Research Papers: Forced Convection

Effect of Uncertainty in Blowing Ratio on Film Cooling Effectiveness

[+] Author and Article Information
Hessam Babaee

Department of Mechanical Engineering,
Louisiana State University,
Baton Rouge, LA 70803
e-mail: hbabae1@lsu.edu

Xiaoliang Wan

Assistant Professor
Department of Mathematics,
Louisiana State University,
Baton Rouge, LA 70803
e-mail: xlwan@math.lsu.edu

Sumanta Acharya

Department of Mechanical Engineering,
Louisiana State University,
Baton Rouge, LA 70803
e-mail: acharya@tigers.lsu.edu

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received April 27, 2013; final manuscript received August 18, 2013; published online November 15, 2013. Assoc. Editor: Phillip M. Ligrani.

J. Heat Transfer 136(3), 031701 (Nov 15, 2013) (11 pages) Paper No: HT-13-1218; doi: 10.1115/1.4025562 History: Received April 27, 2013; Revised August 18, 2013

In this study, the effect of randomness of blowing ratio on film cooling performance is investigated by combining direct numerical simulations with a stochastic collocation approach. The geometry includes a 35-deg inclined jet with a plenum attached to it. The blowing ratio variations are assumed to have a truncated Gaussian distribution with mean of 0.3 and the standard variation of approximately 0.1. The parametric space is discretized using multi-element general polynomial chaos (ME-gPC) with five elements where general polynomial chaos of order 3 is used in each element. Direct numerical simulations were carried out using spectral element method to sample the governing equations in space and time. The probability density function of the film cooling effectiveness was obtained and the standard deviation of the adiabatic film cooling effectiveness on the blade surface was calculated. A maximum of 20% of variation in film cooling effectiveness was observed at 2.2 jet-diameter distance downstream of the exit hole. The spatially-averaged adiabatic film cooling effectiveness was 0.23 ± 0.02. The calculation of all the statistical properties were carried out as off-line post processing. A fast convergence of the polynomial expansion in the random space is observed which shows that the computational strategy is very cost-effective.

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Fig. 1

Three-dimensional schematic of the jet in crossflow along with the boundary conditions

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Fig. 2

Probability density function of blowing ratio; a truncated Gaussian distribution with mean of 0.3 and variance of 0.01. Elemental decomposition (Be,e=1,…,Ne) is shown schematically.

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Fig. 5

Time-averaged streamwise velocity profiles u1¯ at x2 = 0 and BR = 0.15; DNS (solid line), experimental data [4] (triangle)

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Fig. 4

Spectrum of the energy of the velocity signal at location x1 = 2, x2 = 1, and x3 = 0 with blowing ratio of BR = 0.5841

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Fig. 3

Unstructured hexahedral grid; (a) three-dimensional view; (b) x1 − x3 view of the grid in the vicinity of the jet exit, black lines: element boundaries; gray lines: Gauss-Lobatto-Legendre quadrature grid with spectral order of four

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Fig. 9

Probability density function for film cooling effectiveness η˜(ξ)

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Fig. 6

Instantaneous temperature surface in the mid-plane (x3 = 0) for different blowing ratios

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Fig. 7

Time-averaged temperature contours for quadrature points on cooled surface (x2 = 0). In this figure simulations for all Gauss quadrature points are shown.

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Fig. 10

Spanwise-averaged film cooling effectiveness η(x1; ξ) versus random blowing ratio at x1 = 2, 4, 6, and 10

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Fig. 11

The pdf of spanwise-averaged film cooling effectiveness at x1 = 2, 4, 6, and 10. Note that the horizontal axis in (a)–(d) corresponds to the vertical axis in Fig. 10.

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Fig. 12

Uncertainty in the spanwise-averaged film cooling effectiveness η(x1; ξ)

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Fig. 13

Standard deviation of temperature, σθ(x) on the cooled surface of x2 = 0

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Fig. 14

Sensitivity and standard deviation of spanwise-averaged film cooling effectiveness η(x1; ξ)

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Fig. 8

Spatially-averaged film cooling effectiveness η˜M(ξ) with different projection orders M=0,…,3




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