Terms Of Navier-Stokes Equations Research Articles

A high order MOOD method for compressible Navier-Stokes equations: application to hypersonic viscous flows

A very high-order finite volumes numerical method is designed for the simulation of compressible Navier-Stokes equations on 2D unstructured meshes. This scheme is based on the MOOD methods described for Euler's equations, it is an interesting alternative in the design of a scheme adapted to accurate simulations of flows with discontinuities, in all the domain. The main originality of our method is to include the viscosity/diffusion terms of Navier-Stokes equations. These terms may be discretised with the same accuracy of convection terms, though we will restrict ourselves to second-order here. It permits to treat the hypersonic viscous interactions with high accuracy. Numerical experiments are conducted to demonstrate the performance of the proposed method.

A high order MOOD method for compressible Navier-Stokes equations: application to hypersonic viscous flows

A very high-order finite volumes numerical method is designed for the simulation of compressible Navier-Stokes equations on 2D unstructured meshes. This scheme is based on the MOOD methods described for Euler's equations, it is an interesting alternative in the design of a scheme adapted to accurate simulations of flows with discontinuities, in all the domain. The main originality of our method is to include the viscosity/diffusion terms of Navier-Stokes equations. These terms may be discretised with the same accuracy of convection terms, though we will restrict ourselves to second-order here. It permits to treat the hypersonic viscous interactions with high accuracy. Numerical experiments are conducted to demonstrate the performance of the proposed method.

Towards a “moving-point” formulation for the modelling of oscillation-mark formation in the continuous casting of steel

In the continuous casting of steel, solidification begins at a triple point where solid steel, molten steel and molten flux meet; the motion of this point determines how surface defects known as oscillation marks (OSMs) are formed. Here, under a number of simplifying assumptions, we derive an asymptotic model in 15 dimensionless parameters that describes the relevant momentum and heat transfer for the process, and involves both surface tension at the meniscus between molten flux and molten steel and solidification; further development couples an earlier lubrication-theory model for the region below the triple point to a reduced model for the region above that is based around a regular perturbation treatment of the Navier Stokes equations in terms of the capillary number. The resulting problem is then broken up into a hierarchy of four sub-problems; the first one – for the velocity field in the molten flux – is considered in depth. A numerical algorithm is developed for an isothermal situation in which the triple point moves only perpendicular to the casting direction; this involves the solution of a novel “moving point” problem to determine motion of the triple point. Comparison of model and experimental results indicates that this reduced model could produce OSMs having the observed depth if the triple point were not too far below the top of the meniscus; otherwise, the computed marks would be too deep. The analysis also tentatively relates the location of the triple point to the fold- and overflow-type OSMs that generally form in practice.

A complex-valued first integral of Navier-Stokes equations: Unsteady Couette flow in a corrugated channel system

For a two-dimensional incompressible viscous flow, a first integral of the governing equations of motion is constructed based on a reformulation of the unsteady Navier-Stokes equations in terms of complex variables and the subsequent introduction of a complex potential field; complementary solid and free surface boundary conditions are formulated. The methodology is used to solve the challenging problem of unsteady Couette flow between two sinusoidally varying corrugated rigid surfaces utilising two modelling approaches to highlight the versatility of the first integral. In the Stokes flow limit, the results obtained in the case of steady flow are found to be in excellent agreement with corresponding investigations in the open literature. Similarly, for unsteady flow, the results are in accord with related investigations, exploring material transfer between trapped eddies and the associated bulk flow, and vice versa. It is shown how the work relates to the classical complex variable method for solving the biharmonic problem and perspectives are provided as to how the first integral may be further utilised to investigate other fluid flow features.

Local Method of Approximate Particular Solutions for Two-Dimensional Unsteady Incompressible Flow

A meshless local method of approximated particular solutions (LMAPS) is used to analyze incompressible fluid flow in a two dimensionalcavity. The method solves the incompressible Navier-Stokes equations in terms of theprimitive variables using the fractional step scheme. The basic equations are derived via interpolation using integrated multiquadrics radial basis functions. Lid-driven cavity benchmark case for various Reynolds numbers is presented in the article. The procedure produces stable solutions with results comparable to those in literature.

