Terms Of Navier-Stokes Equations Research Articles

A novel CFD-DEM coupling method with moving boundary for simulating one-dimensional consolidation test

The conventional CFD-DEM coupling method fails to dynamically modify the position of the drainage boundary during the simulation of one-dimensional consolidation tests (1d CT), resulting in inaccuracies in the numerical results. In this study, a novel CFD-DEM coupling method with moving boundaries is proposed to simulate 1d CT where the fluid boundary adaption and internal mesh reconstruction are implemented with reference to the real-time morphology of the consolidation specimen. Additionally, the convective terms in Navier-Stokes equations are modified to account for the moving drainage boundary and the equation of state (EOS) is introduced to consider fluid compressibility. A series of 1d CTs based on the traditional fixed boundary and the moving boundary are subsequently conducted for comparison. Moreover, the influence of fluid compressibility and mesh coarseness on the consolidation characteristics is briefly discussed. The proposed method is verified to serve well in revealing the underlying microscopic mechanism of the Mandel-Cryer effect and complementing the traditional consolidation theories.

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

We are concerned with strong axisymmetric solutions to the 3D incompressible Navier-Stokes equations. We show that if 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 of a strong solution u on the time interval [0, T] is bounded by A≫1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A \\gg 1$$\\end{document} then for each k≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 0 $$\\end{document} there exists Ck>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_k>1$$\\end{document} such that ‖Dku(t)‖L∞(R3)≤t-(1+k)/2expexpACk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert D^k u (t) \\Vert _{L^\\infty (\\mathbb {R}^3)} \\le t^{-(1+k)/2}\\exp \\exp A^{C_k}$$\\end{document} for all t∈(0,T]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\in (0,T]$$\\end{document}.

Simulation of incompressible viscous flow using finite element method

This study focused on simulating incompressible viscous flow using the finite element method. This study used velocity and pressure as unknowns known as primitive variable formulations. Simulation of incompressible fluid flow poses numerical challenges due to the presence of nonlinear convective terms in Navier-Stokes equations and the incompressible nature of the fluid. If the connection between velocities and pressure is not discretized correctly, the stable and convergent velocities might be gained, but the obtained pressure will be oscillatory. To avoid these difficulties, continuous quadratic and additional cubic bubble functions will be used for the velocity field and linear functions for the pressure field. This kind of discretization satisfies the Ladyzhenskaya-Babuška-Brezzi (LBB) stability condition. Two cases of different Reynolds numbers were used to test the formulation's effectiveness. In the case of Reynolds number 0.12, no vortices were formed, suggesting that the flow is primarily governed by fluid friction, and fluid inertia has minimal effect. In the case of Reynolds number 120, the vortex formation, which is known as Von Kármán vortex street, appeared. These results concluded that the formulation using the finite element method is correct.

A new improved regularity criterion of solutions to Leray-𝛼-MHD model and Navier-Stokes equation

In this paper, we obtain the n n th-logarithmically improved regularity criterion of smooth solutions for the incompressible Leray- α \alpha -MHD model in terms of the magnetic field B B . Meanwhile, the new logarithmically improved regularity criterion for the 3D Navier-Stokes equation in terms of the pressure π \pi and gradient of velocity ∇ u \nabla u can also be established. Especially, we explore a new logarithmically improved Serrin’s criterion for the 3D Navier-Stokes equations, which improves the results of Lei and Zhou [Commn. Pure Appl. Anal. 12 (2013), pp. 2715–2719].

Regular sets and an 𝜖-regularity theorem in terms of initial data for the Navier–Stokes equations

We prove an $\epsilon$-regularity criterion for the 3D Navier-Stokes equations in terms of initial data. It shows that if a scaled local $L^2$ norm of initial data is sufficiently small around the origin, a suitable weak solution is regular in a set enclosed by a paraboloid started from the origin. The result is applied to the estimate of the regular set for local energy solutions with initial data in weighted $L^2$ spaces. We also apply this result to studying energy concentration near a possible blow-up time and regularity of forward discretely self-similar solutions.

