Algorithm for analysis of flows in ribbed annuli

Abstract

SUMMARYSpectral methods for analyses of steady flows in annuli bounded by walls with either axi-symmetric or longitudinal ribs are developed. The physical boundary conditions are enforced using the immersed boundary conditions concept. In the former case, the Stokes stream function is used to eliminate pressure and to reduce system of field equations to a single fourth-order partial differential equation. The ribs are assumed to be periodic in the axial direction and this permits representation of the solution in terms of the Fourier expansion. In the latter case, the problem is reduced to the Laplace equations for the flow modifications that can be expressed in terms of the Fourier expansions. The modal functions, which are functions of the radial coordinate, are discretized using Chebyshev polynomials. The problem formulations are closed using either the fixed volume flow rate constraint or the fixed pressure gradient constraint. Various tests have been carried out in order to demonstrate the spectral accuracy of the discretizations, as well as the spectral accuracy of the enforcement of the flow boundary conditions at the ribbed walls using the immersed boundary conditions concept. Special linear solver that takes advantage of the matrix structure has been implemented in order to reduce computational time and memory requirements. It is shown that the algorithm has superior performance when one is interested in the analysis of a large number of geometries, as part of the coefficient matrix that corresponds to the field equation is always the same and one needs to change only the part of the matrix that corresponds to the boundary relations when changing geometry of the flow domain. Copyright

Stability of flow in a channel with longitudinal grooves

“…A systematic analysis of a large number of configurations is possible using either the immersed boundary conditions method (IBC) (Szumbarski & Floryan 1999;Husain, Floryan & Szumbarski 2009;Husain & Floryan 2010;Mohammadi & Floryan 2012;Moradi & Floryan 2012) or the domain transformation method (DT) (Husain & Floryan 2010;Mohammadi & Floryan 2012). Both techniques permit the determination of the flow details with spectral accuracy for the complete range of topographies of practical interest and a seamless transition between different topographic forms.…”

Flows in annuli with longitudinal grooves

“…The algorithm relies on a mapping that transforms the complex geometry into a straight strip suitable for implementation of spectral discretization. An alternative method proposed by Moradi & Floryan (2012) uses the physical domain and enforces boundary conditions on the corrugated boundaries using the concept of immersed boundary conditions. The former method provides better access to configurations with extreme geometries.…”

Spectrally-accurate algorithm for the analysis of flows in two-dimensional vibrating channels

“…The additional attractiveness of the IBC method is associated with the precise mathematical formalism, high accuracy and sharp identification of the location of time-dependent physical boundaries. The method has been extended to two-dimensional unsteady problems [28], moving boundary problems involving Laplace [29] and biharmonic [30] operators, the complete Navier-Stokes system [31], to operators involving different classes of non-Newtonian fluids [32,33], to three-dimensional operators [34,35] as well as to operators expressed in cylindrical coordinate systems [36]. Its accuracy has been improved through the use of the overdetermined formulation [37].…”