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.

Transient Flow Patterns and Pressures Characteristics in a Hydrostatic Pocket

1994 ◽  
Vol 116 (1) ◽  
pp. 139-146 ◽  
Author(s):  
M. J. Braun ◽  
Y. M. Zhou ◽  
F. K. Choy
Keyword(s):  
Transient Flow ◽  
Stokes Equations ◽  
Flow Patterns ◽  
Time Dependent ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Computational Error ◽  
Convergence Criteria ◽  
Navier Stokes Equations ◽  
Recirculating Flow

The flow in an open lid, shear flow driven cavity with a jet penetrating at its bottom is described by a mathematical model that uses the Navier-Stokes equations written in terms of the primary variables, u, v, and p. Using a time dependent conservative formulation, a finite difference method is applied through a staggered grid. The power law scheme is used in the treatment of the convective terms for this highly recirculating flow. The time dependent numerical experiments use both geometric (α = d/l, λ = c/l and γ = b/l), and dynamic similarity parameters [Reynolds number (R) and jet strength (F)] to study the development of the flow patterns, velocities, pressures and shear forces when the top plate impulsively entrains the cavity driving shear layer. The experiments are performed in two-dimensional cavities of square geometry (α = 1). 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 pressures were successfully satisfied.

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.

Differential Formulation of Discontinuous Galerkin and Related Methods for the Navier-Stokes Equations

2013 ◽  
Vol 13 (4) ◽  
pp. 1013-1044 ◽  
Author(s):  
Haiyang Gao ◽  
Z. J. Wang ◽  
H. T. Huynh
Keyword(s):  
Discontinuous Galerkin ◽  
Stokes Equations ◽  
High Order Accuracy ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Correction Procedure ◽  
Spectral Difference ◽  
Navier Stokes Equations ◽  
Differential Formulation ◽  
Continuous Scheme

AbstractA new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by Wang and Gao is renamed CPR (correction procedure or collocation penalty via reconstruction). The CPR approach employs the differential form of the equation and accounts for the jumps in flux values at the cell boundaries by a correction procedure. In addition to being simple and economical, it unifies several existing methods including discontinuous Galerkin, staggered grid, spectral volume, and spectral difference. To discretize the dif-fusion terms, we use the BR2 (Bassi and Rebay), interior penalty, compact DG (CDG), and I-continuous approaches. The first three of these approaches, originally derived using the integral formulation, were recast here in the CPR framework, whereas the I-continuous scheme, originally derived for a quadrilateral mesh, was extended to a triangular mesh. Fourier stability and accuracy analyses for these schemes on quadrilateral and triangular meshes are carried out. Finally, results for the Navier-Stokes equations are shown to compare the various schemes as well as to demonstrate the capability of the CPR approach.

Dynamic simulation of a coated microbubble in an unbounded flow: response to a step change in pressure

2017 ◽  
Vol 822 ◽  
pp. 717-761 ◽  
Author(s):  
M. Vlachomitrou ◽  
N. Pelekasis
Keyword(s):  
Stokes Equations ◽  
Step Change ◽  
Basis Functions ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Viscous Forces ◽  
Compressive Stresses ◽  
Post Buckling ◽  
Flow Domain

A numerical method is developed to study the dynamic behaviour of an encapsulated bubble when the viscous forces of the surrounding liquid are accounted for. The continuity and Navier–Stokes equations are solved for the liquid, whereas the coating is described as a viscoelastic shell with bending resistance. The Galerkin Finite Element Methodology is employed for the spatial discretization of the flow domain surrounding the bubble, with the standard staggered grid arrangement that uses biquadratic and bilinear Lagrangian basis functions for the velocity and pressure in the liquid, respectively, coupled with a superparametric scheme with $B$-cubic splines as basis functions pertaining to the location of the interface. The spine method and the elliptic mesh generation technique are used for updating the mesh points in the interior of the flow domain as the shape of the interface evolves with time, with the latter being distinctly superior in capturing severely distorted shapes. The stabilizing effect of the liquid viscosity is demonstrated, as it alters the amplitude of the disturbance for which a bubble deforms and/or collapses. For a step change in the far-field pressure the dynamic evolution of the microbubble is captured until a static equilibrium is achieved. Static shapes that are significantly compressed are captured in the post-buckling regime, leading to symmetric or asymmetric shapes, depending on the relative dilatation to bending stiffness ratio. As the external overpressure increases, shapes corresponding to all the solution families that were captured evolve to exhibit contact as the two poles approach each other. Shell viscosity prevents jet formation by relaxing compressive stresses and bending moments around the indentation generated at the poles due to shell buckling. This behaviour is conjectured to be the inception process leading to static shapes with contact regions.

