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RESEARCH PAPERS: Forced Convection

# A General Scheme for the Boundary Conditions in Convective and Diffusive Heat Transfer With Immersed Boundary Methods

[+] Author and Article Information
Arturo Pacheco-Vega1

CIEP-Facultad de Ciencias Químicas, Universidad Autónoma de San Luis Potosí, San Luis Potosí 78210, Méxicoapacheco@uaslp.mx

J. Rafael Pacheco, Tamara Rodić

Mechanical and Aerospace Engineering Department, Arizona State University, Tempe, AZ 85287-6106

1

Corresponding author.

J. Heat Transfer 129(11), 1506-1516 (Feb 12, 2007) (11 pages) doi:10.1115/1.2764083 History: Received June 19, 2006; Revised February 12, 2007

## Abstract

We describe the implementation of an interpolation technique, which allows the accurate imposition of the Dirichlet, Neumann, and mixed (Robin) boundary conditions on complex geometries using the immersed-boundary technique on Cartesian grids, where the interface effects are transmitted through forcing functions. The scheme is general in that it does not involve any special treatment to handle either one of the three types of boundary conditions. The accuracy of the interpolation algorithm on the boundary is assessed using several two- and three-dimensional heat transfer problems: (1) forced convection over cylinders placed in an unbounded flow, (2) natural convection on a cylinder placed inside a cavity, (3) heat diffusion inside an annulus, and (4) forced convection around a stationary sphere. The results show that the scheme preserves the second-order accuracy of the equations solver and are in agreement with analytical and/or numerical data.

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## Figures

Figure 2

General interpolation scheme for Dirichlet–Neumann boundary conditions: (a) three nodes outside the boundary; (b) two nodes outside the boundary

Figure 3

Bilinear interpolation

Figure 4

Streamlines and temperature contours of flow around cylinder for Re=80: (a) streamlines; (b) isotherms

Figure 5

Nu number along cylinder surface for Re=120. 엯, present scheme; ◇, experiments by Eckert and Soehngen (29).

Figure 6

Streamlines and temperature contours of flow around two cylinders for Re=80: (a) streamlines; (b) isotherms

Figure 7

Cylinder placed eccentrically in a square cavity

Figure 8

Streamlines on a 200×200 grid for (a) Ra=104 and (b) Ra=106

Figure 9

Isotherms on a 200×200 grid for (a) Ra=104 and (b) Ra=106

Figure 10

Nu number along cold wall for Ra=106 and Pr=10. ◇, present scheme; 엯, Demirdžić (35).

Figure 1

Interpolation scheme at nodes (a), and (b) or (c)

Figure 11

Nu number along cylinder surface for Ra=106 and Pr=10. Angle φ is measured from top of cylinder. ◇, present scheme; 엯, Demirdžić (35).

Figure 12

Concentric cylindrical annulus

Figure 13

((a)–(c)) L2 norm and maximum norm (ϵmax) as functions of the mesh size δ for different inner boundary conditions. (d) Error versus Δt for inner Robin boundary conditions. (a) Dirichlet (a=1, b=0, c=1). (b) Neumann (a=0, b=1, c=1). (c) Robin (a=1, b=1, c=1). (d) Radii locations: ◇, PA; 엯, PB; ▵, PC.

Figure 14

Numerical (solid lines) versus analytical (symbols) time-dependent solutions. -▵-, t=0.15; -◇-, t=0.25; -엯-, t=0.35; -▿-, t=1.40 (steady state).

Figure 15

Streamlines and temperature contours for flow around a sphere in the 50⩽Re⩽220 range. (a) Re=50, (b) Re=100, (c) Re=150, (d) Re=200, and (e) Re=220.

Figure 16

Oblique view of vortical structures of flow for Re=300

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