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Research Papers: Forced Convection

# Heat Transfer Resulting From the Interaction of a Vortex Pair With a Heated Wall

[+] Author and Article Information
Roland Martin

Laboratoire de Modélisation et d’Imagerie en Géosciences, CNRS UMR 5212 and INRIA Futurs Magique3D, Université de Pau et des Pays de l’Adour, Bâtiment IPRA, Avenue de l’Université, 64013 Pau, France

Roberto Zenit

Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México, Apartado Postal 70-360, Distrito Federal 04510, México

J. Heat Transfer 130(5), 051701 (Apr 08, 2008) (8 pages) doi:10.1115/1.2885182 History: Received December 04, 2006; Revised July 02, 2007; Published April 08, 2008

## Abstract

The motion of a two-dimensional vortex pair moving toward a wall is studied numerically. The case for which the wall is heated is analyzed. The equations of momentum and energy conservation are solved using a finite volume scheme. In this manner, the instantaneous heat transfer from the wall is obtained and is related to the dynamics of the fluid vortex interacting with the wall. It was found that, as expected, when the fluid vortex approaches the wall, the heat transfer increases significantly. The heat transfer changes in a nonmonotonic manner as a function of time: When the vortex first reaches the wall, a volume of heated fluid is convected from the wall; this fluid volume circulates in the vicinity of the wall, causing the rate of heat transfer to decrease slightly, to then increase again. A wide range of Prandtl and Reynolds numbers were tested. A measure of the effective heat transfer coefficient, or Nusselt number, is proposed.

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## Figures

Figure 1

The interaction of a vortex pair (or ring) with a heated wall

Figure 2

Mean Nusselt number as a function of time. Simulations for two mesh sizes, considering Re=250 and Pr=0.7.

Figure 3

Evolution of the vorticity field. Black and white levels on the figure represent values of 0.5s−1 and −0.5s−1 of vorticity, respectively. The solid lines show contours of isovorticity, which are used to visualize the vortical structures. The case shown is for Re=250. The time is shown in dimensionless terms: t*=t∕(a∕U). A unit of t* represents the time it takes for a vortex to move, at a constant velocity U, a distance equivalent to its size. The distances are normalized by the size of the heated plate: x∕L0 and y∕L0. Note that only the right side of the simulation is shown. The mesh shown on the first image is indicative of the computational grid.

Figure 6

Nusselt number as a function of position over the plate for different time instants. The case shown is for Re=250 and Pr=1.4. Note that only the right side of the profile is shown.

Figure 7

Mean Nusselt number as a function of time. Simulations for two typical values of Re and Pr are shown. Also, simulations for a purely conductive system (no fluid motion) are shown for comparison.

Figure 8

Mean Nusselt number as a function of time. (a) Simulations keeping Re fixed and Pr varied. (b) Simulations keeping Pr fixed and Re varied.

Figure 9

Ratio of the mean convective to the purely conductive Nusselt numbers. Simulations for two typical Re numbers are shown for four values of Pr.

Figure 10

Maximum of the mean Nusselt number against the Reynolds number for several values of Pr between 1.4 and 13.26. Additional results are also shown for Pr=0.7 and 100 and for Re=250 and 1000.

Figure 11

Time integral of the mean Nusselt number against the Reynolds number for several values of Pr between 1.4 and 13.26

Figure 5

Evolution of the temperature field. Black and white levels in the figure represent values of 0.9 and 1.0 of the dimensionless temperature (T−Tw)∕(Tinf−Tw), respectively. The solid lines show contours of isovorticity, which are used to visualize the vortical structures. The case shown is for Re=250 and Pr=1.4. Time and size are scaled, as in Fig. 3. Note also that only the right side of the simulation is shown.

Figure 4

Trajectory of the center of the main vortex. Two simulation cases are shown: (○) Re=250; (+) Re=1000. The dashed line is the trajectory from Eq. 8, considering C=22.9. The results of Orlandi (10) are also shown.

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