3D Navier-Stokes Equations Research Articles

Uniqueness of the invariant measure and asymptotic stability for the 2D Navier-Stokes equations with multiplicative noise

Uniqueness of the invariant measure and asymptotic stability for the 2D Navier-Stokes equations with multiplicative noise

Some regularity criteria via the signed curl operators for the 3D Navier-Stokes equations

Some regularity criteria via the signed curl operators for the 3D Navier-Stokes equations

Pullback Exponential Attractors with Explicit Fractal Dimensions for Non-Autonomous Partial Functional Differential Equations

The aim of this paper is to propose a new method to construct pullback exponential attractors with explicit fractal dimensions for non-autonomous infinite-dimensional dynamical systems in Banach spaces. The approach is established by combining the squeezing properties and the covering of finite subspace of Banach spaces, which generalize the method established for autonomous systems in Hilbert spaces (Eden A, Foias C, Nicolaenko B, and Temam R Exponential attractors for dissipative evolution equations, Wiley, New York, 1994). The method is especially effective for non-autonomous partial functional differential equations for which phase space decomposition based on the exponential dichotomy of the linear part or variation techniques are available for proving squeezing property. The theoretical results are illustrated by applications to several specific non-autonomous partial functional differential equations, including a retarded reaction–diffusion equation, a retarded 2D Navier–Stokes equation and a retarded semilinear wave equation. The constructed exponential attractors possess explicit fractal dimensions which do not depend on the entropy number but only on some inner characteristics of the studied equations including the spectra of the linear part and the Lipschitz constants of the nonlinear terms and hence do not require the smooth embedding between two spaces in the previous work.

Geometric regularity criteria for the Navier–Stokes equations in terms of velocity direction

In this paper, we are inspired by a famous result by Constantin and Fefferman who proved that a simple geometrical assumption on the direction of the vorticity leads to the regularity of weak solutions of the 3D Navier–Stokes equations. We show that the same result can be achieved if the vorticity direction is replaced by the velocity direction. We further strengthen this result and prove that in fact it is not necessary to consider the velocity direction in all close space points but only in the points whose distance equals to a small positive number dependant on the data. In the second part of the paper, we extend a result by Berselli and C rdoba concerning the role of the helicity for the regularity of the weak solutions of the Navier–Stokes equations.

Effect of discrete heating on the key parameters and entropy generation in a corrugated enclosure filled with hybrid nanofluid

This research work aims to study the effect of discrete heating on natural convective heat transfer, fluid flow, and entropy generation within a corrugated enclosure filled with water-based molybdenum disulfide (MoS2)–silicon dioxide (SiO2) hybrid nanofluid. In the study, various heights of the discrete heaters in terms of aspect ratios ( AR = 0.2 , 0.4 , 0.6 , 0.8 ) are considered for various Rayleigh numbers ( Ra = 10 3 − 10 6 ) with varying nanoparticle volume fractions ( ϕ = 0 % − 4 % ). Stream function-vorticity formulations of two-dimensional (2D) Navier-Stokes equations and energy equation are used as the governing equations, and a higher-order compact (HOC) finite difference scheme is employed to discretize those equations. The results obtained from the current developed code are first validated with the existing numerical and experimental data. Finally, the present computed results are studied, both qualitatively and quantitatively, in terms of streamlines, isotherms, entropy generations, average Nusselt numbers, and Bejan numbers. At Ra = 10 6 and the aspect ratio of heater length AR = 0.2 , a different and interesting flow and heat transfer phenomenon is observed. The results show that an increase in the aspect ratio of heater length from 0.2 to 0.8 decreases the average Nusselt number and total entropy generation by 57.49% and 13.76%, respectively, for Ra = 10 6 and ϕ = 4 % . In addition, the effect of nanoparticle concentration on total entropy generation increases up to 73.40% when the aspect ratio of heater length increases to 0.8.

Gromov–Hausdorff stability of global attractors for the 3D Navier–Stokes equations with damping

Gromov–Hausdorff stability of global attractors for the 3D Navier–Stokes equations with damping

Logistic Regression Analytically Solves the 3D Navier Stokes Equations

The velocity $\mathbf{u=}2\nu P\left(  1-P\right)  \left(
 \begin{array}
     [c]{c}%
     \mu_{1}\\
     -\mu_{2}\\
     \mu_{3}%
 \end{array}
 \right)  $\ and pressure $p,$ where $p,\mathbf{u}\in C^{\infty}(\mathbb{R}^{3}\times\lbrack0,\infty)),P=\frac{1}{1+e^{\mu_{1}x+\mu_{2}y+\mu_{3}z-kt}}$
 have been verified and validated in commercial softwares that has implemented
 Vector Calculus.
 
