The vorticity equation as an angular momentum equation

1973 ◽  
Vol 74 (2) ◽  
pp. 365-367 ◽  
Author(s):  
P. C. Chatwin
Keyword(s):  
Angular Momentum ◽  
Stokes Equations ◽  
Momentum Equation ◽  
Rate Of Change ◽  
The Body ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Material Volume ◽  
The Moment ◽  
Image Position

In a Newtonian fluid, in which the body forces are conservative, in which the pressure is a function only of density and in which the kinematic viscosity v is uniform, the vorticity ω satisfies the equationwhere u is the velocity field, so that ω = ▿ × u. This equation is normally derived by taking the curl of the Navier–Stokes equations. However, the vorticity has many interpretations in terms of the angular velocity of elements of fluid and it is natural to expect that (1) can be derived by equating the rate of change of the angular momentum of a small material volume element about its centre of mass with the moment of the forces acting on the element. Such a derivation is presented here in the hope that it may be of pedagogic interest.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

Studies on Natural Convection Induced Flow and Thermal Behavior Inside Electronic Equipment Cabinet Model

2001 ◽  
Author(s):  
Masaru Ishizuka ◽  
Guoyi Peng ◽  
Shinji Hayama
Keyword(s):  
Natural Convection ◽  
Thermal Behavior ◽  
Body Surface ◽  
Stokes Equations ◽  
The Body ◽  
Electronic Equipment ◽  
Navier Stokes ◽  
Surface Boundary ◽  
Navier Stokes Equations ◽  
Induced Flow

Abstract In the present work, an important basic flow phenomena, the natural convection induced flow, is studied numerically. Three-dimensional Navier-Stokes equations along with the temperature equation are solved on the basis of finite difference method. Generalized coordinate system is used so that sufficient grid resolution could be achieved in the body surface boundary layer region. Differential terms with respect to time are approximated by forward differences, diffusions terms are approximated by the implicit Euler form, convection terms in the Navier-Stokes equations are approximated by the third order upwind difference scheme. The heat flux at the body surface of heater is specified. The results of calculation showed a satisfactory agreement with the measured data and led to a good understanding of the overall flow and thermal behavior inside electronic equipment cabinet model which is very difficult, if not impossible, to gather by experiment.

scholarly journals Plane poloidal-toroidal decomposition of doubly periodic vector fields. Part 2. The Stokes equations

2005 ◽  
Vol 47 (1) ◽  
pp. 39-50 ◽  
Author(s):  
G. D. McBain
Keyword(s):  
Vector Fields ◽  
Stokes Equations ◽  
Momentum Equation ◽  
Integral Representations ◽  
Navier Stokes ◽  
Field Equations ◽  
Navier Stokes Equations ◽  
Time Discretisation ◽  
Mass Constraint ◽  
Conservation Of Momentum

AbstractWe continue our study of the adaptation from spherical to doubly periodic slot domains of the poloidal-toroidal representation of vector fields. Building on the successful construction of an orthogonal quinquepartite decomposition of doubly periodic vector fields of arbitrary divergence with integral representations for the projections of known vector fields and equivalent scalar representations for unknown vector fields (Part 1), we now present a decomposition of vector field equations into an equivalent set of scalar field equations. The Stokes equations for slow viscous incompressible fluid flow in an arbitrary force field are treated as an example, and for them the application of the decomposition uncouples the conservation of momentum equation from the conservation of mass constraint. The resulting scalar equations are then solved by elementary methods. The extension to generalised Stokes equations resulting from the application of various time discretisation schemes to the Navier-Stokes equations is also solved.

