Numerical simulation of shock wave propagation in microchannels using continuum and kinetic approaches

Numerical simulations of shock wave propagation in microchannels and microtubes (viscous shock tube problem) have been performed using three different approaches: the Navier–Stokes equations with the velocity slip and temperature jump boundary conditions, the statistical Direct Simulation Monte Carlo method for the Boltzmann equation, and the model kinetic Bhatnagar–Gross–Krook equation with the Shakhov equilibrium distribution function. Effects of flow rarefaction and dissipation are investigated and the results obtained with different approaches are compared. A parametric study of the problem for different Knudsen numbers and initial shock strengths is carried out using the Navier–Stokes computations.      

Asymptotic Behavior of Solutions to the Compressible Navier–Stokes Equation Around a Parallel Flow

The initial boundary value problem for the compressible Navier–Stokes equation is considered in an infinite layer of . It is proved that if the Reynolds and Mach numbers are sufficiently small, then strong solutions to the compressible Navier–Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations. The large time behavior of the solution is described by a solution of a one-dimensional viscous Burgers equation. The proof is given by a combination of spectral analysis of the linearized operator and a variant of the Matsumura–Nishida energy method.     

Large Eddy Simulations Using the Subgrid-Scale Estimation Model and Truncated Navier–Stokes Dynamics

We describe a procedure for large eddy simulations of turbulence which uses the subgrid-scale estimation model and truncated Navier–Stokes dynamics. In the procedure the large eddy simulation equations are advanced in time with the subgrid-scale stress tensor calculated from the parallel solution of the truncated Navier–Stokes equations on a mesh two times smaller in each Cartesian direction than the mesh employed for a discretization of the resolved quantities. The truncated Navier–Stokes equations are solved through a sequence of runs, each initialized using the subgrid-scale estimation model. The modeling procedure is evaluated by comparing results of large eddy simulations for isotropic turbulence and turbulent channel flow with the corresponding results of experiments, theory, direct numerical simulations, and other large eddy simulations. Subsequently, simplifications of the general procedure are discussed and evaluated. In particular, it is possible to formulate the procedure entirely in terms of the truncated Navier–Stokes equation and a periodic processing of the small-scale component of its solution. Received 27 April 2001 and accepted 16 December 2001     

Finite Element-Based Characterization of Pore-Scale Geometry and Its Impact on Fluid Flow

We present a finite element (FEM) simulation method for pore geometry fluid flow. Within the pore space, we solve the single-phase Reynold’s lubrication equation—a simplified form of the incompressible Navier–Stokes equation yielding the velocity field in a two-step solution approach. (1) Laplace’s equation is solved with homogeneous boundary conditions and a right-hand source term, (2) pore pressure is computed, and the velocity field obtained for no slip conditions at the grain boundaries. From the computed velocity field, we estimate the effective permeability of porous media samples characterized by section micrographs or micro-CT scans. This two-step process is much simpler than solving the full Navier–Stokes equation and, therefore, provides the opportunity to study pore geometries with hundreds of thousands of pores in a computationally more cost effective manner than solving the full Navier–Stokes’ equation. Given the realistic laminar flow field, dispersion in the medium can also be estimated. Our numerical model is verified with an analytical solution and validated on two 2D micro-CT scans from samples, the permeabilities, and porosities of which were pre-determined in laboratory experiments. Comparisons were also made with published experimental, approximate, and exact permeability data. With the future aim to simulate multiphase flow within the pore space, we also compute the radii and derive capillary pressure from the Young–Laplace’s equation. This permits the determination of model parameters for the classical Brooks–Corey and van-Genuchten models, so that relative permeabilities can be estimated.     

Quasi-Neutral Limit for a Model of Viscous Plasma

We perform a rigorous analysis of the quasi-neutral limit for a model of viscous plasma represented by the Navier–Stokes–Poisson system of equations. It is shown that the limit problem is the Navier–Stokes system describing a barotropic fluid flow, with the pressure augmented by a component related to the nonlinearity in the original Poisson equation.     

