System Of Navier-Stokes Equations Research Articles

Navier–Stokes Equation in a Cone with Cross-Sections in the Form of 3D Spheres, Depending on Time, and the Corresponding Basis

The work establishes the unique solvability of a boundary value problem for a 3D linearized system of Navier–Stokes equations in a degenerate domain represented by a cone. The domain degenerates at the vertex of the cone at the initial moment of time, and, as a consequence of this fact, there are no initial conditions in the problem under consideration. First, the unique solvability of the initial-boundary value problem for the 3D linearized Navier–Stokes equations system in a truncated cone is established. Then, the original problem for the cone is approximated by a countable family of initial-boundary value problems in domains represented by truncated cones, which are constructed in a specially chosen manner. In the limit, the truncated cones will tend toward the original cone. The Faedo–Galerkin method is used to prove the unique solvability of initial-boundary value problems in each of the truncated cones. By carrying out the passage to the limit, we obtain the main result regarding the solvability of the boundary value problem in a cone.

Numerical investigations on flow and heat transfer characteristics of a high-enthalpy double-cone in thermal and chemical non-equilibrium for hypersonic propulsion

There are strong interests in hypersonic propulsion communities in predicting the flow and heat transfer characteristics over a hypersonic vehicle. However, this is quite challenging due to the emergence of thermal and chemical out-of-equilibrium effects as prominent in a high-temperature flow. In this work, we conduct 2D numerical investigations on the hypersonic double cone flow in a surrounding of the stagnation enthalpy of 15.23 MJ/kg by solving the laminar Navier–Stokes equation systems with the vibrational non-equilibrium processes and seven species air reactions. We then conduct the sensitivities analyses on the flow and heat transfer features, as different transport models and different thermochemical models are applied. It is shown that the aerodynamic heat is more sensitive to the transport models than the flow, and the transport coefficients computed by the Gupta-Yos model are more suitable for predicting the double cone flow at the high enthalpy. On the other hand, the flow field of the high enthalpy double cone is more sensitive to the thermochemical models than the heat transfer on the wall excluding the peak position. Behind the detached shock, chemical reactions are found to enhance thermal or vibrational out-of-equilibrium effects. However, vibrational non-equilibrium effects are revealed to weaken the degree of molecule dissociation reactions. Two flow separations occur in the flow field computed by using the thermal non-equilibrium gas models, but the flows of thermal equilibrium gas models separate only once. The maximum values of wall parameters without chemical reactions are found to deviate greatly from the experimental measurements. The thermochemical out-of-equilibrium gas model can predict the separation region size closer to the experiment than the two reported numerical results. The wall static pressure and heat flux are also shown to well match the experimental results. The present work should shed lights on better understanding thermochemical non-equilibrium effects of complex flow phenomenon like the shock wave/boundary layer interaction in hypersonic propulsion with high enthalpy surroundings.

Numerical modeling of essentially subsonic flows of compressible gas

A new method for solving essentially subsonic flows is proposed, which represents a significant step in the field of numerical modeling of flows based on the Navier-Stokes system of equations. The method uses an ENO (Essentially Non-Oscillatory) scheme of third order accuracy, which provides higher accuracy when calculating flows with low speed of sound. One of the key features of this method is the introduction of nondimensionalization parameters. These parameters make it possible to adapt the Navier-Stokes equations to different physical conditions and avoid the rigidity of the equations, which is often encountered in numerical modeling problems. This makes the method more flexible and applicable to a variety of engineering and physical problems. To check and approbate this technique, calculations are carried out for two important problems - flow inside a cavern and Poiseuille flow. The value of the Reynolds number, Re=100, as well as various sizes of computational grids are considered. The obtained results are compared with experimental data, and a high degree of agreement between the model and real phenomena is observed. This indicates the effectiveness and accuracy of the proposed method in solving complex flows in various engineering and physical problems.

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.

