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We introduce a new approach for adaptive mesh refinement in which adaptivity is driven by low rank decomposition and optimal sensing of the dynamically evolving flow field. This method seeks an ordered set of locations for mesh adaptation from the instantaneous data-driven basis of an online proper orthogonal decomposition of the velocity, which organizes features into sparse optimal orthogonal modes based on an energy norm. The sensing is achieved via a computationally expedient discrete empirical interpolation method using rank-revealing QR factorization (Drmac and Gugercin SIAM J Sci Comput 38(2):A631–A648, 2016). The methodology is applicable to a wide range of numerical discretizations, and is tested on a spatiotemporally evolving incompressible turbulent jet, a complex wind turbine wake, and supersonic flow over a forward-facing step. The proposed approach is demonstrated to predict accurate velocity statistics and yield significantly smaller grids in comparison to gradient-based methods. The algorithm is seen to focus refinement in the vicinity of dynamically significant regions such as those characterized by high turbulence kinetic energy, coherent structures and shock interactions. Moreover, the approach does not require parameters or thresholds, which may be difficult to obtain for complex flows, to be known a priori to facilitate mesh adaptation.

The Koopman operator has emerged as a powerful tool for the analysis of nonlinear dynamical systems as it provides coordinate transformations to globally linearize the dynamics. While recent deep learning approaches have been useful in extracting the Koopman operator from a data-driven perspective, several challenges remain. In this work, we formalize the problem of learning the continuous-time Koopman operator with deep neural networks in a measure-theoretic framework. Our approach induces two types of models: differential and recurrent form, the choice of which depends on the availability of the governing equations and data. We then enforce a structural parameterization that renders the realization of the Koopman operator provably stable. A new autoencoder architecture is constructed, such that only the residual of the dynamic mode decomposition is learned. Finally, we employ mean-field variational inference (MFVI) on the aforementioned framework in a hierarchical Bayesian setting to quantify uncertainties in the characterization and prediction of the dynamics of observables. The framework is evaluated on a simple polynomial system, the Duffing oscillator, and an unstable cylinder wake flow with noisy measurements., Comment: 31 pages

Koopman decomposition is a non-linear generalization of eigen-decomposition, and is being increasingly utilized in the analysis of spatio-temporal dynamics. Well-known techniques such as the dynamic mode decomposition (DMD) and its linear variants provide approximations to the Koopman operator, and have been applied extensively in many fluid dynamic problems. Despite being endowed with a richer dictionary of nonlinear observables, nonlinear variants of the DMD, such as extended/kernel dynamic mode decomposition (EDMD/KDMD) are seldom applied to large-scale problems primarily due to the difficulty of discerning the Koopman invariant subspace from thousands of resulting Koopman eigenmodes. To address this issue, we propose a framework based on multi-task feature learning to extract the most informative Koopman invariant subspace by removing redundant and spurious Koopman triplets. In particular, we develop a pruning procedure that penalizes departure from linear evolution. These algorithms can be viewed as sparsity promoting extensions of EDMD/KDMD. Further, we extend KDMD to a continuous-time setting and show a relationship between the present algorithm, sparsity-promoting DMD, and an empirical criterion from the viewpoint of non-convex optimization. The effectiveness of our algorithm is demonstrated on examples ranging from simple dynamical systems to two-dimensional cylinder wake flows at different Reynolds numbers and a three-dimensional turbulent ship air-wake flow. The latter two problems are designed such that very strong nonlinear transients are present, thus requiring an accurate approximation of the Koopman operator. Underlying physical mechanisms are analyzed, with an emphasis on characterizing transient dynamics. The results are compared to existing theoretical expositions and numerical approximations., 48 pages

A projection-based formulation is presented for non-linear model reduction of problems with extreme scale disparity. The approach allows for the selection of an arbitrary, but complete, set of solution variables while preserving the structure of the governing equations. Least-squares-based minimization is leveraged to guarantee symmetrization and discrete consistency with the full-order model (FOM). Two levels of scaling are used to achieve the conditioning required to effectively handle problems with extremely disparate physical phenomena, characterized by extreme stiffness in the system of equations. The formulation -- referred to as model-form preserving least-squares with variable transformation (MP-LSVT) -- provides global stabilization for both implicit and explicit time integration schemes. To achieve computational efficiency, a pivoted QR decomposition is used with oversampling, and adapted to the MP-LSVT method. The framework is demonstrated in representative two- and three-dimensional reacting flow problems, and the MP-LSVT is shown to exhibit improved stability and accuracy over standard projection-based ROM techniques. Physical realizability and local stability are promoted by enforcing limiters in both temperature and species mass fractions. These limiters are demonstrated to be important in eliminating regions of spurious burning, thus enabling the ROMs to provide accurate representations of the heat release rate and flame propagation speed. In the 3D application, it is shown that more than two orders of magnitude acceleration in computational efficiency can be achieved, while also providing reasonable future-state predictions. A key contribution of this work is the development and demonstration of a comprehensive ROM formulation that targets highly challenging multi-scale transport-dominated problems., 47 pages, 26 figures

