Navier-Stokes Equations Of Motion Research Articles

Two Dimensional Hydrodynamics Model with Evaporation for Coastal Systems

Introduction. The use of two-dimensional (2D) hydrodynamic models is relevant, despite the development of numerical methods of marine hydrodynamics focused on the use of three-dimensional spatial models. This is due to the modelling of hydrodynamic processes in shallow and coastal systems in solving practically important problems of predicting the transport of pollutants in suspended and dissolved forms. Evaporation for the Southern of Russia marine coastal systems (the Azov Sea, the Northern Caspian, etc.), and even more so in the coastal areas of the Red Sea, is a significant factor that affects not only the balance of water masses, but also makes changes in the momentum of the system and the distribution of the velocity vector of the aquatic environment. This effect is significant for coastal currents and shallow-water systems.Materials and Methods. The traditional method of converting the terms of the Navier-Stokes equations containing differentiation by horizontal spatial variables was used, involving the rearrangement of differentiation operations by horizontal spatial coordinates and integration by vertical coordinate when constructing a spatially two-dimensional model of hydrodynamics of marine coastal systems when integrated by vertical coordinate. This made it possible to avoid the appearance of non-physical sources of energy and momentum in the spatially two-dimensional model, which can be essential in traditional 2D models with significant depth differences characteristic of coastal systems. The implementation of the analogue of the law of conservation of the total mechanical energy of the system for the constructed 2D model is investigated.Results. Using the correct transformation of the 3D model (integration of the Navier-Stokes equations and continuity along a vertical coordinate, taking into account evaporation from a free surface), spatially two-dimensional models of hydrodynamics are constructed, for which the basic conservation laws, including mass and total mechanical energy of the system, are fulfilled. The implementation of an analogue of the law of conservation of total mechanical energy for various types of boundary conditions, including at the bottom, is investigated. The evaporation from the free surface is correctly accounted for not only in the continuity equation, but also in the equations of motion taking into account wind and waves.Discussion and Conclusion. 2D model of hydrodynamics has been constructed and studied, taking into account evaporation not only in the mass balance equation (continuity), but also in the Navier-Stokes equations of motion. The proposed model can be used for predictive modelling of hydrophysical processes, including the spread of pollutants in the aquatic environment of coastal systems and shallow reservoirs in relation to marine systems such as the Sea of Azov, the Northern Caspian Sea, coastal areas of the Red Sea, etc. Spatially two-dimensional models of marine hydrodynamics, without replacing three-dimensional models, can serve as a model basis for operational forecasting of situations in coastal systems and shallow-water objects using computing systems with relatively low performance and a moderate amount of RAM (5–10 Tflops, 2–4 TB, respectively).

The shortfall and rise in energy deposition and combustion via OpenFOAM

In the energy deposition process, a heat source is used to melt a powder or wire as it is being deposited onto the face of a material. While powder provides for greater deposition accuracy, wire appears to be more resource efficient than powder. The procedure involves layering materials and allowing them to harden in a melt pool to create new features. Experiments that explore energy deposition are expensive. As a result, Computational Fluid Dynamics (CFD) may be used to simulate the flame or energy deposition. Fick's law is used to describe the transmission of mass fraction species and energy deposition in this model. Fourier's law is used to calculate the temperature field in the CFD model. For the flow of flame, the Navier-Stokes equations of motion are used. Based on the combustion and energy deposition, this verified CFD model may be used to optimize the operational parameters for regulating the size of the flame and the temperature distribution of the flame. The rise in energy deposition near thin channel and combustion tank is examined in the study. The simulation of the issue is done using the Finite Volume Method (FVM). OpenFOAM is being used in producing the graphical results.

