Hierarchical Divergence-Free Bases and Their Application to Particulate Flows

2003 ◽  
Vol 70 (1) ◽  
pp. 44-49 ◽  
Author(s):  
V. Sarin ◽  
A. H. Sameh
Keyword(s):  
Stokes Equations ◽  
Equations Of Motion ◽  
Navier Stokes ◽  
Particulate Flow ◽  
Particulate Flows ◽  
Navier Stokes Equations ◽  
Flow Problems ◽  
Rigid Particles ◽  
The Matrix ◽  
Divergence Free

The paper presents an algebraic scheme to construct hierarchical divergence-free basis for velocity in incompressible fluids. A reduced system of equations is solved in the corresponding subspace by an appropriate iterative method. The basis is constructed from the matrix representing the incompressibility constraints by computing algebraic decompositions of local constraint matrices. A recursive strategy leads to a hierarchical basis with desirable properties such as fast matrix-vector products, a well-conditioned reduced system, and efficient parallelization of the computation. The scheme has been extended to particulate flow problems in which the Navier-Stokes equations for fluid are coupled with equations of motion for rigid particles suspended in the fluid. Experimental results of particulate flow simulations have been reported for the SGI Origin 2000.

Numerical Investigation of the Coulter Principle in a Microfluidic Device

2013 ◽  
Author(s):  
Muheng Zhang ◽  
Yongsheng Lian
Keyword(s):  
Electric Field ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Navier Stokes ◽  
Working Mechanism ◽  
Navier Stokes Equations ◽  
Sheath Flow ◽  
Coulter Counters ◽  
Pass Through ◽  
Cell Motion

Coulter counters are analytical microfluidic instrument used to measure the size and concentration of biological cells or colloid particles suspended in electrolyte. The underlying working mechanism of Coulter counters is the Coulter principle which relies on the fact that when low-conductive cells pass through an electric field these cells cause disturbances in the measurement (current or voltage). Useful information about these cells can be obtained by analyzing these disturbances if an accurate correlation between the measured disturbances and cell characteristics. In this paper we use computational fluid dynamics method to investigate this correlation. The flow field is described by solving the Navier-Stokes equations, the electric field is represented by a Laplace’s equation in which the conductivity is calculated from the Navier-Stokes equations, and the cell motion is calculated by solving the equations of motion. The accuracy of the code is validated by comparing with analytical solutions. The study is based on a coplanar Coulter counter with three inlets that consist of two sheath flow inlet and one conductive flow inlet. The effects of diffusivity, cell size, sheath flow rate, and cell geometry are discussed in details. The impacts of electrode size, gap between electrodes and electrode location on the measured distribution are also studied.

Front Tracking Approach and Application to Multi-Fluid Flow

2006 ◽  
Author(s):  
Manasa Ranjan Behera ◽  
K. Murali
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Computational Domain ◽  
Navier Stokes Equations ◽  
Flow Problems ◽  
Fixed Grid ◽  
Moving Interface ◽  
Fluid Properties ◽  
Volume Method ◽  
Transfer Of Information

Multiphase flows simulations using a robust interface-tracking method, are presented. The method is based on writing one set of governing equations for the whole computational domain and treating the different phases as single fluid domain with variable material properties. Interfacial terms are accounted for by adding the appropriate sources as δ functions at the boundary separating the phases. The unsteady Navier-Stokes equations are solved by finite volume method on a fixed, structured grid and the interface, or front, is tracked explicitly by a lower dimensional grid. Interfacial source terms are computed on the front and transferred to the fixed grid. Advection of fluid properties such as density and viscosity is done by following the motion of the front. The method has been implemented for interfacial flow problems, depicting the interface and topology change capturing capability. The representation of the moving interface and its dynamic restructuring, as well as the transfer of information between the moving front and the fixed grid, is discussed. Extensions of the method to density stratified flows, and interfacial movements are then presented.

scholarly journals Vortex motion of a continuous medium depending on the pressure change

2020 ◽  
Vol 2020 (6) ◽  
Author(s):  
Alexander Braginsky
Keyword(s):  
Angular Velocity ◽  
Continuous Medium ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Vortex Motion ◽  
Pressure Change ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Least Action ◽  
Force Components

