scholarly journals A numerical model based on the curvilinear coordinate system for the MAC method simplified

2018 ◽  
Vol 39 (2) ◽  
pp. 87
Author(s):  
Eliandro Rodrigues Cirilo ◽  
Alessandra Negrini Dalla Barba ◽  
Paulo Laerte Natti ◽  
Neyva Maria Lopes Romeiro
Keyword(s):  
Coordinate System ◽  
Navier Stokes ◽  
Parallel Plates ◽  
Curvilinear Coordinate System ◽  
Driven Cavity ◽  
Continuity Equations ◽  
Finite Differences Method ◽  
Curvilinear Coordinate ◽  
Marker And Cell Method ◽  
Incompressible Fluid Flows

In this paper we developed a numerical methodology to study some incompressible fluid flows without free surface, using the curvilinear coordinate system and whose edge geometry is constructed via parametrized spline. First, we discussed the representation of the Navier-Stokes and continuity equations on the curvilinear coordinate system, along with the auxiliary conditions. Then, we presented the numerical method – a simplified version of MAC (Marker and Cell) method – along with the discretization of the governing equations, which is carried out using the finite differences method and the implementation of the FOU (First Order Upwind) scheme. Finally, we applied the numerical methodology to the parallel plates problem, lid-driven cavity problem and atherosclerosis problem, and then we compare the results obtained with those presented in the literature.

scholarly journals A NOTE ON NAVIER–STOKES EQUATIONS WITH NONORTHOGONAL COORDINATES

2018 ◽  
Vol 59 (3) ◽  
pp. 335-348 ◽  
Author(s):  
J. M. HILL ◽  
Y. M. STOKES
Keyword(s):  
Coordinate System ◽  
Stokes Equations ◽  
Flat Space ◽  
Curvilinear Coordinates ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Curvilinear Coordinate System ◽  
Navier Stokes Equations ◽  
Curvilinear Coordinate ◽  
Stress Boundary Conditions

There are many fluid flow problems involving geometries for which a nonorthogonal curvilinear coordinate system may be the most suitable. To the authors’ knowledge, the Navier–Stokes equations for an incompressible fluid formulated in terms of an arbitrary nonorthogonal curvilinear coordinate system have not been given explicitly in the literature in the simplified form obtained herein. The specific novelty in the equations derived here is the use of the general Laplacian in arbitrary nonorthogonal curvilinear coordinates and the simplification arising from a Ricci identity for Christoffel symbols of the second kind for flat space. Evidently, however, the derived equations must be consistent with the various general forms given previously by others. The general equations derived here admit the well-known formulae for cylindrical and spherical polars, and for the purposes of illustration, the procedure is presented for spherical polar coordinates. Further, the procedure is illustrated for a nonorthogonal helical coordinate system. For a slow flow for which the inertial terms may be neglected, we give the harmonic equation for the pressure function, and the corresponding equation if the inertial effects are included. We also note the general stress boundary conditions for a free surface with surface tension. For completeness, the equations for a compressible flow are derived in an appendix.

Flow Over a Cylinder at a Plane Boundary—A Model Based Upon (k–ε) Turbulence

1989 ◽  
Vol 111 (4) ◽  
pp. 414-419 ◽  
Author(s):  
T. Solberg ◽  
K. J. Eidsvik
Keyword(s):  
Pressure Gradient ◽  
Coordinate System ◽  
Adverse Pressure Gradient ◽  
Plane Boundary ◽  
Length Scale ◽  
Two Dimensional ◽  
Curvilinear Coordinate System ◽  
Recirculating Flow ◽  
Model Based ◽  
Curvilinear Coordinate

A model for two-dimensional flows over a cylinder at a plane boundary is developed. The model, based upon a (k-ε) turbulence closure, is formulated in a curvilinear coordinate system based upon frictionless flow. A length scale modification in areas of adverse pressure gradient and recirculating flow appears to be more realistic than the standard (k-ε) model. The main features of the predicted flow do not depend critically upon the details of the grid or model, which means that a well defined solution is obtained. The solution appears to be reasonable and validated to the extent that the data permits.

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