Evolutionary Navier Stokes Equations Research Articles

Numerical Analysis of Penalty Stabilized Finite Element Discretizations of Evolution Navier–Stokes Equations

We perform in this paper the numerical analysis of some penalty stabilized solvers for the unsteady Navier---Stokes equations. We consider low-order and high-order methods. The low-order method is a pure penalty method, while the high-order one is a projection-stabilized method. We perform their numerical analysis (stability and convergence) for solutions that only need to bear the natural regularity. In this analysis, the stability is based upon specific inf-sup conditions. No local orthogonality properties are needed for the projection-interpolation operator. The convergence is based upon the representation of the stabilizing terms by means of bubble finite element spaces. We include some numerical tests for realistic flows that confirm the theoretical expectations.

ERROR ESTIMATES FOR THE FULLY DISCRETE STABILIZED GAUGE-UZAWA METHOD -PART I: THE NAVIER-STOKES EQUATIONS

The stabilized Gauge-Uzawa method (SGUM), which is a second order projection type algorithm to solve the time-dependent Navier-Stokes equations, has been newly constructed in 2013 Pyo's paper. The accuracy of SGUM has been proved only for time discrete scheme in the same paper, but it is crucial to study for fully discrete scheme, because the numerical errors depend on discretizations for both space and time, and because discrete spaces between velocity and pressure can not be chosen arbitrary. In this paper, we find out properties of the fully discrete SGUM and estimate its errors and stability to solve the evolution Navier-Stokes equations. The main difficulty in this estimation arises from losing some cancellation laws due to failing divergence free condition of the discrete velocity function. This result will be extended to Boussinesq equations in the continuous research (part II) and is essential in the study of part II.

A missed persistence property for the Euler equations and its effect on inviscid limits

We consider the problem of strong convergence, as the viscosity goes to zero, of the solutions to the three-dimensional evolutionary Navier–Stokes equations, under the Navier slip-type boundary condition (1.4), to the solution of the Euler equations under the no-penetration condition. In two dimensions, the above strong convergence holds in any smooth domain. Furthermore, in three dimensions, arbitrarily strong convergence results hold in the half-space case. In spite of the above results, recently we presented an explicit family of smooth initial data in the 3D sphere, for which the result fails. The result was proved as a by-product of the lack of time persistency for the above boundary condition under the Euler flow. Our aim here is to show a more general, and simpler proof, displayed in arbitrary, smooth domains.

Direction of Vorticity and Regularity up to the Boundary: On the Lipschitz-Continuous Case

In their famous 1993 paper, Constantin and Fefferman consider the evolution Navier-Stokes equations in the whole space R 3 and prove, essentially, that if the direction of the vorticity is Lipschitz continuous in the space variables, during a given time-interval, then the corresponding solution is regular. Since Lipschitz-continuity is a very natural, basic, property, it looks interesting to go further in this particular direction. In this paper, we consider the initial-boundary value problem for the Navier-Stokes equations in a regular, bounded, domain under a slip boundary condition, and prove regularity of the solution, up to the boundary, under a weakened Lipschitz-continuity assumption on the direction of the vorticity. The interest of our result highly relies on the fact that the Lipschitz-continuity coefficient g(x, t )i ssharp. This means, in a sense, that our finding possesses the same level of accuracy as that of the classical "Prodi-Serrin" type conditions; see the introductory section. It should be remarked that a similar result was already obtained in the 2009 paper by Beirao da Veiga and Berselli. In the latter, the proof of an analogous sharp result was shown under the assumption of 1 -H˝ older continuity on the direction of vorticity. The authors also claimed, correctly, that by the same ideas the proof of such a result could be extended to H˝ older exponents β ∈ )0 , 1 ). However the proofs would be extremely involved. On the contrary, the proof followed in this paper treat the Lipschitz case is definitely more elementary than any other proof, even if restricted to the whole space case.