On the stability and extension of reduced-order Galerkin models in incompressible flows

Proper orthogonal decomposition (POD) has been used to develop a reduced-order model of the hydrodynamic forces acting on a circular cylinder. Direct numerical simulations of the incompressible Navier–Stokes equations have been performed using a parallel computational fluid dynamics (CFD) code to simulate the flow past a circular cylinder. Snapshots of the velocity and pressure fields are used to calculate the divergence-free velocity and pressure modes, respectively. We use the dominant of these velocity POD modes (a small number of eigenfunctions or modes) in a Galerkin procedure to project the Navier–Stokes equations onto a low-dimensional space, thereby reducing the distributed-parameter problem into a finite-dimensional nonlinear dynamical system in time. The solution of the reduced dynamical system is a limit cycle corresponding to vortex shedding. We investigate the stability of the limit cycle by using long-time integration and propose to use a shooting technique to home on the system limit cycle. We obtain the pressure-Poisson equation by taking the divergence of the Navier–Stokes equation and then projecting it onto the pressure POD modes. The pressure is then decomposed into lift and drag components and compared with the CFD results.     

2D Navier–Stokes Equations in a Time Dependent Domain with Neumann Type Boundary Conditions

In this paper the two-dimensional Navier–Stokes system for incompressible fluid coupled with a parabolic equation through the Neumann type boundary condition for the second component of the velocity is considered. Navier–Stokes equations are defined on a given time dependent domain. We prove the existence of a weak solution for this system. In addition, we prove the continuous dependence of solutions on the data for a regularized version of this system. For a special case of this regularized system also a problem with an unknown interface is solved.     

Random Kick-Forced 3D Navier–Stokes Equations in a Thin Domain

We consider the Navier–Stokes equations in the thin 3D domain , where is a two-dimensional torus. The equation is perturbed by a non-degenerate random kick force. We establish that, firstly, when ε ≪ 1, the equation has a unique stationary measure and, secondly, after averaging in the thin direction this measure converges (as ε → 0) to a unique stationary measure for the Navier–Stokes equation on . Thus, the 2D Navier–Stokes equations on surfaces describe asymptotic in time, and limiting in ε, statistical properties of 3D solutions in thin 3D domains.     

Determining modes and Grashof number in 2D turbulence: a numerical case study

We study how the number of numerically determining modes in the Navier–Stokes equations depends on the Grashof number. Consider the two-dimensional incompressible Navier–Stokes equations in a periodic domain with a fixed time-independent forcing function. We increase the Grashof number by rescaling the forcing and observe through numerical computation that the number of numerically determining modes stabilizes at some finite value as the Grashof number increases. This unexpected result implies that our theoretical understanding of continuous data assimilation is incomplete until an analytic proof which makes use of the non-linear term in the Navier–Stokes equations is found.      

Polynomial Filtration Laws for Low Reynolds Number Flows Through Porous Media

In this study, we use the method of homogenization to develop a filtration law in porous media that includes the effects of inertia at finite Reynolds numbers. The result is much different than the empirically observed quadratic Forchheimer equation. First, the correction to Darcy’s law is initially cubic (not quadratic) for isotropic media. This is consistent with several other authors (Mei and Auriault, J Fluid Mech 222:647–663, 1991; Wodié and Levy, CR Acad Sci Paris t.312:157–161, 1991; Couland et al. J Fluid Mech 190:393–407, 1988; Rojas and Koplik, Phys Rev 58:4776–4782, 1988) who have solved the Navier–Stokes equations analytically and numerically. Second, the resulting filtration model is an infinite series polynomial in velocity, instead of a single corrective term to Darcy’s law. Although the model is only valid up to the local Reynolds number, at the most, of order 1, the findings are important from a fundamental perspective because it shows that the often-used quadratic Forchheimer equation is not a universal law for laminar flow, but rather an empirical one that is useful in a limited range of velocities. Moreover, as stated by Mei and Auriault (J Fluid Mech 222:647–663, 1991) and Barree and Conway (SPE Annual technical conference and exhibition, 2004), even if the quadratic model were valid at moderate Reynolds numbers in the laminar flow regime, then the permeability extrapolated on a Forchheimer plot would not be the intrinsic Darcy permeability. A major contribution of this study is that the coefficients of the polynomial law can be derived a priori, by solving sequential Stokes problems. In each case, the solution to the Stokes problem is used to calculate a coefficient in the polynomial, and the velocity field is an input of the forcing function, F, to subsequent problems. While numerical solutions must be utilized to compute each coefficient in the polynomial, these problems are much simpler and robust than solving the full Navier–Stokes equations.     

