Instationary Navier-Stokes Equations Research Articles

Error estimates for finite element discretizations of the instationary Navier–Stokes equations

In this work we consider the two dimensional instationary Navier–Stokes equations with homogeneous Dirichlet/no-slip boundary conditions. We show error estimates for the fully discrete problem, where a discontinuous Galerkin method in time and inf-sup stable finite elements in space are used. Recently, best approximation type error estimates for the Stokes problem in the L∞(I; L2(Ω)), L2(I; H1(Ω)) and L2(I; L2(Ω)) norms have been shown. The main result of the present work extends the error estimate in the L∞(I; L2(Ω)) norm to the Navier–Stokes equations, by pursuing an error splitting approach and an appropriate duality argument. In order to discuss the stability of solutions to the discrete primal and dual equations, a specially tailored discrete Gronwall lemma is presented. The techniques developed towards showing the L∞(I; L2(Ω)) error estimate, also allow us to show best approximation type error estimates in the L2(I; H1(Ω)) and L2(I; L2(Ω)) norms, which complement this work.

A neural network multigrid solver for the Navier-Stokes equations

We present the deep neural network multigrid solver (DNN-MG) that we develop for the instationary Navier-Stokes equations. DNN-MG improves computational efficiency using a judicious combination of a geometric multigrid solver and a recurrent neural network with memory. DNN-MG uses the multi-grid method to classically solve on coarse levels while the neural network corrects interpolated solutions on fine ones, thus avoiding the increasingly expensive computations that would have to be performed there. This results in a reduction in computation time through DNN-MG's highly compact neural network. The compactness results from its design for local patches and the available coarse multigrid solutions that provides a “guide” for the corrections. A compact neural network with a small number of parameters also reduces training time and data. Furthermore, the network's locality facilitates generalizability and allows one to use DNN-MG trained on one mesh domain also on different ones. We demonstrate the efficacy of DNN-MG for variations of the 2D laminar flow around an obstacle. For these, our method significantly improves the solutions as well as lift and drag functionals while requiring only about half the computation time of a full multigrid solution. We also show that DNN-MG trained for the configuration with one obstacle can be generalized to other time dependent problems that can be solved efficiently using a geometric multigrid method.

On regularity of weak solutions for the Navier–Stokes equations in general domains

AbstractLet u be a weak solution of the instationary Navier–Stokes equations in a completely general domain which additionally satisfies the strong energy inequality. Firstly, we prove that u is regular if the kinetic energy is left‐side Hölder continuous with Hölder exponent and with a sufficiently small Hölder seminorm. This result extends the previous ones by several authors [5, 6, 7, 8] in which the domain Ω is additionally supposed to be bounded or have the uniform C2‐boundary . Secondly, we show that if and for all with a sufficiently small positive constant C then u is regular in [0, T). Our proofs use the theory about the existence of local strong solutions and uniqueness arguments in the general domain.

From Jean Leray to the millennium problem: the Navier\u2013Stokes equations

One of the Millennium problems of Clay Mathematics Institute from 2000 concerns the regularity of solutions to the instationary Navier–Stokes equations in three dimensions. For an official problem description, we refer to an article by Charles Fefferman (Existence and smoothness of the Navier–Stokes equation. http://www.claymath.org/sites/default/files/navierstokes.pdf) on the whole space problem as well as the periodic setting in mathbb {R}^3. In this article, we will introduce the Navier–Stokes equations, describe their main mathematical problems, discuss several of the most important results, starting from 1934 with the seminal work by Jean Leray, and proceeding to very recent results on non-uniqueness and examples involving singularities. On this tour de force we will explain the open problem of regularity.

Numerical resolution of optimal control problem for the in-stationary Navier–Stokes equations

Abstract In this paper we present a study of optimal control problem for the unsteady Navier–Stokes equations. We discuss the existence of the solution, adopt a new numerical resolution for this problem and combine Euler explicit scheme in time and both of methods spectral and finite elements in space. Finally, we give some numerical results proving the effectiveness of our approach.

