Dimensional Incompressible Navier-Stokes Equations Research Articles

Topological asymptotic expansion for the full Navier–Stokes equations

This paper is concerned with a topological sensitivity analysis for the two dimensional incompressible Navier–Stokes equations. We derive a topological asymptotic expansion for a shape functional with respect to the creation of a small geometric perturbation inside the fluid flow domain. The geometric perturbation is modeled as a small obstacle. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is discussed. The obtained results are valid for a large class of shape fonctions and arbitrarily shaped geometric perturbations. The established topological asymptotic expansion provides a useful tool for shape and topology optimization in fluid mechanics.

Pressure, Intermittency, Singularity

We give conditions for regularity of solutions of three dimensional incompressible Navier-Stokes equations based on the pressure and on structure functions.

Global existence of strong solutions to the kinetic Cucker-Smale model coupled with the two dimensional incompressible Navier-Stokes equations

<p style='text-indent:20px;'>In this paper, we investigate existence of global-in-time strong solutions to the Cauchy problem of the kinetic Cucker–Smale model coupled with the incompressible Navier–Stokes equations in the two dimensional space. By introducing a weighted Sobolev space and using the maximal regularity estimate on the linear non-stationary Stokes equations, we present a complete analysis on existence of global-in-time strong solutions to the coupled model, without any smallness assumptions on initial data.</p>

On the Existence of Leray-Hopf Weak Solutions to the Navier-Stokes Equations

We give a rather short and self-contained presentation of the global existence for Leray-Hopf weak solutions to the three dimensional incompressible Navier-Stokes equations, with constant density. We give a unified treatment in terms of the domains and the relative boundary conditions and in terms of the approximation methods. More precisely, we consider the case of the whole space, the flat torus, and the case of a general bounded domain with a smooth boundary (the latter supplemented with homogeneous Dirichlet conditions). We consider as approximation schemes the Leray approximation method, the Faedo-Galerkin method, the semi-discretization in time and the approximation by adding a Smagorinsky-Ladyžhenskaya term. We mainly focus on developing a unified treatment especially in the compactness argument needed to show that approximations converge to the weak solutions.

Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations

<p style='text-indent:20px;'>The stability and convergence of the Fourier pseudo-spectral method are analyzed for the three dimensional incompressible Navier-Stokes equation, coupled with a variety of time-stepping methods, of up to fourth order temporal accuracy. An aliasing error control technique is applied in the error estimate for the nonlinear convection term, while an a-priori assumption for the numerical solution at the previous time steps will also play an important role in the analysis. In addition, a few multi-step temporal discretization is applied to achieve higher order temporal accuracy, while the numerical stability is preserved. These semi-implicit numerical schemes use a combination of explicit Adams-Bashforth extrapolation for the nonlinear convection term, as well as the pressure gradient term, and implicit Adams-Moulton interpolation for the viscous diffusion term, up to the fourth order accuracy in time. Optimal rate convergence analysis and error estimates are established in details. It is proved that, the Fourier pseudo-spectral method coupled with the carefully designed time-discretization is stable provided only that the time-step and spatial grid-size are bounded by two constants over a finite time. Some numerical results are also presented to verify the established convergence rates of the proposed schemes.</p>

Selection of Quasi-stationary States in the Stochastically Forced Navier–Stokes Equation on the Torus

The stochastically forced vorticity equation associated with the two dimensional incompressible Navier-Stokes equation on $D_\delta:=[0,2\pi\delta]\times [0,2\pi]$ is considered for $\delta\approx 1$, periodic boundary conditions, and viscocity $0<\nu\ll 1$. An explicit family of quasi-stationary states of the deterministic vorticity equation is known to play an important role in the long-time evolution of solutions both in the presence of and without noise. Recent results show the parameter $\delta$ plays a central role in selecting which of the quasi-stationary states is most important. In this paper, we aim to develop a finite dimensional model that captures this selection mechanism for the stochastic vorticity equation. This is done by projecting the vorticity equation in Fourier space onto a center manifold corresponding to the lowest eight Fourier modes. Through Monte Carlo simulation, the vorticity equation and the model are shown to be in agreement regarding key aspects of the long-time dynamics. Following this comparison, perturbation analysis is performed on the model via averaging and homogenization techniques to determine the leading order dynamics for statistics of interest for $\delta\approx1$.

Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces

We study regularity criteria for the $d$-dimensional incompressible Navier-Stokes equations. We prove if $u\in L_{\infty}^tL_d^x((0,T)\times \mathbb{R}^d_+)$ is a Leray-Hopf weak solution vanishing on the boundary and the pressure $p$ satisfies a local condition $\|p\|_{L_{2-1/d}(Q(z_0,1)\cap (0,T)\times \mathbb{R}^d_+)}\leq K$ for some constant $K>0$ uniformly in $z_0$, then $u$ is regular up to the boundary in $(0,T)\times \mathbb{R}^d_+$. Furthermore, when $T=\infty$, $u$ tends to zero as $t\rightarrow \infty$. We also study the local problem in half unit cylinder $Q^+$ and prove that if $u\in L^t_{\infty}L^x_d(Q^+)$ and $ p\in L_{2-1/d}(Q^+)$, then $u$ is H\"{o}lder continuous in the closure of the set $Q^+(1/4)$. This generalizes a result by Escauriaza, Seregin, and \v{S}ver\'{a}k to higher dimensions and domains with boundary.

On the Reynolds time-averaged equations and the long-time behavior of Leray–Hopf weak solutions, with applications to ensemble averages

We consider the three dimensional incompressible Navier–Stokes equations with a non stationary source term , chosen in a suitable space. We prove the existence of global Leray–Hopf weak solutions and also that it is possible to characterize (up to sub-sequences) their long-time averages, which satisfy the Reynolds averaged equations, involving the additional Reynolds stress. Moreover, we show that the turbulent dissipation is bounded by the sum of the Reynolds stress work and the external turbulent fluxes, without any additional assumption, other than that of using Leray–Hopf weak solutions.Finally, in the last section we consider ensemble averages of solutions, associated to a set of different forces and we prove that the fluctuations continue to have a dissipative effect on the mean flow.

Regularity Criteria of The Incompressible Navier-Stokes Equations via Only One Entry of Velocity Gradient

In this paper we establish regularity conditions for the three dimensional incompressible Navier-Stokes equations in terms of one entry of the velocity gradient tensor, say for example, $$ \partial _{3}u_{3}$$ . We show that if $$\partial _{3}u_{3}$$ satisfies certain integrable conditions with respect to time and space variables in anisotropic Lebesgue spaces, then a Leray-Hopf weak solution is actually regular. The anisotropic Lebesgue space helps us to almost reach the Prodi-Serrin level 2 in certain special case. Moreover, regularity conditions on non-diagonal element of gradient tensor $$\partial _1 u_3$$ are also established, which covers some previous literature.

Vanishing viscosity limit for homogeneous axisymmetric no-swirl solutions of stationary Navier-Stokes equations

(−1)-homogeneous axisymmetric no-swirl solutions of three dimensional incompressible stationary Navier-Stokes equations which are smooth on the unit sphere minus the north and south poles have been classified. In this paper we study the vanishing viscosity limit of sequences of these solutions. As the viscosity tends to zero, some sequences of solutions Clocm converge to solutions of Euler equations on the sphere minus the poles, while for other sequences of solutions, transition layer behaviors occur. For every latitude circle, there are sequences which Clocm converge respectively to different solutions of the Euler equations on the spherical caps above and below the latitude circle. We give detailed analysis of these convergence and transition layer behaviors.

