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RESEARCH PAPERS: Forced Convection

# Accurate Boundary Element Solutions for Highly Convective Unsteady Heat Flows

[+] Author and Article Information
M. M. Grigoriev1

Department of Civil Engineering,  State University of New York at Buffalo, Buffalo, NY 14260

G. F. Dargush

Department of Civil Engineering,  State University of New York at Buffalo, Buffalo, NY 14260

1

Author to whom correspondence should be addressed: M. M. Grigoriev, Department of Civil Engineering, 212 Ketter Hall, State University of New York at Buffalo, Buffalo, NY 14260.

J. Heat Transfer 127(10), 1138-1150 (May 10, 2005) (13 pages) doi:10.1115/1.2035109 History: Received August 08, 2004; Revised May 10, 2005

## Abstract

Several recently developed boundary element formulations for time-dependent convective heat diffusion appear to provide very efficient computational tools for transient linear heat flows. More importantly, these new approaches hold much promise for the numerical solution of related nonlinear problems, e.g., Navier–Stokes flows. However, the robustness of these methods has not been examined, particularly for high Peclet number regimes. Here, we focus on these regimes for two-dimensional problems and develop the necessary temporal and spatial integration strategies. The algorithm takes advantage of the nature of the time-dependent convective kernels, and combines analytic integration over the singular portion of the time interval with numerical integration over the remaining nonsingular portion. Furthermore, the character of the kernels lets us define an influence domain and then localize the surface and volume integrations only within this domain. We show that the localization of the convective kernels becomes more prominent as the Peclet number of the flow increases. This leads to increasing sparsity and in most cases improved conditioning of the global matrix. Thus, iterative solvers become the primary choice. We consider two representative example problems of heat propagation, and perform numerical investigations of the accuracy and stability of the proposed higher-order boundary element formulations for Peclet numbers up to $105$.

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## Figures

Figure 1

Variation of the kernel Φ(d,t) with respect to time

Figure 2

Domain of influence due to time-integrated kernel originated at point O

Figure 3

Surface plots of the instantaneous convective diffusion g kernel: (a) κ=10−3; (b) κ=10−5. Source point located at ξi=(1,0.5), velocity of the media vi=(1,0.5).

Figure 4

Influence domains due to time-integrated kernel: (a) Pe=1250 and (b) Pe=125,000. Time t varies from 0.01 to 0.64 with increments in factors of 2; ϵs=10−10.

Figure 5

Global sparse matrix A for Problem 1, boundary element mesh 40×40, pt=4, ph=4; (a) Pe=1250, density of the matrix 0.01755; (b) Pe=125,000, density of the matrix 0.00934

Figure 6

Temperature profiles for Problem 1 at Pe=1250; Δt=0.05 Lines—exact solutions; Symbols—BEM solutions for pt=4, ph=4; (a) x2=0.25, t=0.5; (b) x2=0.5, t=1, Peh=62.5 for mesh 5×5, and Peh=31.25 for mesh 10×10

Figure 7

Surface plots of the thermal temperatures for Pe=125,000; Δt=0.005, pt=4, ph=4, t=1.0. Problem 1: (a) Mesh 90×90, Peh=347.2; (b) mesh 100×100, Peh=312.5; (c) mesh 154×154, Peh=202.9.

Figure 8

Temperature profiles for Problem 1 at Pe=125,000; Δt=0.005; x2=0.5; t=1; Line—exact solution; Symbols—BEM solutions for pt=4, ph=4 (a) Peh=347.2 for mesh 90×90 and Peh=312.5 for mesh 100×100; (b) Peh=202.9 for mesh 154×154 and Peh=195.3 for mesh 160×160

Figure 9

BEM solution stability chart for Pe=125,000; pt=4, ph=4; Problem 1

Figure 10

Convergence of the BEM solution error with respect to the mesh size h; pt=4 Problem 1: Pe=125,000, Δt=0.002

Figure 11

Surface plots of the temperatures for Pe=20,000; Δt=0.005, pt=4, ph=4 Problem 2, mesh 80×80, Peh=62.5: (a) t=0.25; (b) t=1.0

Figure 12

Temperature profiles for Problem 2 at Pe=20,000; mesh 80×80, Δt=0.005. Lines—exact solutions; Symbols—BEM solutions for pt=4, ph=4 (a) x2=0.2; (b) x2=0.8.

Figure 13

Time evolution of the BEM solution error for Problem 2: Pe=20,000, pt=4, ph=4 (a) variable time step size, mesh 100×100; (b) variable mesh size, Δt=0.0025

Figure 14

Convergence of the BEM solution error with respect to the mesh size h. Problem 2: Pe=20,000, Δt=0.0025; pt=4.

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