3D Stokes Equations Research Articles

An Adaptive Two-Grid Solver for DPG Formulation of Compressible Navier–Stokes Equations in 3D

Abstract We present an overlapping domain decomposition iterative solver for linear systems resulting from the discretization of compressible viscous flows with the Discontinuous Petrov–Galerkin (DPG) method in three dimensions. It is a two-grid solver utilizing the solution on the auxiliary coarse grid and the standard block-Jacobi iteration on patches of elements defined by supports of the coarse mesh base shape functions. The simple iteration defined in this way is used as a preconditioner for the conjugate gradient procedure. Theoretical analysis indicates that the condition number of the preconditioned system should be independent of the actual finite element mesh and the auxiliary coarse mesh, provided that they are quasiuniform. Numerical tests confirm this result. Moreover, they show that presence of strongly flattened or elongated elements does not slow the convergence. The finite element mesh is subject to adaptivity, i.e. dividing the elements with large errors until a required accuracy is reached. The auxiliary coarse mesh is adjusting to the nonuniform actual mesh.

Hydrodynamic Processes in Angular Fitting Connections of a Transport Machine’s Hydraulic Drive

The article presents the theoretical research on hydraulic processes occurring in hydraulic drives of transport machines. The research analysed the influence of hydrodynamic processes on the fluid flow characteristics following installation of angular fitting connections in pipeline systems. The analysis was based on the numerical simulations by Reynolds-averaged Navier–Stokes equations in 3D. The fluid flow in the hydraulic system pipeline connections were investigated for fluid flow rate up to 100 l/min. For numerical simulation, a mesh independence study was performed. By modeling. we obtained results of pressure drops, turbulence models, flow coefficients and energy losses at 45° and 90° angular fitting connections. The article compares the results obtained using the calculation method based on the standard of equivalent length fitting with the findings presented. The research conducted indicated that using the equivalent length method is not appropriate for studying angular fitting connections. It was found that additional investigations are necessary for each type of angular fitting connection.

Numerical and convergence analysis of the stochastic Lagrangian averaged Navier–Stokes equations

The primary emphasis of this work is the development of a finite element based space–time discretization for solving the stochastic Lagrangian averaged Navier–Stokes (LANS-α) equations of incompressible fluid turbulence with multiplicative random forcing, under nonperiodic boundary conditions within a bounded polygonal (or polyhedral) domain of Rd, d∈{2,3}. The convergence analysis of a fully discretized numerical scheme is investigated and split into two cases according to the spacial scale α, namely we first assume α to be controlled by the step size of the space discretization so that it vanishes when passing to the limit, then we provide an alternative study when α is fixed. A preparatory analysis of uniform estimates in both α and discretization parameters is carried out. Starting out from the stochastic LANS-α model, we achieve convergence toward the continuous strong solutions of the stochastic Navier–Stokes equations in 2D when α vanishes at the limit. Additionally, convergence toward the continuous strong solutions of the stochastic LANS-α model is accomplished if α is fixed. Neither of the mentioned convergences involves the Skorokhod theorem.

A separation between the boundary shape and the boundary functions in the parametric integral equation system for the 3D Stokes equation

The paper introduces the analytical modification of the classic boundary integral equation (BIE) for Stokes equation in 3D. The performed modification allows us to obtain separation of the approximation boundary shape from the approximation of the boundary functions. As a result, the equations, called the parametric integral equation system (PIES) with formal separation between the boundary geometry and the boundary functions, are obtained. It enables us to independently choose the most convenient methods of boundary geometry modeling depending on its complexity without any intrusion into the approximation of boundary functions and vice versa. Furthermore, we investigated the possibility of the modeling of the domains bounded by rectangular and triangular parametric Bézier patches. The boundary functions are approximated by generalized Chebyshev series. In addition, numerical techniques for solving the obtained PIES have been developed. The effectiveness of the presented strategy for boundary representation by surface patches in connection with PIES has been studied in numerical examples.