Conservative semi-Lagrangian method for continuity equation with various time steps in different parts of computational domain

The system of Navier Stokes differential equations describes gas flow near rocket nozzle. To find solution of this system in general case, scientists and engineers use numerical methods. Modern numerical methods do not allow engineers to model all features of gas flow. It happened because of sophisticated physical processes and limitations of computational hardware. So, at least where are two ways to improve it: to enlarge hardware or reduce computational complicacy. The global aim of many science investigations is to develop numerical approach with reduced computational complicacy and at the same time without loss of computational accuracy. In 1959, Aksel C. Wiin-Nielsen proposed a new and effective trajectories method for problem of numerical forecasting. In 1966, K. M. Magomedov developed similar approach (method of characteristics) for numerical modelling of space (three dimensional on space) gas flow. In 1982, O. Pironneau showed a new and effective approach for two dimensional approximation of the Navier Stokes problem. It was based on method of characteristics also. Nowadays, these methods are called semi-Lagrangian or Eulerian Lagrangian methods. They use Lagrangian nature of the transport process. To apply this advantage, scientists decompose each equation of Navier Stokes system into three parts: convective part (hyperbolic type), elliptic part and part of right-hand side. Scientists use semi-Lagrangian approach to approximate the convective parts of equations. To develop and test modern algorithms from family of semi-Lagrangian methods, we use continuity equation from the system of Navier Stokes equations. Conservative versions of semi-Lagrangian approach are based on Gauss Ostrogradsky (divergence) theorem. It allows scientists to get conservation low (balance equation) for numerical solution in norm of L1 space. Aim of our investigation is to use different time steps in different parts of computation domain. It enables us to obtain at the same time three advantages: convergence of numerical solution to exact solution, reduction of computational complicacy, implementation of conservation low (balance equation) without weight coefficients. For this purpose we decompose computational domain into two parts (subdomains) and use different time steps in them. Main complication is design algorithm in the boundaries of computational subdomains. We use one dimensional (on space) problem to demonstrate ability of developing numerical method with described advantages. Generalization of considered approach for two- or even three-dimensional cases allows engineers to model gas flow more accurately and without artificial viscosity. It is essentially important in the parts of computational domain with high level of solution gradient.

PENYELESAIAN SISTEM PERSAMAAN STOKES MENGGUNAKAN PARTIAL FOURIER TRANSFORM PADA FLUIDA TERMAMPATKAN YANG DISERTAI TEGANGAN PERMUKAAN

Fluid is a substance that flows because of the pressure. Based on their ability to resisting pressure, compressible fluid is one of the fluid which volume can be compressed so they have a constant density. The basic equation of fluid is Navier Stokes equation system that have a non linear partial differential form. Nevertheless, to obtain the solution of non linear partial differential problem is not easy. Therefore, Navier Stokes equation system are linearized into Stokes equation system. In this research, we investigate the solution of the Stokes equation system by partial Fourier transform for compressible fluid with surface tension in half-space.

Analysis of the Multi-Dimensional Navier–Stokes Equation by Caputo Fractional Operator

In this article, we investigate the solution of the fractional multidimensional Navier–Stokes equation based on the Caputo fractional derivative operator. The behavior of the solution regarding the Navier–Stokes equation system using the Sumudu transform approach is discussed analytically and further discussed graphically.

Monitoring of the Behaviour and State of Nanoscale Particles in a Gas Cleaning System of an Ore-Thermal Furnace

The aim of this paper is to define and select stable zones in the off-gas duct of an ore-thermal furnace using a mathematical model. This is needed to increase the effectiveness of exhaust gas composition control in metallurgical silicon production. Methods. The goals of this study were achieved by means of computational fluid dynamics. A model with a water-cooled furnace roof as well as a model comprising steel gas passes with a sliding shutter was developed using ANSYS Fluent software. Both models were symmetrical to ensure a uniform gas-dust distribution, which allowed us to test the adequacy of the obtained models. The models were based on the Navier–Stokes equations system as well as on a discrete phase model (DPM) that was developed using the Euler–Lagrange method. Results. As a result of the modelling, a transition flow mode (Re 0-7437) was revealed behind the sliding shutter. As such, it can be assumed that the most suitable place for measuring equipment to be installed is directly behind the closed part of the sliding shutter.

