A numerical simulation of steady and unsteady two-dimensional flows past cylinder with dimples based on highly accurate pseudospectral method is the subject of the present thesis. The vorticity-streamfunction formulation of two-dimensional incompressible Navier-Stokes equations with no-slip boundary conditions is used. The system is formulated on a unit disk using curvilinear body fitted coordinate system. Key issues of the curvilinear coordinate transformation are discussed, to show its importance in properly defined node distribution. For the space discretization of the governing system the Fourier-Chebyshev pseudospectral approximation on a unit disk is implemented. To handle the singularity at the pole of the unit disk the approach of defining the computational grid proposed by Fornberg was implemented. Two algorithms for solving steady and unsteady problems are presented. For steady flow simulations the non-linear time-independent Navier-Stokes problem is solved using the Newton's method. For the time-dependent problem the semi-implicit third order Adams-Bashforth/Backward Differentiation scheme is used. In both algorithms the fully coupled system with two no-slip boundary conditions is solved. Finally numerical result for both steady and unsteady solvers are presented. A comparison of results for the smooth cylinder with those from other authors shows good agreement. Spectral accuracy is demonstrated using the steady solver. / February 2007

A numerical simulation of steady and unsteady two-dimensional flows past cylinder with dimples based on highly accurate pseudospectral method is the subject of the present thesis. The vorticity-streamfunction formulation of two-dimensional incompressible Navier-Stokes equations with no-slip boundary conditions is used. The system is formulated on a unit disk using curvilinear body fitted coordinate system. Key issues of the curvilinear coordinate transformation are discussed, to show its importance in properly defined node distribution. For the space discretization of the governing system the Fourier-Chebyshev pseudospectral approximation on a unit disk is implemented. To handle the singularity at the pole of the unit disk the approach of defining the computational grid proposed by Fornberg was implemented. Two algorithms for solving steady and unsteady problems are presented. For steady flow simulations the non-linear time-independent Navier-Stokes problem is solved using the Newton's method. For the time-dependent problem the semi-implicit third order Adams-Bashforth/Backward Differentiation scheme is used. In both algorithms the fully coupled system with two no-slip boundary conditions is solved. Finally numerical result for both steady and unsteady solvers are presented. A comparison of results for the smooth cylinder with those from other authors shows good agreement. Spectral accuracy is demonstrated using the steady solver.

A numerical simulation of steady and unsteady two-dimensional flows past cylinder with dimples based on highly accurate pseudospectral method is the subject of the present thesis. The vorticity-streamfunction formulation of two-dimensional incompressible Navier-Stokes equations with no-slip boundary conditions is used. The system is formulated on a unit disk using curvilinear body fitted coordinate system. Key issues of the curvilinear coordinate transformation are discussed, to show its importance in properly defined node distribution. For the space discretization of the governing system the Fourier-Chebyshev pseudospectral approximation on a unit disk is implemented. To handle the singularity at the pole of the unit disk the approach of defining the computational grid proposed by Fornberg was implemented. Two algorithms for solving steady and unsteady problems are presented. For steady flow simulations the non-linear time-independent Navier-Stokes problem is solved using the Newton's method. For the time-dependent problem the semi-implicit third order Adams-Bashforth/Backward Differentiation scheme is used. In both algorithms the fully coupled system with two no-slip boundary conditions is solved. Finally numerical result for both steady and unsteady solvers are presented. A comparison of results for the smooth cylinder with those from other authors shows good agreement. Spectral accuracy is demonstrated using the steady solver.

Many problems in fluid mechanics and material sciences deal with liquid-vapour flows. In these flows, the ideal gas assumption is not accurate and the van der Waals equation of state is usually used. This equation of state is non-convex and causes the solution domain to have two hyperbolic regions separated by an elliptic region. Therefore, the governing equations of these flows have a mixed elliptic-hyperbolic nature. Numerical oscillations usually appear with standard finite-difference space discretization schemes, and they persist when the order of accuracy of the semi-discrete scheme is increased. In this study, we propose to use a Chebyshev pseudospectral method for solving the governing equations. A comparison of the results of this method with very high-order (up to tenth-order accurate) finite difference schemes is presented, which shows that the proposed method leads to a lower level of numerical oscillations than other high-order finite difference schemes, and also does not exhibit fast-traveling packages of short waves which are usually observed in high-order finite difference methods. The proposed method can thus successfully capture various complex regimes of waves and phase transitions in both elliptic and hyperbolic regimes

Many problems in fluid mechanics and material sciences deal with liquid-vapour flows. In these flows, the ideal gas assumption is not accurate and the van der Waals equation of state is usually used. This equation of state is non-convex and causes the solution domain to have two hyperbolic regions separated by an elliptic region. Therefore, the governing equations of these flows have a mixed elliptic-hyperbolic nature. Numerical oscillations usually appear with standard finite-difference space discretization schemes, and they persist when the order of accuracy of the semi-discrete scheme is increased. In this study, we propose to use a Chebyshev pseudospectral method for solving the governing equations. A comparison of the results of this method with very high-order (up to tenth-order accurate) finite difference schemes is presented, which shows that the proposed method leads to a lower level of numerical oscillations than other high-order finite difference schemes, and also does not exhibit fast-traveling packages of short waves which are usually observed in high-order finite difference methods. The proposed method can thus successfully capture various complex regimes of waves and phase transitions in both elliptic and hyperbolic regimes

In this thesis we present a numerical algorithm to solve the singular Inverse Sturm-Liouville problems with symmetric potential functions. The singularity, which comes from the unbounded domain of the problem, is treated by considering the limiting case of the associated problem on the symmetric finite interval. In contrast to regular problems which are considered on a finite interval the singular inverse problem has an ill-conditioned structure despite of the limiting treatment. We use the regularization techniques to overcome the ill-posedness difficulty. Moreover, since the problem is nonlinear the iterative solution procedures are needed. Direct computation of the eigenvalues in iterative solution is handled via psoudespectral methods. The numerical examples of the considered problem are given to illustrate the accuracy and convergence behaviour.

