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Hall, Brenton TaylorUsing the Non-Uniform Dynamic Mode Decomposition to Reduce the Storage Required for PDE Simulations
Master of Mathematical Sciences, The Ohio State University, 2017, Mathematical Sciences
Partial Differential Equation simulations can produce large amounts of data that are very slow to transfer. There have been many model reduction techniques that have been proposed and utilized over the past three decades. Two popular techniques Proper Orthogonal Decomposition and Dynamic Mode Decomposition have some hindrances. Non-Uniform Dynamic Mode Decomposition (NU-DMD), which was introduced in 2015 by Gueniat et al., that overcomes some of these hindrances. In this thesis, the NU-DMD's mathematics are explained in detail, and three versions of the NU-DMD's algorithm are outlined. Furthermore, different numerical experiments were performed on the NU-DMD to ascertain its behavior with repect to errors, memory usage, and computational efficiency. It was shown that the NU-DMD could reduce an advection-diffusion simulation to 6.0075% of its original memory storage size. The NU-DMD was also applied to a computational fluid dynamics simulation of a NASA single-stage compressor rotor, which resulted in a reduced model of the simulation (using only three of the five simulation variables) that used only about 4.67% of the full simulation's storage with an overall average percent error of 8.90%. It was concluded that the NU-DMD, if used appropriately, could be used to possibly reduce a model that uses 400GB of memory to a model that uses as little as 18.67GB with less than 9% error. Further conclusions were made about how to best implement the NU-DMD.

Committee:

Ching-Shan Chou (Advisor); Jen-Ping Chen (Committee Member)

Subjects:

Aerospace Engineering; Applied Mathematics; Computer Science; Mathematics; Mechanical Engineering

Keywords:

Fluid Dynamics; Fluid Flow; Model Reduction; Partial Differential Equations; reducing memory; Dynamic Mode Decomposition; Decomposition; memory; Non-Uniform Dynamic Mode Decomposition

Deshmukh, RohitModel Order Reduction of Incompressible Turbulent Flows
Doctor of Philosophy, The Ohio State University, 2016, Aero/Astro Engineering
Galerkin projection is a commonly used reduced order modeling approach; however, stability and accuracy of the resulting reduced order models are highly dependent on the modal decomposition technique used. In particular, deriving stable and accurate reduced order models from highly turbulent flow fields is challenging due to the presence of multi-scale phenomenon that cannot be ignored and are not well captured using the ubiquitous Proper Orthogonal Decomposition (POD). A truncated set of proper orthogonal modes is biased towards energy dominant, large-scale structures and results in over-prediction of kinetic energy from the corresponding reduced order model. The accumulation of energy during time-integration of a reduced order model may even cause instabilities. A modal decomposition technique that captures both the energy dominant structures and energy dissipating small scale structures is desired in order to achieve a power balance. The goal of this dissertation is to address the stability and accuracy issues by developing and examining alternative basis identification techniques. In particular, two modal decomposition methods are explored namely, sparse coding and Dynamic Mode Decomposition (DMD). Compared to Proper Orthogonal Decomposition, which seeks to truncate the basis spanning an observed data set into a small set of dominant modes, sparse coding is used to identify a compact representation that span all scales of the observed data. Dynamic mode decomposition seeks to identity bases that capture the underlying dynamics of a full order system. Each of the modal decomposition techniques (POD, Sparse, and DMD) are demonstrated for two canonical problems of an incompressible flow inside a two-dimensional lid-driven cavity and past a stationary cylinder. The constructed reduced order models are compared against the high-fidelity solutions. The sparse coding based reduced order models were found to outperform those developed using the dynamic mode and proper orthogonal decompositions. Furthermore, energy component analyses of high-fidelity and reduced order solutions indicate that the sparse models capture the rate of energy production and dissipation with greater accuracy compared to the dynamic mode and proper orthogonal decomposition based approaches. Significant computational speedups in the fluid flow predictions are obtained using the computed reduced order models as compared to the high-fidelity solvers.

Committee:

Jack McNamara (Advisor); Datta Gaitonde (Committee Member); Ryan Gosse (Committee Member); Joseph Hollkamp (Committee Member); Mohammad Samimy (Committee Member)

Subjects:

Aerospace Engineering

Keywords:

Turbulent flows; reduced order modeling; Navier-Stokes equations; nonlinear dynamics; Galerkin projection; modal decomposition; proper orthogonal decomposition; dynamic mode decomposition; sparse coding