Numerical Simulation of Turbulent Flow in Bends and Confluences Considering Free Surface Changes Using the Volume of Fluid Method

The impact of secondary flows on the flow velocity in open channel bends and confluences was simulated using three-dimensional (3D) numerical models. The Reynolds-averaged Navier–Stokes equation system was utilized as the governing equations and two different turbulence models were employed in this study: the standard k-ε model and the realizable k–ε model. In a recent study by the authors, the rigid lid approach was used, which does not allow for vertical displacement of the water surface. In this study, the simulation of free surface displacements was simulated using the volume of fluid free-surface tracking method. The numerical models were evaluated and validated by using the experimental data of a sharply curved channel and a confluent channel. The accuracies of the two turbulence models were evaluated and discussed. This study found that both models can satisfactorily reproduce the experimental data. However, the standard k–ε model performed better for the curved channel case while the Realizable k–ε model performed better for the confluent channel case.

Numerical simulation of a low Prandtl number flow over a backward facing step with an anisotropic four-equation turbulence model

In recent years the use of liquid metals has become more and more popular for heat transfer applications in many fields ranging from IV generation fast nuclear reactors to solar power plants. Due to their low Prandtl number values, the similarity between dynamical and thermal fields cannot be assumed and sophisticated heat turbulence models are required to take into account the anisotropy of the turbulent heat transfer involving liquid metals. In the present work, we solve an anisotropic four-equation turbulence model coupled with the Reynolds Averaged Navier Stokes system of equations to simulate a turbulent flow of liquid sodium over a vertical backward-facing step. We implement an explicit algebraic model for Reynolds stress tensor and turbulent heat flux that takes into account flow anisotropic behavior. We study forced and mixed convection regimes when a uniform heat flux is applied on the wall behind the step. Linear isotropic approximations for eddy viscosity and eddy thermal diffusivity underestimate the turbulent heat flux components while this anisotropic model shows a better agreement with DNS results.

Predicting ion concentration polarization and analyte stacking/focusing at nanofluidic interfaces.

We report on the investigation of electropreconcentration phenomena in micro-/nanofluidic devices integrating 100μm long nanochannels using 2D COMSOL simulations based on the coupled Poisson-Nernst-Planck and Navier-Stokes system of equations. Our numerical model is used to demonstrate the influence of key governing parameters such as electrolyte concentration, surface charge density, and applied axial electric field on ion concentration polarization (ICP) dynamics in our system. Under sufficiently extreme surface-charge-governed transport conditions, ICP propagation is shown to enable various transient and stationary stacking and counter-flow gradient focusing mechanisms of anionic analytes. We resolve these spatiotemporal dynamics of analytes and electrolyte ICP over disparate time and length scales, and confirm previous findings that the greatest enhancement is observed when a system is tuned for analyte focusing at the charge, excluding microchannel, nanochannel electrical double layer (EDL) interface. Moreover, we demonstrate that such tuning can readily be achieved by including additional nanochannels oriented parallel to the electric field between two microchannels, effectively increasing the overall perm-selectivity and leading to enhanced focusing at the EDL interfaces. This approach shows promise in providing added control over the extent of ICP in electrokinetic systems, particularly under circumstances in which relatively weak ICP effects are observed using only a single channel.

Redistribution of Energy during Interaction of a Shock Wave with a Temperature Layered Plasma Region at Hypersonic Speeds

The paper is devoted to the problem of the interaction between a shock wave and a thermally stratified energy source for the purpose of supersonic/hypersonic flow control realization. The effect of the thermally stratified energy source on a shock wave with the Mach number in the range of 6–12 is researched numerically based on the Navier-Stokes system of equations. Redistribution of specific internal energy and volume density of kinetic energy behind the wave front is investigated. Multiple manifestations of the Richtmyer-Meshkov instability has been obtained which has caused the blurring and disappearance of shock wave and contact discontinuity fronts in density fields. A study of the efficiency of using a stratified energy source instead of a homogeneous one with the same value of the full energy is carried out. The agreement with the available experimental data for the shock wave Mach number 6 has been obtained.