Global strong solutions to 3-D Navier-Stokes system with strong dissipation in one direction

We consider three dimensional incompressible Navier-Stokes equation $(NS)$ with different viscous coefficient in the vertical and horizontal variables. In particular, when one of these viscous coefficients is large enough compared to the initial data, we prove the global well-posedness of this system. In fact, we obtain the existence of a global strong solution to $(NS)$ when the initial data verify an anisotropic smallness condition which takes into account the different roles of the horizontal and vertical viscosity.

Selection of quasi-stationary states in the Navier–Stokes equation on the torus

The two dimensional incompressible Navier–Stokes equation on with , periodic boundary conditions, and viscosity is considered. Bars and dipoles, two explicitly given quasi-stationary states of the system, evolve on the time scale and have been shown to play a key role in its long-time evolution. Of particular interest is the role that δ plays in selecting which of these two states is observed. Recent numerical studies suggest that, after a transient period of rapid decay of the high Fourier modes, the bar state will be selected if , while the dipole will be selected if . Our results support this claim and seek to mathematically formalize it. We consider the system in Fourier space, project it onto a center manifold consisting of the lowest eight Fourier modes, and use this as a model to study the selection of bars and dipoles. It is shown for this ODE model that the value of δ controls the behavior of the asymptotic ratio of the low modes, thus determining the likelihood of observing a bar state or dipole after an initial transient period. Moreover, in our model, for all , there is an initial time period in which the high modes decay at the rapid rate , while the low modes evolve at the slower rate. The results for the ODE model are proven using energy estimates and invariant manifolds and further supported by formal asymptotic expansions and numerics.

On the global existence of columnar solutions of the Navier–Stokes equations

We consider columnar solutions of the three dimensional incompressible Navier–Stokes equations in a slab type domain. We call the solution u a columnar flow when for some scalar function γ and φ. In this paper, we obtain the global existence of columnar flows in Sobolev spaces if is nonnegative. We also show that a solution blows up in finite time if the initial data is radial and the negative part of is dominant. Furthermore, we present nontrivial exact blow-up solutions. These exact solutions blow up at every point of the domain and the blow-up rate of these solutions in the W1,p norm for is (T − t)−1 near the blow-up time t = T > 0.

Lagrangian Coherent Structures in Tandem Flapping Wing Hovering

Lagrangian Coherent Structures (LCS) of tandem wings hovering in an inclined stroke plane is studied using ImmersedBoundary Method (IBM) by solving two dimensional (2D) incompressible Navier-Stokes equations. Coherent structures responsible for the force variation are visualized by calculating Finite Time Lyapunov Exponents (FTLE), and vorticity contours. LCS is effective in determining the vortex boundaries, flow separation, and the wing-vortex interactions accurately. The effects of inter-wing distance and phase difference on the force generation are studied. Results show that in-phase stroking generates maximum vertical force and counter-stroking generates the least vertical force. In-phase stroking generates a wake with swirl, and counter stroking generates a wake with predominant vertical velocity. Counter stroking aids the stability of the body in hovering. As the hindwing operates in the wake of the forewing, due to the reduction in the effective Angle of Attack (AoA), the hindwing generates lesser force than that of a single flapping wing.

A new local regularity criterion for suitable weak solutions of the Navier–Stokes equations in terms of the velocity gradient

We study the partial regularity of suitable weak solutions to the three dimensional incompressible Navier–Stokes equations. There have been several attempts to refine the Caffarelli–Kohn–Nirenberg criterion (1982). We present an improved version of the CKN criterion with a direct method, which also provides the quantitative relation in Seregin’s criterion (2007).

On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ

One particular metric that generates the weak topology on the weak global attractor $\mathcal{A}_w$ of three dimensional incompressible Navier-Stokes equations is introduced and used to obtain an upper bound for the Kolmogorov entropy of $\mathcal{A}_w$. This bound is expressed explicitly in terms of the physical parameters of the fluid flow.

