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Preconditioning based on Momentum Interpolation for the coupled solution of the Navier-Stokes Equations | Grupo de Fluidodinámica Numérica

Preconditioning based on Momentum Interpolation for the coupled solution of the Navier-Stokes Equations

  • user warning: Table './drupal_gfn_web/cache_filter' is marked as crashed and should be repaired query: SELECT data, created, headers, expire, serialized FROM cache_filter WHERE cid = '3:3ed5d76f74cc758dbe5464911cc2476a' in /srv/www/gfn/includes/cache.inc on line 27.
  • user warning: Table './drupal_gfn_web/cache_filter' is marked as crashed and should be repaired query: UPDATE cache_filter SET data = '\n<p>(<strong>Ana Cubero</strong>, February 2008)</p>\n\n<p>Tesis available <a href=\"http://zaguan.unizar.es/record/3372\">here</a></p>\n\n<h1><strong>Poisson Equation</strong></h1>\n\n<p>In this work, a new algorithm for solving the Navier-Stokes equations in a\ncoupled and implicit manner is proposed; it combines techniques alternatively\nused in the two traditional approaches in the field (i.e. methods based on\npressure, implicit and segregated, and methods based on density, explicit and\ncoupled). In order to improve the conditioning of the matrix of the coupled\nsystem, an equation for pressure is derived from the equations of continuity and\nmomentum, and a term of artificial compressibility is added to the resulting\nPoisson equation.</p>\n\n<h1><strong>Compact momentum interpolation</strong></h1>\n\n<p>The Navier-Stokes equations are discretized according to the finite volume\nmethod and the co-located grid scheme. To avoid the well-known phenomenon of\nvelocity-pressure decoupling that occurs in this kind of grids, the velocity\ncomponents of the mass flow through the corresponding faces are calculated\nfollowing the Momentum Interpolation method, which adds a corrective term to the\nstandard lineal interpolation between nodes; such term involves pressure\ngradients between each node and its corresponding neighbour. This work makes use\nof a new formulation for this interpolation which avoids some of the\ninconsistencies in the original expression; it also allows to obtain a compact\nPoisson equation (for pressure), as well as its implementation in a partially\nimplicit way, thus achieving the velocity-pressure coupling (u_P-p_P) in the\nsame (current) iteration, without increasing the size of the computational\nmolecule.</p>\n\n<div><img src=\"http://gfn.unizar.es/sites/default/files/cavity_OMIvCMI.jpg\"\nalt=\"\" /></div>\n\n<h1><strong>System solution</strong></h1>\n\n<p>The coupled algebraic system is linearized using a successive substitution\n(or Picard) procedure, and the convergence of the iterative method is xxx in the\ndiagonal dominance (a sufficient condition). To this end, a differed\nimplementation of the convective interpolation schemes for the transported\nvariables is used; and inertial relaxation is added to compensate the extra,\noff-diagonal terms arising out of the coupling of the physical variables.</p>\n\n<h1><strong>Results</strong></h1>\n\n<p>The solution of a wide range of benchmark cases, by means of a parallel CFD\ncode also developed along this research, confirms the good behaviour of the\nalgorithm regarding accuracy, robustness, and numerical stability. It is found\nadequate for both incompressible, highly convection-dominated flows and those\nwhere strong temperature gradients and variable density are present. The\ntransient terms are discretized in an implicit manner; the efficiency and\naccuracy of the implicit method for non-stationary flows have been verified\nsolving, as a test case, the laminar vortex shedding after a square\ncylinder.</p>\n\n<p>Finally, the capacity of the proposed algorithm for solving a computationally\ndemanding problem is shown: the simulation of the three dimensional, transient\nand turbulent vortex shedding at a moderately high Reynolds number (Re = 21400)\nis performed using Large Eddy Simulation (LES).</p>\n\n<div><img src=\"http://gfn.unizar.es/sites/default/files/buoy_Ra1e7.jpg\" alt=\"\"\nwidth=\"300\" height=\"300\" /></div>\n\n<div><img src=\"http://gfn.unizar.es/sites/default/files/LES_vortz3D.jpg\" alt=\"\"\n/></div>\n\n<!-- by Texy2! -->', created = 1501210037, expire = 1501296437, headers = '', serialized = 0 WHERE cid = '3:3ed5d76f74cc758dbe5464911cc2476a' in /srv/www/gfn/includes/cache.inc on line 112.
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(Ana Cubero, February 2008)

Tesis available here

Poisson Equation

In this work, a new algorithm for solving the Navier-Stokes equations in a coupled and implicit manner is proposed; it combines techniques alternatively used in the two traditional approaches in the field (i.e. methods based on pressure, implicit and segregated, and methods based on density, explicit and coupled). In order to improve the conditioning of the matrix of the coupled system, an equation for pressure is derived from the equations of continuity and momentum, and a term of artificial compressibility is added to the resulting Poisson equation.

Compact momentum interpolation

The Navier-Stokes equations are discretized according to the finite volume method and the co-located grid scheme. To avoid the well-known phenomenon of velocity-pressure decoupling that occurs in this kind of grids, the velocity components of the mass flow through the corresponding faces are calculated following the Momentum Interpolation method, which adds a corrective term to the standard lineal interpolation between nodes; such term involves pressure gradients between each node and its corresponding neighbour. This work makes use of a new formulation for this interpolation which avoids some of the inconsistencies in the original expression; it also allows to obtain a compact Poisson equation (for pressure), as well as its implementation in a partially implicit way, thus achieving the velocity-pressure coupling (u_P-p_P) in the same (current) iteration, without increasing the size of the computational molecule.

System solution

The coupled algebraic system is linearized using a successive substitution (or Picard) procedure, and the convergence of the iterative method is xxx in the diagonal dominance (a sufficient condition). To this end, a differed implementation of the convective interpolation schemes for the transported variables is used; and inertial relaxation is added to compensate the extra, off-diagonal terms arising out of the coupling of the physical variables.

Results

The solution of a wide range of benchmark cases, by means of a parallel CFD code also developed along this research, confirms the good behaviour of the algorithm regarding accuracy, robustness, and numerical stability. It is found adequate for both incompressible, highly convection-dominated flows and those where strong temperature gradients and variable density are present. The transient terms are discretized in an implicit manner; the efficiency and accuracy of the implicit method for non-stationary flows have been verified solving, as a test case, the laminar vortex shedding after a square cylinder.

Finally, the capacity of the proposed algorithm for solving a computationally demanding problem is shown: the simulation of the three dimensional, transient and turbulent vortex shedding at a moderately high Reynolds number (Re = 21400) is performed using Large Eddy Simulation (LES).