Reduced-order modelling of unsteady small-disturbance flows using a frequency-domain proper orthogonal decomposition technique

Hall, Kenneth C.; Thomas, Jeffrey P.; Dowell, Earl H.

doi:10.2514/6.1999-655

Cited by 82 publications

(53 citation statements)

References 18 publications

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“…Writing s = s 0 + s , the Taylor series expansion of the transfer function (15) about some complex point s 0 yields…”

Section: B Multiple Interpolation Point Arnoldi Methodsmentioning

confidence: 99%

“…As discussed by Willcox et al [10], as the number of frequency points is increased and the number of moments matched at each point is reduced to one, the method becomes a frequency-domain POD approach, which uses SVD on a set of complex responses obtained at selected frequency sample points to construct a basis [15], [16], [17]. Figure 8 compares the accuracy provided by the multiplepoint Arnoldi method and the POD method.…”

Section: From Multiple-point Arnoldi To Proper Orthogonal Decomposmentioning

confidence: 99%

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Model Reduction for Active Control Design Using Multiple-Point Arnoldi Methods

Lassaux

Willcox

2003

41st Aerospace Sciences Meeting and Exhibit

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I. ABSTRACT A multiple-point Arnoldi method is derived for model reduction of computational fluid dynamic systems. By choosing the number of frequency interpolation points and the number of Arnoldi vectors at each frequency point, the user can select the accuracy and range of validity of the resulting reduced-order model while balancing computational expense. The multiplepoint Arnoldi approach is combined with a singular value decomposition approach similar to that used in the proper orthogonal decomposition method. This additional processing of the basis allows a further reduction in the number of states to be obtained, while retaining a significant computational cost advantage over the proper orthogonal decomposition. Results are presented for a supersonic diffuser subject to mass flow bleed at the wall and perturbations in the incoming flow. The resulting reduced-order models capture the required dynamics accurately while providing a significant reduction in the number of states. The reduced-order models are used to generate transfer function data, which are then used to design a simple feedforward controller. The controller is shown to work effectively at maintaining the average diffuser throat Mach number. II. INTRODUCTIONComputational fluid dynamics (CFD) has reached a considerable level of maturity and is now routinely used in many applications for both external and internal flows. Euler and NavierStokes solvers enjoy widespread use for aerodynamic design and analysis, and provide accurate answers for a variety of complex flows. However, despite ever increasing computational power, unsteady problems are computationally very expensive and time-consuming. More efficient methods for time-varying flow can be obtained if the disturbances are small, and the unsteady solution can be considered to be a small perturbation about a steady-state flow [1]. In this case, a set of linearized equations is obtained which can be time-marched to obtain the flow solution at each instant.Even under the linearization assumption, any CFD-based technique will generate models with a prohibitively high number of states. For this reason, CFD models are not appropriate for many applications where model size and cost are issues. For example, when the aerodynamic solver must be coupled to another disciplinary model, as in aeroelastic analysis or multidisciplinary optimization, CFD models cannot be used. Another application which requires low-order models is control design.The concept of using active control to enhance the stability properties of an unsteady flow has been addressed for several applications [2], [3]. In order to derive control models that will be effective, it is vital that the relevant unsteady flow dynamics are captured accurately. A model is required that will capture not only the dynamics of the disturbance to be controlled, but also the visibility offered by the sensing and the effect on the flow of the actuation mechanism. A high-fidelity CFD code can offer the degree of flow resolution that is required; however, ...

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“…Writing s = s 0 + s , the Taylor series expansion of the transfer function (15) about some complex point s 0 yields…”

Section: B Multiple Interpolation Point Arnoldi Methodsmentioning

confidence: 99%

Section: From Multiple-point Arnoldi To Proper Orthogonal Decomposmentioning

confidence: 99%

Model Reduction for Active Control Design Using Multiple-Point Arnoldi Methods

Lassaux

Willcox

2003

41st Aerospace Sciences Meeting and Exhibit

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“…It computes a set of empirical eigenfunctions using flow solutions, which are more commonly called snapshots [50]. These snapshots can either be obtained from time domain simulations [45], or derived in the frequency domain by exploiting the linearity of the governing equations [19,24]. Snapshots of the dual CFD system can also be included in the POD process, yielding an approximate balanced truncation [56].…”

Section: Reduced-order Linear Modelsmentioning

confidence: 99%

Reduced-Order, Trajectory Piecewise-Linear Models for Nonlinear Computational Fluid Dynamics