Waindim, MbuOn Unsteadiness in 2-D and 3-D Shock Wave/Turbulent Boundary Layer Interactions
Doctor of Philosophy, The Ohio State University, 2017, Aero/Astro Engineering
Shock-boundary layer interactions (SBLIs) are ubiquitous occurrences in supersonic and hypersonic vehicles and have the tendency to inhibit their structural and aerodynamic performance. For example, in the inlets and isolators of such vehicles, the shock wave generated by one surface interacts with the boundary layer on an adjacent one. They are also present on the exterior of the vehicles, e.g. at the fuselage/vertical stabilizer junctions. These interactions cause unsteady separation, resulting in reduced air in-take efficiency, or unstart in extreme cases; unsteady vortex shedding which yields undesirable broadband noise; and significant pressure fluctuations which compromise the structural integrity of the vehicle and which can lead to loss of control authority. Mitigating these issues is therefore an important part of optimizing aerodynamic and structural design of high speed vehicles. The first step in this respect is obtaining a better understanding of the interaction unsteadiness. Nominally 2-D interactions have been studied extensively and have identified low-frequency shock motions which lead to undesirable pressure loads. The particular frequencies associated with the motions have been characterized using time resolved experiments and computations, and shown to depend on the mean size of the separation. The physical processes responsible for these frequencies are however still under investigation and the physical relationship between the shock motions and pulsations of the separation bubble remains obscure. For flow fields where the shock is swept, a complex 3-D interaction is encountered whose unsteady features are even less well understood. The mean structure of these 3-D interactions has been obtained experimentally and using RANS simulations, and shown to be profoundly different from the 2-D flow field indicating that progress in understanding 2-D interactions cannot be directly translated to 3-D. Specifically, there is no recirculating region in the 3-D interaction contrasting with the 2-D case where a closed separation bubble essentially drives the dynamics. This effort seeks to understand the unsteadiness in such 3-D interactions. Although Large Eddy Simulations (LES) are invaluable for characterizing time evolving features, they are prohibitively expensive for swept interactions. A stability analysis based method, the Mean Flow Perturbation (MFP) technique appears as a better alternative to LES in terms of computational cost. It involves tracking the evolution of disturbances through the interaction in space and time to identify the low frequency and stability properties of the chaotic 3-D dynamical system. A systematic verification and validation of the technique is presented to provide a solid basis for its employment. By implementing the technique on canonical problems with similar properties as the swept interaction, its applicability for flows with strong gradients and viscous effects is established. The method is then implemented on the simpler 2-D interaction with two specific goals: to ensure that MFP accurately captures well known unsteady features of the flow and to learn relevant lessons and establish best practices for SBLI application. By definition, the implementation of MFP requires the knowledge of a mean base state. For the nominally 2-D interactions, the mean of a previously obtained LES is used as the basis. In addition to providing the input to the MFP technique, the LES is used as a testbed to: (i) Improve effectiveness of a technique for generating spatially developing supersonic turbulent boundary layers (TBLs) as correctly characterizing the inflow boundary condition is a critical component of simulating SBLIs. (ii) Generate insight on possible physical mechanisms responsible for the low frequency unsteadiness in 2-D by exploiting asymmetry in the shock motions. Besides the LES supplied mean base state, the MFP technique is also implemented with a RANS supplied base flow. No significant differences in the results are observed indicating a relative independence of the MFP technique to the base flow generation procedure. Due to the highly prohibitive nature of implementing LES for 3-D interactions, the above observation acts as a substantial endorsement for using RANS generated base state for studying 3-D interactions with the MFP technique. It also provides specific insight into ways to correctly implement the technique when the base flow comes from RANS. Finally, the dynamics of the swept interaction are explored to characterize its unsteady and stability features. Here the basic state is obtained by tailoring RANS calculations to experiments at Florida State University, allowing the separation and other relevant mean features to be accurately captured. The flow field obtained by perturbing this mean is post processed to identify the length and time scales relevant to the flow field. This effort accomplishes three things: (i) a tripping technique is generated to efficiently specify turbulent boundary layer appropriate for use as the inflow condition, (ii) LES of 2-D interactions are obtained and analyzed to identify the cause of energetic low frequency and (iii) the relevant frequencies in the swept interaction are characterized using MFP and the stability properties of the interaction are identified, distinguishing its dynamics from the 2-D. Each step involves a myriad of statistical tools that can be adapted for other applications. The findings of each effort are presented. It is found that the effectiveness of tripping a laminar boundary layer to yield its turbulent counterpart is dependent on a range of factors: (i) grid resolution, (ii) strength of the force associated with the trip, (iii) wall thermal condition near the trip and (iv) Mach number. By characterizing the stability of the boundary layer in the trip region, a precursor for transition to turbulence is identified. This makes the method efficient for generating turbulent boundary layers at alternative desired flow conditions (Mach and Reynolds numbers) as appropriate trip parameters can be obtained a priori. Statistical analyses of the shock motions in 2-D interactions reveal an asymmetry whose quality appears to be dependent on the separation bubble size. For massive separation bubbles, where the shear layer is detached, the collapse phase is a rapid process, possibly owing to the ease of formation of Kelvin-Helmholtz (K-H) structures which convect mass and momentum out of the bubble, leading to its collapse. For moderate separation bubbles, the collapse phase is instead the slower process. It is found that the quality of the asymmetry is linked to the linearly stable position of the shock, which has implications for control. In addition, there is evidence of modulation of the frequencies associated with K-H shedding by the low frequencies characteristic of the shock motions, establishing a link between the two physical phenomena and further reinforcing the role of eddies in bubble collapse. The most substantial outcome of this work is the insight obtained regarding the dynamics of the swept interaction. The results show that: (i) The shock generated by the fin is anchored unlike the oscillating reflected shock in 2-D; consequently, the low frequencies observed in 2-D are not present in this swept interaction. (ii) A convective inviscid instability is identified at a frequency an order of magnitude higher than the characteristic frequency of shock motions in 2-D. It is shown to be a consequence of the crossflow and analogous to the mid frequencies associated with K-H shedding in the 2-D interaction. (iii) The absolute instability observed in 2-D does not persist here as the absence of a closed separation reduces the interaction’s ability to perpetually self-sustain introduced perturbations.

Committee:

Datta Gaitonde (Advisor); Jen-Ping Chen (Committee Member); Jack McNamara (Committee Member); Mo Samimy (Committee Member)

Subjects:

Aerospace Engineering; Fluid Dynamics

Keywords:

shock waves; fluid dynamics; turbulence; boundary layers; computational fluid dynamics; CFD; unsteadiness; SBLI; dynamic mode decomposition; DMD; empirical mode decomposition; EMD; stability analysis