 ...

Mean attractors and invariant measures of locally monotone and generally coercive SPDEs driven by superlinear noise

We study the global solvability, mean attractors and invariant measures for an abstract locally monotone and generally coercive SPDEs driven by infinite-dimensional superlinear noise defined in a dual space of intersection of finitely many Banach spaces. The main feature of this abstract system is that it covers a larger class of fundamental models which are included or not included previously. Under an extended locally monotone variational setting, we establish the global well-posedness, Itô's energy equality and existence of mean random attractors in some high-order Bochner spaces. The existence, uniqueness, support, (high-order and exponential) moment estimates, ergodicity, (pointwise and Wasserstein-type) exponentially mixing and asymptotic stability of invariant measures and evolution systems of measures are discussed for autonomous and nonautonomous stochastic equations. A stopping time technique is used to prove the convergence of solutions in probability in order to overcome the difficulty caused by the local monotonicity and superlinear growth of the coefficients. Our abstract results and unified methods are expected to be applied to various types of SPDEs like 2D Navier-Stokes equations, 2D MHD equations, 2D magnetic Bénard problem, Burgers type equations, 3D Leray α-model, convective Brinkman-Forchheimer equations, fractional (s,p)-Laplacian equations with monotone nonlinearities of polynomial growth of arbitrary order, and others.

Uniform approximation of 2D Navier-Stokes equations with vorticity creation by stochastic interacting particle systems

We consider a stochastic interacting particle system in a bounded domain with reflecting boundary, including creation of new particles on the boundary prescribed by a given source term. We show that such particle system approximates 2D Navier–Stokes equations in vorticity form and impermeable boundary, the creation of particles modeling vorticity creation at the boundary. Kernel smoothing, more specifically smoothing by means of the Neumann heat semigroup on the space domain, allows to establish uniform convergence of regularized empirical measures to (weak solutions of) Navier–Stokes equations.

Exponential stability and stabilization of a stochastic 2D nonlocal Cahn-Hilliard-Navier-Stokes equations with multiplicative noise

In this article, we study the asymptotic stability of the unique strong solution to a stochastic version of a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. The model consists of the 2D Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. We establish the existence, uniqueness and regularity results for the stationary solution of the stochastic 2D nonlocal Cahn-Hilliard Navier-Stokes equations. Furthermore, we prove that under some conditions on the forcing terms, the strong solution converges exponentially in the mean square and almost surely exponentially to the stationary solution. Finally, we also prove a result related to the stabilization of the stochastic 2D nonlocal Cahn-Hilliard-Navier-Stokes equations.

Numerical Study of Turbulent Wake of Offshore Wind Turbines and Retention Time of Larval Dispersion

Offshore Wind Farm (OWF) foundations are considered to have a potential impact on the larval dispersion of benthic species. This study focused on OWFs’ impacts on larval dispersion, considering factors such as the foundation type, flow velocity, flow direction, and release type using numerical modelling. At the scale of monopile and gravity-based foundations, a combination of two numerical models was used: the Eulerian model (OpenFOAM), solving the 3D Navier–Stokes equations for computing the hydrodynamics, and the Lagrangian model (Ichthyop), solving the advection–diffusion equation for the larval dispersion simulations. The validation model tests were evaluated with experimental data as a first step of the study. Accurate results were achieved, yielding a Turbulent Kinetic Energy (TKE) Root-Mean-Squared Error (RMSE) in the range of 6.82–8.27 ×10−5kg/m·s2 within the refined mesh, with a coefficient of determination (R2) approaching unity. For the second phase, more-realistic simulations were modelled. Those simulations demonstrated turbulent wakes downstream of the foundations and horseshoe vortex formations near the bottom. A larval dispersion was simulated using passive particles’ motion. Vertical flumes in the wake with particles experiencing both upward and downward motions, impacting the fall velocities of the particles, were observed. The influence of gravity-based foundations might lead to a stepping-stone effect with a retention time of up to 9 min, potentially allowing the settlement of competent larvae. In a similar geometry with an angular spring tide velocity, 0.4% of particles were trapped.

