible Navier–Stokes Equations

The slip boundary conditions for the compressible Navier–Stokes equations are derived systematically from the Boltzmann equation on the basis of the Chapman–Enskog solution of the Boltzmann equation and the analysis of the Knudsen layer adjacent to the boundary. The resulting formulas of the slip boundary conditions are summarized with explicit values of the slip coefficients for hard-sphere molecules as well as the Bhatnagar–Gross–Krook model. These formulas, which can be applied to specific problems immediately, help to prevent the use of often used slip boundary conditions that are either incorrect or without theoretical basis.     

Non-Formation of Vacuum States for Compressible Navier–Stokes Equations

We prove that weak solutions of the Navier–Stokes equations for compressible fluid flow in one space dimension do not exhibit vacuum states, provided that no vacuum states are present initially. The solutions and external forces that we consider are quite general: the essential requirements are that the mass and energy densities of the fluid be locally integrable at each time, and that the L ² _loc-norm of the velocity gradient be locally integrable in time. Our analysis shows that, if a vacuum state were to occur, the viscous force would impose an impulse of infinite magnitude on the adjacent fluid, thus violating the hypothesis that the momentum remains locally finite. Received: 20 March 2000 / Accepted: 16 July 2000     

Probabilistic Estimates for the Two-Dimensional Stochastic Navier–Stokes Equations

We consider the Navier–Stokes equation on a two-dimensional torus with a random force, white noise in time, and analytic in space, for arbitrary Reynolds number R. We prove probabilistic estimates for the long-time behavior of the solutions that imply bounds for the dissipation scale and energy spectrum as R.     

Spectrally-accurate algorithm for moving boundary problems for the Navier–Stokes equations

A spectral algorithm based on the immersed boundary conditions (IBC) concept is developed for simulations of viscous flows with moving boundaries. The algorithm uses a fixed computational domain with flow domain immersed inside the computational domain. Boundary conditions along the edges of the time-dependent flow domain enter the algorithm in the form of internal constraints. Spectral spatial discretization uses Fourier expansions in the stream-wise direction and Chebyshev expansions in the normal-to-the-wall direction. Up to fourth-order implicit temporal discretization methods have been implemented. It has been demonstrated that the algorithm delivers the theoretically predicted accuracy in both time and space. Performances of various linear solvers employed in the solution process have been evaluated and a new class of solver that takes advantage of the structure of the coefficient matrix has been proposed. The new solver results in a significant acceleration of computations as well as in a substantial reduction in memory requirements.     

Cosmological and Quantum Solutions of the Navier–Stokes Equations

Russian Physics Journal - It is shown that the vector Navier–Stokes equation has a variety of quantum solutions, so the scope of this equation is not limited to the field of classical...     

Blow-up of Critical Besov Norms at a Potential Navier–Stokes Singularity

We prove that if an initial datum to the incompressible Navier–Stokes equations in any critical Besov space \({\dot B^{-1+\frac 3p}_{p,q}({\mathbb {R}}^{3})}\), with \({3 < p, q < \infty}\), gives rise to a strong solution with a singularity at a finite time \({T > 0}\), then the norm of the solution in that Besov space becomes unbounded at time T. This result, which treats all critical Besov spaces where local existence is known, generalizes the result of Escauriaza et al. (Uspekhi Mat Nauk 58(2(350)):3–44, 2003) concerning suitable weak solutions blowing up in \({L^{3}({\mathbb R}^{3})}\). Our proof uses profile decompositions and is based on our previous work (Gallagher et al., Math. Ann. 355(4):1527–1559, 2013), which provided an alternative proof of the \({L^{3}({\mathbb R}^{3})}\) result. For very large values of p, an iterative method, which may be of independent interest, enables us to use some techniques from the \({L^{3}({\mathbb R}^{3})}\) setting.     

The Incompressible Navier–Stokes for the Nonlinear Discrete Velocity Models

Abstract

We establish the incompressible Navier–Stokes limit for the discrete velocity model of the Boltzmann equation in any dimension of the physical space, for densities which remain in a suitable small neighborhood of the global Maxwellian. Appropriately scaled families solutions of discrete Boltzmann equation are shown to have fluctuations that locally in time converge strongly to a limit governed by a solution of Incompressible Navier–Stokes provided that the initial fluctuation is smooth, and converges to appropriate initial data. As applications of our results, we study the Carleman model and the one-dimensional Broadwell model.