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Review Articles

Heat Transfer Scaling Close to the Wall for Turbulent Channel Flows

[+] Author and Article Information
Dimitrios V. Papavassiliou

e-mail address: dvpapava@ou.edu
School of Chemical,
Biological and Materials Engineering,
The University of Oklahoma,
Norman, OK 73019

1Corresponding author.

Manuscript received November 4, 2012; final manuscript received April 27, 2013; published online July 15, 2013. Assoc. Editor: Herman J.H. Clercx.

Appl. Mech. Rev 65(3), 031002 (Jul 15, 2013) (20 pages) Paper No: AMR-12-1060; doi: 10.1115/1.4024428 History: Received November 04, 2012; Revised April 27, 2013

This work serves a two-fold purpose of briefly reviewing the currently existing literature on the scaling of thermal turbulent fields and, in addition, proposing a new scaling framework and testing its applicability. An extensive set of turbulent scalar transport data for turbulent flow in infinitely long channels is obtained using a Lagrangian scalar tracking approach combined with direct numerical simulation of turbulent flow. Two cases of Poiseuille channel flow, with friction Reynolds numbers 150 and 300, and different types of fluids with Prandtl number ranging from 0.7 to 50,000 are studied. Based on analysis of this database, it is argued that the value and the location of the maximum normal turbulent heat flux are important scaling parameters in turbulent heat transfer. Implementing such scaling on the mean temperature profile for different fluids and Reynolds number cases shows a collapse of the mean temperature profiles onto a single universal profile in the near wall region of the channel. In addition, the profiles of normal turbulent heat flux and the root mean square of the temperature fluctuations appear to collapse on one profile, respectively. The maximum normal turbulent heat flux is thus established as a turbulence thermal scaling parameter for both mean and fluctuating temperature statistics.

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Figures

Grahic Jump Location
Fig. 1

Configuration of the Poiseuille flow channel depicting the two cases of heat flux boundary conditions applied to the channel walls in this study. For Case 1 (uniform heat flux applied from bottom wall of the channel only), qwB is constant while qwT is zero. For Case 2 (uniform heat flux applied from both walls of the channel), both qwB and qwT are constant with qwB=-qwT.

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Fig. 2

Values of the normal turbulent heat flux as a function of the wall-normal distance with a constant heat flux applied to one channel wall while maintaining the other adiabatic, in flow with Reτ = 150, obtained from the DNS/LST data and Kader's equation [34]: (a) for small Pr, Pr = 0.7, 3, 6, 10, and 200 and (b) high Pr, Pr = 500, 2400, 7000, 15,000, and 50000

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Fig. 3

Values of the normal turbulent heat flux as a function of the wall-normal distance with a constant heat flux applied to one channel wall while maintaining the other adiabatic, in flow with Reτ = 300, obtained from the DNS/LST data and Kader's equation [34]: (a) for small Pr, Pr = 0.7, 6, 20, and 50 and (b) high Pr, Pr = 200, 2400, 7000, 15,000, and 50,000

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Fig. 4

Values of the normal turbulent heat flux as a function of the wall-normal distance with a constant and uniform heat flux applied to both channel walls in flow with Reτ = 150, obtained from the DNS/LST data and Kader's equation [34]: (a) for small Pr, Pr = 0.7, 3, 6, 10, and 200, and (b) high Pr, Pr = 500, 2400, 7000, 15,000, and 50,000

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Fig. 5

Values of the normal turbulent heat flux as a function of the wall-normal distance with a constant and uniform heat flux applied to both channel walls in flow with Reτ = 300, obtained from the DNS/LST data and Kader's equation [34]: (a) for small Pr, Pr = 0.7, 6, 20, and 50 and (b) high Pr, Pr = 200, 2400, 7000, 15,000, and 50,000

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Fig. 6

Comparisons of the values of the normal turbulent heat flux as a function of the wall-normal distance for two different Re, Reτ = 150 and 300, for different Pr, Pr = 0.7, 6, 200, 2400, 7000, 15,000, and 50,000 with: (a) constant uniform heat flux applied to only bottom wall and (b) constant uniform heat flux applied to both walls

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Fig. 7

Location and values of the peak normal turbulent heat flux plotted as a function of the fluid Pr in two different Reτ, Reτ = 150 and 300 cases, obtained using three different methods, namely, the DNS/LST, Kader's [34] and the theoretical correlations of Eqs. (33) and (34), for the uniform constant heat flux boundary condition applied to one channel wall represented as (a) peak location and (b) peak value

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Fig. 8

Location and values of the peak normal turbulent heat flux plotted as a function of the fluid Pr in two different Reτ, Reτ = 150 and 300 cases, obtained using four different methods, namely, the DNS/LST, Kader's [34] and the theoretical correlations of Srinivasan and Papavassiliou given in Eqs. (35) and (36), and theoretical correlations of Kawamura et al. [35] given in Eqs. (37) and (38), for the uniform constant heat flux boundary condition applied to both channel walls represented as (a) peak location and (b) peak value

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Fig. 9

Mean temperature profile plotted as a function of the wall-normal distance for the case of one wall of the channel heated with constant heat flux. Results from DNS/LST and Kader's equation [34] at Reτ = 150, obtained for different Pr: (a) Pr = 0.7, 3, 6, 10, and 200, and (b) Pr = 500, 2400, 7000, 15,000, and 50,000.

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Fig. 10

Mean temperature profile plotted as a function of the wall-normal distance for the case of one wall of the channel heated with constant heat flux. Results from DNS/LST and Kader's equation [34] at Reτ = 300, obtained for different Pr: (a) Pr = 0.7, 6, 20, and 50, and (b) Pr = 200, 2400, 7000, 15,000, and 50,000.

