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Research Papers: Forced Convection

# Temperature Scalings and Profiles in Forced Convection Turbulent Boundary Layers

[+] Author and Article Information
Xia Wang1

Department of Mechanical Engineering, Oakland University, Rochester, MI 48309wang@oakland.edu

Luciano Castillo, Guillermo Araya

Department of Mechanical Engineering, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180

1

Corresponding author.

J. Heat Transfer 130(2), 021701 (Feb 04, 2008) (17 pages) doi:10.1115/1.2813781 History: Received June 13, 2006; Revised May 30, 2007; Published February 04, 2008

## Abstract

Based on the theory of similarity analysis and the analogy between momentum and energy transport equations, the temperature scalings have been derived for forced convection turbulent boundary layers. These scalings are shown to be able to remove the effects of Reynolds number and the pressure gradient on the temperature profile. Furthermore, using the near-asymptotic method and the scalings from the similarity analysis, a power law solution is obtained for the temperature profile in the overlap region. Subsequently, a composite temperature profile is found by further introducing the functions in the wake region and in the near-the-wall region. The proposed composite temperature profile can describe the entire boundary layer from the wall all the way to the outer edge of the turbulent boundary layer at finite Re number. The experimental data and direct numerical simulation (DNS) data with zero pressure gradient and adverse pressure gradient are used to confirm the accuracy of the scalings and the proposed composite temperature profiles. Comparison with the theoretical profiles by Kader (1981, “Temperature and Concentration Profiles in Fully Turbulent Boundary Layers  ,” Int. J. Heat Mass Transfer, 24, pp. 1541–1544; 1991, “Heat and Mass Transfer in Pressure-Gradient Boundary Layers  ,” Int. J. Heat Mass Transfer, 34, pp. 2837–2857) shows that the current theory yields a higher accuracy. The error in the mean temperature profile is within 5% when the present theory is compared to the experimental data. Meanwhile, the Stanton number is calculated using the energy and momentum integral equations and the newly proposed composite temperature profile. The calculated Stanton number is consistent with previous experimental results and the DNS data, and the error of the present prediction is less than 5%. In addition, the growth of the thermal boundary layer is obtained from the theory and the average error is less than 5% for the range of Reynolds numbers between $5×105$ and $5×106$ when compared with the empirical correlation for the experimental data of isothermal boundary layer conditions.

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## Figures

Figure 1

Schematic showing various regions inside the thermal boundary layer

Figure 2

Comparisons of the temperature profiles in inner variables using the classical scaling and present scaling for ZPG and APG flows

Figure 3

Comparisons of the temperature profiles in outer variables using the classical scaling and present scaling for ZPG and APG flows

Figure 4

The composite temperature profile of ZPG flows: Blackwell (18)

Figure 5

The composite temperature profile of ZPG flows: Reynolds (19)

Figure 6

The composite temperature profile of DNS data for ZPG flows: Kong (20)

Figure 7

The new composite temperature profile for APG flow: Blackwell (18)m=−0.15

Figure 8

The new composite temperature profile for APG flow: Blackwell (18)m=−0.2

Figure 9

Kader’s composite temperature profile for APG flows: Blackwell (18)m=−0.15

Figure 10

Reynolds number based on the thermal boundary layer thickness versus Reynolds number based on the x coordinate

Figure 11

Stanton number calculated by the composite temperature profile and the integral energy equation for the isothermal data

Figure 12

Stanton number calculated by the composite temperature profile and the integral energy equation for the isoflux data

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