State of art of different kinds of fluid flow interactions with piezo for energy harvesting considering experimental, simulations and mathematical modeling

In this work, the different kinds of fluid flow interactions with piezo smart materials have been discussed for energy harvesting. The present work has been classified into the following categories: (i) experimental investigations (ii) simulation and (iii) mathematical modelling. In section (i) different experimental set-ups such as harvesting of energy with the help of vortex flow, turbulent flow, cross flow, flow in an open channel and closed channel and flow through nozzles have been examined. In section (ii) simulations studies performed with different tools/software like ANSYS/fluent, COSMOL etc. have been detailed. Lastly, in section (iii) different mathematical equations such as Navier-Stokes equation of motion, Continuity equation, finite element method, numerical methods, transport equations, Bernoulli equation, equation of linear elasticity, fluid structural equations, piezoelectric equations and coupled-wave equations are described for generation of energy with fluid’s interaction. The present work has to fulfil two aims: (i) active engineers can choose the best appropriate methodology for their work in the same field (ii) researchers can know about how the area of energy harvesting has been grown in various decades, what are the practical application of this field with real life and the literature gaps of the field.

State of art of different kinds of fluid flow interactions with piezo for energy harvesting considering experimental, simulations and mathematical modeling

In this work, the different kinds of fluid flow interactions with piezo smart materials have been discussed for energy harvesting. The present work has been classified into the following categories: (i) experimental investigations (ii) simulation and (iii) mathematical modelling. In section (i) different experimental set-ups such as harvesting of energy with the help of vortex flow, turbulent flow, cross flow, flow in an open channel and closed channel and flow through nozzles have been examined. In section (ii) simulations studies performed with different tools/software like ANSYS/fluent, COSMOL etc. have been detailed. Lastly, in section (iii) different mathematical equations such as Navier-Stokes equation of motion, Continuity equation, finite element method, numerical methods, transport equations, Bernoulli equation, equation of linear elasticity, fluid structural equations, piezoelectric equations and coupled-wave equations are described for generation of energy with fluid’s interaction. The present work has to fulfil two aims: (i) active engineers can choose the best appropriate methodology for their work in the same field (ii) researchers can know about how the area of energy harvesting has been grown in various decades, what are the practical application of this field with real life and the literature gaps of the field.

Pressure curves for compressible flows with slip through asymmetric local constrictions

A second-order nonlinear differential equation is derived for the pressure of a compressible flow with slip at the wall through a constricted cylinder. The ideal gas equation of state is used, and the Karman-Pohlhausen method is utilised to derive the pressure differential equation from the Navier-Stokes equations of motion for a Newtonian viscous fluid. The solution for pressure is determined numerically and assessed in various flow geometries. This work is an extension of existing assessments in that nonlinear terms are kept in the differential equation for pressure, as well as second-order derivative terms. Additionally, wall slip and compressibility are incorporated in the equations, as well as geometries that are asymmetric with respect to the location of maximum constriction.

The fundamental solution of linearized nonstationary Navier-Stokes equations of motion around a rotating and translating body

We derive the fundamental solution of the linearized problem of the motion of a viscous uid around a rotating body when the axis of rotation of the body is not parallel to the velocity of the uid at innity.

Flow exergy as a Lagrangian for the Navier-Stokes equations for incompressible flow

A novel variational derivation of the Navier-Stokes equations for incompressible flows is presented and discussed. The Lagrangian density is obtained from the exergy balance equation written for both the (Lagrangian) steady and quasi-stationary isothermal flows of an incompressible fluid. The exergy of a fluid mass (composed of a kinetic, a pressure-work, a diffusive, and a dissipative portion, the latter being the result of viscous irreversibility) is derived first, and it is then shown that a formal minimisation of the exergy variation (i.e. destruction) generates, without recurring to “local potentials”, the Navier-Stokes equations of motion under the given assumptions. The acceleration being held constant, the proposed variational method can be classified as a “restricted” principle. The problem is also briefly discussed both in its historical perspective and in its possible formal and practical implications.

HEAT AND MASS TRANSFER DYNAMICS IN THE MICROCHANNEL ADSORPTION REACTOR

A dynamic model for conjugate heat and mass transfer in a microchannel adsorption reactor is developed. The model is based on transient, two-dimensional, and compressible Navier-Stokes equations of motion as the governing conservation equations. Appropriate boundary conditions for the momentum, heat, and mass transfer at the channel wall in the presence of adsorption for the no-slip and slip flows are formulated and incorporated into the generalized single-equation-based framework for solving conjugate problems. The 500- mu m-long parallel-plate channel with spacing between walls 10 mu m and a wall thickness 2 mu m is considered as a prototype of the unit cell of the adsorption microreactor. Air is taken a carrier gas and water vapor as an adsorbable species. The flow conditions are characterized by the Reynolds and Knudsen numbers equal to 130x10-2 and 6.5x10-3, respectively. The Freundlich adsorption isotherm is utilized to specify the adsorption desorption equilibrium. The theoretical model developed is validated by comparing the predictions with available theoretical results and experimental data for similar systems. The analysis provides new insights and fundamental understanding of the complex physics of dynamic interactions between heat and mass transfer and adsorption desorption on the microscale. The effects of different boundary conditions on the transient heat and mass transfer with and without surface physicochemical interactions are also investigated and reported.