A posteriori error estimations for mixed finite-element approximations to the Navier–Stokes equations

A posteriori estimates for mixed finite element discretizations of the Navier–Stokes equations are derived. We show that the task of estimating the error in the evolutionary Navier–Stokes equations can be reduced to the estimation of the error in a steady Stokes problem. As a consequence, any available procedure to estimate the error in a Stokes problem can be used to estimate the error in the nonlinear evolutionary problem. A practical procedure to estimate the error based on the so-called postprocessed approximation is also considered. Both the semidiscrete (in space) and the fully discrete cases are analyzed. Some numerical experiments are provided.

AN OVERVIEW OF BDF2 GAUGE-UZAWA METHODS FOR INCOMPRESSIBLE FLOWS

The Gauge-Uzawa method (GUM) in (9) which is a projection type al- gorithm to solve evolution Navier-Stokes equations has many advantages and supe- rior performance. But this method has been studied for backward Euler time discrete scheme which is the first order technique, because the classical second order GUM re- quests rather strong stability condition. Recently, the second order time discrete GUM was modified to be unconditionally stable and estimated errors in (12). In this paper, we contemplate several GUMs which can be derived by the same manner within (12), and we dig out properties of them for both stability and accuracy. In addition, we evaluate an stability condition for the classical GUM to construct an adaptive GUM for time to make free from strong stability condition of the classical GUM.

Reducing slip boundary value problems from the half to the whole space. Applications to inviscid limits and to non-Newtonian fluids

The study of a very large class of linear and non-linear, stationary and evolutive partial differential problems in the half-space (or similar) under the slip boundary condition is reduced here to the much simpler study of the corresponding results for the same problem in the whole space. The approach is particularly suitable for proving new results in strong norms. To determine whether this extension is available, turns out to be a simple exercise. The verification depends on a few general features of the functional space X related to the space variables. Hence, we present an approach as much as possible independent of the particular space X. We appeal to a reflection technique. Hence a crucial assumption is to be in the presence of flat boundaries (see below). Instead of stating “general theorems” we rather prefer to illustrate how to apply our results by considering a couple of interesting problems. As a main example, we show that sharp vanishing viscosity limit results that hold for the evolution Navier–Stokes equations in the whole space can be extended to the slip boundary value problem in the half-space. We also show some applications to non-Newtonian fluid problems.

A challenging open problem: The inviscid limit under slip-type boundary conditions.

In these notes we present some results proved in the forthcoming paper [3]. We consider the $\,3-D$ evolutionary Navier-Stokes equations with a Navier slip-type boundary condition, and study the problem of the <i>strong convergence</i> ($ k > 1+\frac{3}{p},$ see below) of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. This problem is still open, except in the case of flat boundaries. However, if we drop the convective terms (Stokes problem), the inviscid, strong limit result holds. The cause of this different behavior is quite subtle.

Concerning the W k,p -Inviscid Limit for 3-D Flows Under a Slip Boundary Condition

We consider the 3-D evolutionary Navier–Stokes equations with a Navier slip-type boundary condition, see (1.2), and study the problem of the strong convergence of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. We prove here, in the flat boundary case, convergence in Sobolev spaces W k, p (Ω), for arbitrarily large k and p (for previous results see Xiao and Xin in Comm Pure Appl Math 60:1027–1055, 2007 and Beirao da Veiga and Crispo in J Math Fluid Mech, 2009, doi: 10.1007/s00021-009-0295-4 ). However this problem is still open for non-flat, arbitrarily smooth, boundaries. The main obstacle consists in some boundary integrals, which vanish on flat portions of the boundary. However, if we drop the convective terms (Stokes problem), the inviscid, strong limit result holds, as shown below. The cause of this different behavior is quite subtle. As a by-product, we set up a very elementary approach to the regularity theory, in L p -spaces, for solutions to the Navier–Stokes equations under slip type boundary conditions.

