An Adaptive Wavelet Method for the Incompressible Navier-Stokes Equations in Complex Domains

2004 ◽  
Author(s):  
Damrongsak Wirasaet ◽  
Samuel Paolucci
Keyword(s):  
Stokes Equations ◽  
Momentum Equation ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Adaptive Wavelet ◽  
Brinkman Equations ◽  
Wavelet Collocation Method ◽  
Alternative Means ◽  
Interpolating Wavelets ◽  
Solid Obstacles

An adaptive wavelet-based method provides an alternative means to refine grids according to local demands of the physical solution. One of the prominent challenges of such a method is the application to problems defined on complex domains. In the case of incompressible flow, the application to problems with complicated domains is made possible by the use of the Navier-Stokes/Brinkman equations. These equations take into account solid obstacles by adding a penalized velocity term in the momentum equation. In this study, an adaptive wavelet collocation method, based on interpolating wavelets, is first applied to a benchmark problem defined on a simple domain to demonstrate the accuracy and efficiency of the method. Then the penalty technique is used to simulate flows over obstacles. The numerical results are compared with those obtained by other computational approaches as well as with experiments.

Adaptive Wavelet Method for Incompressible Flows in Complex Domains

2005 ◽  
Vol 127 (4) ◽  
pp. 656-665 ◽  
Author(s):  
Damrongsak Wirasaet ◽  
Samuel Paolucci
Keyword(s):  
Incompressible Flows ◽  
Momentum Equation ◽  
Navier Stokes ◽  
Adaptive Wavelet ◽  
Brinkman Equations ◽  
Wavelet Collocation Method ◽  
Alternative Means ◽  
Interpolating Wavelets ◽  
Solid Obstacles ◽  
Penalty Technique

An adaptive wavelet-based method provides an alternative means to refine grids according to local demands of the physical solution. One of the prominent challenges of such a method is the application to problems defined on complex domains. In the case of incompressible flow, the application to problems with complicated domains is made possible by the use of the Navier-Stokes–Brinkman equations. These equations take into account solid obstacles by adding a penalized velocity term in the momentum equation. In this study, an adaptive wavelet collocation method, based on interpolating wavelets, is first applied to a benchmark problem defined on a simple domain to demonstrate the accuracy and efficiency of the method. Then the penalty technique is used to simulate flows over obstacles. The numerical results are compared to those obtained by other computational approaches as well as to experiments.

Adaptive Wavelet Methods for the Navier-Stokes Equations

2001 ◽  
pp. 303-318 ◽  
Author(s):  
K. Schneider ◽  
M. Farge ◽  
F. Koster ◽  
M. Griebel
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Adaptive Wavelet ◽  
Wavelet Methods

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.

Analysis of Unsteadiness on Casing Treatment Mechanisms in an Axial Compressor

2011 ◽  
Author(s):  
G. Legras ◽  
N. Gourdain ◽  
I. Tre´binjac ◽  
X. Ottavy
Keyword(s):  
Stokes Equations ◽  
Flow Analysis ◽  
Axial Compressor ◽  
Momentum Equation ◽  
Navier Stokes ◽  
Present Contribution ◽  
Navier Stokes Equations ◽  
Casing Treatment ◽  
Budget Analysis ◽  
Flow Phenomena

Control devices based on casing treatments have already shown their capability to improve the flow stability in compressors. However their optimization remains complex due to a partial understanding of the related physical mechanisms. The present paper proposes to use a budget analysis of the Navier Stokes equations to support the understanding of such flow phenomena. Based on the original work of Shabbir and Adamczyk (2005), the strength of the present contribution is to generalize the flow analysis method to all Navier-Stokes equations, including unsteady terms. A high-pressure multistage compressor equipped with circumferential casing grooves is chosen to demonstrate the potential of this approach. Steady and unsteady Reynolds-Averaged Navier-Stokes (URANS) equations are solved with a structured multi-blocks solver. Results are then briefly compared to experimental data to validate the numerical method. The analysis of the unsteady axial momentum equation for configurations with and without casing treatment points out some of the mechanisms responsible for the stability improvement. The analysis also indicates that the flow unsteadiness generated by upstream stator wakes (stator/rotor interaction) reduces viscous efforts and increases convective forces, significantly modifying the compressor stability. Finally, the proposed post processing method shows very interesting results for the understanding of circumferential grooves and it should be also used for non-axisymmetric casing treatment configurations.