Effects of bulk viscosity, heat capacity ratio, and Prandtl number on the dispersion relationship of compressible flows

Here, we present the variation of the dispersion characteristics of the three-dimensional (3D) linearized compressible Navier–Stokes equation (NSE) to bulk viscosity ratio, specific heat ratio (γ), and Prandtl number (Pr). The 3D compressible NSE supports five types of waves, two vortical, one entropic, and two acoustic modes. While the vortical and entropic modes are non-dispersive, the acoustic modes are dispersive only up to a specific bifurcation wavenumber. We illustrate the characteristics and variation of relative (with respect to the vortical mode) diffusion coefficient for entropic and acoustic modes and a specially designed dispersion function for acoustic modes with depressed wavenumber η=KM/Re, the bulk viscosity ratio, γ, and Prandtl number Pr of the flow. Here, K, M, and Re denote the absolute wavenumber of disturbances, Mach number, and Reynolds number of the flow, respectively. At lower wavenumber components, the deviation of the dispersion function from the inviscid and adiabatic case is proportional to η2 at the leading order, and the relative diffusion coefficients increase linearly with bulk viscosity ratio and γ while varying inversely with Pr. With the increase in the bulk viscosity ratio, the shape and extent of the dispersion function alter significantly, and the change is more substantial for higher wavenumber components. The relative diffusion coefficient for entropic and acoustic modes shows contrasting variation with wavenumber depending upon bulk viscosity ratio, γ, and Pr. We also show by solving linearized compressible NSE that relatively significant evolution and radiation of acoustic and entropic disturbances occur when the bulk viscosity ratio is close to the corresponding critical value of maximum bifurcation wavenumber. Based on this criterion, we have presented an empirical relation for estimating bulk viscosity ratio depending upon γ and Pr, giving the corresponding range for obtaining relatively significant disturbance evolution.

Numerical Approach Based on Solving 3D Navier–Stokes Equations for Simulation of the Marine Propeller Flow Problems

The report presents the approach implemented in the Russian LOGOS software package for the numerical simulation of the marine propeller flow problems using unstructured computational meshes automatically generated by the mesh generator. This approach includes a computational model based on the Navier–Stokes equation system and written with respect to the physical process: the turbulent nature of flow with transient points is accounted using the Reynolds Averaged Navier–Stokes method and the k–ω SST model of turbulence by Menter along with the γ–Reθ (Gamma Re Theta) laminar-turbulent transition model; the Volume of Fluid method supplemented with the Schnerr–Sauer cavitation model is used to simulate the cavitation processes; a rotating propeller is simulated by a moving computational mesh and the GGI method to provide conformity of the solutions on adjacent boundaries of arbitrarily-shaped unstructured meshes of the two domains. The specific features of the numerical algorithms in use are described. The method validation results are given; they were obtained because of the problems of finding the performance curves of model-scale propellers in open water, namely the problems of finding the performance of propellers KP505 and IB without consideration of cavitation and the performance of propellers VP1304 and C5 under cavitation conditions. The paper demonstrates that the numerical simulation method presented allows for obtaining sufficiently accurate results to predict the main hydrodynamic characteristics for most modes of operation of the propellers.

Global existence and non-uniqueness for the Cauchy problem associated to 3D Navier–Stokes equations perturbed by transport noise

We show global existence and non-uniqueness of probabilistically strong, analytically weak solutions of the three-dimensional Navier–Stokes equations perturbed by Stratonovich transport noise. We can prescribe either: (i) any divergence-free, square integrable intial condition; or (ii) the kinetic energy of solutions up to a stopping time, which can be chosen arbitrarily large with high probability. Solutions enjoy some Sobolev regularity in space but are not Leray–Hopf.

