© 2021 Elsevier B.V.One of the strengths of the discontinuous Galerkin (DG) method has been its balance between accuracy and robustness, which stems from DG's intrinsic (upwind) dissipation being biased towards high frequencies/wavenumbers. This is particularly useful in high Reynolds-number flow simulations where limitations on mesh resolution typically lead to potentially unstable under-resolved scales. In continuous Galerkin (CG) discretisations, similar properties are achievable through the addition of artificial diffusion such as spectral vanishing viscosity (SVV). However although SVV is recognised as very useful in CG-based high-fidelity turbulence simulations, this approach has been observed to be sub-optimal when compared to DG at intermediate polynomials orders (P≈3). In this paper we explore an alternative stabilisation approach through the introduction of a continuous interior penalty on the gradient discontinuity at elemental boundaries, which we refer to as a gradient jump penalisation (GJP). Analogous to DG methods, this introduces a penalisation at the elemental interfaces as opposed to the interior element stabilisation of SVV. Detailed eigenanalysis of the GJP approach shows its potential as equivalent (sometimes superior) to DG dissipation and hence superior to previous SVV approaches. Through eigenanalysis, a judicious choice of GJP's P-dependent scaling parameter is made and found to be consistent with previous a-priori error analysis. The favourable properties of the GJP stabilisation approach are also supported by turbulent flow simulations of the incompressible Navier–Stokes equation, as we achieve higher quality flow solutions at P=3 using GJP, whereas SVV performs marginally worse at P=5 with twice as many degrees of freedom in total.

© 2021 American Physical Society.A method is proposed in order to optimally decompose the trace of cross-covariances of flow fluctuations, such as Reynolds stresses. Such method, referred to as cross proper orthogonal decomposition (CPOD), leads to a basis of modes extracted from a flow database that are optimal in representing an inner product related to the cross-covariance of interest. A sample application is shown for the representation of Reynolds shear stress in a turbulent channel flow with friction Reynolds number equal to 179. Leading modes are shown to comprise streamwise vortices and streaks with phase opposition between streamwise (u′) and wall-normal (v′) velocities, representing ejections and sweeps, and higher-order modes show similar structures, but with u′ and v′ in phase. A combination of such structures leads to an accurate reconstruction of the Reynolds stress, and consequently of the mean flow, with a reasonable near-wall reconstruction with the leading CPOD mode pair (even and odd modes) for each considered wave number, and a close match of the profiles with the five leading CPOD mode pairs. The present method is thus a valuable modal decomposition technique targeting cross-covariances of flow quantities such as Reynolds stresses.

© 2020 The Author(s). Published by Cambridge University Press.Ensemble data assimilation methods integrate measurement data and computational flow models to estimate the state of fluid systems in a robust, scalable way. However, discretization errors in the dynamical and observation models lead to biased forecasts and poor estimator performance. We propose a low-rank representation for this bias, whose dynamics is modelled by data-informed, time-correlated processes. State and bias parameters are simultaneously corrected online with the ensemble Kalman filter. The proposed methodology is then applied to the problem of estimating the state of a two-dimensional flow at modest Reynolds number using an ensemble of coarse-mesh simulations and pressure measurements at the surface of an immersed body in a synthetic experiment framework. Using an ensemble size of 60, the bias-aware estimator is demonstrated to achieve at least 70 % error reduction when compared to its bias-blind counterpart. Strategies to determine the bias statistics and their impact on the estimator performance are discussed.

