Showing papers on "Navier–Stokes equations published in 2012"

Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models

Abstract: The objective of this self-contained book is two-fold. First, the reader is introduced to the modelling and mathematical analysis used in fluid mechanics, especially concerning the Navier-Stokes equations which is the basic model for the flow of incompressible viscous fluids. Authors introduce mathematical tools so that the reader is able to use them for studying many other kinds of partial differential equations, in particular nonlinear evolution problems. The background needed are basic results in calculus, integration, and functional analysis. Some sections certainly contain more advanced topics than others. Nevertheless, the authors' aim is that graduate or PhD students, as well as researchers who are not specialized in nonlinear analysis or in mathematical fluid mechanics, can find a detailed introduction to this subject.

Decay of Dissipative Equations and Negative Sobolev Spaces

Abstract: We develop a general energy method for proving the optimal time decay rates of the solutions to the dissipative equations in the whole space. Our method is applied to classical examples such as the heat equation, the compressible Navier-Stokes equations and the Boltzmann equation. In particular, the optimal decay rates of the higher-order spatial derivatives of solutions are obtained. The negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. We use a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis.

A synthesis of numerical methods for modeling wave energy converter-point absorbers

Abstract: During the past few decades, wave energy has received significant attention for harnessing ocean energy. Industry has proposed many technologies and, based on their working principle, these technologies generally can be categorized into oscillating water columns, point absorbers, overtopping systems, and bottom-hinged systems. In particular, many researchers have focused on modeling the point absorber, which is thought to be the most cost-efficient technology to extract wave energy. To model such devices, several modeling methods have been used such as analytical methods, boundary integral equation methods and Navier–Stokes equation methods. The first two are generally combined with the use of empirical solution to represent the viscous damping effect, while the last one is directly included in the solution. To assist the development of wave energy conversion (WEC) technologies, this paper extensively reviews the methods for modeling point absorbers.

A Lagrangian Approach for the Incompressible Navier-Stokes Equations with Variable Density

Abstract: We investigate the Cauchy problem for the inhomogeneous Navier-Stokes equations in the whole n-dimensional space. Under some smallness assumption on the data, we show the existence of global-in-time unique solutions in a critical functional framework. The initial density is required to belong to the multiplier space of \input amssym $\dot {B}^{n/p-1}_{p,1}({\Bbb R}^n)$. In particular, piecewise-constant initial densities are admissible data provided the jump at the interface is small enough and generate global unique solutions with piecewise constant densities. Using Lagrangian coordinates is the key to our results, as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence. © 2012 Wiley Periodicals, Inc.

Hydroelastic response and energy harvesting potential of flexible piezoelectric beams in viscous flow

Abstract: Electroactive polymers such as piezoelectric elements are able to generate electric potential differences from induced mechanical deformations. They can be used to build devices to harvest ambient energy from natural flow-induced deformations, e.g., as flapping flags subject to flowing wind or artificial seaweed subject to waves or underwater currents. The objectives of this study are to (1) investigate the transient hydroelastic response and energy harvesting potential of flexible piezoelectric beams fluttering in incompressible, viscous flow, and (2) identify critical non-dimensional parameters that govern the response of piezoelectric beams fluttering in viscous flow. The fluid-structure interaction response is simulated using an immersed boundary approach coupled with a finite volume solver for incompressible, viscous flow. The effects of large beam deformation, membrane tension, and coupled electromechanical responses are all considered. Validation studies are shown for the motion of a flexible filam...

A Novel Model Order Reduction Approach for Navier-Stokes Equations at High Reynolds Number

Abstract: A new approach to model order reduction of the Navier-Stokes equations at high Reynolds number is proposed. Unlike traditional approaches, this method does not rely on empirical turbulence modeling or modification of the Navier-Stokes equations. It provides spatial basis functions different from the usual proper orthogonal decomposition basis function in that, in addition to optimally representing the training data set, the new basis functions also provide stable and accurate reduced-order models. The proposed approach is illustrated with two test cases: two-dimensional flow inside a square lid-driven cavity and a two-dimensional mixing layer.

