Identifying Inefficiencies in Unsteady Pin Fin Heat Transfer Using Orthogonal Decomposition

[+] Author and Article Information
Markus Schwänen1

Department of Mechanical Engineering,  Texas A&M University, College Station, TX 77843schwaenen@tamu.edu

Andrew Duggleby

Department of Mechanical Engineering,  Texas A&M University, College Station, TX 77843schwaenen@tamu.edu


Corresponding author.

J. Heat Transfer 134(2), 020904 (Dec 13, 2011) (10 pages) doi:10.1115/1.4004873 History: Received November 05, 2010; Revised August 03, 2011; Published December 13, 2011; Online December 13, 2011

Internal cooling of the trailing edge region in a gas turbine blade is typically achieved with an array of pin fins. In order to better understand the effectiveness of this configuration, high performance computations are performed on cylindrical pin fins with a spanwise distance to fin diameter ratio of 2 and height over fin diameter ratio of one. For validation purposes, the flow Reynolds number based on hydraulic channel diameter and bulk velocity (Re = 12,800) was set to match experiments available in the open literature. Simulations included a URANS and LES on a single row of pin fins where the URANS domain was 1 pin wide versus the LES with 3 pins. The resulting time-dependent flow field was analyzed using a variation of bi-orthogonal decomposition (BOD), where the correlation matrices were built using the internal energy in addition to the three velocity components. This enables a detailed comparison of URANS and LES to assess the URANS modeling assumptions as well as a flow decomposition with respect to the flow structure’s influence on surface heat transfer. This analysis shows low order modes which do not contribute to turbulent heat flux, but instead increase the heat exchanger’s global inefficiency. In the URANS study, the forth mode showed the first nonzero temperature basis function, which means that a considerable amount of energy is contained in flow structures that do not contribute to increasing endwall heat transfer. In the LES, the first non zero temperature basis function was the seventh mode. Both orthogonal basis function sets were evaluated with respect to each mode’s contribution to turbulent heat exchange with the surface. This analysis showed that there exists one distinct, high energy mode that contributes to wall heat flux, whereas all others do not. Modifying this mode could potentially be used to improve the heat exchanger’s efficiency with respect to pressure loss.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

Schematic of a turbine blade with a pin fin cooling array in the trailing edge region from Kindlmann [2]

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Figure 2

The URANS domain mesh close to the pin fin is displayed (top). A top view of the bottom wall mesh is shown in the lower left figure. Grid refinement in regions of high velocity gradients on the pin fin surface viewed from the front are visible in the lower right figure.

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Figure 3

The computational domain of the URANS study with applied boundary conditions is shown schematically

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Figure 4

The deviation of Nusselt number augmentation for three coarser grids compared to the finest grid solution is plotted for the URANS simulation

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Figure 5

The plot shows normalized lift force on the pin for the URANS study (top) and lift and drag for the LES on the middle pin (bottom). The vertical lines indicate the time span of the smaller sample used for orthogonal decomposition. The lift and drag coefficients vary in magnitude and slightly in frequency. Even though the shedding breaks down for some time in the LES, it recovers to its former extent.

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Figure 6

The local, span-wise averaged friction factor augmentation from the LES study is plotted as a function of streamwise distance. The pressure data is normalized with different baseline correlations.

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Figure 7

The local, span-wise averaged Nusselt number augmentation from the URANS (top, black solid line labeled “computations”) and LES (bottom) study compared to experiments is plotted as a function of streamwise distance. For the LES (one temperature field for NuLES ), the temperature data from the simulation is normalized with three different baseline correlations for Nu0 .

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Figure 8

The contour plots of the average bottom wall temperature (left) and the scalar spatial eigenfunction ϕk(x→) of mode 0 at the bottom surface (right) from the URANS computation are identical. The data are scaled to a common color range.

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Figure 9

Time eigenfunctions Ψik(t) from the URANS computation of the first 6 modes for velocity (3 left) and temperature (right). The constant average streamwise velocity component (top left), two lift modes oscillating in the spanwise velocity ν and three shearing modes 3-5 can be identified.

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Figure 10

Different modes k ≠ 0 have been used to evaluate the integrals in Eq. 12 for the URANS and show that the dominant mode combination in terms of surface heat flux is 3/3. This mode combination is the most useful in meeting the device’s design goal of augmenting heat transfer. Modes 1 and 2 can be considered parasitic since they contain drag-producing flow structures but have no impact on turbulent heat flux.

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Figure 11

The figure shows contours of the scalar spatial eigenfunction ϕk(x→)at 10%D. The plots from left to right are: Mode 2 URANS, mode 2 LES, mode 3 URANS, mode 3 LES, mode 4 URANS, mode 7 LES.

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Figure 12

Time eigenfunctions Ψik(t) of 8 modes for velocity (3 left) and temperature (right) from the LES. The constant average streamwise velocity component (top left), two lift modes oscillating in the spanwise velocity v and two mainly shearing modes 4-5 can be identified. The first mode with nonzero temperature function is mode 7. The functions are scaled for better visualization.

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Figure 13

The figure shows isosurfaces at −4 (red) and 4 (blue) of the scalar spatial eigenfunction ϕk(x→) of the LES. Upper left: Mode 0, lower left: Mode 2, not contributing to heat transfer. Upper right: Mode 3, a shearing mode. Lower right: Mode 7, useful mode in terms of heat transfer.

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Figure 14

Different modes k ≠ 0 have been used to evaluate the integrals in Eq. 12 for the LES and show that the dominant mode combination in terms of positive surface heat flux is 7/7. There are 6 lower order modes (energetically greater) that do not contribute to the heat transfer.




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