Asymptotic Stability of Combination of Viscous Contact Wave with Rarefaction Waves for One-Dimensional Compressible Navier–Stokes System

We are concerned with the large-time behavior of solutions of the Cauchy problem to the one-dimensional compressible Navier–Stokes system for ideal polytropic fluids, where the far field states are prescribed. When the corresponding Riemann problem for the compressible Euler system admits the solution consisting of contact discontinuity and rarefaction waves, it is proved that for the one-dimensional compressible Navier–Stokes system, the combination wave of a “viscous contact wave”, which corresponds to the contact discontinuity, with rarefaction waves is asymptotically stable, provided the strength of the combination wave is suitably small. This result is proved by using elementary energy methods.     

The Low Mach Number Limit for the Full Navier–Stokes–Fourier System

We study the low Mach number asymptotic limit for solutions to the full Navier–Stokes–Fourier system, supplemented with ill-prepared data and considered on an arbitrary time interval. Convergencetowards the incompressible Navier–Stokes equations is shown.     

The flow field near the triple point in steady shock reflection

This paper investigates the flow field near three intersecting shock waves appearing in steady Mach reflection. Results of numerical computations reveal a “von Neumann Paradox”—like feature for weak shock waves, in which the flow field between the reflected and the Mach stem is smooth with no distinct slip flow region and changes rather smoothly. An analytical solution of the Navier–Stokes equations constructed using a polar–coordinate system gives a flow field with the same properties as the numerical simulation.     

Global Weak Solutions to Equations of Motion for Magnetic Fluids

We study the differential system governing the flow of an incompressible ferrofluid under the action of a magnetic field. The system consists of the Navier–Stokes equations, the angular momentum equation, the magnetization equation, and the magnetostatic equations. We prove, by using the Galerkin method, a global in time existence of weak solutions with finite energy of an initial boundary-value problem and establish the long-time behavior of such solutions. The main difficulty is due to the singularity of the gradient magnetic force.      

Optimal Results for the Nonhomogeneous Initial-Boundary Value Problem for the Two-Dimensional Navier–Stokes Equations

In this work we study the fully nonhomogeneous initial boundary value problem for the two-dimensional time-dependent Navier–Stokes equations in a general open space domain in R² with low regularity assumptions on the initial and the boundary value data. We show that the perturbed Navier–Stokes operator is a diffeomorphism from a suitable function space onto its own dual and as a corollary we get that the Navier–Stokes equations are uniquely solvable in these spaces and that the solution depends smoothly on all involved data. Our source data space and solution space are in complete natural duality and in this sense, without any smallness assumptions on the data, we solve the equations for data with optimally low regularity in both space and time.     

Global Regularity for a Class of Generalized Magnetohydrodynamic Equations

It remains unknown whether or not smooth solutions of the 3D incompressible MHD equations can develop finite-time singularities. One major difficulty is due to the fact that the dissipation given by the Laplacian operator is insufficient to control the nonlinearity and for this reason the 3D MHD equations are sometimes regarded as “supercritical”. This paper presents a global regularity result for the generalized MHD equations with a class of hyperdissipation. This result is inspired by a recent work of Terence Tao on a generalized Navier–Stokes equations (T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equations, arXiv: 0906.3070v3 [math.AP] 20 June 2009), but the result for the MHD equations is not completely parallel to that for the Navier–Stokes equations. Besov space techniques are employed to establish the result for the MHD equations.     

Vanishing Viscosity Limit of the Compressible Navier–Stokes Equations for Solutions to a Riemann Problem

We study the vanishing viscosity limit of the compressible Navier–Stokes equations to the Riemann solution of the Euler equations that consists of the superposition of a shock wave and a rarefaction wave. In particular, it is shown that there exists a family of smooth solutions to the compressible Navier–Stokes equations that converges to the Riemann solution away from the initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients. This gives the first mathematical justification of this limit for the Navier–Stokes equations to the Riemann solution that contains these two typical nonlinear hyperbolic waves.     

Weak in Space, Log in Time Improvement of the Lady?enskaja�CProdi�CSerrin Criteria

In this article we present a Ladyženskaja–Prodi–Serrin Criteria for regularity of solutions for the Navier–Stokes equation in three dimensions which incorporates weak L ^p norms in the space variables and log improvement in the time variable.