AIAA JOURNAL Vol. 44, No. 5, May 2006 Technical Notes TECHNICAL NOTES are short manuscripts describing new developments or important results of a preliminary nature. These Notes should not exceed 2500 words (where a figure or table counts as 200 words). Following informal review by the Editors, they may be published within a few months of the date of receipt. Style requirements are the same as for regular contributions (see inside back cover). Geometrical Description of Subgrid-Scale Stress Tensor Based on Euler Axis/Angle Bing-Chen Wang ∗ and Eugene Yee † Defence Research and Development Canada–Suffield, Medicine Hat, Alberta T1A 8K6, Canada and Donald J. Bergstrom ‡ University of Saskatchewan, Saskatoon, Saskatchewan S7N 5A9, Canada I. Introduction T HE rotational dynamics between the fixed triads of two or more rigid bodies, or attitude dynamics, 1,2 is of fundamental inter- est to scientists and engineers working in the area of astronomy, spacecraft, satellites, and robotics. Recently, the methodology of attitude dynamics has been incorporated into fluids studies and pro- vides some new insights into the local geometrical property of fluid tensors, construction of novel local structure-based constitutive rela- tions for turbulence modelling, 3−5 and elucidation of the mechanism of a variety of physical processes such as dissipation and backscatter of the turbulence kinetic energy (TKE) between the resolved scale and subgrid-scale (SGS) motions. 6−10 In the research areas related to rigid bodies, for example, maneu- ver of airplanes, motion of joints, and satellite control, there are mul- tiple different and equivalent parameterization methods, each with its own merits and demerits, that can be used for investigating the at- titude of a three-dimensional frame (or, in particular, the relative ro- tation of one frame with respect to another frame). Earlier milestones related to the topic of attitude dynamics include the famous work M´ ecanique Analytique by Joeseph-Louis Lagrange (1736–1813) and the quaternion introduced by Sir William R. Hamilton (1805– 1865). Despite the long research history of the subject, modern advances and applications of the frame attitudes of rigid bodies still keep growing at an impressive rate in a number of diverse areas. 1,2,6−13 A fundamental difference in attitude representation between the area of fluid dynamics and those of astrodynamics, 12 aircraft engineering, 2,11 and robotic manipulation 13 is that the latter applications concern attitude control, whereas those in fluid dynam- ics have focused on the physical interpretation and computational consistency of the parameters of the representation. Received 31 August 2005; revision received 7 December 2005; accepted for publication 14 December 2005. Copyright c  2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/06 $10.00 in correspondence with the CCC. ∗ NSERC Visiting Fellow to Canadian Government Laboratory, P.O. Box 4000, Station Main; bingchen.wang@drdc-rddc.gc.ca. † Defence Scientist, P.O. Box 4000, Station Main. ‡ Professor, Department of Mechanical Engineering, 57 Campus Drive; don bergstrom@engr.usask.ca. Only two types of methods for attitude representation have, so far, been utilized in the literature of fluid dynamics. Betchov 14 pioneered the application of three Euler angles to study the geometrical prop- erties of the velocity gradient tensor u i, j def = ∂ j u i . This method was later systematically developed in Refs. 3–5 for investigation of the eigenframe attitude of the velocity gradient, Reynolds stress, SGS Reynolds stress, and SGS stress (τ ij def = u i u j −¯ u i ¯ u j ) tensors. Here, an overbar indicates a resolved/filtered quantity following the con- vention of large-eddy simulation (LES), and an eigenframe refers to an orthonormal triad formed by the three independent normalized eigenvectors of a second-order real symmetric tensor (such as τ ij ). For investigating the eigenframe attitude of −τ ij with respect to the eigenframe of the resolved strain rate tensor [ ¯ S ij def = ( ¯ u i, j +¯ u j,i )/2], Tao et al., 9 Horiuti, 8 and Higgins et al. 10 adopted the axis-azimuth representation, which includes three angles, namely, the colatitude θ , longitude φ, and azimuthal angle ζ . (For details of the geometri- cal description, the readers are referred to the papers by Tao et al. 9 and Shuster. 11 ) As just reviewed, to parameterize the attitude of the eigenframe of −τ ij with respect to that of ¯ S ij , the conventional method is to use the Euler angles 3−5 or the axis-azimuthal angles. 8−10 However, note that the parameters used in both of these methods are not uniquely defined. The set of three Euler angles can be parameterized using in total 12 different equivalent ways (6 symmetric sets and 6 asymmet- ric sets) depending on the method that is adopted for decomposing the relative rotational motion between the two frames, and a sim- ilar situation holds for the axis-azimuth representation. 2,11 In this Technical Note, we propose a simple method to describe the atti- tude of the eigenframe of −τ ij with respect to that of ¯ S ij , based on the Euler axis/angle. In contrast with the two mentioned methods of attitude parameterization through Euler angles and axis-azimuthal angles, the proposed method of Euler axis/angle is uniquely defined: It utilizes only one special angle, that is, Euler rotation angle χ , to quantify the rotation and only one special vector, that is, Euler axis q to define the central axis of the rotation. Both the Euler axis and Eu- ler rotation angle are the so-called natural invariants 1 of the rotation matrix R and provide an elegant method for a simple decomposi- tion and invariant representation of the relative rotation between two eigenframes through the use of Euler’s theorem. II. Rotational Motion in ℜ 3 and Euler Axis/Angle Let E = [e 1 , e 2 , e 3 ] and E ′ = [e ′ 1 , e ′ 2 , e ′ 3 ] be the basis vectors for the orthonormal observer-referenced frame (or the absolute frame) and the orthonormal object-referenced frame (or the relative frame), respectively. Let ℜ be the set of real numbers. The direction-cosine matrix R ∈ℜ 3 × 3 is formed from the bases E and E ′ as R ij = e ′ i · e j = cos(e ′ i , e j ) (1) which is referred to as the rotation matrix. This matrix links the frame E and E ′ through an orthogonal transformation. In consequence, we have R T R = RR T = I, R −1 = R T , det(R) =±1 (2) and e ′ i = R ij e j and e i = R ji e ′ j . An orthogonal matrix corresponding to det(R) =+1 is called proper or special orthogonal (SO); oth- erwise, it is referred to as improper. A proper R represents pure 1106 Downloaded by UNIVERSITY OF WATERLOO on January 4, 2015 | http://arc.aiaa.org | DOI: 10.2514/1.19803