OPTIMAL ERROR ESTIMATE FOR SEMI-DISCRETE GAUGE-UZAWA METHOD FOR THE NAVIER-STOKES EQUATIONS

The gauge-Uzawa method which has been constructed in [11] is a projection type method to solve the evolution Navier-Stokes equations. The method overcomes many shortcomings of projection methods and displays superior numerical performance [11, 12, 15, 16]. However, we have obtained only suboptimal accuracy via the energy estimate in [11]. In this paper, we study semi-discrete gauge-Uzawa method to prove optimal accuracy via energy estimate. The main key in this proof is to construct the intermediate equation which is formed to gauge-Uzawa algorithm. We will estimate velocity errors via comparing with the intermediate equation and then evaluate pressure errors via subtracting gauge-Uzawa algorithm from Navier-Stokes equations.

A stabilized semi-implicit Galerkin scheme for Navier–Stokes equations

By introducing a time relaxation term for the time derivative of higher frequency components, we proposed a stabilized semi-implicit Galerkin scheme for evolutionary Navier–Stokes equations in this paper. Analysis shows that such a scheme has weaker stability conditions than that of a classical semi-implicit Galerkin scheme and, when a suitable relaxation parameter σ is chosen, it generates an approximate solution with the same accuracy as the classical one. That means the proposed scheme might use a larger time step to generate a bounded approximate solution. Thus it is more suitable for long time simulations.

Sharp Inviscid Limit Results under Navier Type Boundary Conditions. An Lp Theory

We consider the evolutionary Navier–Stokes equations with a Navier slip-type boundary condition, and study the convergence of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. We obtain quite sharp results in the 2-D and 3-D cases. However, in the 3-D case, we need to assume that the boundary is flat.

Turbulence models, $p-$fluid flows, and $W^{2, L}$ regularity of solutions

In this article we prove some sharp regularity results for thestationary and the evolution Navier-Stokes equations with sheardependent viscosity, under the no-slip boundary condition. This is aclassical turbulence model, considered by von Neumann and Richtmeyerin the 50's, and by Smagorinski in the beginning of the 60's (for$p=3$). The model was extended to other physical situations,and deeply studied from a mathematical point of view, byLadyzhenskaya in the second half of the 60's. In the sequel weconsider the case $ p> 2 $. We are interested in regularityresults in Sobolev spaces, up to the boundary, in dimension$n= 3$, for the second order derivatives of the velocity and thefirst order derivatives of the pressure. In spite of the very richliterature on this subject, sharp regularity results up to theboundary are quite new. In the sequel we improve in a verysubstantial way all the known results in the literature. In order toemphasize the very new ideas, we consider a flat boundary (the socalled 'cubic-domain' case). However, all the regularity resultsstated here hold in the presence of smooth boundaries, by following [3].

Analysis of a multistate control problem related to food technology

This paper is concerned with an optimal control problem related to the determination of an optimal profile for the steam temperature into the autoclave along the processing of canned foods. The problem studies a system coupling the evolution Navier–Stokes equations with the heat transfer equation by natural convection (the so-called Boussinesq equations), and with the microorganisms removal equation. The essential difficulties in the study of this multistate control problem arise from the lack of uniqueness for the solution of the state system. Here we obtain—after a careful analysis of the problem mathematical formulation—the uniqueness of part of the state, and the existence of optimal solutions.

Navier–Stokes Equations with Shear Thinning Viscosity. Regularity up to the Boundary

In this article we prove some sharp regularity results for the stationary and the evolution Navier–Stokes equations with shear dependent viscosity, see (1.1), under the no-slip boundary condition(1.4). We are interested in regularity results for the second order derivatives of the velocity and for the first order derivatives of the pressure up to the boundary, in dimension n ≥ 3. In reference [4] we consider the stationary problem in the half space \({\mathbb{R}}_+^n\) under slip and no-slip boundary conditions. Here, by working in a simpler context, we concentrate on the basic ideas of proofs. We consider a cubic domain and impose our boundary condition (1.4) only on two opposite faces. On the other faces we assume periodicity, as a device to avoid unessential technical difficulties. This choice is made so that we work in a bounded domain Ω and, at the same time, with a flat boundary. In the last section we provide the extension of the results from the stationary to the evolution problem.

