Showing papers on "Navier–Stokes equations published in 2016"

The Navier-Stokes Problem in the 21st Century

Abstract: Presentation of the Clay Millennium Prizes Regularity of the three-dimensional fluid flows: a mathematical challenge for the 21st century The Clay Millennium Prizes The Clay Millennium Prize for the Navier-Stokes equations Boundaries and the Navier-Stokes Clay Millennium Problem The physical meaning of the Navier-Stokes equations Frames of references The convection theorem Conservation of mass Newton's second law Pressure Strain Stress The equations of hydrodynamics The Navier-Stokes equations Vorticity Boundary terms Blow up Turbulence History of the equation Mechanics in the Scientific Revolution era Bernoulli's Hydrodymica D'Alembert Euler Laplacian physics Navier, Cauchy, Poisson, Saint-Venant, and Stokes Reynolds Oseen, Leray, Hopf, and Ladyzhenskaya Turbulence models Classical solutions The heat kernel The Poisson equation The Helmholtz decomposition The Stokes equation The Oseen tensor Classical solutions for the Navier-Stokes problem Small data and global solutions Time asymptotics for global solutions Steady solutions Spatial asymptotics Spatial asymptotics for the vorticity Intermediate conclusion A capacitary approach of the Navier-Stokes integral equations The integral Navier-Stokes problem Quadratic equations in Banach spaces A capacitary approach of quadratic integral equations Generalized Riesz potentials on spaces of homogeneous type Dominating functions for the Navier-Stokes integral equations A proof of Oseen's theorem through dominating functions Functional spaces and multipliers The differential and the integral Navier-Stokes equations Uniform local estimates Heat equation Stokes equations Oseen equations Very weak solutions for the Navier-Stokes equations Mild solutions for the Navier-Stokes equations Suitable solutions for the Navier-Stokes equations Mild solutions in Lebesgue or Sobolev spaces Kato's mild solutions Local solutions in the Hilbertian setting Global solutions in the Hilbertian setting Sobolev spaces A commutator estimate Lebesgue spaces Maximal functions Basic lemmas on real interpolation spaces Uniqueness of L3 solutions Mild solutions in Besov or Morrey spaces Morrey spaces Morrey spaces and maximal functions Uniqueness of Morrey solutions Besov spaces Regular Besov spaces Triebel-Lizorkin spaces Fourier transform and Navier-Stokes equations The space BMO-1 and the Koch and Tataru theorem Koch and Tataru's theorem Q-spaces A special subclass of BMO-1 Ill-posedness Further results on ill-posedness Large data for mild solutions Stability of global solutions Analyticity Small data Special examples of solutions Symmetries for the Navier-Stokes equations Two-and-a-half dimensional flows Axisymmetrical solutions Helical solutions Brandolese's symmetrical solutions Self-similar solutions Stationary solutions Landau's solutions of the Navier-Stokes equations Time-periodic solutions Beltrami flows Blow up? First criteria Blow up for the cheap Navier-Stokes equation Serrin's criterion Some further generalizations of Serrin's criterion Vorticity Squirts Leray's weak solutions The Rellich lemma Leray's weak solutions Weak-strong uniqueness: the Prodi-Serrin criterion Weak-strong uniqueness and Morrey spaces on the product space R x R3 Almost strong solutions Weak perturbations of mild solutions Partial regularity results for weak solutions Interior regularity Serrin's theorem on interior regularity O'Leary's theorem on interior regularity Further results on parabolic Morrey spaces Hausdorff measures Singular times The local energy inequality The Caffarelli-Kohn-Nirenberg theorem on partial regularity Proof of the Caffarelli-Kohn-Nirenberg criterion Parabolic Hausdorff dimension of the set of singular points On the role of the pressure in the Caffarelli, Kohn, and Nirenberg regularity theorem A theory of uniformly locally L2 solutions Uniformly locally square integrable solutions Local inequalities for local Leray solutions The Caffarelli, Kohn, and Nirenberg epsilon-regularity criterion A weak-strong uniqueness result The L3 theory of suitable solutions Local Leray solutions with an initial value in L3 Critical elements for the blow up of the Cauchy problem in L3 Backward uniqueness for local Leray solutions Seregin's theorem Known results on the Cauchy problem for the Navier-Stokes equations in presence of a force Local estimates for suitable solutions Uniqueness for suitable solutions A quantitative one-scale estimate for the Caffarelli-Kohn-Nirenberg regularity criterion The topological structure of the set of suitable solutions Escauriaza, Seregin, and Sverak's theorem Self-similarity and the Leray-Schauder principle The Leray-Schauder principle Steady-state solutions Self-similarity Statement of Jia and Sverak's theorem The case of locally bounded initial data The case of rough data Non-existence of backward self-similar solutions alpha-models Global existence, uniqueness and convergence issues for approximated equations Leray's mollification and the Leray-alpha model The Navier-Stokes alpha -model The Clark- alpha model The simplified Bardina model Reynolds tensor Other approximations of the Navier-Stokes equations Faedo-Galerkin approximations Frequency cut-off Hyperviscosity Ladyzhenskaya's model Damped Navier-Stokes equations Artificial compressibility Temam's model Vishik and Fursikov's model Hyperbolic approximation Conclusion Energy inequalities Critical spaces for mild solutions Models for the (potential) blow up The method of critical elements Notations and glossary Bibliography Index