Abstract In this paper, we study the vortex motion of a continuous medium, which is described by forces obtained from the principle of least action. It is shown that in a continuous medium the vortex force components are proportional to the velocity and pressure gradient components. This article gives a description of the 2D vortex motion of air in zones of high and low pressure. If the pressure decreases, the angular velocity of rotation of the continuous medium increases, whereas if the pressure increases, the angular velocity fades. The lifting force is obtained due to the vortex movement of air in the form of a funnel. It is shown that the vortex force contains a vortex term of the Euler hydrodynamic equations with a relative factor equal to the velocity of the continuous medium squared divided by the sound velocity squared. To describe the motion of a continuous medium correctly it is necessary to replace the forces obtained by Euler with the forces obtained from the minimum of action in the equations of motion. It is concluded that vortex motions and turbulence are described by the obtained equations of motion, and not by the Navier–Stokes equations. Most likely, this is related to the Problem of the Millennium description of turbulence announced at the International Congress of Mathematics in 2000.

scholarly journals Energy decay of vortices in viscous fluids: an applied mathematics view

2012 ◽  
Vol 709 ◽  
pp. 593-609 ◽  
Author(s):  
Jan Nordström ◽  
Björn Lönn
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Applied Mathematics ◽  
Energy Decay ◽  
Viscous Fluids ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Matrix ◽  
High Order Finite Difference ◽  
Dissipative Terms

AbstractThe energy decay of vortices in viscous fluids governed by the compressible Navier–Stokes equations is investigated. It is shown that the main reason for the slow decay is that zero eigenvalues exist in the matrix related to the dissipative terms. The theoretical analysis is purely mathematical and based on the energy method. To check the validity of the theoretical result in practice, numerical solutions to the Navier–Stokes equations are computed using a stable high-order finite difference method. The numerical computations corroborate the theoretical conclusion.

Mechanics of thermohaline interleaving: beyond the empirical flux laws

2011 ◽  
Vol 675 ◽  
pp. 117-140 ◽  
Author(s):  
TIMOUR RADKO
Keyword(s):  
Large Scale ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Field Measurements ◽  
Navier Stokes ◽  
Homogeneous Field ◽  
Spontaneous Generation ◽  
Navier Stokes Equations ◽  
Salt Fingers ◽  
Stratified Ocean

An analytical theory is developed which illustrates the dynamics of the spontaneous generation of thermohaline intrusions in the stratified ocean with density compensated lateral temperature and salinity gradients. Intrusions in the model are driven by the interaction with the initially homogeneous field of salt fingers, whose amplitude and spatial orientation is weakly modulated by the long wavelength perturbations introduced into the system. The asymptotic multiscale analysis makes it possible to identify intrusive instabilities resulting from the positive feedback of salt fingers on large-scale perturbations and analyse the resulting patterns. The novelty of the proposed analysis is related to our ability to avoid using empirical double-diffusive flux laws – an approach taken by earlier models. Instead, we base our analytical explorations directly on the governing (Navier–Stokes) equations of motion. The model predictions of the growth rates and preferred slopes of intrusions are in general agreement with the laboratory and field measurements.

Development of a Pressure-Based Coupled CFD Solver for Turbulent and Compressible Flows in Turbomachinery Applications

2014 ◽  
Author(s):  
Luca Mangani ◽  
Marwan Darwish ◽  
Fadl Moukalled
Keyword(s):  
Compressible Flows ◽  
Stokes Equations ◽  
Computational Cost ◽  
Simple Algorithm ◽  
Numerical Data ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Analysis And Design ◽  
Flow Problems ◽  
Matrix Coefficients

In this paper we present a fully coupled algorithm for the resolution of compressible flows at all speed. The pressure-velocity coupling at the heart of the Navier Stokes equations is accomplished by deriving a pressure equation in similar fashion to what is done in the segregated SIMPLE algorithm except that the influence of the velocity fields is treated implicitly. In a similar way, the assembly of the momentum equations is modified to treat the pressure gradient implicitly. The resulting extended system of equations, now formed of matrix coefficients that couples the momentum and pressure equations, is solved using an algebraic multigrid solver. The performance of the coupled approach and the improved efficiency of the novel developed code was validated comparing results with experimental and numerical data available from reference literature test cases as well as with segregated solver as exemplified by the SIMPLE algorithm. Moreover the reference geometries considered in the validation process cover the typical aerodynamics applications in gas turbine analysis and design, considering Euler to turbulent flow problems and clearly indicating the substantial improvements in terms of computational cost and robustness.

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