0
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

Professor
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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Han, J., Dutta, S., and Ekkad, S., 2001, Gas Turbine Heat Transfer and Cooling Technology, Taylor & Francis, London.
Bogard, D. G., and Thole, K. A., 2006, “Gas Turbine Film Cooling,” J. Propul. Power, 22(2), pp. 249–270. [CrossRef]
Baldauf, S., Schulz, A., and Wittig, S., 2001, “High-Resolution Measurements of Local Effectiveness From Discrete Hole Film Cooling,” ASME J. Turbomach., 123(4), pp. 758–765. [CrossRef]
Bidan, G., Vezier, C., and Nikitopoulos, D. E., 2013, “Study of Unforced and Modulated Film-Cooling Jets Using Proper Orthogonal Decomposition—Part I: Unforced Jets,” ASME J. Turbomach., 135(2), p. 021037. [CrossRef]
Abhari, R. S., 1996, “Impact of Rotor–Stator Interaction on Turbine Blade Film Cooling,” ASME J. Turbomach., 118(1), pp. 123–133. [CrossRef]
Womack, K. M., Volino, R. J., and Schultz, M. P., 2008, “Combined Effects of Wakes and Jet Pulsing on Film Cooling,” ASME J. Turbomach., 130(4), p. 041010. [CrossRef]
Xiu, D., 2009, “Fast Numerical Methods for Stochastic Computations: A Review,” Comm. Comp. Phys., 5(2–4), pp. 242–272.
Ghanem, R. G., and Spanos, P. D., 1991, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York.
Xiu, D., and Karniadakis, G. E., 2002, “The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations,” SIAM J. Sci. Comput. (USA), 24(2), pp. 619–644. [CrossRef]
Venturi, D., Wan, X., and Karniadakis, G. E., 2008, “Stochastic Low-Dimensional Modelling of a Random Laminar Wake Past a Circular Cylinder,” J. Fluid Mech., 606, pp. 339–367. [CrossRef]
Wan, X., and Karniadakis, G. E., 2005, “An Adaptive Multi-Element Generalized Polynomial Chaos Method for Stochastic Differential Equations,” J. Comput. Phys., 209(2), pp. 617–642. [CrossRef]
Wan, X., and Karniadakis, G. E., 2006, “Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures,” SIAM J. Sci. Comput. (USA), 28(3), pp. 901–928. [CrossRef]
Acharya, S., Tyagi, M., and Hoda, A., 2006, “Flow and Heat Transfer Predictions for Film Cooling,” Ann. N. Y Acad. Sci., 934(1), pp. 110–125. [CrossRef]
Acharya, S., and Tyagi, M., 2003, “Large Eddy Simulation of Film Cooling Flow From an Inclined Cylindrical Jet,” ASME Conference Proceedings, 2003(36886), pp. 517–526.
Peet, Y., and Lele, S. K., 2008, “Near Field of Film Cooling Jet Issued Into a Flat Plate Boundary Layer: LES Study, ASME Conference Proceedings, 2008(43147), pp. 409–418.
Iourokina, I. V., and Lele, S. K., 2005, “Towards Large Eddy Simulation of Film-Cooling Flows on a Model Turbine Blade Leading Edge,” AIAA Paper No. 670.
Guo, X., Schroder, W., and Meinke, M., 2006, “Large-Eddy Simulations of Film Cooling Flows,” Comput. Fluids, 35(6), pp. 587–606. [CrossRef]
Renze, P., Schroder, W., and Meinke, M., 2008, “Large-Eddy Simulation of Film Cooling Flows at Density Gradients,” Int. J. Heat Fluid Flow, 29(1), pp. 18–34. [CrossRef]
Babaee, H., Acharya, S., and Wan, X., 2013, “Optimization of Forcing Parameters of Film Cooling Effectiveness,” ASME Conference Proceedings, ASME.
Muldoon, F., and Acharya, S., 2009, “DNS Study of Pulsed Film Cooling for Enhanced Cooling Effectiveness,” Int. J. Heat Mass Transfer, 52(13–14), pp. 3118–3127. [CrossRef]
Smirnov, A., Shi, S., and Celik, I., 2001, “Random Flow Generation Technique for Large Eddy Simulations and Particle-Dynamics Modeling,” J. Fluids Eng., 123(2), pp. 359–371. [CrossRef]
Xiu, D., 2007, “Efficient Collocational Approach for Parametric Uncertainty Analysis,” Comm. Comp. Phys., 2(2), pp. 293–309.
Xiu, D., 2010, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University, Princeton, NJ.
Warburton, T., 1998, “Spectral/hp Element Methods on Polymorphic Domains,” Ph.D. thesis, Brown University, Providence, RI.
Karniadakis, G. E., and Sherwin, S. J., 2005, Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford University, New York.

Figures

Grahic Jump Location
Fig. 1

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

Grahic Jump Location
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.

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
Fig. 9

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

Grahic Jump Location
Fig. 5

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

Grahic Jump Location
Fig. 10

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

Grahic Jump Location
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.

Grahic Jump Location
Fig. 12

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

Grahic Jump Location
Fig. 6

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

Grahic Jump Location
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.

Grahic Jump Location
Fig. 8

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

Grahic Jump Location
Fig. 13

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

Grahic Jump Location
Fig. 14

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

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In