Continuous Crystallization in a Helically Coiled Flow Tube: Analysis of Flow Field, Residence Time Behavior, and Crystal Growth

A continuously operated helically coiled flow tube (HCT) crystallizer is investigated for crystal growth. Inline video-imaging is used for crystal shape analysis and residence time estimation of potash alum. The main finding is that there is a size-dependent particle residence time. Large particles move faster through the HCT than small particles. Consequently, small crystals have more time to grow in the HCT. Physical reasons for this behavior are proposed, for example, small-scale flow characteristics. In a direct numerical simulation of the instationary Navier–Stokes equations, velocity fluctuations and a secondary flow are identified. The presented flow field may have a different impact on the particles and cause the size-dependent particle residence time. A particle size dependent residence time may potentially narrow the crystal size and shape distribution in such a process, frequently a desired feature in solids’ production.

Incompressible Navier–Stokes flows with time-dependent prescribed fluxes in domains with cylindrical outlets to infinity

We consider instationary Navier–Stokes equations with prescribed time-dependent fluxes in a cylindrical domain \(\Omega \subset {\mathbb R}^n,n\ge 3,\) with several exits to infinity. First, we prove existence and uniqueness of time-dependent Poiseuille flow in \(L^q\)-spaces on infinite straight cylinders over time interval \((0,T)\), \(0 2, q\in (\frac{n-1}{2},\infty )\), with an exponential weight along the axial directions of \(\Omega \).

Weak solutions of the Navier–Stokes equations with non-zero boundary values in an exterior domain satisfying the strong energy inequality

In an exterior domain Ω⊂R3 and a time interval [0,T), 0<T⩽∞, consider the instationary Navier–Stokes equations with initial value u0∈Lσ2(Ω) and external force f=divF, F∈L2(0,T;L2(Ω)). As is well-known there exists at least one weak solution in the sense of J. Leray and E. Hopf with vanishing boundary values satisfying the strong energy inequality. In this paper, we extend the class of global in time Leray–Hopf weak solutions to the case when u▪=g with non-zero time-dependent boundary values g. Although uniqueness for these solutions cannot be proved, we show the existence of at least one weak solution satisfying the strong energy inequality and a related energy estimate.

The instationary Stokes equations in weighted Bessel-potential spaces

We investigate the solvability of the instationary Navier–Stokes equations with fully inhomogeneous data in a bounded domain \(\Omega \subset {{\mathbb{R}}^{n}} \). The class of solutions is contained in \(L^{r}(0, T; H^{\beta, q}_{w} (\Omega))\), where \({H^{\beta, q}_{w}} (\Omega)\) is a Bessel-Potential space with a Muckenhoupt weight w. In this context we derive solvability for small data, where this smallness can be realized by the restriction to a short time interval. Depending on the order of this Bessel-Potential space we are dealing with strong solutions, weak solutions, or with very weak solutions.

The Instationary Navier–Stokes Equations in Weighted Bessel-Potential Spaces

We investigate the solvability of the instationary Navier–Stokes equations with fully inhomogeneous data in a bounded domain \(\Omega \subset {{\mathbb{R}}^{n}} \). The class of solutions is contained in \(L^{r}(0, T; H^{\beta, q}_{w} (\Omega))\), where \({H^{\beta, q}_{w}} (\Omega)\) is a Bessel-Potential space with a Muckenhoupt weight w. In this context we derive solvability for small data, where this smallness can be realized by the restriction to a short time interval. Depending on the order of this Bessel-Potential space we are dealing with strong solutions, weak solutions, or with very weak solutions.

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.

Sufficient second-order optimality conditions for convex control constraints

In this article sufficient optimality conditions are established for optimal control problems with pointwise convex control constraints. Here, the control is a function with values in R n . The constraint is of the form u ( x ) ∈ U ( x ) , where U is a set-valued mapping that is assumed to be measurable with convex and closed images. The second-order condition requires coercivity of the Lagrange function on a suitable subspace, which excludes strongly active constraints, together with first-order necessary conditions. It ensures local optimality of a reference function in an L ∞ -neighborhood. The analysis is done for a model problem namely the optimal distributed control of the instationary Navier–Stokes equations.

Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations

In this paper sufficient optimality conditions are established for optimal control of both steady-state and instationary Navier-Stokes equations. The second-order condition requires coercivity of the Lagrange function on a suitable subspace together with first-order necessary conditions. It ensures local optimality of a reference function in a Ls -neighborhood, whereby the underlying analysis allows to use weaker norms than L∞ .

A-revolve: an adaptive memory-reduced procedure for calculating adjoints; with an application to computing adjoints of the instationary Navier–Stokes system

This article presents a low-storage and low-run-time approach for calculating numerical approximations of adjoint equations for the instationary Navier–Stokes equations with adaptive evaluation of the discretization step. It utilizes adaptive checkpointing. The amounts of memory reduction due to checkpointing and of the increase in run-time caused by repeated forward integration of the Navier–Stokes equations are reported. It is one result that memory reduction of two orders of magnitude only causes a slow down factor of 2–3 in run-time. It is a further result that the adaptive checkpointing for the instationary Navier–Stokes equations causes only a slight increase of the run-time compared to the static optimal checkpointing. The amounts of run-time increase caused by the adaptive checkpoint allocations compared with the static optimal checkpointing are reported as well.

Regularity of the Adjoint State for the Instationary Navier-Stokes Equations

We are considering imbeddings of abstract functions in spaces of functions being continuous in time. A family of functions depending on certain parameters is discussed in detail. In particular, this example shows that such functions do not belong to the space C([0, T], H). In the second part, we investigate an optimal control problem for the instationary Navier-Stokes equation. We will answer the question, in what sense the initial value problem for the adjoint equation can be solved.

Parabolic Dirac operators and the Navier-Stokes equations over time-varying domains

We consider parabolic Dirac operators which do not involve fractional derivatives and use them to show the solvability of the in-stationary Navier–Stokes equations over time-varying domains. Copyright © 2005 John Wiley & Sons, Ltd.

Semidiscretization and error estimates for distributed control of the instationary Navier-Stokes equations

Optimal control problems governed by the two-dimensional instationary Navier–Stokes equations and their spatial discretizations with finite elements are investigated. A concept of semi–discrete solutions to the control problem is introduced which is utilized to prove existence and uniqueness of discrete controls in neighborhoods of regular continuous solutions. Furthermore, an optimal error estimate in terms of the spatial discretization parameter is given.

Globalization of SQP-Methods in Control of the Instationary Navier-Stokes Equations

A numerically inexpensive globalization strategy of sequential quadratic programming methods (SQP-methods) for control of the instationary Navier Stokes equations is investigated. Based on the proper functional analytic setting a convergence analysis for the globalized method is given. It is argued that the a priori formidable SQP-step can be decomposed into linear primal and linear adjoint systems, which is amenable for existing CFL-software. A report on a numerical test demonstrates the feasibility of the approach.

Finite element discretization of the Navier–Stokes equations with a free capillary surface

The instationary Navier–Stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions. A stable finite element discretization is presented. The key idea is the treatment of the curvature terms by a variational formulation. In the context of a discontinuous in time space–time element discretization stability in (weak) energy norms can be proved. Numerical examples in 2 and 3 space dimensions are given.

SOME EXISTENCE AND UNIQUENESS RESULTS FOR A TIME-DEPENDENT COUPLED PROBLEM OF THE NAVIER–STOKES KIND

In this paper, we consider some systems which are close to the instationary Navier–Stokes equations. The structure of these systems is the following: An (N + 1)-dimensional equation for motion (including the incompressibility condition) and a scalar equation involving an additional unknown, k = k(x,t). Among other things, they serve to model the behavior of certain turbulent flows. We are mainly concerned with existence and uniqueness results. The main difficulties are due to the scalar equation. In particular, the right-hand side is typically in L1; furthermore, there are nonlinear terms of the kind ∇·(μ(k)∇ k) and ∇·(B(k)), where μ and B are general continuous functions (no growth condition at infinity is imposed). Following the previous work of other authors, it is crucial to introduce the notion of weak-renormalized solution. Our results give the existence in the two-dimensional case, as well as the uniqueness of regular solution in both the two- and three-dimensional cases.