SIMULATION OF WASTE WATER TREATMENT BASED ON CFD MODEL: EXPRESS CALCULATION

Purpose. Development of CFD model to evaluate the efficiency of wastewater treatment in a horizontal settler. The CFD model can be used to calculate flow hydrodynamics and mass transfer in settlers with complex geometric shape in the area of wastewater flow. Methodology. For numerical simulation of the process of wastewater movement in a horizontal settler, two mathematical models are used. The first model is based on the motion equations of a viscous incompressible fluid – the Navier-Stokes equations. The Navier-Stokes equations are written in the variables «vorticity - flow function». A two-dimensional mass transfer equation is used to calculate the concentration of a pollutant in a horizontal settler. To numerically integrate the two-dimensional mass transfer equation, a finite-difference splitting scheme is used. The splitting of the modeling equation of mass transfer is carried out so that at each fractional step to determine the unknown value of the pollutant concentration by an explicit formula. For numerical integration of the vortex transfer equation and the equation for the flow function (the Navier-Stokes system of equations), finite-difference splitting schemes are used. Findings. Based on the developed CFD model, a complex of computer programs has been developed, which makes it possible to determine the efficiency of water treatment in a horizontal settler with additional elements. The results of a computational experiment to assess the efficiency of water treatment in a horizontal settler with additional elements in the form of plates are presented. Originality. An efficient CFD model has been created, which allows to quickly evaluate the efficiency of wastewater treatment in a horizontal settler with additional elements. The developed CFD model takes into account the geometric shape of the facility and the most significant physical factors, that influence the efficiency of the waste water treatment in horizontal settler: non-uniform flow velocity, diffusion, different position of inlet and outlet openings. Practical value. The developed CFD model belongs to the class of «diagnostic models» and can be used to assess the efficiency of treatment facilities at the stage of their preliminary design.

Investigation of the Motion of Viscous Fluid between Rotating Disks

The issue of the motion of fluid between two rotating disks often arises in practice and it has been little investigated yet. The practical instances of this issue are designing distributors of disk-type fuel pumps, lubrication of smooth disk surfaces during rotation etc.The present article studies the process of the leakage of fluid between disks when the fluid is supplied to the disks under a certain pressure P. The objective of the research is to obtain mathematical models for determining the amount of the fluid leaking at the pin and the amount of the fluid leaking out. It is shown that the leakage of the fluid between the disks can be described using the Navier–Stokes equation system. The mathematical models for determining the amount of the fluid leaking at the pin and the amount of the fluid leaking out have been obtained.

Method of effective parameters for calculations of turbulent swirling flows in engineering constructions

The presented article is devoted to the application of the effective parameter method for calculating vortex structures in a turbulent swirling flow. The Navier-Stokes system of equations is used to study flow hydrodynamics, in which the Reynolds number and the swirl ratio are determined based on the concept of turbulent viscosity. The obtained solutions are compared with experimental data.

Investigation of strength of shaped elements of the main gas pipeline

The research has been carried out for the purpose of a complex numerical three-dimensional modeling of the stressed state of taps and tees of main gas pipelines taking into account the gas-dynamic processes occurring in these shaped elements and the temperature difference in their walls. A 3D modeling of the elbow with a 90° angle and a reinforcing pad on the main line and the drainage of the passage line of the trunk of the main gas pipeline has been carried out. There has been studied the gas flow with 3D models of shaped elements of the main gas pipeline by means of the CFD modeling. The simulation has been рerformed for the equidistant tees in which the entire flow from the main stream flows into its branch. The mathematical model is based on the solution of the Navier–Stokes equation system, continuity equation, closed by a two-parametric k -e model of the Launder–Sharma turbulence with corresponding initial and boundary conditions. The simulation results are visualized in the ANSYS Fluent R18.2 Academic Postprocessor by constructing the pressure fields on the contours and in the longitudinal and transverse sections of shaped elements. The exact values of pressure at different points of the inner cavity of the shaped elements have been determined, the places of rise and fall of pressure identified. There have been performed the simulation of the temperature difference in the walls of the drainage, the trunk of the main gas pipeline in the module ANSYS Transient Thermal. The results of CFD and temperature modeling were imported into the mechanical module ANSYS Static Structural, where the finite element method was used to simulate the stressed state of the shaped elements of the main gas pipeline, taking into account the gas-dynamic processes occurring in their internal cavity and the temperature difference in the walls. The results of the simulation have been visualized by constructing a three-dimensional color fields of equivalent von Mises stresses in the tee and in the elbow. The places of the maximum equivalent stresses in the wall of the studied shaped elements have been revealed.

