Hakan I. Tarman

An Eigenfunction Analysis of Turbulent Thermal Convection

Turbulent convection is considered for the case of stress-free boundary conditions. The Boussinesq equations are numerically integrated for Pr = 0.72 and Ra = 46 000. This results in a data base which is analyzed by decomposing the flow in terms of eigenfunctions generated from the spatial covariance tensor. It is shown that this leads to a relatively simple dynamical description in terms of relatively few modes. Explicit representations of the major modes and their evolution are presented. As another application it is shown that this approach leads to a substantial compression of the turbulence data.

A KARHUNEN-LOEVE-BASED APPROACH TO NUMERICAL SIMULATION OF TRANSITION IN RAYLEIGH-BENARD CONVECTION

A Karhunen-Loève (K-L) basis is generated empirically, using a database obtained by numerical integration of Boussinesq equations representing Rayleigh-Benard convection in a weakly turbulent state in a periodic convective box with free upper and lower surfaces. This basis is then used to reduce the governing partial differential equation (PDE) into a truncated system of amplitude equations under Galerkin projection. In the generation and implementation of the basis, the symmetries of the PDE and the geometry are fully exploited. The resulting amplitude equations are integrated numerically and it is shown that, with the use of the K-L basis in the present formulation, the known dynamics of the flow in the transition region is completely captured.

Efficient Numerical Algorithm for Cascaded Raman Fiber Lasers Using a Spectral Method

Over recent decades, fiber Raman lasers (FRLs) have received much attention from researchers and have become a challenge for them both numerically and experimentally. The equations governing the FRLs are in the form of a first-order system of nonlinear two-point boundary-value ordinary differential equations. In this paper, an algorithm for solving this system of differential equations using a spectral method, namely Chebyshev pseudospectral method is presented in detail and then numerical simulations are performed. The main advantage of the spectral methods is in their optimality in achieving high accuracy by using fewer degrees of freedom under suitable conditions. It is shown that the proposed spectral method in combination with the Newton method results in a considerable reduction in the size of the discretized problem and in the computational effort to achieve high accuracy. In this paper, a new approach for constructing an initial approximate solution for the Newton iteration is also presented.

On the Accuracy of Spectral Element Method in Electromagnetic Scattering Problems

Spectral element method (SEM), which is known of its high accuracy, has been recently applied in solving electromagnetic problems governed by Maxwells equations. This paper investigates the accuracy of SEM in two- dimensional, frequency-domain electromagnetic scattering problems where Helmholtz equation acts as the governing partial differential equation (PDE). As experience in meshing a problem in finite element method is important to obtain accurate results, the choice of elements in SEM, on the other hand, is important too. The aspect ratio in this paper is taken into account while studying the accuracy in a single element by utilizing the Greens function. In addition, the scalar field scattered by a circular cylinder placed in front of an incident plane wave is solved after truncating the domain by perfectly matched layer. Numerical results show that one should carefully discretize the problem and keeping the aspect ratio close to unity as much as possible to guarantee accurate results.

Solenoidal bases for numerical studies of?transition in pipe flow

In this paper, solenoidal basis functions are employed in numerical studies of transition in incompressible pipe flow. The bases are formulated in terms of Legendre polynomials, which are more favorable both for the functional form of the basis functions and for the inner product integrals arising in the Galerkin projection scheme. The projection is performed onto the dual solenoidal basis set to eliminate the pressure gradient term from the governing equations. This simplifies the numerical approach to the problem and the resulting scheme is used to study the nonlinear stability of pipe flow. Some preliminary results are compared with those found in the literature.

Drag Reduction via Phase Randomization in Turbulent Pipe Flow

In this study, possibility of reducing drag in turbulent pipe flow via phase randomization is investigated. Phase randomization is a passive drag reduction mechanism, the main idea behind which is, reduction in drag can be obtained via distrupting the wave-like structures present in the flow. To facilitate the investigation flow in a circular cylindrical pipe is simulated numerically. DNS (direct numerical simulation) approach is used with a solenoidal spectral formulation, hence the continuity equation is automatically satisfied (Tugluk and Tarman, Acta Mech 223(5):921–935, 2012). Simulations are performed for flow driven by a constant mass flux, at a bulk Reynolds number (Re) of 4900. Legendre polynomials are used in constructing the solenoidal basis functions employed in the numerical method.

A Spectral Solenoidal-Galerkin Method for Rotating Thermal Convection between Rigid Plates

The problem of thermal convection between rotating rigid plates under the influence of gravity is treated numerically. The approach uses solenoidal basis functions and their duals which are divergence free. The representation in terms of the solenoidal bases provides ease in the implementation by a reduction in the number of dependent variables and equations. A Galerkin procedure onto the dual solenoidal bases is utilized in order to reduce the governing system of partial differential equations to a system of ordinary differential equations for subsequent parametric study. The Galerkin procedure results in the elimination of the pressure and is facilitated by the use of Fourier-Legendre spectral representation. Numerical experiments on the linear stability of rotating thermal convection and nonlinear simulations are performed and satisfactorily compared with the literature.

Variational Approach to Power Evolution in Cascaded Fiber Raman Laser

A variational approach is formulated and implemented for numerically solving a system of nonlinear two-point boundary value problem (BVP) with coupled boundary conditions modeling the power evolution in cascaded fiber Raman laser with the fiber Bragg gratings at the ends of the cavity. The nonlinearity is treated by successive linearization and the coupled boundary conditions are naturally incorporated into the system through integration in the variational setting. A global approximation of the dependent variables in terms of Legendre polynomials is used to provide a stable Lagrangian interpolation representation as well as the Legendre-Gauss quadrature for accurate numerical evaluation of integrals in the variational formulation. An initial approximate solution is constructed for the delicate convergence to the solution. The approach is validated against an approximate analytic solution and some exact integrals of the variables. The numerical experiments show exponential (spectral) accuracy achieved with much lower resolution in comparison to a widely available BVP solver. Further numerical experiments are performed to reveal the physical characteristics of the underlying model.

Numerical simulation of flow through stenotic artery

Numerical simulation of flow through stenotic artery is performed using spectral element method in an axisymmetric geometry.This method, under suitable conditions, provides high accuracy. The use of the weak form of the governing model equations, brings flexibility in treating solution domains with nonstandard geometry. The solenoidal character of the flow is preserved by the treatment of the pressure in a natural way in the numerical formulation. The use of Legendre-Lobatto grid points with its denser distribution near the boundaries increases the ability of the grid to resolve the flow boundary layer. This study is aimed as the first step in the development of the technique.

Dynamics of wall bounded flow

There are various scenarios proposed in literature for transition in plane channel (Poiseuille) flow. In this work, one of these scenarios, namely, streak break-down, is tested numerically using a Karhunen-Loeve (K-L) based model. The K-L basis was empirically generated earlier using a numerical database representing the flow. This basis is modified in this work to include the mean flow. A K-L basis provides an optimal parametrization of the underlying flow in energy norm. Since it is specific to the flow, each basis element carries an independent characteristic of the flow and has physical interpretation. A system of model amplitude equations is then obtained by Galerkin projection of the governing equations onto the space spanned by the K-L basis. The physical interpretation of the basis elements is used to truncate the resulting system to obtain a low dimensional model.