Gratton

Willcox

2004

34th AIAA Fluid Dynamics Conference and Exhibit

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Computational fluid dynamics (CFD) is now widely used throughout the fluid dynamics community and yields accurate models for problems of interest. However, due to its high computational cost, CFD is limited for some applications. Therefore, model reduction has been used to derive low-order models that replicate CFD behavior over a restricted range of inputs, and various frameworks have been developed. Unfortunately, the majority of those methods are limited to linear cases and do not properly handle reduction of nonlinear systems.In order to overcome restrictions of weak nonlinearity and the costly representation of the system's nonlinearity found in other nonlinear reduction approaches, a trajectory piecewise-linear (TPWL) scheme is developed for a CFD model of the twodimensional Euler equations. The approach uses a weighted combination of linearized models to represent the nonlinear CFD system. Using a set of training trajectories obtained via a simulation of the nonlinear CFD model, algorithms are presented for linearization point selection and weighting of the models. Using the same training trajectories to provide a snapshot ensemble, the proper orthogonal decomposition (POD) is used to create a reduced-space basis, onto which the TPWL model is projected. This projection yields an efficient reduced-order model of the nonlinear system, which does not require the evaluation of any full-order system residuals, while capturing a large portion of the nonlinear space.The method is applied to the case of flow through an actively controlled supersonic diffuser. Convergence of the TPWL approach is presented for both full-order and reduced-order cases. The TPWL approach and the POD combine naturally to form an efficient reduction procedure and the methodology is found to yield accurate results, including cases with significant shock motion. Reduced-order PWL models are shown to be three orders of magnitude more efficient than the nonlinear CFD for simulation of a representative test case. AcknowledgmentsA number of peoples contributed in various ways to the last two years of my life here, in Boston, and made them more than enjoyable and memorable.Foremost on my list is my advisor, Karen, who helped me more than anything else in getting through MIT. Even though a communication problem initially restricted the flow of information between us, and despite the fact that I thought for a long time that she was English, she has always shown me a great deal of patience. Also, the availability, guidance and responsiveness she provided during all that time went beyond any expectations I possibly could have had for a research advisor. I will never thank her enough for the opportunities to go to Rotterdam and Singapore.Thanks to Jim Paduano and all the member of the supersonic project for their initial interest toward me and allowing me such an easy switch to model order reduction. Thanks to Mark Drela who provided me with precious hints in debugging mtflow. I must also acknowledge Michal Rewienski and Alexandre Megretsk...

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“…The frequency domain models are obtained from matching transfer functions computed from the measured input-output data [27]. Examples of the frequency domain ROMs are the indicial response method [3,62,61] and a frequency-domain model based on Proper Orthogonal Decomposition (POD) [25,66]. The time domain models are based on the state space representation by matching time histories of measured data.…”

Section: Reduced-order Aerodynamic Modelsmentioning

confidence: 99%

Challenges in the aerodynamics modeling of an oscillating and translating airfoil at large incidence angles

Ghoreyshi

Cummings

2013

Aerospace Science and Technology

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The challenges in the modeling of the unsteady lift of a two-dimensional NACA 0012 airfoil oscillating and translating at large incidence angles are investigated in this paper. Forced oscillation motions with two reduced frequency values of 0.1 and 1.0 were used. The CFD results show that a hysteresis loop was developed in the lift variation with angle of attack; shapes of the loops change significantly with reduced frequency value. A dynamic stall vortex was identified in the pitching and plunging motions with k = 0.1.This vortex delays the onset of flow separation over the upper surface to a higher incidence than would occur in steady conditions. The unsteady lift of the very fast motion is enhanced significantly due to the formation of a compression wave on the lower surface and an expansion wave on the upper surface. The results show clear limitations of unsteady aerodynamic theories for modeling motions at high incidences. Also, the CFD solution of indicial functions show a large oscillation at angles of attack beyond the stall angle, but the predictions of a model based on Radial Basis Function matches CFD values quite well.

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Reduced-order modelling of unsteady small-disturbance flows using a frequency-domain proper orthogonal decomposition technique

Cited by 82 publications

References 18 publications

Model Reduction for Active Control Design Using Multiple-Point Arnoldi Methods

Model Reduction for Active Control Design Using Multiple-Point Arnoldi Methods

Reduced-Order, Trajectory Piecewise-Linear Models for Nonlinear Computational Fluid Dynamics

Challenges in the aerodynamics modeling of an oscillating and translating airfoil at large incidence angles

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