Grahic Jump Location
Fig. 11

Mean temperature profile plotted as a function of the wall-normal distance for the case of both the walls of the channel heated with constant heat flux. Results from DNS/LST and Kader's equation [34] at Reτ = 150, obtained for different Pr: (a) Pr = 0.7, 3, 6, 10, and 200, and (b) Pr = 500, 2400, 7000, 15,000, and 50,000.

Grahic Jump Location
Fig. 12

Mean temperature profile plotted as a function of the wall-normal distance for the case of both the walls of the channel heated with constant heat flux. Results from DNS/LST and Kader's equation [34] at Reτ = 300, obtained for different Pr: (a) Pr = 0.7, 6, 20, and 50, and (b) Pr = 200, 2400, 7000, 15,000, and 50,000.

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Fig. 13

Values of the mean temperature scaled using the Wang et al. [63] scaling plotted as a function of the scaled wall-normal location for the case where one channel wall is heated with constant heat flux, plotted for different Pr in flow cases with different Reτ: (a) Reτ = 150 and (b) Reτ = 300

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Fig. 14

Values of the mean temperature scaled using the Wang et al. [63] scaling plotted as a function of the scaled wall-normal location for the case where both the channel walls are heated with constant heat flux, plotted for different Pr in flow cases with different Reτ: (a) Reτ = 150 and (b) Reτ = 300

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Fig. 15

Values of the mean temperature scaled using the Srinivasan and Papavassiliou scaling plotted as a function of the scaled wall-normal location with scaling values of the maximum normal turbulent heat flux obtained from DNS/LST, for the case where one channel wall is heated with constant heat flux, plotted for different Pr in flow cases with different Reτ: (a) Reτ = 150 and (b) Reτ = 300. The thick, solid line indicates the average obtained for all the Pr data while the error bars represent the error with one standard deviation.

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Fig. 16

Values of the mean temperature scaled using the Srinivasan and Papavassiliou scaling plotted as a function of the scaled wall-normal location with scaling values of the maximum normal turbulent heat flux obtained from DNS/LST, for the case where both the channel walls are heated with constant heat flux, plotted for different Pr in flow cases with different Reτ: (a) Reτ = 150 and (b) Reτ = 300. The thick, solid line indicates the average obtained for all the Pr data while the error bars represent the error with one standard deviation.

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Fig. 17

Values of the mean temperature scaled using the Srinivasan and Papavassiliou scaling plotted as a function of the scaled wall-normal location with scaling values of the maximum normal turbulent heat flux obtained from theoretical correlations of Srinivasan and Papavassiliou presented in Eqs. (35) and (36), for the case where both the channel walls are heated with constant heat flux, plotted for different Pr in flow cases with different Reτ: (a) Reτ = 150 and (b) Reτ = 300

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Fig. 18

Values of the mean temperature scaled using the Srinivasan and Papavassiliou scaling plotted as a function of the scaled wall-normal location with scaling values of the maximum normal turbulent heat flux obtained from theoretical correlations of Kawamura et al. [70] presented in Eqs. (37) and (38), for the case where both the channel walls are heated with constant heat flux, plotted for different Pr in flow cases with different Reτ: (a) Reτ = 150 and (b) Reτ = 300

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Fig. 19

Value of the locations of the peak normal turbulent heat flux plotted as a function of the Peτ in both the wall heating cases

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Fig. 20

Values of the mean temperature scaled using the Srinivasan and Papavassiliou scaling plotted as a function of the scaled wall-normal location with scaling values of the maximum normal turbulent heat flux used from Table 7, for the case where one channel wall is heated with constant heat flux, plotted for different Pr in flow cases with different Reτ: (a) Reτ = 150 and (b) Reτ = 300. The thick, solid line indicates the average obtained for all the Pr data while the error bars represent the error with one standard deviation.

Grahic Jump Location
Fig. 21

Values of the mean temperature scaled using the Srinivasan and Papavassiliou scaling with the MNTHF obtained from Eq. (36), plotted as a function of the scaled wall-normal location with scaling values taken from Table 8 (power law) for the case where both channel walls are heated with constant heat flux. Different Pr are presented in flow cases with different Reτ: (a) Reτ = 150 and (b) Reτ = 300. The thick, solid line indicates the average obtained for all the Pr data while the error bars represent the error with one standard deviation.

Grahic Jump Location
Fig. 22

Values of the normal turbulent heat flux scaled using the Srinivasan and Papavassiliou scaling plotted as a function of the scaled wall-normal location with scaling values of the maximum normal turbulent heat flux obtained from DNS/LST, for the case where one channel wall is heated with constant heat flux. Different Pr are presented in flow cases with different Reτ: (a) Reτ = 150 and (b) Reτ = 300. The thick, solid line indicates the average obtained for all the Pr data while the error bars represent the error with one standard deviation.

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Fig. 23

Values of the normal turbulent heat flux scaled using the Srinivasan and Papavassiliou scaling plotted as a function of the scaled wall-normal location with scaling values of the maximum normal turbulent heat flux obtained from DNS/LST, for the case where both the channel walls are heated with constant heat flux. Different Pr are presented in flow cases with different Reτ: (a) Reτ = 150 and (b) Reτ = 300. The thick, solid line indicates the average obtained for all the Pr data while the error bars represent the error with one standard deviation.

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Fig. 24

Values of the root mean square of the temperature varying as a function of the wall normal distance obtained from Refs. [61-63] for different Pr at Reτ = 180

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Fig. 25

Scaled values of the root mean square of the temperature varying as a function of the scaled wall normal distance obtained using the scaling of Srinivasan and Papavassiliou shown for different Pr at Reτ = 180

Tables

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