IMMERSED BOUNDARY CONDITIONS METHOD FOR COMPUTATIONAL FLUID DYNAMICS PROBLEMS

This dissertation presents implicit spectrally-accurate algorithms based on the concept of immersed boundary conditions (IBC) for solving a range of computational fluid dynamics (CFD) problems where the physical domains involve boundary irregularities. Both fixed and moving irregularities are considered with particular emphasis placed on the two-dimensional moving boundary problems. The physical model problems considered are comprised of the Laplace operator, the biharmonic operator and the Navier-Stokes equations, and thus cover the most commonly encountered types of operators in CFD analyses. The IBC algorithm uses a fixed and regular computational domain with flow domain immersed inside the computational domain. Boundary conditions along the edges of the time-dependent flow domain enter the algorithm in the form of internal constraints. Spectral spatial discretization for two-dimensional problems is based on Fourier expansions in the stream-wise direction and Chebyshev expansions in the normal-to-the-wall direction. Up to fourth-order implicit temporal discretization methods have been implemented. The IBC algorithm is shown to deliver the theoretically predicted accuracy in both time and space. Construction of the boundary constraints in the IBC algorithm provides degrees of freedom in excess of that required to formulate a closed system of algebraic equations. The ‘classical IBC formulation’ works by retaining number boundary constraints that are just sufficient to form a closed system of equations. The use of additional boundary constraints leads to the ‘over-determined formulation’ of the IBC algorithm. Over- determined systems are explored in order to improve the accuracy o f the IBC method and to expand its applicability to more extreme geometries. Standard direct over-determined solvers based on evaluation of pseudo-inverses of the complete coefficient matrices have been tested on three model problems, namely, the Laplace equation, the biharmonic equation and the Navier-Stokes equations. In all cases tested the over-determined formulations based on standard solvers were found to improve the accuracy and the range o f applicability o f the IBC method. Efficient linear solvers suitable for the spectral implementation of the IBC method have been developed and tested in the context of two-dimensional steady and unsteady Stokes flow in the presence of fixed boundary irregularities. These solvers can work with the classical as well as the over-determined formulations of the method. Significant acceleration of the computations as well as significant reduction of the memory requirements have been accomplished by taking advantage of the structure of the coefficient matrix resulting from the implementation of the IBC algorithm. Performances of the new solvers have been compared with the standard direct solvers and are shown to be of up to two orders of magnitude better. It has been determined that the new methods are by at least an order of magnitude faster than the iterative methods while removing restrictions based on the convergence criteria and thus expanding the severity of the geometries that can be dealt with using the IBC algorithm. The performance of the IBC method combined with the new solvers has been compared with the performance of a method based on the generation of the boundary conforming grids, and is found to be better by at least two orders of magnitude. Application of the new solvers to the unsteady problems also results in performance improvement of up to two orders of magnitude. The specialized solvers applied to the over-determined formulation is shown to be at least two orders of magnitude faster than their standard counterparts while capable of extending the range of applicability of the IBC algorithm by 50%-70% for the Stokes flow problem. The concept of the specialized solvers has been extended to solve two-dimensional moving boundary problems described by the Navier-Stokes equations, where the new solver has been shown to result in a significant acceleration of computations as well as substantial reduction in memory requirements. The conceptual aspects of extending the IBC algorithm for solving three-dimensional problems have been presented using the vorticity-velocity formulation of the three- dimensional Navier-Stokes equations. Test results on the implementation of the IBC algorithm for three-dimensional problems are discussed in the context of heat diffusion IV problems in the presence of fixed as well as moving boundaries. The algorithm is shown to be spectrally-accurate in space and capable of delivering theoretically predicted accuracy in time for the different test problems. Given a potentially large size of the resultant linear algebraic system, various methods that take advantage of the special structure of the coefficient matrix have been explored in search for an efficient solver, including two versions of the specialized direct solver as well as serial and parallel iterative solvers. Both versions of the specialized direct solver have been shown to be more computationally efficient than the other solution methods. Possible applications of the IBC algorithm for analyzing physical problems have also been presented. The advantage of using IBC algorithm is illustrated by considering its application to two physical problems, which are - i) analysis of the effects of distributed roughness on friction factor and ii) analysis of traveling wave instability in wavy channels. These examples clearly show the attractiveness of the IBC algorithm for studying effects of a large array of boundary geometries on the flow field