Error estimates for the approximation of the velocity tracking problem with Bang-Bang controls

The velocity tracking problem for the evolutionary Navier–Stokes equations in 2d is studied. The controls are of distributed type but the cost functional does not involve the usual quadratic term for the control. As a consequence the resulting controls can be of bang-bang type. First and second order necessary and sufficient conditions are proved. A fully-discrete scheme based on discontinuous (in time) Galerkin approach combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, τ and h respectively, satisfy τ ≤ C h 2 , then L 2 error estimates are proved for the difference between the states corresponding to locally optimal controls and their discrete approximations.

Higher-order in time “quasi-unconditionally stable” ADI solvers for the compressible Navier–Stokes equations in 2D and 3D curvilinear domains

This paper introduces alternating-direction implicit (ADI) solvers of higher order of time-accuracy (orders two to six) for the compressible Navier-Stokes equations in two- and three-dimensional curvilinear domains. The higher-order accuracy in time results from 1) An application of the backward differentiation formulae time-stepping algorithm (BDF) in conjunction with 2) A BDF-like extrapolation technique for certain components of the nonlinear terms (which makes use of nonlinear solves unnecessary), as well as 3) A novel application of the Douglas-Gunn splitting (which greatly facilitates handling of boundary conditions while preserving higher-order accuracy in time). As suggested by our theoretical analysis of the algorithms for a variety of special cases, an extensive set of numerical experiments clearly indicate that all of the BDF-based ADI algorithms proposed in this paper are "quasi-unconditionally stable" in the following sense: each algorithm is stable for all couples $(h,\Delta t)$ of spatial and temporal mesh sizes in a problem-dependent rectangular neighborhood of the form $(0,M_h)\times (0,M_t)$. In other words, for each fixed value of $\Delta t$ below a certain threshold, the Navier-Stokes solvers presented in this paper are stable for arbitrarily small spatial mesh-sizes. The second order formulation has further been rigorously shown to be unconditionally stable for linear hyperbolic and parabolic equations in two-dimensional space. Although implicit ADI solvers for the Navier-Stokes equations with nominal second-order of temporal accuracy have been proposed in the past, the algorithms presented in this paper are the first ADI-based Navier-Stokes solvers for which second order or better accuracy has been verified in practice under non-trivial (non-periodic) boundary conditions.

Existence and Uniqueness of Solutions to Heat Equations with Hysteresis Coupled with Navier–Stokes Equations in 2D and 3D

This paper is concerned with a system of heat equations with hysteresis and Navier–Stokes equations. In Tsuzuki (J Math Anal Appl 423:877–897, 2015) an existence result is obtained for the problem in a 2-dimensional domain with the Navier–Stokes equation in a weak sense. However the result does not include uniqueness for the problem due to the low regularity for solutions. This paper establishes existence and uniqueness in 2- and 3-dimensional domains with the Navier–Stokes equation in a stronger sense. Moreover this work decides required height of regularity for the initial data by introducing the fractional power of the Stokes operator.

Direct numerical simulation of electroconvective instability and hydrodynamic chaos near an ion-selective surface

We present a comprehensive analysis of transport processes associated with electrohydrodynamic chaos in electrokinetic systems containing an ion-selective surface. The system considered is an aqueous symmetric binary electrolyte between an ion-selective surface and a stationary reservoir. Transport is driven by an external electric field. Using direct numerical simulations (DNS) of the coupled Poisson–Nernst–Planck and Navier–Stokes equations in 2D we show significant transitions in flow behavior from coherent vortex pairs to fully chaotic multi-layer vortex structures with a broadband energy spectrum. Additionally, we demonstrate that these vortices can eject both positive and negative free charge density into the bulk of the domain and completely disrupt the structure of the traditionally described extended space charge region. The resulting dynamical behavior poses a challenge for traditional asymptotic modeling that relies on the quasi-electroneutral bulk assumption. Furthermore, we quantify for the first time the relative importance of energy dissipation due to viscous effects in various transport regimes. Finally, we present a framework for the development of ensemble-averaged models (similar to Reynolds Averaged Navier–Stokes equations) and assess the importance of the unclosed terms based on our DNS data.

