Simultaneous Variable Solutions of the Incompressible Steady Navier-Stokes Equations in General Curvilinear Coordinate Systems

1992 ◽  
Vol 114 (3) ◽  
pp. 299-305 ◽  
Author(s):  
G. Vradis ◽  
V. Zalak ◽  
J. Bentson
Keyword(s):  
Convergence Rates ◽  
Stokes Equations ◽  
Flow Direction ◽  
Solution Algorithm ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Solution Technique ◽  
Algebraic Equations ◽  
Coordinate Systems ◽  
Navier Stokes Equations

A simultaneous variable solution technique for the incompressible, steady, two-dimensional Navier-Stokes equations in primitive formulation and general curvilinear orthogonal and nonorthogonal coordinate systems has been developed. The governing equations are discretized using finite difference approximations. The formulation is fully second order accurate and the well-known staggered grid of Welch and Harlow is used. The solution algorithm is based on an iterative marching technique in which the algebraic equations are linearized by evaluating the coefficients at the previous iteration level. The resulting system of linear equations is solved in a marching fashion by employing a block tridiagonal solution algorithm to obtain the solution along lines transverse to the main flow direction. The strong pressure-velocity coupling inherent in the present formulation results in high convergence rates. Flows in channels of different geometries have been computed and the results have been compared to available data in the literature. In all cases the method has demonstrated to be accurate, robust and computationally efficient.

A HYBRID FINITE-ANALYTIC ALGORITHM FOR SOLVING THREE-DIMENSIONAL ADVECTION-DIFFUSION EQUATIONS

2004 ◽  
Vol 01 (03) ◽  
pp. 407-430 ◽  
Author(s):  
H. M. HU ◽  
K.-H. WANG
Keyword(s):  
Convergence Rates ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Diffusion Equations ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
The Stability

The hybrid finite-analytic (HFA) method for discretization of a three-dimensional advection-diffusion equation is developed using the superposition of the HFA solutions of locally linearized one-dimensional advection-diffusion equations. An example calculation of a system of three-dimensional nonlinear equations is conducted to test the convergence and accuracy of the 7-point numerical scheme. Good agreements between calculated and analytical solutions are obtained. An algorithm based on the HFA method with multigrid technique and Gauss-Seidel iteration is also developed to solve the three-dimensional Navier-Stokes equations in a staggered grid system. The stability and efficiency of the method are demonstrated by performing calculations of the fluid flow in a three-dimensional cubic cavity with a moving top wall. The proposed procedure is observed to exhibit good rates of smoothing and almost grid-independent convergence rates in comparison with a single-grid iteration method. The results are in excellent agreement with other published computational results.

The Effects of a Hydrostatic Pocket Aspect Ratio, Supply Orifice Position, and Attack Angle on Steady-State Flow Patterns, Pressure, and Shear Characteristics

1993 ◽  
Vol 115 (4) ◽  
pp. 678-685 ◽  
Author(s):  
M. J. Braun ◽  
F. K. Choy ◽  
Y. M. Zhou
Keyword(s):  
Stokes Equations ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Computational Error ◽  
Convergence Criteria ◽  
Steady State Flow ◽  
Shear Forces ◽  
Navier Stokes Equations ◽  
Recirculating Flow ◽  
Shear Characteristics

The flow in a hydrostatic pocket is described by a mathematical model that uses the Navier-Stokes equations written in terms of the primary variables, u, v, and p. Using the conservative formulation, a finite difference method is applied through a staggered grid. The power law scheme is applied in the treatment of the convective terms for this highly recirculating flow. The discussion pertaining to the convergence of the numerical scheme and the computational error, shows that the strict convergence criteria applied to both velocities and pressure were successfully statisfied. The numerical model is applied in a parametric mode to the study of the velocities, the pressure patterns, and shear forces that characterize the flow in a square (α = 1), deep (α>1), and shallow (α≪1) hydrostatic pocket. The effects of the variation of the location and angle of the hydrostatic jet are also investigated.

On Offset Placement of a Compound Droplet in a Channel Flow

2021 ◽  
Author(s):  
Jagannath Mahato ◽  
Dhananjay Kumar Srivastava ◽  
Dinesh Kumar Chandraker ◽  
Rajaram Lakkaraju
Keyword(s):  
Viscosity Ratio ◽  
Stokes Equations ◽  
Temporal Dynamics ◽  
Flow Direction ◽  
Navier Stokes ◽  
Volume Of Fluid Method ◽  
Lateral Migration ◽  
Navier Stokes Equations ◽  
Deformation Patterns ◽  
Compound Droplet

Abstract Investigations on flow dynamics of a compound droplet have been carried out in a two-dimensional fully-developed Poiseuille flow by solving the Navier-Stokes equations with the evolution of the droplet using the volume of fluid method with interface compression. The outer droplet undergoes elongation similar to a simple droplet of same size placed under similar ambient condition in the flow direction, but, the inner droplet evolves in compressed form. The compound droplet is varied starting from the centerline towards the walls of the channel. The simulations showed that on applying an offset, asymmetric slipper-like shapes are observed as opposed to symmetric bullet-like shapes through the centerline. Temporal dynamics, deformation patterns, and droplet shell pinch-off mode vary with the offset, with induction of lateral migration. Also, investigations are done on the effect of various parameters like droplet size, Capillary number, and viscosity ratio on the deformation magnitude and lateral migration.

