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Research Papers: Fundamental Issues and Canonical Flows

# Entropy Generation in the Viscous Parts of Turbulent Boundary Layers

[+] Author and Article Information
Donald M. McEligot1

Aerospace and Mechanical Engineering Department, University of Arizona, Tucson, AZ 85721; Institute für Kernenergetik und Energiesysteme (IKE), University of Stuttgart, D-70569 Stuttgart, Deutschland; Idaho National Laboratory (INL), Idaho Falls, ID 83415-3885

Edmond J. Walsh

Stokes Research Institute, Mechanical and Aeronautical Engineering Department, University of Limerick, Limerick, Ireland

Eckart Laurien

Institute für Kernenergetik und Energiesysteme (IKE), University of Stuttgart, D-70569 Stuttgart, Deutschland

Philippe R. Spalart

Boeing Commercial Airplanes, Seattle, WA 98124-2207

1

Corresponding author.

J. Fluids Eng 130(6), 061205 (Jun 05, 2008) (12 pages) doi:10.1115/1.2928376 History: Received October 30, 2006; Revised February 04, 2008; Published June 05, 2008

## Abstract

The local (pointwise) entropy generation rate per unit volume $S‴$ is a key to improving many energy processes and applications. Consequently, in the present study, the objectives are to examine the effects of Reynolds number and favorable streamwise pressure gradients on entropy generation rates across turbulent boundary layers on flat plates and—secondarily—to assess a popular approximate technique for their evaluation. About two-thirds or more of the entropy generation occurs in the viscous part, known as the viscous layer. Fundamental new results for entropy generation in turbulent boundary layers are provided by extending available direct numerical simulations. It was found that, with negligible pressure gradients, results presented in wall coordinates are predicted to be near “universal” in the viscous layer. This apparent universality disappears when a significant pressure gradient is applied; increasing the pressure gradient decreases the entropy generation rate. Within the viscous layer, the approximate evaluation of $S‴$ differs significantly from the “proper” value but its integral, the entropy generation rate per unit surface area $Sap″$, agrees within 5% at its edge.

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## Figures

Figure 1

Streamwise mean velocity profiles for turbulent boundary layers with negligible pressure gradients; the Reynolds number is based on momentum thickness (13)

Figure 2

Relative magnitudes of contributions to entropy generation in turbulent boundary layers with negligible pressure gradients (Reθ≈1410, solid curves; Reθ≈670, dashes; Reθ≈300, short dashes)

Figure 3

Pointwise and areal entropy generation rates across turbulent boundary layers with negligible pressure gradients (curves denoted as in Fig. 2)

Figure 4

Streamwise mean velocity profiles for turbulent boundary layers with favorable pressure gradients (12-13): −Kp=0, solid curve; −Kp≈0.0120, long dashes; −Kp≈0.0187, short dashes; −Kp≈0.0202, “centerline” curve

Figure 5

Relative magnitudes of contributions to entropy generation in turbulent boundary layers with favorable pressure gradients; symbols as in Fig. 4

Figure 6

Pointwise and areal entropy generation rates across turbulent boundary layers with favorable pressure gradients; symbols as in Fig. 4

Figure 7

Examples of approximate and exact approaches to calculating entropy generation rates in turbulent boundary layers from DNS with (a) a zero pressure gradient (Reθ≈1410) and (b) a favorable pressure gradient (−Kp≈0.012, Kv≈1.5×10−6, Reθ≈690)

Figure 8

Comparisons of entropy generation rates predicted by approximate and exact approaches near the edges of the viscous layers: (a) variation with Reynolds number based on momentum thickness and (b) variation with streamwise pressure gradient (Sap‴∕S‴: zpg, open squares; favorable pressure gradient, crossed squares; Sap‴∕S‴: zpg, open circles; favorable pressure gradient, solid circles)

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