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Research Papers: Techniques and Procedures

Improving the Spatial Resolution and Stability by Optimizing Compact Finite Differencing Templates

[+] Author and Article Information
Stephen A. Jordan

 Naval Undersea Warfare Center, Newport, RI 02842stephen.jordan@navy.mil

J. Fluids Eng 131(12), 121402 (Nov 30, 2009) (11 pages) doi:10.1115/1.4000576 History: Received February 09, 2009; Revised October 20, 2009; Published November 30, 2009; Online November 30, 2009

Parameter optimization is an excellent path for easily raising the resolution efficiency of compact finite differencing schemes. Their low-resolution errors are attractive for resolving the fine-scale turbulent physics even in complex flow domains with difficult boundary conditions. Most schemes require optimizing closure stencils at and adjacent to the domain boundaries. But these constituents can potentially degrade the local resolution errors and destabilize the final solution scheme. Current practices optimize and analyze each participating stencil separately, which incorrectly quantifies their local resolution errors. The proposed process optimizes each participant simultaneously. The result is a composite template that owns consistent spatial resolution properties throughout the entire computational domain. Additionally, the optimization technique leads to templates that are numerically stable as understood by an eigenvalue analysis. Finally, the predictive accuracy of the optimized schemes are evaluated using four canonical test problems that involve resolving linear convection, nonlinear Burger wave, turbulence along a flat plate, and circular cylinder wall pressure.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Dispersive errors of the tenth-order standard compact stencil (10c) as compared to optimized two-parameter (42) and three-parameter (43) fourth-order families

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Figure 2

Resolution errors of the fourth-order standard boundary stencil as coupled with a fourth-order Padé scheme: template notation 4-4-4. (a) Dispersive and (b) dissipative.

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Figure 3

Dispersive errors of optimized templates: notation −42− signifies fourth-order two-parameter family. (a) Template 42−42−42−42−42 and (b) template 42−42−6−42−42.

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Figure 4

Stability characteristics of the fourth-order standard (4-4-4) and optimized templates 42−4−42 and 43−4−43

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Figure 5

Stability characteristics of various optimized templates: three-parameter fourth-order template (43−43−43−43−43−43−43)2. (a) Neutrally stable and (b) strictly stable.

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Figure 6

Solutions of linear convection using fourth-order (4c) and sixth-order (6c) interior schemes as well as standard (4-4-4) and two optimized templates

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Figure 7

Solution of burgers nonlinear viscous equation using standard fourth-order scheme (4-4-4) and various optimized templates. (a) Error norm and (b) percent improvement.

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Figure 8

Schematic of LES computations of the flat plate flow

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Figure 9

Turbulent boundary layer growth and energy spectra using standard and optimized templates 42−42−5−6−5−42−42 (streamwise), 42−42−5−42−42 (normal) and -6- (spanwise periodic). (a) Stremwise growth and (b) energy spectra.

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Figure 10

Turbulent flat plate streamwise intensity and convective velocity using standard and various optimized templates: experimental data Reθ=670(23), Reθ=1400(24), and Reθ=8100(25). (a) Stremwise intensity and (b) convective velocity.

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Figure 11

Mean and fluctuating coefficients, drag, and base pressure of flow past a circular cylinder using standard and various optimized templates: experimental data ReD=720(26). (a) Mean and (b) fluctuating.

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