Remarks on the Navier–Stokes evolution equations under slip type boundary conditions with linear friction

In general terms, we show that the regularity results proved in [2] (see also [7]) for strong solutions to the linear stationary Stokes system under slip type boundary conditions allow to extend to these boundary conditions many of the well known results for solutions to the nonlinear evolution Navier–Stokes equations under the classical non-slip boundary condition.

THE GAUGE-UZAWA FINITE ELEMENT METHOD PART II: THE BOUSSINESQ EQUATIONS

The Gauge-Uzawa finite element method (GU-FEM)13 is a fully discrete projection type method to solve the evolution Navier–Stokes equations, which overcomes many shortcomings of projection methods and displays superior numerical performance. In this second part, we apply the GU-FEM to the evolution Boussinesq equations, which model the thermal driven motion of incompressible fluids. We show unconditional stability and error estimates for velocity, pressure and temperature, the three physical unknowns. We use a new variational approach13 and realistic regularity assumptions. We conclude with two physically relevant numerical simulations, the Benard convection problem and the thermal driven cavity flow.

Vorticity and Regularity for Viscous Incompressible Flows under the Dirichlet Boundary Condition. Results and Related Open Problems

In reference [7] it is proved that the solution of the evolution Navier–Stokes equations in the whole of R 3 must be smooth if the direction of the vorticity is Lipschitz continuous with respect to the space variables. In reference [5] the authors improve the above result by showing that Lipschitz continuity may be replaced by 1/2-Holder continuity. A central point in the proofs is to estimate the integral of the term (ω · ∇)u · ω, where u is the velocity and ω = ∇ × u is the vorticity. In reference [4] we extend the main estimates on the above integral term to solutions under the slip boundary condition in the half-space R + 3 . This allows an immediate extension to this problem of the 1/2-Holder sufficient condition. The aim of these notes is to show that under the non-slip boundary condition the above integral term may be estimated as well in a similar, even simpler, way. Nevertheless, without further hypotheses, we are not able now to extend to the non slip (or adherence) boundary condition the 1/2-Holder sufficient condition. This is not due to the “nonlinear" term (ω · ∇)u · ω but to a boundary integral which is due to the combination of viscosity and adherence to the boundary. On the other hand, by appealing to the properties of Green functions, we are able to consider here a regular, arbitrary open set Ω.

Bivariate spline method for numerical solution of time evolution Navier‐Stokes equations over polygons in stream function formulation

AbstractWe use a bivariate spline method to solve the time evolution Navier‐Stokes equations numerically. The bivariate splines we use in this article are in the spline space of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the Navier‐Stokes equations is employed. Galerkin's method is applied to discretize the space variables of the nonlinear fourth‐order equation, Crank‐Nicholson's method is applied to discretize the time variable, and Newton's iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in L2(0, T; H2(Ω)) ∩ L∞(0, T; H1(Ω)) of the 2D nonlinear fourth‐order problem and give an estimate of how fast the numerical solution converges to the weak solution. The C1 cubic splines are implemented in MATLAB for solving the Navier‐Stokes equations numerically. Our numerical experiments show that the method is effective and efficient. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 776–827, 2003.

Least‐squares mixed finite element methods for the incompressible Navier‐Stokes equations

AbstractLeast‐squares mixed finite element schemes are formulated to solve the evolutionary Navier‐Stokes equations and the convergence is analyzed. We recast the Navier‐Stokes equations as a first‐order system by introducing a vorticity flux variable, and show that a least‐squares principle based on L2 norms applied to this system yields optimal discretization error estimates. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 441–453, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10015