Simulation of landslide tsunami in the Russian Far East based on 3D Navier–Stokes equations

The paper presents the results of modeling landslide tsunamis near the Kamchatka Peninsula in part of the Pacific Ocean. The paper describes the model based on the three-dimensional (3D) Navier–Stokes equations. The model is supplemented with the rheological relation based on the Bingham model to account for the rheology of landslide masses. The paper proposes a modification of the classical Bingham model with a non-zero yield strength, which implies that the medium is at rest or it moves as a solid (body) in the absence of a tension in the medium exceeding this limit. The application of the classical model is impossible within the framework of the used equation system. The paper proposes its modification, which consists in the possibility of changing the yield strength to zero value by adding a linear function to a given shear rate, until which the fluid flows as Newtonian and after reaching it, the flow conditions of the substance obeys Bingham’s law. An original algorithm is used to simulate waves in real water areas, it implements open boundary conditions. The algorithm is based on the use of a damping boundary layer that absorbs the kinetic energy of the incoming wave, which is taken into account using an additional source in the angular momentum equation. A method is proposed for determining the resistance coefficient, the value of which determines the intensity of absorption of the kinetic energy of the wave. The used mathematical model makes it possible to model in a single way the occurrence, propagation and rolling on shore of tsunami waves of landslide origin. The results of modeling of an underwater landslide in the waters of the Kamchatka Bay near the city of Ust-Kamchatsk are presented, taking into account bathymetric date. The paper includes the analysis of dependence of wave heights on the volume of landslide in the zone of its initial position and at several points of the coast, as well as sections of the coast (in particular on Bering Island), which may be most severely affected by the occurrence of landslide tsunamis in this water area.

Topological sensitivity analysis for the 3D nonlinear Navier–Stokes equations

This work is devoted to a topological asymptotic expansion for the nonlinear Navier–Stokes operator. We consider the 3D Navier–Stokes equations as a model problem and we derive a topological sensitivity analysis for a design function with respect to the insertion of a small obstacle inside the fluid flow domain. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is examined. The performed mathematical framework can be applied for a large class of design functions and arbitrarily shaped geometric perturbations. The obtained asymptotic formula can serve as a useful tool for solving a variety of topology optimization problems in fluid mechanics.

Bifurcation Analysis Software and Chaotic Dynamics for Some Problems in Fluid Dynamics Laminar–Turbulent Transition

The analysis of bifurcations and chaotic dynamics for nonlinear systems of a large size is a difficult problem. Analytical and numerical approaches must be used to deal with this problem. Numerical methods include solving some of the hardest problems in computational mathematics, which include system spectral and algebraic problems, specific nonlinear numerical methods, and computational implementation on parallel architectures. The software structure that is required to perform numerical bifurcation analysis for large-scale systems was considered in the paper. The software structure, specific features that are used for successful bifurcation analysis, globalization strategies, stabilization, and high-precision implementations are discussed. We considered the bifurcation analysis in the initial boundary value problem for a system of partial differential equations that describes the dynamics of incompressible ABC flow (3D Navier–Stokes equations). The initial stationary solution is characterized by the stability and connectivity to the main solutions branches. Periodic solutions were considered in view of instability transition problems. Finally, some questions of higher dimensional attractors and chaotic regimes are discussed.

Dynamics and robustness for the 2D Navier–Stokes equations with multi-delays in Lipschitz-like domains

This paper is concerned with the dynamics of the two-dimensional Navier–Stokes equations with multi-delays in a Lipschitz-like domain, subject to inhomogeneous Dirichlet boundary conditions. The regularity of global solutions and of pullback attractors, based on tempered universes, is established, extending the results of Yang, Wang, Yan and Miranville (Discrete Contin. Dyn. Syst. 41 (2021) 3343–3366). Furthermore, the robustness of pullback attractors when the delays, considered as small perturbations, disappear is also derived. The key technique in the proofs is the application of a retarded Gronwall inequality and a variable index for the tempered pullback dynamics, allowing to obtain uniform estimates and the compactness of the process.

Simulation of Drag Crisis in Flow Past a Circular Cylinder using 2D Computations

The unsteady incompressible 2D Navier-Stokes equations are solved using a finite element method with Streamline-Upwind/Petrov-Galerkin (SUPG) and Pressure-Stabilizing/Petrov- Galerkin (PSPG) stabilization terms for high Reynolds number (105 to I06) flow past a cylinder to stimulate drag-crisis. The mean drag coefficient for various Re are compared with experimental data. It is observed that the low at Re =106 separates much later, from the cylinder surface as compared to that at Re =105. The wake at Re =106 is narrower than that at Re = 105. The time-averaged flow field shows a standing bubble just downstream of the flow separation point for the Re = 105 flow. The bubble is not observed for the flow at Re = 106. The energy spectrum of the flow exhibits the characteristics of 2D isotropic turbulence.

Quantitative Control of Solutions to the Axisymmetric Navier-Stokes Equations in Terms of the Weak L3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^3$$\\end{document} Norm

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