Blow-up conditions of the incompressible Navier-Stokes equations in terms of sequentially defined Besov spaces

In this paper, we introduce sequentially defined Besov spaces adapted to the scaling invariant property of the 3D incompressible Navier-Stokes equations. We first show that these spaces are Banach spaces. We then establish regularity conditions in these spaces.

Study Of The Optimum Conditions And Parameters To Perform Simulation Of Low Temperature Capacitive Radio Frequency Argon Discharge

Capacitively coupled plasma (CCP) is used extensively in plasma processing, such as, plasma etching, deposition, and sputtering. Particle-in-Cell (PIC) is used to explore the discharge dynamics microscopically. Considering all species in PIC simulation is challenging. Also, considering various atomic transitions, e.g., ionization, recombination, excitation, and deexcitation is not doable. The state of the art is to carry out fluid simulation. We compare between PIC and fluid simulation for radio frequency CCP to reveal the effect of various approximations as assuming constant temperature, simplifying Navier-Stokes equations in terms of particles mobility and diffusivity, and considering artificial boundary conditions at the electrodes. The fluid model predicts qualitatively PIC results in few minutes. For Argon discharge in geometrically symmetric CCP, the plasma bulk is quasineutral. Over the electrodes, two sheaths are built up due to the difference between ion and electron fluxes. The dynamics of two sheaths are out of phase. In the fluid model, chemical reactions and atomic processes can be considered. Argon metastable states density is maximum in the plasma bulk. In semi dark sheaths, metastable states concentration is small. The proposed fluid model could be used as a simulation platform to find the optimum conditions and to interpret experimental results.

Regularity criteria via one directional derivative of the velocity in anisotropic Lebesgue spaces to the 3D Navier-Stokes equations

In this paper, we consider the regularity criterion for 3D incompressible Navier-Stokes equations in terms of one directional derivative of the velocity in anisotropic Lebesgue spaces. More precisely, it is proved that u becomes a regular solution if the ∂3u satisfies∫0T‖‖‖∂3u(t)‖Lx1p‖Lx2q‖Lx3rβ1+ln⁡(‖∂3u(t)‖L2+e)dt<∞, where 2β+1p+1q+1r=1 and 2<p,q,r≤∞,1−(1p+1q+1r)≥0.

A time averaged steady state method for the Navier–Stokes equations

AbstractThis work derives an incompressible variational multiscales time‐averaged Navier–Stokes (NS) formulation that aims at obtaining accurate steady state solutions. Rather than using the standard time instantaneous velocity and pressure, the new formulation devises a time averaging procedure based on rewriting and solving the NS equations in terms of the newly defined time‐averaged velocity and pressure. Hence, the method could be understood as a convenient change of variable so that the problem is rewritten directly in terms of the steady state quantities. The important advantage of such a point of view is that it can in principle be applied to any other formulation. Such time averaging procedure is complemented by two time step modification strategies in order to accelerate the convergence to the steady state. The guidelines of an integrated framework are presented in the article, starting with the description of the proposed numerical technique applied to general incompressible flows. The explanation is enhanced with a one‐dimensional (1D) nonlinear oscillator example. Several results are presented concerning analytical benchmarks, simulation of flows in laminar, transitional and turbulent regimes with and without an inherently steady solution.

High-fidelity simulations of gravity currents using a high-order finite-difference spectral vanishing viscosity approach