IMMERSED BOUNDARY CONDITIONS METHOD FOR COMPUTATIONAL FLUID DYNAMICS PROBLEMS

This dissertation presents implicit spectrally-accurate algorithms based on the concept of immersed boundary conditions (IBC) for solving a range of computational fluid dynamics (CFD) problems where the physical domains involve boundary irregularities. Both fixed and moving irregularities are considered with particular emphasis placed on the two-dimensional moving boundary problems. The physical model problems considered are comprised of the Laplace operator, the biharmonic operator and the Navier-Stokes equations, and thus cover the most commonly encountered types of operators in CFD analyses. The IBC algorithm uses a fixed and regular computational domain with flow domain immersed inside the computational domain. Boundary conditions along the edges of the time-dependent flow domain enter the algorithm in the form of internal constraints. Spectral spatial discretization for two-dimensional problems is based on Fourier expansions in the stream-wise direction and Chebyshev expansions in the normal-to-the-wall direction. Up to fourth-order implicit temporal discretization methods have been implemented. The IBC algorithm is shown to deliver the theoretically predicted accuracy in both time and space. Construction of the boundary constraints in the IBC algorithm provides degrees of freedom in excess of that required to formulate a closed system of algebraic equations. The ‘classical BBC formulation’ works by retaining number boundary constraints that are just sufficient to form a closed system of equations. The use of additional boundary constraints leads to the ‘over-determined formulation’ of die IBC algorithm. Over- determined systems are explored in order to improve the accuracy of the IBC method and to expand its applicability to more extreme geometries. Standard direct over-determined solvers based on evaluation of pseudo-inverses of the complete coefficient matrices have been tested on three model problems, namely, the Laplace equation, the biharmonic equation and the Navier-Stokes equations. In all cases tested the over-determined m formulations based on standard solvers were found to improve the accuracy and the range o f applicability o f the IBC method. Efficient linear solvers suitable for the spectral implementation of the IBC method have been developed and tested in the context of two-dimensional steady and unsteady Stokes flow in the presence of fixed boundary irregularities. These solvers can work with the classical as well as the over-determined formulations of the method. Significant acceleration of the computations as well as significant reduction of the memory requirements have been accomplished by taking advantage of the structure of the coefficient matrix resulting from the implementation of the IBC algorithm. Performances o f the new solvers have been compared with the standard direct solvers and are shown to be of up to two orders of magnitude better. It has been determined that the new methods are by at least an order of magnitude faster than the iterative methods while removing restrictions based on the convergence criteria and thus expanding the severity of the geometries that can be dealt with using the IBC algorithm. The performance of the IBC method combined with the new solvers has been compared with the performance of a method based on the generation of the boundary conforming grids, and is found to be better by at least two orders of magnitude. Application of the new solvers to the unsteady problems also results in performance improvement of up to two orders of magnitude. The specialized solvers applied to the over-determined formulation is shown to be at least two orders of magnitude faster than their standard counterparts while capable of extending the range of applicability of the IBC algorithm by 50%-70% for the Stokes flow problem. The concept of the specialized solvers has been extended to solve two-dimensional moving boundary problems described by the Navier-Stokes equations, where the new solver has been shown to result in a significant acceleration of computations as well as substantial reduction in memory requirements. The conceptual aspects of extending the IBC algorithm for solving three-dimensional problems have been presented using the vorticity-velocity formulation of the three- dimensional Navier-Stokes equations. Test results on the implementation of the IBC algorithm for three-dimensional problems are discussed in the context of heat diffusion IV problems in the presence of fixed as well as moving boundaries. The algorithm is shown to be spectrally-accurate in space and capable of delivering theoretically predicted accuracy in time for the different test problems. Given a potentially large size of the resultant linear algebraic system, various methods that take advantage of the special structure of the coefficient matrix have been explored in search for an efficient solver, including two versions of the specialized direct solver as well as serial and parallel iterative solvers. Both versions of the specialized direct solver have been shown to be more computationally efficient than the other solution methods. Possible applications of the IBC algorithm for analyzing physical problems have also been presented. The advantage of using IBC algorithm is illustrated by considering its application to two physical problems, which are - i) analysis of the effects of distributed roughness on friction factor and ii) analysis of traveling wave instability in wavy channels. These examples clearly show the attractiveness of the IBC algorithm for studying effects of a large array of boundary geometries on the flow field