CFD Analysis of Vortex Shedding in a Circular Cylinder: Flow Induced Vibration Design

2006 ◽  
Author(s):  
Nadeem Ahmed Sheikh ◽  
M. Afzaal Malik ◽  
Arshad Hussain Qureshi ◽  
M. Anwar Khan ◽  
Shahab Khushnood
Keyword(s):  
Circular Cylinder ◽  
Vortex Shedding ◽  
Stokes Equations ◽  
Boundary Layer Separation ◽  
The Body ◽  
Navier Stokes ◽  
Structural Vibrations ◽  
Flow Induced Vibration ◽  
Navier Stokes Equations ◽  
Implicit Approach

Flow past a blunt body, such as a circular cylinder, usually experiences boundary layer separation and very strong flow oscillations in the wake region behind the body at a discrete frequency that is correlated to the Reynolds number of the flow. The periodic nature of the vortex shedding phenomenon can sometimes lead to unwanted structural vibrations. The effect of vibrating instability of a single cylinder is investigated in a uniform flow using the power of computational methods. Fluid structure coupling procedure predicts the fluid forces responsible for structural vibrations. An implicit approach to the solution of the unsteady two-dimensional Navier-Stokes equations is used for computation of flow parameters. Calculations are performed in parallel using a domain re-meshing/deforming technique with efficient communication requirements. Results for the unsteady shedding flow behind a circular cylinder are presented with experimental comparisons, showing the feasibility of accurate, efficient, time-dependent estimation of shedding frequency and resulting vibrations.

scholarly journals Physical and numerical analysis of the front of a gravity current on a horizontal bottom

1998 ◽  
Vol 26 ◽  
pp. 289-295
Author(s):  
Mohamed Naaim ◽  
Thierry Pellarin
Keyword(s):  
Numerical Model ◽  
Stokes Equations ◽  
Gravity Current ◽  
Processing System ◽  
The Body ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Experimental Approaches ◽  
Two Parameters ◽  
Two Stages

In this paper, numerical and experimental approaches are applied to analyse the dynamics of the front of a gravity current. This study focused on two parameters: internal density and velocity fields. The salt concentration was determined by a potentiometric process. The internal velocities were determined using an optical device and an image-processing system. The structure of the head of the gravity current was analysed. Its density was measured and two stages of evolution were observed. This analysis allows us to coufirm the existence of two important stages. Forxf<xs, where the dynamics depend on the initial condition, the flow consists of a head and body and the front density is constant. Forxf>xs, we show that the density of the front decreases and evolves towards the Hallworth and others (1993) law. From a comparison between the experiments and the numerical model, we show that the numerical model, which is based on Navier–Stokes equations and on thek−Lturbulence model (whereLis the height of the gravity current), can predict well flow in the slump regime and in the inertia–buoyancy regime with smoothed results in the transition from the head to the body of the gravity current.

scholarly journals Dynamic Domain Decomposition Strategy Coupling Lattice Boltzmann Methods With Finite Differences Approximations of the Navier-Stokes Equations to Study Bodies in Current

2015 ◽  
Author(s):  
Giuseppina Colicchio ◽  
Claudio Lugni ◽  
Marilena Greco ◽  
Odd M. Faltinsen
Keyword(s):  
Domain Decomposition ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Dynamic Change ◽  
Accurate Solution ◽  
The Body ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Coupling Strategy ◽  
Hybrid Solver

A Domain-Decomposition (DD) strategy is proposed for problems involving regions with slow variations of the flow (A) and others where the fluid features undergo rapid changes (B), like in the case of steady current past bodies with pronounced local unsteadiness connected with the vortex shedding from the structures. For an efficient and accurate solution of such problems, the DD couples a Finite Difference solver of the Navier-Stokes equations (FD-NS) with a Multiple Relaxation Time Lattice Boltzmann method (MRT-LBM). Regions A are handled by FD-NS, while zones B are solved by MRT-LBM and the two solvers exchange information within a strong coupling strategy. Present DD strategy is able to deal with a dynamic change of the sub-domains topology. This feature is needed when regions with vorticity shed from the body vary in time for a more flexible and reliable solution strategy. Its performances in terms of accuracy and efficiency have been successfully assessed by comparing the hybrid solver against a full FD-NS solution and experimental data for a 2D circular cylinder in an impulsively started flow.