This numerical work investigates the potential of a high-order finite-difference spectral vanishing viscosity approach to simulate gravity currents at high Reynolds numbers. The method introduces targeted numerical dissipation at small scales through altering the discretisation of the second derivatives of the viscous terms in the incompressible Navier-Stokes equations to mimic the spectral vanishing viscosity (SVV) operator, originally designed for the regularisation of spectral element method (SEM) solutions of pure advection problems. Using a sixth-order accurate finite-difference scheme, the adoption of the SVV method is straightforward and comes with a negligible additional computational cost. In order to assess the ability of this high-order finite-difference spectral vanishing viscosity approach, we performed large-eddy simulations (LES) of a gravity current in a channelised lock-exchange set-up with our SVV model and with the well-known explicit static and dynamic Smagorinsky sub-grid scale (SGS) models. The obtained data are compared with a direct numerical simulation (DNS) based on more than 800 million mesh nodes, and with experimental measurements. A framework for the energy budget is introduced to investigate the behaviour of the gravity current. First, it is found that the DNS is in good agreement with the experimental data for the evolution of the front location and velocity field as well as for the stirring and mixing inside the gravity current. Secondly, the LES performed with less than 0.4% of the total number of mesh nodes compared to the DNS, can reproduce the main features of the gravity currents, with the SVV model yielding slightly more accurate results. It is also found that the dynamic Smagorinsky model performs better than its static version. For the present study, the static and dynamic Smagorinsky models are 1.8 and 2.5 times more expensive than the SVV model, because the latter does not require the calculation of explicit SGS terms in the Navier-Stokes equations nor spatial filtering operations.

Liutex-based vortex dynamics: A preliminary study

Vortex dynamics, with the possibility of efficient flow control, is explored in this study based on the new introduced vortex definition and identification system of Liutex. With the six core elements of vortex identification, including (1) absolute strength, (2) relative strength, (3) local rotational axis, (4) global rotational axis, (5) vortex core size and (6) vortex boundary, provided by the Liutex system, it is possible to numerically devise strategies, primarily by introducing additional source terms in Navier-Stokes equations, which we call Liutex force field model here, to control the vortex regions. Two methodologies of centripetal force model and counter-rotation force model are preliminarily investigated in a cavitating flow around two-dimensional Clark-Y hydrofoil. It is found that Liutex based models are capable of illustrating the vortex dynamics and possibly strengthening or weakening the vortices.

Conditional regularity for the 3D Navier-Stokes equations in terms of the middle eigenvalue of the strain tensor

In this paper, we consider the regularity criteria for the 3D incompressible Navier-Stokes equations involving the middle eigenvalue ( \begin{document}$ \lambda_2 $\end{document} ) of the strain tensor. It is proved that if \begin{document}$ \lambda^+_2 $\end{document} belongs to Multiplier space or Besov space, then the weak solution remains smooth on \begin{document}$ [0, T] $\end{document} , where \begin{document}$ \lambda^{+}_2 = \max\{\lambda_2, 0\} $\end{document} . These regularity conditions allows us to improve the result obtained by Miller [ 7 ].

Two-dimensional turbulent flow in a channel of constant width

The paper deals with the classical problem about a flow of viscous incompressible fluid in an infinite channel of constant width. The problem is solved in the two-dimensional case, both in the laminar and turbulent flow regimes, using as a mathematical model the Navier-Stokes equations in terms of vorticity-stream functions. The presented algorithm is a semi-analytical iterative method, based on the construction of special Green’s functions in the form of expansion in a Fourier series over the transverse coordinate for the channel.

Numerical simulation of air flow in needle-to-cylinder electrohydrodynamic device

The paper presents the methodology and results of numerical simulation of the flow induced by corona discharge obtained with open-source environment Multiphysics Object-Oriented Simulation Environment (MOOSE) framework released by Idaho National Laboratory. Coupled system of governing partial differential equations is solved for the values of electric potential φ and space charge density ρq. The solution is used to calculate the spatial distribution of body force acting on the fluid as FEHD = ρq∇φ. The body force is subsequently introduced as a source term in Navier-Stokes equations solved in MOOSE for the value of fluid velocity. The results are presented as velocity profiles at the outlet of the flow generator and integrated to yield volumetric flow rate and the efficiency of the device. Comparison of the results with data available in literature reference for similar configuration shows satisfactory agreement.