Linear Feedback Control of Boundary Layer Using Electromagnetic Microtiles

This paper presents a system-theory approach to control of a two-dimensional turbulent flow of saltwater on a flat plate using Lorentz forces produced by microtiles of small magnets and electrodes. Beginning with the two-dimensional Navier-Stokes equations of motion, a finite, dimensional, linear state variable, approximate model is obtained using Galerkin’s procedure. Based on this model, linear feedback control laws are obtained to achieve stabilization of the perturbed flow to the base flow. It is shown that spatially distributed longitudinal or surface-normal forces stabilize the flow perturbations. However, for lower wave numbers, longitudinal forces are more effective because surface-normal forces require larger electrode voltages for the same response characteristics. Simulation results are presented to show how stabilization is accomplished in the closed-loop system.

Use of local time stepping with full pseudospectral solutions to the Navier-Stokes equations of motion for flows with shock waves

Use of local time stepping with full pseudospectral solutions to the Navier-Stokes equations of motion for flows with shock waves

COMPUTATION OF CONJUGATE THREE-DIMENSIONAL NATURAL CONVECTION HEAT TRANSFER FROM A TRANSVERSELY FINNED HORIZONTAL CYLINDER

A vectorized finite-difference code has been developed for the Cray-2 supercomputer which has the capability of simulating a wide class of three-dimensional coupled conduction-convection flows. This program numerically solves the transient form of the Navier-Stokes equations of motion for laminar flow using the vorticity-vector potential method. One problem that has been considered is three-dimensional natural convection from a heated horizontal cylinder with an attached circular conducting fin array. Numerical solutions have been obtained for a Rayleigh number of 106 andaPrandtl number of 5. A parametric study has been performed by varying the fin length, fin spacing, fin thickness, and the fin conductivity parameter. Plots showing the local surface heal flux distribution, temperature distributions, and flow fields were developed using color graphics packages available on the Cray-2 that revealed the existence of very complex fin-cylinder interactions. The local heat flux in the upper section of the fins...

The Flow around an Oscillating sphere : 1st Report, The Case of Periodic Oscillation

The numerical solution of the flow around an oscillating sphere in a uniform flow is presented. The stream function and the vorticity are expanded in Fourier series with respect to time, which are substituted into the Navier-Stokes equations of motion and the continuity equation. Partial differential equations are obtained from the Fourier coefficients as functions of the spatial coordinates. These equations are solved numerically by finite-difference method. Consequently the plots of the streamlines, the equi-vorticity lines, the hydrodynamic factors and the unsteady drag are obtained.

A Contribution to the Coanda Effects

Digital simulation of a discontinuous diffusing flow with particular reference to the Coanda effects is carried out. We give an initial flow pattern arbitrarily and modify it step by step so that it satisfies the Navier-Stokes equations of motion. Reynolds number is assumed to be low enough to keep the flow within the laminar region. A symmetrical initial pattern results in a symmetrical solution anyway in all cases but it is supposedly an unstable flow pattern for Reynolds numbers higher than a critical value. We suppose so because: (1) an unsymmetrical initial pattern results in an unsymmetrical solution is deformed by a small disturbance imposed and eventually settles in the above mentioned unsymmetrical pattern, only for a Reynolds number higher than the critical value. It may thus be said that the Coanda effects occur also in the laminar region and, when they occur, solution of the Navier-Stokes equations is not unique.