Numerical Simulation and Modeling of Laminar Developing Flow in Channels and Tubes With Slip

2013 ◽  
Vol 135 (10) ◽  
Author(s):  
Y. S. Muzychka ◽  
R. Enright
Keyword(s):  
Stokes Equations ◽  
Rarefied Gas ◽  
Slip Boundary Condition ◽  
Momentum Equation ◽  
Navier Stokes ◽  
Cfd Simulations ◽  
Navier Stokes Equations ◽  
Navier Slip ◽  
Slip Boundary ◽  
Rarefied Gas Flows

Analytical solutions for slip flows in the hydrodynamic entrance region of tubes and channels are examined. These solutions employ a linearized axial momentum equation using Targ's method. The momentum equation is subjected to a first order Navier slip boundary condition. The accuracy of these solutions is examined using computational fluid dynamics (CFD) simulations. CFD simulations utilized the full Navier–Stokes equations, so that the implications of the approximate linearized axial momentum equation could be fully assessed. Results are presented in terms of the dimensionless mean wall shear stress, τ⋆, as a function of local dimensionless axial coordinate, ξ, and relative slip parameter, β. These solutions can be applied to either rarefied gas flows when compressibility effects are small or apparent liquid slip over hydrophobic and superhydrophobic surfaces. It has been found that, under slip conditions, the minimum Reynolds number should be ReDh>100 in order for the approximate linearized solution to remain valid.

Computation of Inviscid Incompressible Flow Using the Primitive Variable Formulation

1986 ◽  
Vol 108 (1) ◽  
pp. 68-75 ◽  
Author(s):  
S. Abdallah ◽  
H. G. Smith
Keyword(s):  
Boundary Conditions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Type Equation ◽  
Momentum Equation ◽  
Navier Stokes ◽  
Fine Grid ◽  
Navier Stokes Equations ◽  
Pressure Poisson Equation ◽  
Fine Mesh

The primitive variable formulation originally developed for the incompressible Navier–Stokes equations is applied for the solution of the incompressible Euler equations. The unsteady momentum equation is solved for the velocity field and the continuity equation is satisfied indirectly in a Poisson-type equation for the pressure (divergence of the momentum equation). Solutions for the pressure Poisson equation with derivative boundary conditions exist only if a compatibility condition is satisfied (Green’s theorem). This condition is not automatically satisfied on nonstaggered grids. A new method for the solution of the pressure equation with derivative boundary conditions on a nonstaggered grid [25] is used here for the calculation of the pressure. Three-dimensional solutions for the inviscid rotational flow in a 90 deg curved duct are obtained on a very fine mesh (17 × 17 × 29). The use of a fine grid mesh allows for the accurate prediction of the development of the secondary flow. The computed results are in good agreement with the experimental data of Joy [15].

scholarly journals Numerical Modeling of Blood Flow in Irregular Stenosed Artery with the Effects of Gravity

2013 ◽  
Vol 62 (3) ◽  
Author(s):  
Tan Yan Bin ◽  
Norzieha Mustapha
Keyword(s):  
Blood Flow ◽  
Gravitational Force ◽  
Stokes Equations ◽  
Numerical Study ◽  
Momentum Equation ◽  
Navier Stokes ◽  
Governing Equations ◽  
Navier Stokes Equations ◽  
Stenosed Artery ◽  
Flow Through

A numerical study on the influences of gravitational force on an unsteady two–dimensional nonlinear model of blood flow through a stenosed artery is presented. Blood flow through the constricted region with an irregular stenosis is considered as incompressible Newtonian fluid. The governing equations are derived from the Navier–Stokes equations, which also comprise a significant term for gravitational force in the axial momentum equation. The numerical method chosen in this study is the finite difference approximations based on Marker and Cell (MAC) method at which governing equations are develop in staggered grids for discretization. The Poisson equation of pressure is solved by successive–over–relaxation (S.O.R.) method. Pressure–velocity corrector is imposed to increase accuracy. Streamlines, wall shear stress and axial velocity profiles are plotted.

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