Input-output analysis of high-speed axisymmetric isothermal jet noise

Abstract: We use input-output analysis to predict and understand the aeroacoustics of high-speed isothermal turbulent jets. We consider axisymmetric linear perturbations about Reynolds-averaged Navier-Stokes solutions of ideally expanded turbulent jets with jet Mach numbers 0.6 < Mj < 1.8. For each base flow, we compute the optimal harmonic forcing function and the corresponding linear response using singular value decomposition of the resolvent operator. In addition to the optimal mode, input-output analysis also yields sub-optimal modes associated with smaller singular values. For supersonic jets, the optimal response closely resembles a wavepacket in both the near-field and the far-field such as those obtained by the parabolized stability equations (PSE), and this mode dominates the response. For subsonic jets, however, the singular values indicate that the contributions of sub-optimal modes to noise generation are nearly equal to that of the optimal mode, explaining why the PSE do not fully capture the far-fiel...

Enhanced Dissipation and Inviscid Damping in the Inviscid Limit of the Navier–Stokes Equations Near the Two Dimensional Couette Flow

Abstract: In this work we study the long time inviscid limit of the two dimensional Navier–Stokes equations near the periodic Couette flow. In particular, we confirm at the nonlinear level the qualitative behavior predicted by Kelvin’s 1887 linear analysis. At high Reynolds number Re, we prove that the solution behaves qualitatively like two dimensional Euler for times \({{t \lesssim Re^{1/3}}}\), and in particular exhibits inviscid damping (for example the vorticity weakly approaches a shear flow). For times \({{t \gtrsim Re^{1/3}}}\), which is sooner than the natural dissipative time scale O(Re), the viscosity becomes dominant and the streamwise dependence of the vorticity is rapidly eliminated by an enhanced dissipation effect. Afterwards, the remaining shear flow decays on very long time scales \({{t \gtrsim Re}}\) back to the Couette flow. When properly defined, the dissipative length-scale in this setting is \({{\ell_D \sim Re^{-1/3}}}\), larger than the scale \({{\ell_D \sim Re^{-1/2}}}\) predicted in classical Batchelor–Kraichnan two dimensional turbulence theory. The class of initial data we study is the sum of a sufficiently smooth function and a small (with respect to Re−1) L2 function.

A hyperbolic model for viscous Newtonian flows

Abstract: We discuss a pure hyperbolic alternative to the Navier–Stokes equations, which are of parabolic type. As a result of the substitution of the concept of the viscosity coefficient by a microphysics-based temporal characteristic, particle settled life (PSL) time, it becomes possible to formulate a model for viscous fluids in a form of first-order hyperbolic partial differential equations. Moreover, the concept of PSL time allows the use of the same model for flows of viscous fluids (Newtonian or non-Newtonian) as well as irreversible deformation of solids. In the theory presented, a continuum is interpreted as a system of material particles connected by bonds; the internal resistance to flow is interpreted as elastic stretching of the particle bonds; and a flow is a result of bond destructions and rearrangements of particles. Finally, we examine the model for simple shear flows, arbitrary incompressible and compressible flows of Newtonian fluids and demonstrate that Newton’s viscous law can be obtained in the framework of the developed hyperbolic theory as a steady-state limit. A basic relation between the viscosity coefficient, PSL time, and the shear sound velocity is also obtained.

Flow and heat simultaneously induced by two stretchable rotating disks

Abstract: An exact solution for the steady state Navier-Stokes equations in cylindrical coordinates is presented in this work. It serves to investigate the fluid flow and heat transfer occurring between two stretchable disks rotating co-axially at constant distance apart. The governing equations of motion and energy are first transformed into a set of nonlinear differential equation system by the use of von Karman similarity transformations, which are later solved numerically. The small Reynolds number case allows us to extract closed-form solutions for the physical phenomenon. The effects of the same or opposite direction rotation, as well as the stretching parameter and the Reynolds number, are discussed on the flow and heat characteristics. The main physical implication of the results is that stretching action of a disk surface alters considerably the classical flow behavior occurring between two disks and the physically interesting quantities like the torque and heat transfer are elucidated in the presence of a new physical mechanism; that is the surface stretching in the current research.

