Computation of unsteady momentum and heat transfer from a fixed circular cylinder in laminar flow

This paper presents a finite difference solution for 2D, low Reynolds number, unsteady flow around and heat transfer from a stationary circular cylinder placed in a uniform flow. The fluid is assumed to be incompressible and of constant property. The governing equations are the Navier-Stokes equations, the continuity equation, a Poisson equation for pressure and the energy equation. The temperature of the cylinder wall is kept constant and the viscous energy dissipation term is neglected in the energy equation. The computed Strouhal numbers, time-mean drag and base pressure coefficients, as well as the average Nusselt numbers compare well with existing experimental results

Stability and Vortex Shedding of Bluff Body Arrays

The primary purpose of this study was to develop an understanding of the stability of laminar flow through bluff body arrays, and investigate the nature of the unsteady vortex shedding regime that follows. The flow was numerically investigated using a specially developed multi-domain spectral element solver. Important criteria in the solver development were flexibility, efficiency, and accuracy. Flexibility was critical to the functionality of the code, as arrays of varying geometry were investigated. Efficiency with a high degree of accuracy was also of primary importance, with the code implemented to run efficiently on today's massively parallel architectures. Numerical two-dimensional stability analysis of the flow in several configurations of inline and staggered array geometries was performed. The growth rate, eigenfunction, and frequency of the disturbances were determined. The critical Reynolds number for flow transition in each case was identified and compared to that of flow over a single body. Based on the solutions of the laminar flow, a one-dimensional analytical analysis was performed on selected velocity profiles in the wake region. The results of this analysis were used to guide the interpretation of the two dimensional results and formulate a general theory of stability of inline and staggered bluff body arrays. The nature of the flow in the unsteady regime following the onset of instability was examined for an inline and a staggered arrangement. Particular attention was focused on the vortex shedding which was visualized and quantified through computation of the flow swirl, a quantity which identifies regions of rotary motion. The conditions required for the generation of leading edge vortex shedding were identified and discussed. Finally, a third geometry related to the inline and staggered arrays was considered. Flow solution data for this geometry is presented and its suitability as a model for louvered arrays was discussed.Air Conditioning and Refrigeration Project 11

A "poor man's" approach to topology optimization of natural convection problems

Topology optimization of natural convection problems is computationally expensive, due to the large number of degrees of freedom (DOFs) in the model and its two-way coupled nature. Herein, a method is presented to reduce the computational effort by use of a reduced-order model governed by simplified physics. The proposed method models the fluid flow using a potential flow model, which introduces an additional fluid property. This material property currently requires tuning of the model by comparison to numerical Navier-Stokes based solutions. Topology optimization based on the reduced-order model is shown to provide qualitatively similar designs, as those obtained using a full Navier-Stokes based model. The number of DOFs is reduced by 50% in two dimensions and the computational complexity is evaluated to be approximately 12.5% of the full model. We further compare to optimized designs obtained utilizing Newton's convection law.Comment: Preprint version. Please refer to final version in Structural Multidisciplinary Optimization https://doi.org/10.1007/s00158-019-02215-

Thermal receptivity of free convective flow from a heated vertical surface: linear waves

Numerical techniques are used to study the receptivity to small-amplitude thermal disturbances of the boundary layer flow of air which is induced by a heated vertical flat plate. The fully elliptic nonlinear, time-dependent Navier–Stokes and energy equations are first solved to determine the steady state boundary-layer flow, while a linearised version of the same code is used to determine the stability characteristics. In particular we investigate (i) the ultimate fate of a localised thermal disturbance placed in the region near the leading edge and (ii) the effect of small-scale surface temperature oscillations as means of understanding the stability characteristics of the boundary layer. We show that there is a favoured frequency of excitation for the time-periodic disturbance which maximises the local response in terms of the local rate of heat transfer. However the magnitude of the favoured frequency depends on precisely how far from the leading edge the local response is measured. We also find that the instability is advective in nature and that the response of the boundary layer consists of a starting transient which eventually leaves the computational domain, leaving behind the large-time time-periodic asymptotic state. Our detailed numerical results are compared with those obtained using parallel flow theory

A Method for Geometry Optimization in a Simple Model of Two-Dimensional Heat Transfer

This investigation is motivated by the problem of optimal design of cooling elements in modern battery systems. We consider a simple model of two-dimensional steady-state heat conduction described by elliptic partial differential equations and involving a one-dimensional cooling element represented by a contour on which interface boundary conditions are specified. The problem consists in finding an optimal shape of the cooling element which will ensure that the solution in a given region is close (in the least squares sense) to some prescribed target distribution. We formulate this problem as PDE-constrained optimization and the locally optimal contour shapes are found using a gradient-based descent algorithm in which the Sobolev shape gradients are obtained using methods of the shape-differential calculus. The main novelty of this work is an accurate and efficient approach to the evaluation of the shape gradients based on a boundary-integral formulation which exploits certain analytical properties of the solution and does not require grids adapted to the contour. This approach is thoroughly validated and optimization results obtained in different test problems exhibit nontrivial shapes of the computed optimal contours.Comment: Accepted for publication in "SIAM Journal on Scientific Computing" (31 pages, 9 figures

On deterministic approximation of the Boltzmann equation in a bounded domain

In this paper we present a fully deterministic method for the numerical solution to the Boltzmann equation of rarefied gas dynamics in a bounded domain for multi-scale problems. Periodic, specular reflection and diffusive boundary conditions are discussed and investigated numerically. The collision operator is treated by a Fourier approximation of the collision integral, which guarantees spectral accuracy in velocity with a computational cost of $N\,\log(N)$, where $N$ is the number of degree of freedom in velocity space. This algorithm is coupled with a second order finite volume scheme in space and a time discretization allowing to deal for rarefied regimes as well as their hydrodynamic limit. Finally, several numerical tests illustrate the efficiency and accuracy of the method for unsteady flows (Poiseuille flows, ghost effects, trend to equilibrium)

Invariant Discretization Schemes Using Evolution-Projection Techniques

Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the one-dimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for non-invariant schemes of the heat equation in computational coordinates. We propose a new methodology for handling moving discretization grids which are generally indispensable for invariant numerical schemes. The idea is to use the invariant grid equation, which determines the locations of the grid point at the next time level only for a single integration step and then to project the obtained solution to the regular grid using invariant interpolation schemes. This guarantees that the scheme is invariant and allows one to work on the simpler stationary grids. The discretization errors of the invariant schemes are established and their convergence rates are estimated. Numerical tests are carried out to shed some light on the numerical properties of invariant discretization schemes using the proposed evolution-projection strategy