Heat Transfer From a Heated Cylinder Rotating Around Its Axis in Laminar and Turbulent Crossflows of Liquid

Numerical simulation of the three-dimensional flow in the laminar and turbulent regimes around of the heated cylinder rotating around its axis was carried out. Laminar flow of the viscous incompressible fluid is described by the Navier-Stokes system of equations in natural variables. The unsteady Reynolds-averaged Navier-Stokes equations together with the Shear-Stress Transport model are chosen for the simulation of the turbulent fluid flow. The results of numerical simulation are compared with experimental and numerical data of other authors in the case of the flow past of the stationary cylinder. The dependences of the drag coefficient of the cylinder and the Nusselt number from the rotation speed are established.

Numerical simulation of forced and mixed convection turbulent liquid sodium flow over a vertical backward facing step with a four parameter turbulence model

Liquid metals are promising fluids for engineering applications since they have interesting physical properties such as high thermal conductivity and low kinematic viscosity with respect to the values of common operating fluids, i.e. air and water. This implies low Prandtl number values and therefore the similarity assumption between velocity and temperature fields does not hold. In the present work we solve a four parameter turbulence model coupled with the Reynolds Averaged Navier Stokes system of equations to simulate a turbulent flow of liquid sodium over a vertical backward facing step. On the wall behind the step a uniform heat flux is applied in order to study both forced and mixed convection flow regimes. This model for turbulent heat transfer, taking into account different boundary conditions, is analyzed and compared with the Kays correlation and Direct Numerical Simulation data. Profiles of velocity, turbulent kinetic energy, turbulent heat flux, temperature and its squared fluctuations are provided for two simulated Richardson number values, namely 0 and 0.338. Obtained results show a reasonable agreement with reference Direct Numerical Simulation values.

Realization of novel constant rate of kinetic energy change (CRKEC) supersonic ejector

Ejectors invariably convert momentum and kinetic energy of incoming fluids into enthalpy and hence pressure. Since ejectors are primarily viewed as energy exchange devices, it would be appropriate to design an ejector with its geometry being a function of the rate of energy change. This would shift ejector design approach from conventional geometry based one to that of flow physics-based approach. Hence, the paper presents a unique approach to evolving ejector design considering it primarily to be a function energy change within the system, specifically being constant rate of kinetic energy change (CRKEC). This approach has benefits in terms of mitigating the occurrence of thermodynamic shock, which is a major irreversibility that besets conventional ejector systems. A 1-D gas dynamic model including frictional effects for a more realistic design has been developed for estimating supersonic ejector geometry based on CRKEC approach. The model has been used to predict the geometry of a supersonic air ejector for typical input parameters viz., entrainment ratio (ω) ∼0.53, recovery ratio(ζ) ∼1.4, primary stagnation pressure(Pop) ∼5.7 bar, secondary stagnation pressure(Pos) ∼0.7 bar. The results have been verified through detailed numerical analysis using Navier-Stokes system of equations with turbulence in a 2-D axi-symmetric formulation. Also, the experimental results on the developed prototype, which have also been discussed, are observed to be in close agreement with predictions of 1-D gas dynamic model and the numerical studies.

Investigation of Boundary-Value Problem for Stationary System of Equations of Viscous Non-Isothermal Fluid

In the Stokes approximation at small Reynolds and Peclet numbers, we obtain a solution to the boundary-value problem of flow around of particles of spherical shape for stationary system of equations of a viscous non-isothermal fluid comprising a linearized by speed Navier–Stokes equation system and the equation of heat transfer given an exponential-power law of dependence of viscosity of fluid on temperature.