Analysis of a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier-Stokes equations

Abstract: We present the first a priori error analysis of the hybridizable discontinuous Galerkin method for the approximation of the Navier-Stokes equations proposed in J. Comput. Phys. vol. 230 (2011), pp. 1147-1170. The method is defined on conforming meshes made of simplexes and provides piecewise polynomial approximations of fixed degree k to each of the components of the velocity gradient, velocity and pressure. For the stationary case, and under the usual smallness condition for the source term, we prove that the method is well defined and that the global L2-norm of the error in each of the above-mentioned variables converges with the optimal order of k+1 for k ≥ 0. We also prove a superconvergence property of the velocity which allows us to obtain an elementwise postprocessed approximate velocity, H(div)-conforming and divergence-free, which converges with order k + 2 for k ≥ 1. In addition, we show that these results only depend on the inverse of the stabilization parameter of the jump of the normal component of the velocity. Thus, if we superpenalize those jumps, these converegence results do hold by assuming that the pressure lies in H1(Ω) only. Moreover, by letting such stabilization parameters go to infinity, we obtain new H(div)-conforming methods with the above-mentioned convergence properties.

Liouville type theorem for stationary Navier–Stokes equations

Abstract: It is shown that any smooth solution to the stationary Navier–Stokes system in with the velocity, belonging to L 6 and BMO –1, must be zero.

Three-Dimensional Navier–Stokes Equations

Abstract: Banach algebra, 36 Banach–Alaoglu Theorem, 376 Beale–Kato–Majda criterion, 228 Biot–Savart Law global version derivation, 227 estimating u given ω, 227 local version derivation, 231 estimating u given ω, 233 blowup rate for a simple ODE, 218 in H, 139 in L, 218 BMO, 203 Bochner integral, deinition, 39 Bochner spaces, 38 box-counting dimension and Lebesgue measure, 166 bounds the Hausdorff dimension, 166 deinition (coverings), 162 deinition (disjoint balls), 162 of singular times T , 166 of space–time singular set S, 302 the Burgers equation, 10, 11 C, 22 C , 22 C c , 22 C, 22 C , 22 Calderón–Zygmund decompositions, 378 Calderón–Zygmund Theorem, 380 alternative form (condition on |∇K|), 383 centred parabolic cylinder Qr , 240 Chebyshev inequality (generalised), 373 CKN, 279 compatibility conditions, 154 continuation of strong solutions, 229 continuous and differentiable functions, 22–23 critical spaces, 8, 192, 203, see also BMO, Ḣ, and L curl, 20 integration by parts, 225, 237

A Discrete Data Assimilation Scheme for the Solutions of the Two-Dimensional Navier--Stokes Equations and Their Statistics

Abstract: We adapt a previously introduced continuous in time data assimilation (downscaling) algorithm for the two-dimensional Navier--Stokes equations to the more realistic case when the measurements are obtained discretely in time and may be contaminated by systematic errors. Our algorithm is designed to work with a general class of observables, such as low Fourier modes and local spatial averages over finite volume elements. Under suitable conditions on the relaxation (nudging) parameter, the spatial mesh resolution, and the time step between successive measurements, we obtain an asymptotic in time estimate of the difference between the approximating solution and the unknown reference solution corresponding to the measurements, in an appropriate norm, which shows exponential convergence up to a term which depends on the size of the errors. A stationary statistical analysis of our discrete data assimilation algorithm is also provided.

Existence of Global Strong Solutions to a Beam–Fluid Interaction System

Abstract: We study an unsteady nonlinear fluid–structure interaction problem which is a simplified model to describe blood flow through viscoelastic arteries. We consider a Newtonian incompressible two-dimensional flow described by the Navier–Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear viscoelastic beam equation. The fluid and the structure are fully coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action–reaction principle. We prove that strong solutions to this problem are global-in-time. We obtain, in particular that contact between the viscoelastic wall and the bottom of the fluid cavity does not occur in finite time. To our knowledge, this is the first occurrence of a no-contact result, and of the existence of strong solutions globally in time, in the frame of interactions between a viscous fluid and a deformable structure.