Many problems in fluid mechanics and material sciences deal with liquid-vapour flows. In these flows, the ideal gas assumption is not accurate and the van der Waals equation of state is usually used. This equation of state is non-convex and causes the solution domain to have two hyperbolic regions separated by an elliptic region. Therefore, the governing equations of these flows have a mixed elliptic-hyperbolic nature. Numerical oscillations usually appear with standard finite-difference space discretization schemes, and they persist when the order of accuracy of the semi-discrete scheme is increased. In this study, we propose to use a Chebyshev pseudospectral method for solving the governing equations. A comparison of the results of this method with very high-order (up to tenth-order accurate) finite difference schemes is presented, which shows that the proposed method leads to a lower level of numerical oscillations than other high-order finite difference schemes, and also does not exhibit fast-traveling packages of short waves which are usually observed in high-order finite difference methods. The proposed method can thus successfully capture various complex regimes of waves and phase transitions in both elliptic and hyperbolic regimes

Many problems in fluid mechanics and material sciences deal with liquid-vapour flows. In these flows, the ideal gas assumption is not accurate and the van der Waals equation of state is usually used. This equation of state is non-convex and causes the solution domain to have two hyperbolic regions separated by an elliptic region. Therefore, the governing equations of these flows have a mixed elliptic-hyperbolic nature. Numerical oscillations usually appear with standard finite-difference space discretization schemes, and they persist when the order of accuracy of the semi-discrete scheme is increased. In this study, we propose to use a Chebyshev pseudospectral method for solving the governing equations. A comparison of the results of this method with very high-order (up to tenth-order accurate) finite difference schemes is presented, which shows that the proposed method leads to a lower level of numerical oscillations than other high-order finite difference schemes, and also does not exhibit fast-traveling packages of short waves which are usually observed in high-order finite difference methods. The proposed method can thus successfully capture various complex regimes of waves and phase transitions in both elliptic and hyperbolic regimes

Many known extra-solar giant planets lie close to their host stars. Around 60 have their semi-major axes smaller than 0.05 AU. In contrast to planets further out, the vast majority of these close-in planets have low eccentricity orbits. This suggests that their orbits have been circularized likely due to tidal dissipation inside the planets. These exoplanets share with our own Jupiter at least one trait in common: when they are subject to periodic tidal forcing, they behave like a lossy spring, with a tidal ``quality factor'', Q, of order 10^5. This parameter is the ratio between the energy in the tide and the energy dissipated per period. To explain this, a possible solution is resonantly forced internal oscillation. If the frequency of the tidal forcing happens to land on that of an internal eigenmode, this mode can be resonantly excited to a very large amplitude. The damping of such a mode inside the planet may explain the observed Q value. The only normal modes that fall in the frequency range of the tidal forcing (~ few days) are inertial modes, modes restored by the Coriolis force. We present a new numerical technique to solve for inertial modes in a convective, rotating sphere. This technique combines the use of an ellipsoidal coordinate system with a pseudo-spectral method to solve the partial differential equation that governs the inertial oscillations. We show that, this technique produces highly accurate solutions when the density profile is smooth. In particular, the lines of nodes are roughly parallel to the ellipsoidal coordinate axes. In particular, using these accurate solutions, we estimate the resultant tidal dissipation for giant planets, and find that turbulent dissipation of inertial modes in planets with smooth density profiles do not give rise to dissipation as strong as the one observed. We also study inertial modes in density profiles that exhibit discontinuities, as some recent models of Jupiter show. We found that, in this case, our method could not produce convergent solutions for the inertial modes. Additionally, we propose a way to observe inertial modes inside Saturn indirectly, by observing waves in its rings that may be excited by inertial modes inside Saturn.

Many known extra-solar giant planets lie close to their host stars. Around 60 have their semi-major axes smaller than 0.05 AU. In contrast to planets further out, the vast majority of these close-in planets have low eccentricity orbits. This suggests that their orbits have been circularized likely due to tidal dissipation inside the planets. These exoplanets share with our own Jupiter at least one trait in common: when they are subject to periodic tidal forcing, they behave like a lossy spring, with a tidal ``quality factor'', Q, of order 10^5. This parameter is the ratio between the energy in the tide and the energy dissipated per period. To explain this, a possible solution is resonantly forced internal oscillation. If the frequency of the tidal forcing happens to land on that of an internal eigenmode, this mode can be resonantly excited to a very large amplitude. The damping of such a mode inside the planet may explain the observed Q value. The only normal modes that fall in the frequency range of the tidal forcing (~ few days) are inertial modes, modes restored by the Coriolis force. We present a new numerical technique to solve for inertial modes in a convective, rotating sphere. This technique combines the use of an ellipsoidal coordinate system with a pseudo-spectral method to solve the partial differential equation that governs the inertial oscillations. We show that, this technique produces highly accurate solutions when the density profile is smooth. In particular, the lines of nodes are roughly parallel to the ellipsoidal coordinate axes. In particular, using these accurate solutions, we estimate the resultant tidal dissipation for giant planets, and find that turbulent dissipation of inertial modes in planets with smooth density profiles do not give rise to dissipation as strong as the one observed. We also study inertial modes in density profiles that exhibit discontinuities, as some recent models of Jupiter show. We found that, in this case, our method could not produce convergent solutions for the inertial modes. Additionally, we propose a way to observe inertial modes inside Saturn indirectly, by observing waves in its rings that may be excited by inertial modes inside Saturn.