Free-Surface Flow and Fluid-Object Interaction Modeling With Emphasis on Ship Hydrodynamics

Abstract: : This paper presents our approach for the computation of free-surface/rigid-body interaction phenomena with emphasis on ship hydrodynamics. We adopt the level set approach to capture the free-surface. The rigid body is described using six-degree-of-freedom equations of motion. An interface-tracking method is used to handle the interface between the moving rigid body and the fluid domain. An Arbitrary Lagrangian Eulerian version of the residual-based variational multiscale formulation for the Navier Stokes and level set equations is employed in order to accommodate the fluid domain motion. The free-surface/rigid body problem is formulated and solved in a fully coupled fashion. The numerical results illustrate the accuracy and robustness of the proposed approach.

An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations

Abstract: The computation of compressibleflows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and excessively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incom- pressible flow simulation. In this paper, we develop an all-speed asymptotic preserv- ing (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyper- bolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics, to be solved im- plicitly. This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solu- tion techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompressible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes. AMS subject classifications: 35Q35, 65M08, 65M99, 76M12, 76N99

A finite element approach to incompressible two-phase flow on manifolds

Abstract: A two-phase Newtonian surface fluid is modelled as a surface Cahn–Hilliard–Navier–Stokes equation using a stream function formulation. This allows one to circumvent the subtleties in describing vectorial second-order partial differential equations on curved surfaces and allows for an efficient numerical treatment using parametric finite elements. The approach is validated for various test cases, including a vortex-trapping surface demonstrating the strong interplay of the surface morphology and the flow. Finally the approach is applied to a Rayleigh–Taylor instability and coarsening scenarios on various surfaces.

On the wellposedness of three-dimensional inhomogeneous Navier-Stokes equations in the critical spaces

Abstract: We prove the local wellposedness of three-dimensional incompressible inhomogeneous Navier–Stokes equations with initial data in the critical Besov spaces, without assumptions of small density variation. Furthermore, if the initial velocity field is small enough in the critical Besov space \({\dot B^{1/2}_{2,1}(\mathbb{R}^3)}\) , this system has a unique global solution.

Model reduction techniques for fast blood flow simulation in parametrized geometries

Abstract: In this paper we propose a new model reduction technique aimed at real-time blood flow simulations on a given family of geometrical shapes of arterial vessels. Our approach is based on the combination of a low-dimensional shape parametrization of the computational domain and the reduced basis method to solve the associated parametrized flow equations. We propose a preliminary analysis carried on a set of arterial vessel geometries, described by means of a radial basis functions parametrization. In order to account for patient-specific arterial configurations, we reconstruct the latter by solving a suitable parameter identification problem. Real-time simulation of blood flows are thus performed on each reconstructed parametrized geometry, by means of the reduced basis method. We focus on a family of parametrized carotid artery bifurcations, by modelling blood flows using Navier-Stokes equations and measuring distributed outputs such as viscous energy dissipation or vorticity. The latter are indexes that might be correlated with the assessment of pathological risks. The approach advocated here can be applied to a broad variety of (different) flow problems related with geometry/shape variation, for instance related with shape sensitivity analysis, parametric exploration, and shape design.

Sensitivity of 2-D turbulent flow past a D-shaped cylinder using global stability

Abstract: We use adjoint-based gradients to analyze the sensitivity of turbulent wake past a D-shaped cylinder at Re = 13000. We assess the ability of a much smaller control cylinder in altering the shedding frequency, as predicted by the eigenfrequency of the most unstable global mode to the mean flow. This allows performing beforehand identification of the sensitive regions, i.e., without computing the actually controlled states. Our results obtained in the frame of 2-D, unsteady Reynolds-averaged Navier–Stokes compare favorably with experimental data reported by Parezanovic and Cadot [J. Fluid Mech. 693, 115 (2012)] and suggest that the control cylinder acts primarily through a local modification of the mean flow profiles.