© 2018 American Physical Society.Vortex models have been used for decades as computationally efficient tools to investigate unsteady aerodynamics. However, their utility for separated flows - particularly when such flows are subjected to incident disturbances - has been hindered by the tradeoff between the model's physical fidelity and its expectation for fast prediction (e.g., relative to computational fluid dynamics). In this work, it is shown that physical fidelity and speed can be simultaneously achieved by assimilating measurement data into the model to compensate for unrepresented physics. The underlying inviscid vortex model captures the transport of vortex structures with a standard collection of regularized vortex elements that interact mutually and with an infinitely thin flat plate. In order to maintain a low-dimensional representation, with fewer than O(100) degrees of freedom, an aggregation procedure is developed and utilized in which vortex elements are coalesced at each time step. A flow state vector, composed of vortex element properties as well as the critical leading-edge suction parameter, is advanced within an ensemble Kalman filter (EnKF) framework. In this framework, surface pressure is used to correct the states of an ensemble of randomly initiated vortex models. The overall algorithm is applied to several scenarios of an impulsively started flat plate, in which data from a high-fidelity Navier-Stokes simulation at Reynolds number 500 are used as a surrogate for the measurements. The assimilated vortex model efficiently and accurately predicts the evolving flow as well as the normal force in both the undisturbed case (a separated flow) as well as in the presence of one or more incident gusts, despite lack of a priori knowledge of the gust's characteristics.

© 2018, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.A data-driven vortex model of the unsteady aerodynamics of a two-dimensional separated flow is constructed. The vortex model relies on a standard collection of regularized vortex elements that interact mutually and with an infinitely-thin flat plate. In order to maintain a low-dimensional representation, with fewer than O(100) degrees of freedom, a novel aggregation procedure is developed and utilized in which vortex elements are coalesced at each time step. A flow state vector, composed of vortex elements properties as well as the critical leading-edge suction parameter of Ramesh and Gopalarathnam (J. Fluid Mech., 2014), is advanced within an ensemble Kalman filter (EnKF) framework. In this framework, surface pressure measurements, sampled from a truth case, are used to correct the states of an ensemble of randomly-initiated vortex element models. The estimation algorithm is applied to several scenarios of a flat plate impulsively started at 20 degrees angle of attack at Reynolds number 500, in which the truth case comprises a high-fidelity Navier–Stokes simulation. The algorithm provides a good estimate of the flow as well as the aerodynamic force in both the baseline undisturbed case (a separated flow) as well as in the presence of one or more incident gusts, despite lack of a priori knowledge of the incident gust character.

© 2018, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.Ensemble methods can integrate measurement data and CFD-based models to estimate the state of fluid systems in a robust and cost-efficient way. However, discretization errors can render numerical solutions a biased representation of reality. Left unaccounted for, biased forecast and observation models can lead to poor estimator performance. In this work, we propose a low-rank representation for the bias whose dynamics is represented by a colored-noise process. System state and bias parameters are simultaneously corrected on-line with the Ensemble Kalman Filter (EnKF) algorithm. The proposed methodology is demonstrated to achieve a 70% error reduction for the problem of estimating the state of the two-dimensional low-Re flow past a flat plate at high angle of attack using an ensemble of coarse-mesh simulations and pressure measurements at the surface of the body, compared to a bias-blind estimator. Strategies to determine the bias statistics and to deal with nonlinear observation functions in the context of ensemble methods are discussed.

© 2018 by the American Institute of Aeronautics and Astronautics, Inc.Regardless of the plant model, robust flow estimation based on limited measurements remains a major challenge in successful flow control applications. Aiming to combine the robustness of a high-dimensional representation of the dynamics with the cost efficiency of a low-order approximation of the state covariance matrix, a flow state estimator based on the ensemble Kalman filter is applied to two-dimensional flow past a cylinder and an airfoil at high angle of attack and low Reynolds number. For development purposes, the numerical algorithm is used as both the estimator and as a surrogate for the measurements. Estimation is successful using a reduced number of either pressure sensors on the surface of the body or sparsely placed velocity probes in the wake. Because the most relevant features of these flows are restricted to a low-dimensional manifold of the state space, asymptotic behavior of the estimator is shown to be achieved with a small ensemble size. The relative importance of each sensor location is evaluated by analyzing how they influence the estimated flowfield, and optimal locations for pressure sensors are determined. Covariance inflation is used to enhance the estimator performance in the presence of unmodeled freestream perturbations. Acombination of parametric modeling and augmented state methodology is used to successfully estimate the forces on immersed bodies subjected to deterministic and random gusts.