A GRID OF THREE-DIMENSIONAL STELLAR ATMOSPHERE MODELS OF SOLAR METALLICITY. I. GENERAL PROPERTIES, GRANULATION, AND ATMOSPHERIC EXPANSION

Present grids of stellar atmosphere models are the workhorses in interpreting stellar observations, and determining their fundamental parameters. These models rely on greatly simplified models of convection, however, lending less predictive power to such models of late type stars. We present a grid of improved and more reliable stellar atmosphere models of late type stars, based on deep, 3D, convective, stellar atmosphere simulations. This grid is to be used in general for interpreting observations, and improve stellar and asteroseismic modeling. We solve the Navier Stokes equations in 3D and concurrent with the radiative transfer equation, for a range of atmospheric parameters, covering most of stellar evolution with convection at the surface. We emphasize use of the best available atomic physics for quantitative predictions and comparisons with observations. We present granulation size, convective expansion of the acoustic cavity, asymptotic adiabat, as function of atmospheric parameters. These and other results are also available in electronic form.

Liouville-Type Theorems for Steady Flows of Degenerate Power Law Fluids in the Plane

We extend the Liouville-type theorems of Gilbarg and Weinberger and of Koch, Nadirashvili, Seregin and Sverák valid for the stationary variant of the classical Navier–Stokes equations in 2D to the degenerate power law fluid model.

Existence of a martingale solution of the stochastic Navier–Stokes equations in unbounded 2D and 3D domains

Stochastic Navier–Stokes equations in 2D and 3D possibly unbounded domains driven by a multiplicative Gaussian noise are considered. The noise term depends on the unknown velocity and its spatial derivatives. The existence of a martingale solution is proved. The construction of the solution is based on the classical Faedo–Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. Moreover, some compactness and tightness criteria in nonmetric spaces are proved. Compactness results are based on a certain generalization of the classical Dubinsky Theorem.

Stochastic Navier–Stokes Equations Driven by Lévy Noise in Unbounded 3D Domains

Martingale solutions of the stochastic Navier–Stokes equations in 2D and 3D possibly unbounded domains, driven by the Lévy noise consisting of the compensated time homogeneous Poisson random measure and the Wiener process are considered. Using the classical Faedo–Galerkin approximation and the compactness method we prove existence of a martingale solution. We prove also the compactness and tightness criteria in a certain space contained in some spaces of càdlàg functions, weakly càdlàg functions and some Fréchet spaces. Moreover, we use a version of the Skorokhod Embedding Theorem for nonmetric spaces.

A note on a Liouville type result of Gilbarg and Weinberger for the stationary Navier–Stokes equations in 2D

We show that a velocity field u satisfying the stationary Navier–Stokes equations on the entire plane must be constant under the growth condition lim sup |x|−α |u(x)| < ∞ as |x| → ∞ for some α ∈ [0, 1/7).† Bibliography: 10 titles.

HSL_MI20: An efficient AMG preconditioner for finite element problems in 3D

AbstractAlgebraic multigrid (AMG) is one of the most effective iterative methods for the solution of large, sparse linear systems obtained from the discretization of second‐order scalar elliptic self‐adjoint partial differential equations. It can also be used as a preconditioner for Krylov subspace methods. In this communication, we report on the design and development of a robust, effective and portable Fortran 95 implementation of the classical Ruge–Stüben AMG, which is available as package HSL_MI20 within the HSL mathematical software library. The routine can be used as a ‘black‐box’ preconditioner, but it also offers the user a range of options and parameters. Proper tuning of these parameters for a particular application can significantly enhance the performance of an AMG‐preconditioned Krylov solver. This is illustrated using a number of examples arising in the unstructured finite element discretization of the diffusion, the convection–diffusion, and the Stokes equations, as well as transient thermal convection problems associated with the Boussinesq approximation of the Navier–Stokes equations in 3D. Copyright © 2009 John Wiley &amp; Sons, Ltd.