Decay estimates with sharp rates of global solutions of nonlinear systems of fluid dynamics equations

Consider the Cauchy problems for the $n$-dimensional incompressible Navier-Stokes equations \begin{eqnarray*} \frac{\partial{\bf u}}{\partial t}-\alpha\triangle{\bf u}+({\bf u}\cdot\nabla){\bf u}+\nabla p={\bf f}({\bf x},t),\qquad {\bf u}({\bf x},0)={\bf u}_0({\bf x}). \end{eqnarray*} In this system, the dimension $n\geq 3$, ${\bf u}({\bf x},t)=(u_1({\bf x},t),u_2({\bf x},t),\cdots,u_n({\bf x},t))$ and ${\bf f}({\bf x},t)=(f_1({\bf x},t),f_2({\bf x},t),\cdots,f_n({\bf x},t))$ are real vector valued functions of ${\bf x}=(x_1,x_2,\cdots,x_n)$ and $t$. Additionally, $\alpha>0$ is a positive constant. Suppose that the initial function and the external force satisfy appropriate conditions. &nbsp The main purpose of this paper is to make complete use of the uniform energy estimates of the global smooth solutions and couple together a well known Gronwall's inequality to improve the Fourier splitting method to accomplish the decay estimates with sharp rates. The decay estimates with sharp rates of the global smooth solutions of the Cauchy problems for the $n$-dimensional magnetohydrodynamics equations may be established very similarly.

Vanishing viscosity limits for the 3D Navier-Stokes equations with a slip boundary condition

The solvability and vanishing viscosity limit for three dimensional incompressible Navier-Stokes equations with a slip boundary condition were obtained. The proof of these results is based on standard energy estimates.

The Minkowski dimension of interior singular points in the incompressible Navier–Stokes equations

We study the possible interior singular points of suitable weak solutions to the three dimensional incompressible Navier–Stokes equations. We present an improved parabolic upper Minkowski dimension of the possible singular set, which is bounded by 95/63. The result also continue to hold for the three dimensional incompressible magnetohydrodynamic equations without any difficulty.

A staggered space–time discontinuous Galerkin method for the incompressible Navier–Stokes equations on two-dimensional triangular meshes

In this paper we propose a novel arbitrary high order accurate semi-implicit space–time discontinuous Galerkin method for the solution of the two dimensional incompressible Navier–Stokes equations on staggered unstructured triangular meshes. Isoparametric finite elements are used to take into account curved domain boundaries. The discrete pressure is defined on the primal triangular grid and the discrete velocity field is defined on an edge-based staggered dual grid. While staggered meshes are state of the art in classical finite difference approximations of the incompressible Navier–Stokes equations, their use in the context of high order DG schemes is novel and still quite rare. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse four-point block system for the scalar pressure, which is conveniently solved with a matrix-free GMRES algorithm. Note that the same space–time DG scheme on a collocated grid would lead to ten non-zero blocks per element, since substituting the discrete velocity into the discrete continuity equation on a collocated mesh would involve also neighbors of neighbors. From numerical experiments we find that our linear system is well-behaved and that the GMRES method converges quickly even without the use of any preconditioner, which is a unique feature in the context of high order implicit DG schemes. A very simple and efficient Picard iteration is then used in order to derive a space–time pressure correction algorithm that achieves also high order of accuracy in time, which is in general a non-trivial task in the context of high order discretizations for the incompressible Navier–Stokes equations. The flexibility and accuracy of high order space–time DG methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both, space and time. The use of a staggered grid allows to avoid the use of Riemann solvers in several terms of the discrete equations and significantly reduces the total stencil size of the linear system that needs to be solved for the pressure. The proposed method is verified for approximation polynomials of degree up to p=4 in space and time by solving a series of typical numerical test problems and by comparing the obtained numerical results with available exact analytical solutions or other numerical reference data.To the knowledge of the authors, this is the first time that a space–time discontinuous Galerkin finite element method is presented for the incompressible Navier–Stokes equations on staggered unstructured grids.