© 2017, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.Experimental measurements of the flow structure evolving in the separated flow over an NACA 0009 wing at 12° angle of attack were obtained with particle image velocimetry, surface pressures, and force transducer measurements of the lift coefficient and pitching moment coefficient. Phase-averaged two-dimensional velocity field measurements provide details of the separated shear layer evolution following a four-pulse burst sequence from a synthetic jet actuator. The flow field development is quite similar to the observations made by Brzozowski, et al. (2010), who used a pulsed-combustion actuator that is orders of magnitude stronger than the synthetic jet. Proper orthogonal decomposition of the PIV data sets showed that the combination of the time-varying coefficients modes 1 and 2 correlate with the negative of the lift coefficient response. The surface pressure signals were correlated with the roll up and convection of the large-scale vortex structure that follows the actuator burst input. A spatially localized region of high pressure occurs below and slightly behind a “kink” that forms in the shear layer. A localized region of high surface pressure that follows the kinked region correlates with the lift reversal that occurs within 2.0t+ after the burst signal was triggered.

© 2017, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.Regardless of plant model, robust flow estimation based on limited measurements remains a major obstacle to successful flow control applications. Aiming to combine the robustness of a high-dimensional representation of the dynamics with the cost efficiency of a low-order approximation of the state covariance matrix, a flow state estimator based on the Ensemble Kalman Filter (EnKF) is applied to two-dimensional flow past a cylinder and an airfoil at high angle of attack and low Reynolds number. For the development purposes, we use the numerical algorithm as both the estimator and as a surrogate for the measurements. Estimation is successful using a reduced number of either pressure sensors on the surface of the body or sparsely placed velocity probes in the wake. Because the most relevant features of these flows is restricted to a low-dimensional subspace/manifold of the state space, asymptotic behavior of the estimator is shown to be achieved with a small ensemble size. The relative importance of each sensor location is evaluated by analyzing how they in fluence the estimated flow field. Covariance in flation is used to enhance the estimator performance in the presence of unmodeled free stream perturbations. A combination of parametric modeling and augmented state methodology is used to successfully estimate the forces on immersed bodies.

© 2016 Elsevier Inc.This paper proposes two important improvements to shock-capturing strategies using a discontinuous Galerkin scheme, namely, accurate shock identification via finite-time Lyapunov exponent (FTLE) operators and efficient shock treatment through a point-implicit discretization of a PDE-based artificial viscosity technique. The advocated approach is based on the FTLE operator, originally developed in the context of dynamical systems theory to identify certain types of coherent structures in a flow. We propose the application of FTLEs in the detection of shock waves and demonstrate the operator's ability to identify strong and weak shocks equally well. The detection algorithm is coupled with a mesh refinement procedure and applied to transonic and supersonic flows. While the proposed strategy can be used potentially with any numerical method, a high-order discontinuous Galerkin solver is used in this study. In this context, two artificial viscosity approaches are employed to regularize the solution near shocks: an element-wise constant viscosity technique and a PDE-based smooth viscosity model. As the latter approach is more sophisticated and preferable for complex problems, a point-implicit discretization in time is proposed to reduce the extra stiffness introduced by the PDE-based technique, making it more competitive in terms of computational cost.

© 2015John Wiley & Sons, Ltd.When dealing with high-order numerical methods, an adequate treatment of curved surfaces is required not only to guarantee that the expected high-order is maintained in the vicinity of surfaces but also to avoid steady-state convergence issues. Among the variety of high-order surface treatment techniques that have been proposed, the ones employing NURBS (non-uniform rational B-splines) to describe curved surfaces can be considered superior both in terms of accuracy and compatibility with computer-aided design softwares. The current study describes in detail the integration of NURBS-based geometry description in a high-order solver based on the discontinuous Galerkin formulation. Particularly, this work also discusses how and why NURBS curves of very high order can be employed within standard NURBS-based boundary treatment techniques to yield reduced implementation complexity and computational overhead. Theoretical estimates are provided along with numerical experiments in order to support the proposed approach. Minding engineering applications in the context of compressible aerodynamics, additional simulations are addressed as numerical examples to illustrate the advantages of using higher-order NURBS in practical situations.