A Computational Study of a Data Assimilation Algorithm for the Two-dimensional Navier-Stokes Equations

Abstract: We study the numerical performance of a continuous data assimilation (downscaling) algorithm, based on ideas from feedback control theory, in the context of the two-dimensional incompressible Navier– Stokes equations. Our model problem is to recover an unknown reference solution, asymptotically in time, by using continuous-in-time coarse-mesh nodal-point observational measurements of the velocity field of this reference solution (subsampling), as might be measured by an array of weather vane anemometers. Our calculations show that the required nodal observation density is remarkably less that what is suggested by the analytical study; and is in fact comparable to the number of numerically determining Fourier modes, which was reported in an earlier computational study by the authors. Thus, this method is computationally efficient and performs far better t the analytical estimates suggest.

On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions

Abstract: We consider a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. This model consists of the Navier–Stokes equations coupled with a convective nonlocal Cahn–Hilliard equation. Several results were already proven by two of the present authors. However, in the two-dimensional case, the uniqueness of weak solutions was still open. Here we establish such a result even in the case of degenerate mobility and singular potential. Moreover, we show the weak–strong uniqueness in the case of viscosity depending on the order parameter, provided that either the mobility is constant and the potential is regular or the mobility is degenerate and the potential is singular. In the case of constant viscosity, on account of the uniqueness results, we can deduce the connectedness of the global attractor whose existence was obtained in a previous paper. The uniqueness technique can be adapted to show the validity of a smoothing property for the difference of two trajectories which is crucial to establish the existence of an exponential attractor. The latter is established even in the case of variable viscosity, constant mobility and regular potential.

Global Weak Solutions to the Compressible Quantum Navier--Stokes Equations with Damping

Abstract: In this paper, we proved the existence of global weak solutions of the compressible quantum Navier--Stokes equations with large data in three dimensions (3D). The model consists of the compressible Navier--Stokes equations with degenerate viscosity, a nonlinear third-order differential operator known as the quantum Bohm potential, and some damping terms. The global weak solution is shown by using the Faedo--Galerkin method and the compactness arguments. This system is also an approximation to the compressible Navier--Stokes equations. It will help us to prove the existence of global weak solutions to the compressible Navier--Stokes equations with degenerate viscosity in 3D.

Spectral instability of characteristic boundary layer flows

Abstract: In this paper, we construct growing modes of the linearized Navier–Stokes equations about generic stationary shear flows of the boundary layer type in a regime of a sufficiently large Reynolds number: R→∞. Notably, the shear profiles are allowed to be linearly stable at the infinite Reynolds number limit, and so the instability presented is purely due to the presence of viscosity. The formal construction of approximate modes is well documented in physics literature, going back to the work of Heisenberg, C. C. Lin, Tollmien, Drazin, and Reid, but a rigorous construction requires delicate mathematical details, involving, for instance, a treatment of primitive Airy functions and singular solutions. Our analysis gives exact unstable eigenvalues and eigenfunctions, showing that the solution could grow slowly at the rate of et/R. The proof follows the general iterative approach introduced in our companion paper, avoiding having to deal with matching inner and outer asymptotic expansions, but instead involving a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators. Unlike in the channel flows, the spatial domain in the boundary layers is unbounded and the iterative scheme is likely to diverge due to the linear growth in the vertical variable. We introduce a new iterative scheme to simultaneously treat the singularity near critical layers and the asymptotic behavior of solutions at infinity. The instability of generic boundary layers obtained in this paper is linked to the emergence of Tollmien–Schlichting waves in describing the early stage of the transition from laminar to turbulent flows.

A superconvergent HDG method for the Incompressible Navier-Stokes Equations on general polyhedral meshes

Abstract: We present a superconvergent hybridizable discontinuous Galerkin (HDG) method for the steady-state incompressible Navier-Stokes equations on general polyhedral meshes. For arbitrary conforming polyhedral mesh, we use polynomials of degree k+1, k, k to approximate the velocity, velocity gradient and pressure, respectively. In contrast, we only use polynomials of degree k to approximate the numerical trace of the velocity on the interfaces. Since the numerical trace of the velocity field is the only globally coupled unknown, this scheme allows a very efficient implementation of the method. For the stationary case, and under the usual smallness condition for the source term, we prove that the method is well defined and that the global L2-norm of the error in each of the above-mentioned variables and the discrete H1-norm of the error in the velocity converge with the order of k+1 for k>=0. We also show that for k>=1, the global L2-norm of the error in velocity converges with the order of k+2. From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this method achieves optimal convergence for all the above-mentioned variables in L2-norm for k>=0, superconvergence for the velocity in the discrete H1-norm without postprocessing for k>=0, and superconvergence for the velocity in L2-norm without postprocessing for k>=1.