Model Reduction of Computational Aerothermodynamics for Hypersonic Aerothermoelasticity

Abstract: A primary challenge for aerothermoelastic analysis in hypersonic flow is accurate and efficient computation of unsteady aerothermodynamic loads. This study examines two model reduction strategies with the goal to enable the use of computational fluid dynamics within a long time-record, dynamic, aerothermoelastic analysis. One approach seeks to exploit the quasi-steady nature of the flow by using steady-state computational fluid dynamics to capture primary flow features, and simple analytical approximations to account for unsteady effects. The second approach seeks to minimize the computational cost of steady-state computational fluid dynamics flow analysis using either kriging or proper orthogonal decomposition-based modeling techniques. These model reduction strategies are assessed, both individually and combined, in the context of a three-dimensional hypersonic control surface. Results computed over a wide range of operating conditions and reduced frequencies indicate that when combined, the considered approaches yield an aerothermodynamic model that is tractable within a dynamic aerothermoelastic analysis, and generally has less than 5% maximum error relative to computational fluid dynamics.

Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials

Abstract: Here we consider a Cahn-Hilliard-Navier-Stokes system characterized by a nonlocal Cahn-Hilliard equation with a singular (e.g., logarithmic) potential. This system originates from a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids. We have already analyzed the case of smooth potentials with arbitrary polynomial growth. Here, taking advantage of the previous results, we study this more challenging (and physically relevant) case. We first establish the existence of a global weak solution with no-slip and no-flux boundary conditions. Then we prove the existence of the global attractor for the 2D generalized semiflow (in the sense of J.M. Ball). We recall that uniqueness is still an open issue even in 2D. We also obtain, as byproduct, the existence of a connected global attractor for the (convective) nonlocal Cahn-Hilliard equation. Finally, in the 3D case, we establish the existence of a trajectory attractor (in the sense of V.V. Chepyzhov and M.I. Vishik).

Global Well–Posedness of the 3D Primitive Equations with Partial Vertical Turbulence Mixing Heat Diffusion

Abstract: The three–dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly called the primitive equations. To overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection (or density stratification) equation. In this paper we prove the global regularity of strong solutions to this model in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-penetration and stress-free boundary conditions on the solid, top and bottom boundaries. Specifically, we show that short time strong solutions to the above problem exist globally in time, and that they depend continuously on the initial data.

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

Abstract: 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 non-metric spaces. Moreover, some compactness and tightness criteria in non-metric spaces are proved. Compactness results are based on a certain generalization of the classical Dubinsky Theorem.

Long Time Stability of a Classical Efficient Scheme for Two-dimensional Navier-Stokes Equations

Abstract: This paper considers the long-time stability property of a popular semi-implicit scheme for the two-dimensional incompressible Navier-Stokes equations in a periodic box that treats the viscous term implicitly and the nonlinear advection term explicitly. We consider both the semidiscrete (discrete in time but continuous in space) and fully discrete schemes with either Fourier Galerkin spectral or Fourier pseudospectral (collocation) methods. We prove that in all cases, the scheme is long time stable provided that the timestep is sufficiently small. The long time stability in the $L^2$ and $H^1$ norms further leads to the convergence of the global attractors and invariant measures of the scheme to those of the Navier-Stokes equations at vanishing timestep.

