Dimensional Incompressible Navier-Stokes Research Articles

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.

Singularities of certain finite energy solutions to the Navier-Stokes system

We continue and supplement studies from [G. Karch and X. Zheng, Discrete Contin. Dyn. Syst. 35 (2015), 3039-3057] on solutions to the three dimensional incompressible Navier-Stokes system which are regular outside a curve in \begin{document}$ \big(\gamma(t), t\big)\in \mathbb{R}^3\times [0, \infty) $\end{document} and singular on it. We revisit some of the existence results as well as some of the asymptotic estimates obtained in that work in order prove that those solutions belongs to the space \begin{document}$ C\big([0, \infty), L^2( \mathbb{R}^3)^3\big) $\end{document} .

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.

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.

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.

A BOUNDARY CONTROL PROBLEM FOR VORTICITY MINIMIZATION IN TIME-DEPENDENT 2D NAVIER-STOKES EQUATIONS

We deal with a boundary control problem for the vorticity minimization, in which the flow is governed by the time-dependent two dimensional incompressible Navier-Stokes equations. We derive a mathematical formulation and a process for an appropriate control along the portion of the boundary to minimize the vorticity motion due to the flow in the fluid domain. After showing the existence of an optimal solution, we derive the optimality system for which optimal solutions may be determined. The differentiability of the state solution in regard to the control parameter shall be conjunct with the necessary conditions for the optimal solutions.

A Regularity Criterion for the Navier-Stokes Equations in Terms of One Directional Derivative of the Velocity

We consider the regularity criterion for the three dimensional incompressible Navier-Stokes equations in terms of one directional derivative of the velocity. The result shows that if weak solution u satisfies $$\begin{aligned} \partial_{3}u\in L^{\frac{2}{1-r}}\bigl(0,T;\dot{M}^{p,\frac{3}{r}}\bigl( \mathbb{R}^{3}\bigr)\bigr) \end{aligned}$$ with 0

Sums of large global solutions to the incompressible Navier–Stokes equations

Let G be the (open) set of~$\dot H^{\frac 1 2}$ divergence free vector fields generating a global smooth solution to the three dimensional incompressible Navier-Stokes equations. We prove that any element of G can be perturbed by an arbitrarily large, smooth divergence free vector field which varies slowly in one direction, and the resulting vector field (which remains arbitrarily large) is an element of G if the variation is slow enough. This result implies that through any point in G passes an uncountable number of arbitrarily long segments included in G.

A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes

This paper proposes a hybrid vertex-centered finite volume/finite element method for solution of the two dimensional (2D) incompressible Navier-Stokes equations on unstructured grids. An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling. The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by joining the centroid of cells sharing the common vertex. For the temporal integration of the momentum equations, an implicit second-order scheme is utilized to enhance the computational stability and eliminate the time step limit due to the diffusion term. The momentum equations are discretized by the vertex-centered finite volume method (FVM) and the pressure Poisson equation is solved by the Galerkin finite element method (FEM). The momentum interpolation is used to damp out the spurious pressure wiggles. The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both velocity and pressure. The classic test cases, the lid-driven cavity flow, the skew cavity flow and the backward-facing step flow, show that numerical results are in good agreement with the published benchmark solutions.

Aerodynamic Analysis of a Localized Flexible Airfoil at Low Reynolds Numbers

AbstractA localized flexible airfoil at low Reynolds numbers is modeled and the aerodynamic performance is analyzed numerically. With characteristic based split scheme, a fluid solver for two dimensional incompressible Navier-Stokes equations is developed under the ALE framework, coupled with the theory of shallow arch, which is approximated by Galerkin method. Further, the interactions between the unsteady flow and the shallow arch are studied in detail. In particular, the effect of the self-excited vibration of the structure on aerodynamic performance of the airfoil is investigated deeply at various angles of attack. The results show that the lift-to-drag ratio has been increased greatly compared with the rigid airfoil. Finally, the relationship between the self-excited vibration and the evolution of the flow is analyzed using FFT tools.

Optimal Hydraulic Design and Numerical Simulation of Pumping Systems

Optimal hydraulic design is very important in the selection of pumping station design schemes. Through optimal hydraulic design of pumping system it is possible to generate better flow conditions for pump, to reduce hydraulic losses both in suction box and discharge passage and improve pumping system efficiency. Guiding principles and evaluation indexes for optimal hydraulic design of pumping system was proposed, and a case study was given to illustrate the effect of optimal design on the pumping system performances. The time-averaged three dimensional incompressible Navier-Stokes equations were closed by RNG turbulence model and SIMPLEC arithmetic was adopted to couple pressure and velocity fields. Computed results indicate that with the given water level data and within the limitations of civil construction dimensions, the flow conditions of pump in terms of bias angle and distribution uniformity of axial velocity are effectively improved through optimal design, and hydraulic losses of suction box and discharge passage are decreased by 0.115m and the pumping system efficiency are improved by 2.02%. The validity of optimal method by means of computational fluid dynamics is verified by a model pumping system test.