Immersed Boundary Smooth Extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods

The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet it only achieves first-order spatial accuracy near embedded boundaries. In this paper, we introduce a new high-order numerical method which we call the Immersed Boundary Smooth Extension (IBSE) method. The IBSE method achieves high-order accuracy by smoothly extending the unknown solution of the PDE from a given smooth domain to a larger computational domain, enabling the use of simple Cartesian-grid discretizations (e.g. Fourier spectral methods). The method preserves much of the flexibility and robustness of the original IB method. In particular, it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta-functions to communicate information between the computational grid and the boundary. We present a fast algorithm for solving elliptic equations, which forms the basis for simple, high-order implicit-time methods for parabolic PDE and implicit-explicit methods for related nonlinear PDE. We apply the IBSE method to solve the Poisson, heat, Burgers', and Fitzhugh-Nagumo equations, and demonstrate fourth-order pointwise convergence for Dirichlet problems and third-order pointwise convergence for Neumann problems

Computation of Steady Incompressible Flows in Unbounded Domains

In this study we revisit the problem of computing steady Navier-Stokes flows in two-dimensional unbounded domains. Precise quantitative characterization of such flows in the high-Reynolds number limit remains an open problem of theoretical fluid dynamics. Following a review of key mathematical properties of such solutions related to the slow decay of the velocity field at large distances from the obstacle, we develop and carefully validate a spectrally-accurate computational approach which ensures the correct behavior of the solution at infinity. In the proposed method the numerical solution is defined on the entire unbounded domain without the need to truncate this domain to a finite box with some artificial boundary conditions prescribed at its boundaries. Since our approach relies on the streamfunction-vorticity formulation, the main complication is the presence of a discontinuity in the streamfunction field at infinity which is related to the slow decay of this field. We demonstrate how this difficulty can be overcome by reformulating the problem using a suitable background "skeleton" field expressed in terms of the corresponding Oseen flow combined with spectral filtering. The method is thoroughly validated for Reynolds numbers spanning two orders of magnitude with the results comparing favourably against known theoretical predictions and the data available in the literature.Comment: 39 pages, 12 figures, accepted for publication in "Computers and Fluids

Improved procedure for the computation of Lamb's coefficients in the Physalis method for particle simulation

The Physalis method is suitable for the simulation of flows with suspended spherical particles. It differs from standard immersed boundary methods due to the use of a local spectral representation of the solution in the neighborhood of each particle, which is used to bridge the gap between the particle surface and the underlying fixed Cartesian grid. This analytic solution involves coefficients which are determined by matching with the finite-difference solution farther away from the particle. In the original implementation of the method this step was executed by solving an over-determined linear system via the singular-value decomposition. Here a more efficient method to achieve the same end is described. The basic idea is to use scalar products of the finite-difference solutions with spherical harmonic functions taken over a spherical surface concentric with the particle. The new approach is tested on a number of examples and is found to posses a comparable accuracy to the original one, but to be significantly faster and to require less memory. An unusual test case that we describe demonstrates the accuracy with which the method conserves the fluid angular momentum in the case of a rotating particle

Spectrally-Accurate Algorithm for Flows in 3-Dimensional Rough Channels

In this work a spectrally accurate algorithm has been developed for the simulation of three-dimensional flows bounded by rough walls. The algorithm is based on the velocity-vorticity formulation and uses the concept of Immersed Boundary Conditions (IBC) for the enforcement of the boundary conditions. The flow domain is immersed inside a fixed computational domain. The geometry of the boundaries is expressed in terms of double Fourier expansions and boundary conditions enter the algorithm in the form of constraints. The spatial discretization uses Fourier expansions in the stream-wise and span-wise directions and Chebyshev expansions in the wall-normal direction. The algorithm can use either the fixed flow rate constraint or the fixed pressure gradient constraint; a direct implementation of the former constraint is described. An efficient solver which takes advantage of the structure of the coefficient matrix has been developed. Taking the advantage of the reality conditions enhances the efficiency of the solver both in terms of memory and computational speed. It is demonstrated that the applicability of the algorithm can be extended to more extreme geometries using the over-determined formulation. Various tests confirm the spectral accuracy of the algorithm