Research Papers

Differential Equation Specification of Integral Turbulence Length Scales

[+] Author and Article Information
Richard J. Jefferson-Loveday

e-mail: rjj32@cam.ac.uk

Paul G. Tucker, V. Nagabhushana Rao

Whittle Laboratory,
Department of Engineering,
University of Cambridge,
Cambridge, CB3 ODY, UK

John D. Northall

Rolls-Royce, PLC,
Derby, DE24 8BJ, UK

1Corresponding author.

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received April 10, 2012; final manuscript received August 16, 2012; published online March 25, 2013. Editor: David Wisler.

J. Turbomach 135(3), 031013 (Mar 25, 2013) (8 pages) Paper No: TURBO-12-1030; doi: 10.1115/1.4007479 History: Received April 10, 2012; Revised August 16, 2012

A Hamilton–Jacobi differential equation is used to naturally and smoothly (via Dirichlet boundary conditions) set turbulence length scales in separated flow regions based on traditional expected length scales. Such zones occur for example in rim-seals. The approach is investigated using two test cases, flow over a cylinder at a Reynolds number of 140,000 and flow over a rectangular cavity at a Reynolds number of 50,000. The Nee–Kovasznay turbulence model is investigated using this approach. Predicted drag coefficients for the cylinder test-case show significant (15%) improvement over standard steady RANS and are comparable with URANS results. The mean flow-field also shows a significant improvement over URANS. The error in re-attachment length is improved by 180% compared with the steady RANS k-ω model. The wake velocity profile at a location downstream shows improvement and the URANS profile is inaccurate in comparison. For the cavity case, the HJ–NK approach is generally comparable with the other RANS models for measured velocity profiles. Predicted drag coefficients are compared with large eddy simulation. The new approach shows a 20–30% improvement in predicted drag coefficients compared with standard one and two equation RANS models. The shape of the recirculation region within the cavity is also much improved.

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Fig. 7

Vector plots of cylinder wake region (a) k-ω, (b) time averaged URANS SA, (c) HJ–NK (d) experimental measurements [14]

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Fig. 8

Centerline velocity versus axial distance for URANS-SA: dotted curve, Menter BL model: dashed curve, Wilcox k-ω model: dashed-dotted curve and HJ–NK model: solid curve, Exp [14]: ○

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Fig. 9

Mean velocity at x/D = 1.0, key as in Fig. 8

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Fig. 10

Streamlines for URANS-SA flow field

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Fig. 11

Comparison of u/U, v/U, and −u′v′/U2, experimental measurements of Ref. [15]: ○ LES: solid curve

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Fig. 12

LES instantaneous vorticity magnitude contour plots at different time moments for case (II)

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Fig. 6

Cavity length scale contours (a) standard wall distance, (b) L˜ distribution

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Fig. 5

Length scales along cylinder centerline, standard wall distance: dashed line, L˜ distribution: solid line

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Fig. 4

Contour plots (a) standard wall distance, (b) L˜ distribution

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Fig. 3

Schematics of case (I) and case (II)

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Fig. 2

Schematic of wake-flow behind cube

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Fig. 1

Contours of vorticity magnitude at two cross-stream locations for high-order LES calculation

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Fig. 13

Comparison of u/U, experimental measurements of Ref. [15]: ○, RANS-SA: dashed curve, HJ–NK: solid curve, k-ω: dashed-dotted curve, Menter BL: dashed-double dotted curve

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Fig. 14

Comparison of v/U, experimental measurements of Ref. [15]: ○, RANS-SA: dashed curve, HJ–NK: solid curve, k-ω: dashed-dotted curve, Menter BL: dashed-double dotted curve

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Fig. 15

Streamline plots of cavity mean flow field



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