Double diffusive effects on pressure-driven miscible displacement flows in a channel

2012 ◽  
Vol 712 ◽  
pp. 579-597 ◽  
Author(s):  
Manoranjan Mishra ◽  
A. De Wit ◽  
Kirti Chandra Sahu
Keyword(s):  
Stokes Equations ◽  
Miscible Displacement ◽  
Interfacial Instability ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Finite Volume Approach ◽  
Diffusive Effects ◽  
The Moment ◽  
Double Diffusive ◽  
Pressure Driven

AbstractThe pressure-driven miscible displacement of a less viscous fluid by a more viscous one in a horizontal channel is studied. This is a classically stable system if the more viscous solution is the displacing one. However, we show by numerical simulations based on the finite-volume approach that, in this system, double diffusive effects can be destabilizing. Such effects can appear if the fluid consists of a solvent containing two solutes both influencing the viscosity of the solution and diffusing at different rates. The continuity and Navier–Stokes equations coupled to two convection–diffusion equations for the evolution of the solute concentrations are solved. The viscosity is assumed to depend on the concentrations of both solutes, while density contrast is neglected. The results demonstrate the development of various instability patterns of the miscible ‘interface’ separating the fluids provided the two solutes diffuse at different rates. The intensity of the instability increases when increasing the diffusivity ratio between the faster-diffusing and the slower-diffusing solutes. This brings about fluid mixing and accelerates the displacement of the fluid originally filling the channel. The effects of varying dimensionless parameters, such as the Reynolds number and Schmidt number, on the development of the ‘interfacial’ instability pattern are also studied. The double diffusive instability appears after the moment when the invading fluid penetrates inside the channel. This is attributed to the presence of inertia in the problem.

scholarly journals Three-Dimensional Thermo Fluid Analysis of Large Scale Electric Motor

2000 ◽  
Vol 6 (6) ◽  
pp. 433-444 ◽  
Author(s):  
Debasish Biswas ◽  
Masaru Ishizuka ◽  
Hideo Iwasaki
Keyword(s):  
Electric Motor ◽  
Large Scale ◽  
Stokes Equations ◽  
Flow Behavior ◽  
The Body ◽  
Temperature Fields ◽  
Navier Stokes ◽  
Surface Boundary ◽  
Navier Stokes Equations ◽  
Flow Stagnation

In the present work, the flow and temperature fields in large scale rotating electric motor are studied by solving the Navier–Stokes equations along with the temperature equation on the basis of finite difference method. All the equations are written in terms of relative velocity with respect to the rotating frame of reference. Generalized coordinate system is used so that sufficient grid resolution could be achieved in the body surface boundary layer region. Differential terms with respect to time are approximated by forward differences, diffusion terms are approximated by the implicit Euler form, convection terms in the Navier–Stokes equations are approximated by the third order upwind difference scheme. The results of calculation led to a good understanding of the flow behavior, namely, the rotating cavity flow in between the supporting bar of the motor, the flow stagnation and region of temperature rise due to flow stagnation, etc. Also the measured average temperature of the motor coil wall is predicted quite satisfactorily.

Numerical Investigation of 3D Flow and Torque Transmission in Fluid Couplings Under Unsteady Working Condition

1995 ◽  
Author(s):  
T. Formanski ◽  
H. Huitenga ◽  
N. K. Mitra ◽  
M. Fiebig
Keyword(s):  
Turbulent Flows ◽  
Working Condition ◽  
Stokes Equations ◽  
Time History ◽  
Operating Conditions ◽  
Navier Stokes ◽  
3D Flow ◽  
Navier Stokes Equations ◽  
Torque Transmission ◽  
The Moment

Hydrodynamic couplings transmit torque by fluid circulation due to a speed differential between the impeller on the drive side and the runner on the driven side without mechanical contact. Detailed studies of the 3D flow in fluid couplings working at steady operating point were carried out in the last few years for laminar and turbulent flows. In this paper a study of fluid couplings working under unsteady operating conditions is reported for the first time. The unsteady Reynolds averaged Navier-Stokes equations together with the k-ϵ model have been solved by a finite-volume method. The calculations were done by using contour-fitted grids with non-staggered variable arrangement in a rotating frame of reference. The results give insight into the flow structure inside a coupling under unsteady working condition. An integration of the flow field for the considered operating points yields the transmitted torque. The time history of the change of the moment of momentum gives further insights into the behaviour of a fluid coupling under unsteady operating conditions.

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