A step-by-step global crack-tracking approach in E-FEM simulations of quasi-brittle materials

The numerical simulation of two-dimensional fracture processes of quasi-brittle materials by means of the Embedded Finite Element Method is dealt with. Attention is paid to the coupling with the global crack-tracking strategy, which has been proposed in the literature in the form of a heat conduction-like problem. It turns out that the stiffness-like matrix associated with this formulation is singular and a numerical perturbation has to be introduced in order to overcome the ill-posedness of the problem. The dependence of the solution on this parameter may represent a limitation for the global tracking approach. Furthermore, it is found that if the root of each discontinuity is not updated during an incremental analysis, a loss of continuity of the crack path may appear when principal stress directions rotate. This paper aims to provide a solution to the aforementioned issues. An alternative mathematical formulation of the problem is thus given in terms of Navier-Stokes equations, linking the diffusive contribution to a characteristic mesh length. Additionally, a modified crack-tracking algorithm, considering the evolution of the root for the identification of the crack path, is proposed. The numerical assessment of the proposed tracking strategy is reported by means of benchmark tests at the structural level.

Streamwise-body-force-model for rapid simulation combining internal and external flow fields

A streamwise-body-force-model (SBFM) is developed and applied in the overall flow simulation for the distributed propulsion system, combining internal and external flow fields. In view of axial stage effects, fan or compressor effects could be simplified as body forces along the streamline. These body forces which are functions of local parameters could be added as source terms in Navier-Stokes equations to replace solid boundary conditions of blades and hubs. The validation of SBFM with uniform inlet and distortion inlet of compressors shows that pressure performance characteristics agree well with experimental data. A three-dimensional simulation of the integration configuration, via a blended wing body aircraft with a distributed propulsion system using the SBFM, has been completed. Lift coefficient and drag coefficient agree well with wind tunnel test results. Results show that to reach the goal of rapid integrated simulation combining internal and external flow fields, the computational fluid dynamics method based on SBFM is reasonable.

An Exact Solution of Non-Newtonian Peristaltic Flow in a Tube : Rabinowitsch Fluid Model

With the development of medical science, non-Newtonian fluids have taken on added significance with a variety of applications in real life. The flow of non-Newtonian fluids in tubes and pipes plays a vital role in daily life. The examples of such applications are medical instruments, human body, machines, etc. The study of such mechanisms with the Newtonian fluids has not been found satisfactory due to nonlinear behavior of stress strain relationship. In the present analysis, the study of peristaltic flow in a tube has been carried out taking into account the non-Newtonian fluid : Rabinowitsch fluid model. Considering the significant terms in Navier-Stokes equations, solutions have been derived for fluid flow in axial direction in terms of pressure gradient. Using the continuity of flow, and exact solution has been derived for fluid pressure at wall. To establish the applicability of the solution, results for pressure rise at wall, pressure gradient and streamlines have been presented graphically.

Numerical Analysis of Behavior of High-Viscosity Electromagnetic Fluid Using a Coupled Method of Particle Method and FEM

Electromagnetic fluid, which is usually a suspension of extremely fine particles of metal in carrier fluid, shows unique behaviors such as large deformation or a change of material properties in electromagnetic field. This paper deals with elongation and sharpening of electromagnetic fluid with extremely high viscosity in electrostatic field. This behavior is very difficult to calculate with the finite-element model (FEM) because it includes intensive deformation and a large change of interfacial surface of fluid. Therefore, we propose an improved 3-D coupled method of FEM and moving particle semi-implicit (MPS). Electrostatic field analysis with the FEM and fluid analysis with MPS is weakly coupled. This paper implicitly calculates viscous terms in Navier-Stokes equations and succeeds in accurately calculating deformation of high-viscosity electromagnetic fluid.