Global well-posedness for 2D polymeric fluid models and growth estimate

Motivated by [P. Constantin, N. Masmoudi, Global well-posedness for a Smoluchowski equation coupled with Navier–Stokes equations in 2D, Comm. Math. Phys. 278 (2008) 179–191; F. Lin, Ping Zhang, Zhifei Zhang, On the global existence of smooth solution to the 2-D FENE dumbbell model, Comm. Math. Phys. 277 (2008) 531–553], we prove the global existence of smooth solutions to a coupled microscopic–macroscopic model for polymeric fluid in 2D under the co-rotational assumption. Furthermore, we provide an estimate on the time growth of these solutions. Our method is general and can be applied to the co-rotational FENE model and to the rod-like model without the co-rotational assumption.

Condition number of the BEM matrix arising from the Stokes equations in 2D

We study the condition number of the system matrices that appear in the boundary element method when solving the Stokes equations at a 2D domain. At the boundary of the domain we impose Dirichlet conditions or mixed conditions. We show that for certain critical boundary contours the underlying boundary integral equation is not uniquely solvable. As a consequence, the condition number of the system matrix of the discrete equations is infinitely large. Hence, for these critical contours the Stokes cannot be solved by the boundary element method. To overcome this problem the domain can be rescaled. Several numerical examples are provided to illustrate the solvability problems at the critical contours.

Splitting methods for high order solution of the incompressible Navier–Stokes equations in 3D

AbstractThe incompressible Navier–Stokes equations are discretized in space by a hybrid method and integrated in time by the method of lines. The solution is determined on a staggered curvilinear grid in two space dimensions and by a Fourier expansion in the third dimension. The space derivatives are approximated by a compact finite difference scheme of fourth‐order on the grid. The solution is advanced in time by a semi‐implicit method. In each time step, systems of linear equations have to be solved for the velocity and the pressure. The iterations are split into one outer iteration and three inner iterations. The accuracy and efficiency of the method are demonstrated in a numerical experiment with rotated Poiseuille flow perturbed by Orr–Sommerfeld modes in a channel. Copyright © 2005 John Wiley &amp; Sons, Ltd.

Solving inverse problems involving the Navier–Stokes equations discretized by a Lagrange–Galerkin method

In this article, we are investigating the numerical approximation of an inverse problem involving the evolution of a Newtonian viscous incompressible fluid described by the Navier–Stokes equations in 2D. This system is discretized using a low order finite element in space coupled with a Lagrange–Galerkin scheme for the nonlinear advection operator. We introduce a full discrete linearized scheme that is used to compute the gradient of a given cost function by ensuring its consistency. Using gradient based optimization algorithms, we are able to deal with two fluid flow inverse problems, the drag reduction around a moving cylinder and the identification of a far-field velocity using the knowledge of the fluid load on a rectangular bluff body, for both fixed and prescribed moving configurations.

THE KOLMOGOROV EQUATION ASSOCIATED TO THE STOCHASTIC NAVIER–STOKES EQUATIONS IN 2D

We consider a 2D Navier–Stokes equation with Dirichlet boundary conditions perturbed by a stochastic term of the form [Formula: see text], where Q is a non-negative operator and Ẇ is a spacetime white noise. We solve the corresponding Kolmogorov equation in the space L2(H,ν) where ν is an invariant measure and prove the "carré du champs" identity. The key tool are some sharp estimates on the derivatives of the solution. Our main assumption is that the viscosity is large compared with the norm of Q. As a byproduct, we give a new simple proof of the uniqueness of the invariant measure and obtain an existence result for a Hamilton–Jacobi equation related to a control problem.

Ergodicity for the 3D stochastic Navier–Stokes equations

We consider the Kolmogorov equation associated with the stochastic Navier–Stokes equations in 3D, we prove existence of a solution in the strict or mild sense. The method consists in finding several estimates for the solutions u m of the Galerkin approximations of u and their derivatives. These estimates are obtained with the help of an auxiliary Kolmogorov equation with a very irregular negative potential. Although uniqueness is not proved, we are able to construct a transition semigroup for the 3D Navier–Stokes equations. Furthermore, this transition semigroup has a unique invariant measure, which is ergodic and strongly mixing.