Numerical simulations of three-dimensional plunging breaking waves: generation and evolution of aerated vortex filaments

2015 ◽  
Vol 767 ◽  
pp. 364-393 ◽  
Author(s):  
P. Lubin ◽  
S. Glockner
Keyword(s):  
Stokes Equations ◽  
Early Stage ◽  
Flow Direction ◽  
Three Dimensional ◽  
Air Entrainment ◽  
Breaking Waves ◽  
Navier Stokes ◽  
Vortex Filaments ◽  
Navier Stokes Equations ◽  
Main Flow Direction

AbstractThe scope of this work is to present and discuss the results obtained from simulating three-dimensional plunging breaking waves by solving the Navier–Stokes equations, in air and water. Recent progress in computational capabilities has allowed us to run fine three-dimensional simulations, giving us the opportunity to study for the first time fine vortex filaments generated during the early stage of the wave breaking phenomenon. To date, no experimental observations have been made in laboratories, and these structures have only been visualised in rare documentary footage (e.g. BBC 2009 South Pacific. Available on YouTube, 7BOhDaJH0m4). These fine coherent structures are three-dimensional streamwise vortical tubes, like vortex filaments, connecting the splash-up and the main tube of air, elongated in the main flow direction. The first part of the paper is devoted to the presentation of the model and numerical methods. The air entrainment occurring when waves break is then carefully described. Thanks to the high resolution of the grid, these fine elongated structures are simulated and explained.

scholarly journals Detection of multiple solutions using a mid-cell back substitution technique applied to computational fluid dynamics

2000 ◽  
Vol 214 (11) ◽  
pp. 1401-1407
Author(s):  
S R Kendall ◽  
H V Rao
Keyword(s):  
Multiple Solutions ◽  
Computational Models ◽  
Stokes Equations ◽  
Compressible Fluids ◽  
Navier Stokes ◽  
Algebraic Equations ◽  
Navier Stokes Equations ◽  
Flow Modelling ◽  
Spurious Solutions ◽  
Linear Algebraic Equations

Computational models for fluid flow based on the Navier-Stokes equations for compressible fluids led to numerical procedures requiring the solution of simultaneous non-linear algebraic equations. These give rise to the possibility of multiple solutions, and hence there is a need to monitor convergence towards a physically meaningful flow field. The number of possible solutions that may arise is examined, and a mid-cell back substitution technique (MCBST) is developed to detect and avoid convergence towards apparently spurious solutions. The MCBST was used successfully for flow modelling in micron-sized flow passages, and was found to be particularly useful in the early stages of computation, optimizing the speed of convergence.

scholarly journals The Ocean–Land–Atmosphere Model (OLAM). Part II: Formulation and Tests of the Nonhydrostatic Dynamic Core

2008 ◽  
Vol 136 (11) ◽  
pp. 4045-4062 ◽  
Author(s):  
Robert L. Walko ◽  
Roni Avissar
Keyword(s):  
Stokes Equations ◽  
Vertical Motion ◽  
Triangular Mesh ◽  
Atmospheric Modeling ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Wave Flow ◽  
Navier Stokes Equations ◽  
Cell Method ◽  
Atmosphere Model

Abstract The dynamic core of the Ocean–Land–Atmosphere Model (OLAM), which is a new global model that is partly based on the Regional Atmospheric Modeling System (RAMS), is described and tested. OLAM adopts many features of its predecessor, but its dynamic core is new and incorporates a global geodesic grid with triangular mesh cells and a finite-volume discretization of the nonhydrostatic compressible Navier–Stokes equations. The spatial discretization of horizontal momentum is based on a C-staggered grid and uses a method that has not been previously applied in atmospheric modeling. The temporal discretization uses a unique form of time splitting that enforces consistency of advecting mass flux among all conservation equations. OLAM grid levels are horizontal, and topography is represented by the shaved-cell method. Aspects of the shaved-cell method that pertain to the OLAM discretization on the triangular mesh are described, and a method of conserving momentum in shaved cells on a C-staggered grid is presented. The dynamic core was tested in simulations with multiple vertical model levels and significant vertical motion. The tests include an idealized global circulation simulation, a cold density current, and mountain-wave flow over an orographic barrier, all of which are well-known standard benchmark experiments. OLAM gave acceptable results in all tests, demonstrating that its dynamic core produces accurate and robust solutions.

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