Unsteady heat transfer by natural convection in a closed square cavity is investigated numerically. A new finite-volume approach is developed and applied to the two-dimensional continuity, vorticity, and energy equations. The variation of the field variables is approximated by bi-quadratic interpolation formulas over the space occupied by the finite volume and the region surrounding it. These are used in the integral conservation laws for energy, vorticity and mass. The convective transport is modelled using a new upstream-weighting approach which uses volume averages for the vorticity and the energy transported across the boundaries of the finite volume. The weighting is dependent on the skewness of the velocity field to the surfaces of the finite volume as well as its strength. It is adaptive to local flow conditions. The velocities are obtained from the application of the velocity induction law. Use is made of an image system for the free vorticity of fluid. In this way, the no-penetration condition is enforced at the cavity boundaries, but at the same time it may allow a slip condition to exist. This is not permitted in a viscous flow analysis, and the slip velocity is reduced to zero by the production of free vorticity at the boundaries. Two test cases are treated which have exact solutions. The first is not new and involves a rotating shaft. The errors are less than.06% for this case. The second case is new and involves convection past a source and sink. The maximum error is 2.3%. For both test cases, the maximum error occurs at moderate values of the cell Peclet number and diminishes at the extreme low and high values. The time-development of the profiles of the vorticity, horizontal velocity, and temperature is examined at different locations within the cavity for Rayleigh numbers equal to 10³, 10⁴, and 10⁵. For these calculations, a 21 x 21 grid was used. The flow is found to approach a steady-state condition. The steady-state results are compared with a benchmark solution. In general, the agreement is excellent. The discrepancy is found to be less than 2% for the vast majority of the results for this relatively coarse grid.

Severe stenosis may cause critical flow conditions related to artery collapse, plaque cap rupture which leads directly to stroke and heart attack. In this paper, a nonlinear viscoelastic model and a numerical method are introduced to study dynamic behaviors of the tube wall and viscous flow through a viscoelastic tube with a stenosis simulating blood flow in human carotid arteries. The Mooney-Rivlin material model is used to derive a nonlinear viscoelastic thin-wall model for the stenotic viscoelastic tube wall. The mechanical parameters in the Mooney-Rivlin model are calculated from experimental measurements. Incompressible Navier-Stokes equations in the Arbitrary Lagrangian-Eulerian formulation are used as the governing equation for the fluid flow. Interactions between fluid flow and the viscoelastic axisymmetric tube wall are handled by an incremental boundary iteration method. A Generalized Finite Differences Method (GFD) is used to solve the fluid model. The Fourth-Order Runge-Kutta method is used to deal with the viscoelastic wall model where the viscoelastic parameter is adjusted to match experimental measurements. Our result shows that viscoelasticity of tube wall causes considerable phase lag between the tube radius and input pressure. Severe stenosis causes cyclic pressure changes at the throat of the stenosis, cyclic tube compression and expansions, and shear stress change directions in the region just distal to stenosis under unsteady conditions. Results from our nonlinear viscoelastic wall model are compared with results from previous elastic wall model and experimental data. Clear improvements of our viscoelastic model over previous elastic model were found in simulating the phase lag between the pressure and wall motion as observed in experiments. Numerical solutions are compared with both stationary and dynamic experimental results. Mooney-Rivlin model with proper parameters fits the non-linear experimental stress-strain relationship of wall very well. The phase lags of tube wall motion, flow rate variations with respect to the imposed pulsating pressure are simulated well by choosing the viscoelastic parameter properly. Agreement between numerical results and experimental results is improved over the previous elastic model.

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Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (leaves 152-157).

Thesis (Ph.D.)--University of Wyoming, 2008. / Title from PDF title page (viewed on June 28, 2009). Includes bibliographical references.

Thesis (Ph. D. in Aeronautical and Astronautical Engineering)--Naval Postgraduate School, June 2003. / Dissertation supervisor and advisor: E. Roberts Wood. Includes bibliographical references (p. 205-210). Also available online.

This research presents a new, more efficient computational scheme for complex periodic flows, and brings forward two novel ideas. The first consists in the use of a Fourier space time representation in conjunction with a high-order spatial discretization. The second is based on the efficient treatment of the resulting set of equations using a fast, implicit solver. This thesis describes the formulation and implementation of the proposed framework. Firstly, a high-order spectral difference scheme for the Euler equations is introduced. Secondly, the non-linear frequency domain method resolving the unsteady behavior of the flow is discussed. Thirdly, a mathematical and experimental validation of the proposed algorithm is carried out. Numerical experiments performed in this thesis suggest that the methodology could be an attractive new avenue for large scale time-dependent problems, alleviating the computational cost traditionally associated with such simulations.

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