A new regularity criterion for the Navier-Stokes equations in terms of the two components of the velocity

This paper establishes a new regularity criterion for the Navier-Stokes equa- tions in terms of two velocity components. We show that if the two velocity components e u = (u1, u2, 0) satisfy Z T 0 k ˜ u(s)k 2 ˙ B 0,¥ ds < ¥,

Stochastic PDEs and Lack of Regularity

We review results on the existence and uniqueness for a surface growth model with or without space–time white noise. If the surface is a graph, then this model has striking similarities to the three dimensional Navier-Stokes equations in terms of energy estimates and scaling properties, and in both models the question of uniqueness of global weak solutions remains open. In the physically relevant dimension $d=2$ and with the physically relevant space–time white noise driving the equation, the direct fixed-point argument for mild solutions fails, as there is not sufficient regularity for the stochastic forcing. The situation is the simplest case where the method of regularity structures introduced by Martin Hairer can be applied, although we follow here a significantly simpler approach to highlight the key problems. Using spectral Galerkin method or any other type of regularization of the noise, one can give a rigorous meaning to the stochastic PDE and show existence and uniqueness of local solutions in that setting. Moreover, several types of regularization seem to yield all the same solution. We finally comment briefly on possible blow up phenomena and show with a simple argument that many complex-valued solutions actually do blow up in finite time. This shows that energy estimates alone are not enough to verify global uniqueness of solutions. Results in this direction are known already for the 3D-Navier Stokes by Li and Sinai, treating complex valued solutions, and more recently by Tao by constructing an equation of Navier-Stokes type with blow up.

Intrinsic charge and spin conductivities of doped graphene in the Fermi-liquid regime

The experimental availability of ultra-high-mobility samples of graphene opens the possibility to realize and study experimentally the "hydrodynamic" regime of the electron liquid. In this regime the rate of electron-electron collisions is extremely high and dominates over the electron-impurity and electron-phonon scattering rates, which are therefore neglected. The system is brought to a local quasi-equilibrium described by a set of smoothly varying (in space and time) functions, {\it i.e.} the density, the velocity field and the local temperature. In this paper we calculate the charge and spin conductivities of doped graphene due solely to electron-electron interactions. We show that, in spite of the linear low-energy band dispersion, graphene behaves in a wide range of temperatures as an effectively Galilean invariant system: the charge conductivity diverges in the limit $T \to 0$, while the spin conductivity remains finite. These results pave the way to the description of charge transport in graphene in terms of Navier-Stokes equations.

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

English

In this paper, a second order Adams-Bashforth method is proposed to simulate four-sided square lid-driven cavity flow. The convective term and diffusive term in Navier-Stokes equations are solved using a finite central difference scheme and the unsteady term is solved using the Adams-Bashforth method. All of the flow is simulated at below critical Reynolds numbers in a square cavity with an isothermal condition and the same speed for all walls. In this study, the flow structure for a four-sided lid-driven cavity with its four vortices and a graph of the velocity profile along the center of the cavity are presented. In addition, we also study the effect of the Reynolds number on the development of vortices in the cavity. We find that the Reynolds number has a dominant effect on the flow structure in the cavity. The computed results also show good agreement with the published data.

A regularity criterion for the Navier-Stokes equations in terms of the pressure gradient

Abstract The incompressible three-dimensional Navier-Stokes equations are considered. A new regularity criterion for weak solutions is established in terms of the pressure gradient.

Dynamical Study of a Molten Boundary Layer Ejected by Laminar Gas Flow

We consider in the present work the fusion laser cutting of stainless steel sheets under a nitrogen laminar gas jet. The molten metal is treated as a laminar and steady viscous incompressible fluid. The mathematical model describing our problem is set in terms of Navier-Stokes equations, solved numerically using the finite differences method, where the effect of the gas jet velocity on the molten boundary layer is considered. The generated shear stress occurring on the gas-liquid interface and its contribution in the momentum is carried out, and it is found that when the skin friction and the shear stress decrease, the thickness and the velocity at the edge of the molten boundary layer increase along the kerf surface. The layer thickness reduces when the assisting gas velocity is increased.

