scholarly journals Algebraic Construction of a Strongly Consistent, Permutationally Symmetric and Conservative Difference Scheme for 3D Steady Stokes Flow

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 269 ◽  
Author(s):  
Xiaojing Zhang ◽  
Vladimir Gerdt ◽  
Yury Blinkov
Keyword(s):  
Difference Scheme ◽  
Stokes Flow ◽  
Stokes Equations ◽  
Integrability Condition ◽  
Three Dimensional ◽  
Algebraic Approach ◽  
Second Order ◽  
Algebraic Construction ◽  
Pressure Poisson Equation ◽  
Strongly Consistent

By using symbolic algebraic computation, we construct a strongly-consistent second-order finite difference scheme for steady three-dimensional Stokes flow and a Cartesian solution grid. The scheme has the second order of accuracy and incorporates the pressure Poisson equation. This equation is the integrability condition for the discrete momentum and continuity equations. Our algebraic approach to the construction of difference schemes suggested by the second and the third authors combines the finite volume method, numerical integration, and difference elimination. We make use of the techniques of the differential and difference Janet/Gröbner bases for performing related computations. To prove the strong consistency of the generated scheme, we use these bases to correlate the differential ideal generated by the polynomials in the Stokes equations with the difference ideal generated by the polynomials in the constructed difference scheme. As this takes place, our difference scheme is conservative and inherits permutation symmetry of the differential Stokes flow. For the obtained scheme, we compute the modified differential system and use it to analyze the scheme’s accuracy.

A preliminary numerical study of a swirling jet behind a circular disc

2008 ◽  
Vol 223 (5) ◽  
pp. 1127-1139
Author(s):  
H T Toh ◽  
R F Huang ◽  
M J Chern
Keyword(s):  
Stokes Equations ◽  
Numerical Study ◽  
Saddle Points ◽  
Three Dimensional ◽  
Second Order ◽  
Circular Disc ◽  
Mean Velocity ◽  
Swirling Jet ◽  
First Case ◽  
Axis Of Symmetry

The three-dimensional flow fields behind a circular disc produced by an annular swirling jet alone and by an annular swirling jet with a central jet issuing from the disc centre are studied by solving the three-dimensional incompressible Navier—Stokes equations numerically using the solution algorithm of Hirt et al. ( Los Alamos Scientific Lab. Rept. LA-5852 (1970)). The swirl number and the Reynolds number based on the disc diameter and the volumetric mean axial velocity of the annular swirling jet are S=0.194 and Re=656, respectively. The convective and diffusive terms in the governing equations are discretized using the second-order central difference scheme. The resulting discretized equations are advanced in time using the second-order Runge—Kutta scheme. The simulation shows that the flow field behind the circular disc exhibit periodic oscillating behaviour, with the second case having a higher frequency due to the presence of the central jet. The mechanism responsible for this oscillating behaviour is identified and discussed. An analysis of the mean velocity fields in the mid-plane shows the existence of a stagnation point on the axis of symmetry in the first case and two saddle points off the axis of symmetry in the second case.

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.

scholarly journals Numerical Methods for Simulating the Motion of Porous Balls in Simple 3D Shear Flows Under Creeping Conditions

2017 ◽  
Vol 17 (3) ◽  
pp. 397-412 ◽  
Author(s):  
Aixia Guo ◽  
Tsorng-Whay Pan ◽  
Jiwen He ◽  
Roland Glowinski
Keyword(s):  
Fluid Flow ◽  
Numerical Methods ◽  
Stokes Flow ◽  
Shear Flows ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Numerical Results ◽  
Porous Particle ◽  
Particle Interactions ◽  
New Methods

AbstractIn this article, two novel numerical methods have been developed for simulating fluid/porous particle interactions in three-dimensional (3D) Stokes flow. The Brinkman–Debye–Bueche model is adopted for the fluid flow inside the porous particle, being coupled with the Stokes equations for the fluid flow outside the particle. The rotating motion of a porous ball and the interaction of two porous balls in bounded shear flows have been studied by these two new methods. The numerical results show that the porous particle permeability has a strong effect on the interaction of two porous balls.

Direct numerical simulations of bubbly flows. Part 1. Low Reynolds number arrays

1998 ◽  
Vol 377 ◽  
pp. 313-345 ◽  
Author(s):  
ASGHAR ESMAEELI ◽  
GRÉTAR TRYGGVASON
Keyword(s):  
Reynolds Number ◽  
Numerical Simulations ◽  
Stokes Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Direct Numerical Simulations ◽  
Two Dimensional ◽  
Rise Velocity ◽  
Regular Array ◽  
Average Rise

Direct numerical simulations of the motion of two- and three-dimensional buoyant bubbles in periodic domains are presented. The full Navier–Stokes equations are solved by a finite difference/front tracking method that allows a fully deformable interface between the bubbles and the ambient fluid and the inclusion of surface tension. The governing parameters are selected such that the average rise Reynolds number is O(1) and deformations of the bubbles are small. The rise velocity of a regular array of three-dimensional bubbles at different volume fractions agrees relatively well with the prediction of Sangani (1988) for Stokes flow. A regular array of two- and three-dimensional bubbles, however, is an unstable configuration and the breakup, and the subsequent bubble–bubble interactions take place by ‘drafting, kissing, and tumbling’. A comparison between a finite Reynolds number two-dimensional simulation with sixteen bubbles and a Stokes flow simulation shows that the finite Reynolds number array breaks up much faster. It is found that a freely evolving array of two-dimensional bubbles rises faster than a regular array and simulations with different numbers of two-dimensional bubbles (1–49) show that the rise velocity increases slowly with the size of the system. Computations of four and eight three-dimensional bubbles per period also show a slight increase in the average rise velocity compared to a regular array. The difference between two- and three-dimensional bubbles is discussed.

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