A new class of exact solutions for three-dimensional thermal diffusion equations

Abstract: A new class of exact solutions has been obtained for three-dimensional equations of themal diffusion in a viscous incompressible liquid. This class enables the description of the temperature and concentration distribution at the boundaries of a liquid layer by a quadratic law. It has been shown that the solutions of the linearized set of thermal diffusion equations can describe the motion of a liquid at extreme points of hydrodynamic fields. A generalization of the classic Couette flow with a quadratic temperature and concentration distribution at the lower boundary has been considered as an example. The application of the presented class of solutions enables the modeling of liquid counterflows and the construction of exact solutions describing the flows of dissipative media.

An adjoint-based approach for finding invariant solutions of Navier–Stokes equations

Abstract: We consider the incompressible Navier–Stokes equations with periodic boundary conditions and time-independent forcing. For this type of flow, we derive adjoint equations whose trajectories converge asymptotically to the equilibrium and travelling-wave solutions of the Navier–Stokes equations. Using the adjoint equations, arbitrary initial conditions evolve to the vicinity of a (relative) equilibrium at which point a few Newton-type iterations yield the desired (relative) equilibrium solution. We apply this adjoint-based method to a chaotic two-dimensional Kolmogorov flow. A convergence rate of is observed, leading to the discovery of new steady-state and travelling-wave solutions at Reynolds number . Some of the new invariant solutions have spatially localized structures that were previously believed to exist only on domains with large aspect ratios. We show that one of the newly found steady-state solutions underpins the temporal intermittencies, i.e. high energy dissipation episodes of the flow. More precisely, it is shown that each intermittent episode of a generic turbulent trajectory corresponds to its close passage to this equilibrium solution.

Abridged Continuous Data Assimilation for the 2D Navier–Stokes Equations Utilizing Measurements of Only One Component of the Velocity Field

Abstract: We introduce a continuous data assimilation (downscaling) algorithm for the two-dimensional Navier–Stokes equations employing coarse mesh measurements of only one component of the velocity field. This algorithm can be implemented with a variety of finitely many observables: low Fourier modes, nodal values, finite volume averages, or finite elements. We provide conditions on the spatial resolution of the observed data, under the assumption that the observed data is free of noise, which are sufficient to show that the solution of the algorithm approaches, at an exponential rate asymptotically in time, to the unique exact unknown reference solution, of the 2D Navier–Stokes equations, associated with the observed (finite dimensional projection of) velocity.

A Dual-Porosity-Stokes Model and Finite Element Method for Coupling Dual-Porosity Flow and Free Flow

Abstract: In this paper, we propose and numerically solve a new model considering confined flow in dual-porosity media coupled with free flow in embedded macrofractures and conduits. Such situation arises, for example, for fluid flows in hydraulic fractured tight/shale oil/gas reservoirs. The flow in dual-porosity media, which consists of both matrix and microfractures, is described by a dual-porosity model. And the flow in the macrofractures and conduits is governed by the Stokes equation. Then the two models are coupled through four physically valid interface conditions on the interface between dual-porosity media and macrofractures/conduits, which play a key role in a physically faithful simulation with high accuracy. All the four interface conditions are constructed based on fundamental properties of the traditional dual-porosity model and the well-known Stokes--Darcy model. The weak formulation is derived for the proposed model, and the well-posedness of the model is analyzed. A finite element semidiscretizati...

Stochastic Navier-Stokes equations for compressible fluids

Abstract: We study the Navier-Stokes equations governing the motion of an isentropic compressible fluid in three dimensions driven by a multiplicative stochastic forcing. In particular, we consider a stochastic perturbation of the system as a function of momentum and density. We establish existence of a so-called finite energy weak martingale solution under the condition that the adiabatic exponent satisfies gamma > 3/2. The proof is based on a four-layer approximation scheme together with a refined stochastic compactness method and a careful identification of the limit procedure.

Surface-only liquids

Abstract: We propose a novel surface-only technique for simulating incompressible, inviscid and uniform-density liquids with surface tension in three dimensions. The liquid surface is captured by a triangle mesh on which a Lagrangian velocity field is stored. Because advection of the velocity field may violate the incompressibility condition, we devise an orthogonal projection technique to remove the divergence while requiring the evaluation of only two boundary integrals. The forces of surface tension, gravity, and solid contact are all treated by a boundary element solve, allowing us to perform detailed simulations of a wide range of liquid phenomena, including waterbells, droplet and jet collisions, fluid chains, and crown splashes.