A quaternion approach to the modal analysis and reduced-order modeling of three-dimensional fluid systems

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Issac, Yanal
Mavris, Dimitri N.
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In 1967, Lumley derived the proper orthogonal decomposition (POD) in an attempt to provide a mathematical description of patterns that emerge in turbulent flow, which he called coherent structures. The POD method derived by Lumley has deep mathematical roots, is statistically based, and has analytical foundations that provide a rigorous mathematical framework for the extraction and description of coherent structures. However, it was not until 1987, when Sirovich provided a numerically tractable implementation of the POD method capable of tackling large data sets such as the ones encountered in modern day engineering problems, did the POD gain much attention. In recent years, the wide spread success of the POD method has greatly elevated the status of the POD method. Today, the POD method is considered as benchmark procedure, and is at the bedrock of modal analysis and reduced-order modeling of fluid systems. Since its inception, numerous applications, adaptions and variations of the POD have been devised. However, little attention has been paid to addressing the three-dimensional nature of fluid systems. In fact, George states that the POD is agnostic to the nature of the data, as it does not matter whether the data is velocity, pressure or temperature. The aim of this thesis was to explore a fundamentally different approach to the POD that is better suited for three-dimensional fluid systems; an approach that does not compromise the mathematical rigor associated with the concept of coherent structures defined by Lumley. The approach investigated in this thesis replaces the traditional field of real numbers $\R$, with a four-dimensional non-commutative division algebra $\mathbb{H}$, known as the quaternion division algebra. To the knowledge of the author, this thesis is the first to incorporate quaternions into Lumley's mathematical framework. The introduction of quaternions into Lumley's mathematical framework, generalizes the proper orthogonal decomposition to the quaternion proper orthogonal decomposition (QPOD) while preserving its favorable features and extending the POD to higher dimensional spaces. In the work of this thesis it was shown that a quaternion approach abstracts Lumley's mathematical representation of coherent structures at a fundamental level. These abstracted representations, defined for the first time in the work of this thesis are termed quaternion coherent structures, exhibit interesting properties and result in a fascinating phenomena termed the kaleidoscope effect which is not present in the traditional definition of coherent structures. Furthermore, it was numerically and mathematical shown that the QPOD method can better distill the essential dynamics present in a data set and can create more accurate rank-$m$ approximations as compared the POD method. The results presented in this thesis provide compelling evidence advocating for the use of quaternions in the context of modal analysis and reduced-order modeling of three-dimensional fluid systems. In addition a numerical implementation of the QPOD inspired by the work of Sirovich is also presented. The numerical implementation is termed the quaternion snapshot method and in the work of this thesis is shown to be scalable to large systems. Hence, a quaternion representation of the velocity flow field variables, provides a more natural means of incorporating the flow variables into a single holistic variable, which addresses the three-dimensional nature of the data. Such a quaternion representation provides for a more natural, physics-based framework for the treatment of three-dimensional fluid systems which results in more informative modal analysis and more accurate reduced-order models. The consequences of an improved modal analysis of fluid systems will greatly help scientists and engineers further their understanding of fluid flow. In addition, the QPOD method provides a superior capability to capture, isolate, and distill the complex aerodynamics resulting in faster, and more accurate reduced order models which will aid in many aspects of aircraft analysis and design, particularly aeroelastic analysis and design. These accurate but lower-order representations will also pave the way for surrogate-based optimization, uncertainty quantification, and fluid flow control over flexible structures encountered in modern day and future aircraft designs.
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