A Semi-Staggered Numerical Procedure for the Incompressible Navier-Stokes Equations on Curvilinear Grids

2011 ◽  
Author(s):  
Hessam Babaee ◽  
Sumanta Acharya
Keyword(s):  
Finite Difference ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Second Order ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Benchmark Problems ◽  
Navier Stokes Equations ◽  
Curvilinear Grids

An accurate and efficient finite difference method for solving the three dimensional incompressible Navier-Stokes equations on curvilinear grids is developed. The semi-staggered grid layout has been used in which all three components of velocity are stored on the corner vertices of the cell facilitating a consistent discretization of the momentum equations as the boundaries are approached. Pressure is stored at the cell-center, resulting in the exact satisfaction the discrete continuity. The diffusive terms are discretized using a second-order central finite difference. A third-order biased upwind scheme is used to discretize the convective terms. The momentum equations are integrated in time using a semi-implicit fractional step methodology. The convective and diffusive terms are advanced in time using the second-order Adams-Bashforth method and Crank-Nicolson method respectively. The Pressure-Poisson is discretized in a similar approach to the staggered gird layout and thus leading to the elimination of the spurious pressure eigen-modes. The validity of the method is demonstrated by two standard benchmark problems. The flow in driven cavity is used to show the second-order spatial convergence on an intentionally distorted grid. Finally, the results for flow past a cylinder for several Reynolds numbers in the range of 50–150 are compared with the existing experimental data in the literature.

Segmented Domain Decomposition Multigrid Solutions for Two and Three-Dimensional Viscous Flows

1993 ◽  
Vol 115 (4) ◽  
pp. 608-613
Author(s):  
Kumar Srinivasan ◽  
Stanley G. Rubin
Keyword(s):  
Domain Decomposition ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Axial Flow ◽  
Mass Conservation ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Flow Recirculation ◽  
Flux Vector Splitting

Several viscous incompressible two and three-dimensional flows with strong inviscid interaction and/or axial flow reversal are considered with a segmented domain decomposition multigrid (SDDMG) procedure. Specific examples include the laminar flow recirculation in a trough geometry and in a three-dimensional step channel. For the latter case, there are multiple and three-dimensional recirculation zones. A pressure-based form of flux-vector splitting is applied to the Navier-Stokes equations, which are represented by an implicit, lowest-order reduced Navier-Stokes (RNS) system and a purely diffusive, higher-order, deferred-corrector. A trapezoidal or box-like form of discretization insures that all mass conservation properties are satisfied at interfacial and outflow boundaries, even for this primitive-variable non-staggered grid formulation. The segmented domain strategy is adapted herein for three-dimensional flows and is extended to allow for disjoint subdomains that do not share a common boundary.

Numerical Simulation of an Annular Combustor

1993 ◽  
Author(s):  
P. Di Martino ◽  
G. Cinque
Keyword(s):  
Stokes Equations ◽  
Turbulent Transport ◽  
Three Dimensional ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Turbulent Reactive Flows ◽  
Interpolation Technique ◽  
Annular Combustor ◽  
Grid Points

A numerical model to solve three-dimensional turbulent reactive flows in arbitrary shapes is presented. The conservative form of the primitive-variable formulation of steady density-weighted Navier Stokes equations written for a general curvilinear system is adopted. Turbulent transport is described by the k-ε model. The reactions associated with heat release are assumed sufficiently fast for chemical equilibrium to prevail on an instantaneous basis and the influence of local turbulent fluctuations in mixture strenght accounted for by a β-probability density function. The numerical scheme is based on a non-staggered grid (cartesian velocity components and pressure located at the same grid-points) and a special interpolation technique is used to avoid checkerboard oscillations. The present model was used to simulate an annular combustion chamber for which experimental results were available. The agreement between calculation and experiments ranges from fair to good.

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