Longtime behavior for 3D Navier-Stokes equations with constant delays

This paper investigates the longtime behavior of delayed 3D Navier-Stokes equations in terms of attractors. The study will strongly rely on the investigation of the linearized Navier-Stokes system, and the relationship between the discrete dynamical flow for the linearized system and the continuous flow associated to the original system. Assuming the viscosity to be sufficiently large, there exists a unique attractor for the delayed 3D Navier-Stokes equations. Moreover, the attractor reduces to a singleton set.

The use of dual reciprocity method for 2D laminar viscous flow

The paper presents the use of the dual reciprocity multidomain singular boundary method (SBMDR) for the solution of the laminar viscous flow problem described by Navier-Stokes equations. A homogeneous part of the solution is solved using a singular boundary method with the 2D Stokes fundamental solution - Stokeslet. The dual reciprocity approach has been chosen because it is ideal for the treatment of the nonhomogeneous and nonlinear terms of Navier-Stokes equations. The presented SBMDR approach to the solution of the 2D flow problem is demonstrated on a standard benchmark problem - lid-driven cavity.

Lid-driven cavity flow using dual reciprocity

The paper presents the use of the multi-domain dual reciprocity method of fundamental solutions (MD-MFSDR) for the analysis of the laminar viscous flow problem described by Navier-Stokes equations. A homogeneous part of the solution is solved using the method of fundamental solutions with the 2D Stokes fundamental solution Stokeslet. The dual reciprocity approach has been chosen because it is ideal for the treatment of the non-homogeneous and nonlinear terms of Navier-Stokes equations. The presented DR-MFS approach to the solution of the 2D flow problem is demonstrated on a standard benchmark problem - lid-driven cavity.

Object dependent optimization of hydrodynamic forces in liquid stream: Finite element analysis

The examination of liquids in motion refers to the study of hydrodynamics. The present work reports the extended novelty of the hydrodynamics. To be more specific, the flowing fluid stream admitting Newton's law of viscosity is considered in a smooth rectangular channel. The various typical shaped cylinders are placed fixed in between rectangular channel as an obstacle. The shape of obstacles includes the triangle, square, hexagon, octagon and circle. The no-slip condition is carried at both the upper and lower walls of the channel. The right wall as an outlet is specified with the Neumann condition. The fluid is initiated at an inlet of the channel with the two different class of velocity profiles, namely the constant velocity profile and the parabolic profile. The whole physical designed is controlled mathematically in terms of Navier-Stokes equations. The solution is proposed with the finite element method and for the discretization of flow narrating equations the LBB-stable finite element pair is utilized along with a hybrid meshing scheme. The primitive variables namely, the velocity and pressure are reported for each obstacle. The line integration around the outer surface of the triangle, square, hexagon, octagon and circular cylinders is carried for the evaluation of hydrodynamic forces. The statistical data for such hydrodynamic forces is recorded up-to nine various refinement levels. It is noticed that the assumption of the parabolic velocity profile is more realistic approach in comparison with the constant velocity profile. Further, the hydrodynamic forces namely, the drag and lift are found object dependent in a rectangular channel having viscous fluid. It is trusted that the optimized path from triangle to circular shaped obstacle will serve as a helping source for the proceeding studies of hydrodynamics.

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.

Numerical simulation of air flow in needle-to-cylinder electrohydrodynamic device

The paper presents the methodology and results of numerical simulation of the flow induced by corona discharge obtained with open-source environment Multiphysics Object-Oriented Simulation Environment (MOOSE) framework released by Idaho National Laboratory. Coupled system of governing partial differential equations is solved for the values of electric potential φ and space charge density ρq. The solution is used to calculate the spatial distribution of body force acting on the fluid as FEHD = ρq∇φ. The body force is subsequently introduced as a source term in Navier-Stokes equations solved in MOOSE for the value of fluid velocity. The results are presented as velocity profiles at the outlet of the flow generator and integrated to yield volumetric flow rate and the efficiency of the device. Comparison of the results with data available in literature reference for similar configuration shows satisfactory agreement.