Non-equilibrium Shock Structure Calculation with High-order Modified Navier–Stokes Equations

Abstract High-order modified Navier-Stokes (NS) equations are introduced to describe the transitional flow derived from Burnett equations considering the inaccurate simulation of shock structure using traditional NS continuum methods, particularly at high Mach numbers. This process is performed by defining the effective transportation coefficient, which is associated with Mach number, molecule propriety, and continuum failure criterion. Within this method, the linear Stokes and Fourier relationships are replaced by nonlinear constitutive relationships to describe transitional flow. The traditional NS equation formulation is preserved. The shock structures of both monatomic and diatomic gases are concentrated with different Mach numbers to validate the accuracy and effectiveness of the derived method. The comparison of shock structure with DSMC and Burnett equations shows that the high-order modified NS equations obtain better results than traditional NS equations in terms of shock wave, particularly at high Mach numbers. The modified NS equations also capture the significant features of Burnett equations in shock structure simulation.

Study on Electrohydrodynamic Capillary Instability with Heat and Mass Transfer

The effect of an axial electric field on the capillary instability of a cylindrical interface in the presence of heat and mass transfer has been investigated using viscous potential flow theory. In viscous potential flow, the viscous term in Navier-Stokes equation vanishes as vorticity is zero but viscosity is not zero. Viscosity enters through normal stress balance in the viscous potential flow theory and tangential stresses are not considered. A dispersion relation that accounts for the growth of axisymmetric waves is derived and stability is discussed theoretically as well as numerically. Stability criterion is given by critical value of applied electric field as well as critical wave number. Various graphs have been drawn to show the effect of various physical parameters such as electric field, heat transfer capillary number, conductivity ratio, permittivity ratio on the stability of the system. It has been observed that the axial electric field and heat and mass transfer both have stabilizing effect on the stability of the system. Keywords—Capillary instability, Viscous potential flow, Heat and mass transfer, Axial electric field.

Velocity relaxation of an ellipsoid immersed in a viscous incompressible fluid

The motion of an ellipsoid in a viscous incompressible fluid, caused by a small time-dependent applied force, is studied on the basis of the linearized Navier-Stokes equations in terms of the frequency-dependence of the friction tensor. The asymptotic behavior of the hydrodynamic force at high frequency contains a term linear in frequency, with an added mass coefficient, and a term proportional to the square root of frequency, with a Basset coefficient. The latter is calculated from an expression derived by Batchelor [An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, 1967)]. A simple approximate three-pole expression is proposed for the frequency-dependent admittance for each principal direction, embodying added mass, particle mass, the steady state friction coefficient, and the Basset coefficient. It is suggested that a remaining unknown coefficient in the expression be determined by experiment, computer simulation, or numerical solution of an integral equation derived by Pozrikid...

Model of Heat and Mass Transfer in an Industrial Counter-Current Spray-Drying Tower

The paper presents a unique methodology of CFD modeling of heat and mass transfer during a counter-current spray-drying process based on the concept of a negative heat source reflecting the energy necessary for evaporation of water. The volumetric internal heat source term in Navier-Stokes equations describing heat transfer was used to determine energy transfer from explicit to latent heat. Total power consumption during the drying process was estimated on the basis of extensive industrial measurements of air temperature distribution along the axis and radius during regular tower operation. The probability function to control the distribution of density of power consumption as a function of the distance from the atomizers was determined by the trial and error method to fit experimental temperature measurements in the industrial tower. This approach allowed us to carry out 3D CFD accurate calculations of the counter-current spray-drying process analyzing continuous phase only. General methodology applied in the developed CFD model of heat and mass transfer is universal and can be applied for scaling up of the countercurrent spray-drying process.