A Variational Data Assimilation Procedure for the Incompressible Navier-Stokes Equations in Hemodynamics

Abstract: We propose a data assimilation (DA) technique for including noisy measurements of the velocity field into the simulation of the Navier-Stokes equations (NSE) driven by hemodynamics applications. The technique is formulated as an inverse problem where we use a Discretize-then-Optimize approach to minimize the misfit between the recovered velocity field and the data, subject to the incompressible NSE. The DA procedure for this nonlinear problem is a combination of two approaches: the Newton method for the NSE and the DA procedure we designed and tested for the linearized problem. We discuss conditions on the location of velocity measurements that guarantee the well-posedness of the minimization process for the linearized problem. Numerical results, with both noise-free and noisy data, certify the theoretical analysis. Moreover, we consider 2D non-trivial geometries and 3D axisymmetric geometries. Also, we study the impact of noise on non-primitive variables of medical interest.

Rates of Convergence for Discretizations of the Stochastic Incompressible Navier--Stokes Equations

Abstract: We show strong convergence with rates for an implicit time discretization, a semi-implicit time discretization, and a related finite element based space-time discretization of the incompressible Navier--Stokes equations with multiplicative noise in two space dimensions. We use higher moments of computed iterates to optimally bound the error on a subset $\Omega_\kappa$ of the sample space $\Omega$, where corresponding paths are bounded in a proper function space, and $\mathbb{P}[\Omega_\kappa] \to 1$ holds for vanishing discretization parameters. This implies convergence in probability with rates, and motivates a practicable acception/rejection criterion to overcome possible pathwise explosion behavior caused by the nonlinearity. It turns out that it is the interaction of Lagrange multipliers with the stochastic forcing in the scheme which limits the accuracy of general discretely LBB-stable space discretizations, and strategies to overcome this problem are proposed.

Equation of motion for a sphere in non-uniform compressible flows

Abstract: Linearized viscous compressible Navier–Stokes equations are solved for the transient force on a spherical particle undergoing unsteady motion in an inhomogeneous unsteady ambient flow. The problem is formulated in a reference frame attached to the particle and the force contributions from the undisturbed ambient flow and the perturbation flow are separated. Using a density-weighted velocity transformation and reciprocal relation, the total force is first obtained in the Laplace domain and then transformed to the time domain. The total force is separated into the quasi-steady, inviscid unsteady, and viscous unsteady contributions. The above rigorously derived particle equation of motion can be considered as the compressible extension of the Maxey–Riley–Gatignol equation of motion and it incorporates interesting physics that arises from the combined effects of inhomogeneity and compressibility.

Semigroup Splitting and Cubature Approximations for the Stochastic Navier-Stokes Equations

Abstract: Approximation of the marginal distribution of the solution of the stochastic Navier-Stokes equations on the two-dimensional torus by high order numerical methods is considered. The corresponding rates of convergence are obtained for a splitting scheme and the method of cubature on Wiener space applied to a spectral Galerkin discretization of degree $N$. While the estimates exhibit a strong $N$ dependence, convergence is obtained for appropriately chosen time step sizes. Results of numerical simulations are provided and confirm the applicability of the methods.

Global and Trajectory Attractors for a Nonlocal Cahn-Hilliard-Navier-Stokes System

Abstract: The Cahn–Hilliard–Navier–Stokes system is based on a well-known diffuse interface model and describes the evolution of an incompressible isothermal mixture of binary fluids. A nonlocal variant consists of the Navier–Stokes equations suitably coupled with a nonlocal Cahn–Hilliard equation. The authors, jointly with P. Colli, have already proven the existence of a global weak solution to a nonlocal Cahn–Hilliard–Navier–Stokes system subject to no-slip and no-flux boundary conditions. Uniqueness is still an open issue even in dimension two. However, in this case, the energy identity holds. This property is exploited here to define, following J.M. Ball’s approach, a generalized semiflow which has a global attractor. Through a similar argument, we can also show the existence of a (connected) global attractor for the convective nonlocal Cahn–Hilliard equation with a given velocity field, even in dimension three. Finally, we demonstrate that any weak solution fulfilling the energy inequality also satisfies a dissipative estimate. This allows us to establish the existence of the trajectory attractor also in dimension three with a time dependent external force.