On the hydrodynamic behavior of supercooled water drops interacting with columnar ice crystals

A numerical evaluation of the complete Navier-Stokes equations of motion for steady-state, incompressible flow past an infinite circular cylinder is given in terms of the stream function, vorticity, and pressure distribution past such bodies. A method is described which allows use of these flow characteristics: (1) to approximate the characteristics of air flow past hexagonal columnar ice crystals falling under gravity at terminal velocity in air, (2) to compute the trajectory of supercooled cloud drops relative to such ice crystals, and (3) to determine the efficiency with which short columnar ice crystals and needle shaped ice crystals collide with supercooled cloud drops. It is found that for all columnar type ice crystals riming is negligible if the cloud drop size is less than 5∼ μm, and that for riming to commence short columnar crystals must have diameters larger than ∼50 μm, while needle crystals must have diameters larger than ∼40 μm. It is further shown that the collision efficiency cut-offs at the small drop radius and at the large drop radius end of the collision efficiency diagram can be explained on the basis of the cloud drop trajectories for these drop size ranges.

A Numerical Study of Viscous Flow Past a Thin Oblate Spheroid at Low and Intermediate Reynolds Numbers

Abstract The flow past a thin oblate spheroid falling at terminal velocity in an infinite, viscous fluid was investigated using a numerical solution of the steady-state Navier-Stokes equations of motion. The detailed streamfunction and vorticity yielded the drag, pressure distribution, and the extent of the spheroid's downstream wake. Calculations were performed for spheroids of axis ratios 0.05 and 0.2 and Reynolds numbers between 0.1 and 100. The results were compared with other numerical and analytical solutions to the Navier Stokes equations of motion for viscous flow past oblate spheroids and disks and with experimental results in the literature. Our numerical results for oblate spheroids of axis ratio 0.2 agree well with the numerical results of Masliyah and Epstein and with our own experimental results. Our results for oblate spheroids of axis ratio 0.05 agree well with the numerical computations of Michael and available experimental results on disks, but depart significantly from the numerical res...

The stability of unsteady axisymmetric incompressible pipe flow close to a piston. Part 1. Numerical analysis

A numerical solution of the Navier-Stokes equations of motion by means of finite-difference forms of the vorticity and continuity equations is presented. This is applied to the study of the flow of an incompressible fluid produced by the motion from rest of a piston in a cylindrical tube of circular cross-section.Experiments at high Reynolds number indicated the presence in the starting flow of a ring vortex which was not reproduced by computation. Iteration to determine the stream function was not found to be necessary to achieve 1% accuracy. Omitting iteration is equivalent to only slightly disturbing the flow. An additional random disturbance applied to the flow at each time step was found to result in the production of the ring vortex.

Effects of viscosity on long waves

A study of effects of viscosity on non-linear long waves is made. Beginning with the Navier-Stokes equations of motion, the long wave approximation is achieved by an expansion scheme similar to Friedrichs'. Non-linear solutions are obtained by applying the theory of relatively undistorted waves. It is found that shock formation is delayed by the viscous effect. Various conditions are obtained in determining the viscous, non-linear and radial decay effects on the solution for a shockless expansion wave-front propagating over large distances.

The Steady Slow Motion of a Viscous Fluid through a Tapered Tube

It is the purpose of this paper to investigate the steady slow motion of an incompressible viscous fluid through a tapered tube, and to examine theoretically the dependence of the distribution of shear stress on the semi-angle of the cone. The volume of flow per unit time has also been calculated and a formula corresponding to the Poiseuille's equation has been obtained. In our calculations it is assumed that the velocity of the liquid is so small that the inertia term in the Navier-Stokes equations of motion can be neglected.

A theory of homogeneous isotropic turbulence

Homogeneous and isotropic turbulence has been discussed in the present paper. An attempt has been made to find the simplifying hypothesis for connecting the higher order correlation tensor with the lower ones. Starting from the Navier-Stokes equations of motion for an incompressible fluid and following the usual method of taking the averages, a differential equation in Q and X, the defining scalar of the second order correlation tensor Q x and the defining scalar of a third order isotropic tensor X ijk , has been derived. The tensor X ijk stands for a tensorial expression containing the derivatives of the third and the fourth order tensors. Then the hypothesis is used that X=F(Q), where F is an unknown function. To find the forms of F, Kolmogoroff's similarity principles have been used, and thus two forms for F(Q) corresponding to two regions of the validity of these principles have been deduced.