A finite-volume scheme is developed for the three-dimensional, incompressible vorticity transport equations (VTE) using multi-dimensional upwinding, with the goal of efficient computations of vortex-dominated flows. By modifying the VTE with a term proportional to the divergence of the vorticity, a stable hyperbolic PDE system with a simple structure is revealed. The structure of the resulting eigensystem, including the vortex stretching term, makes the formulation and solution of the generalized Riemann problem and multi-dimensional upwinding more natural, when compared to the Euler equations. To reduce the computational costs of determining the transverse fluxes, a flux-based wave propagation approach is employed. In this approach, the transverse fluxes are computed via direct manipulation of the one-dimensional generalized Riemann problem with no additional Riemann problem solutions needed. The numerical scheme is implemented within an adaptive mesh refinement framework and evaluated on a series of canonical vortex-dominated flows. A translating vortex flow reveals that the multi-dimensional upwinding substantially outperforms one-dimensional schemes in terms of accuracy and computational time, especially when the vortex propagates oblique to cell surfaces. By including transverse fluxes in simulations of propagating vortex rings, key physical attributes including propagation velocity and impulse can be captured with better accuracy compared to one-dimensional schemes. Further vortex ring simulations demonstrate that the proposed multi-dimensional scheme can preserve vorticity with relatively coarse grids compared to simulations employing the incompressible Euler equations. Integrated quantities from leapfrogging vortex ring problems and energy spectra from decaying turbulent flows confirm that the multi-dimensional scheme can accurately reproduce flows with complex vortex stretching and multiple length scales.

A suite of computational tools capable of predicting in-plane low-frequency rotorcraft noise and its control using blade-tip geometry modifications is developed. The combined code, consisting of AV...

Derivation of reduced order representations of dynamical systems requires the modeling of the truncated dynamics on the retained dynamics. In its most general form, this so-called closure model has to account for memory effects. In this work, we present a framework of operator inference to extract the governing dynamics of closure from data in a compact, non-Markovian form. We employ sparse polynomial regression and artificial neural networks to extract the underlying operator. For a special class of non-linear systems, observability of the closure in terms of the resolved dynamics is analyzed and theoretical results are presented on the compactness of the memory. The proposed framework is evaluated on examples consisting of linear to nonlinear systems with and without chaotic dynamics, with an emphasis on predictive performance on unseen data., 33 pages

The development of reduced models for complex multiscale problems remains one of the principal challenges in computational physics. The optimal prediction framework of Chorin et al., which is a reformulation of the Mori-Zwanzig (M-Z) formalism of non-equilibrium statistical mechanics, provides a methodology for the development of mathematically-derived reduced models of dynamical systems. Several promising models have emerged from the optimal prediction community and have found application in molecular dynamics and turbulent flows. In this work, a new M-Z-based closure model that addresses some of the deficiencies of existing methods is developed. The model is constructed by exploiting similarities between two levels of coarse-graining via the Germano identity of fluid mechanics and by assuming that memory effects have a finite temporal support. The appeal of the proposed model, which will be referred to as the `dynamic-MZ-$\tau$' model, is that it is parameter-free and has a structural form imposed by the mathematics of the coarse-graining process (rather than the phenomenological assumptions made by the modeler, such as in classical subgrid scale models). To promote the applicability of M-Z models in general, two procedures are presented to compute the resulting model form, helping to bypass the tedious error-prone algebra that has proven to be a hindrance to the construction of M-Z-based models for complex dynamical systems. While the new formulation is applicable to the solution of general partial differential equations, demonstrations are presented in the context of Large Eddy Simulation closures for Burgers equation, decaying homogeneous turbulence, and turbulent channel flow. The performance of the model and validity of the underlying assumptions are investigated in detail., Comment: 28 pages. This is a pre-print of an article submitted to the Journal of Computational Physics

A modeling paradigm is developed to augment predictive models of turbulence by effectively using limited data generated from physical experiments. The key components of the current approach involve...

In trajectory planning and control design for unmanned air vehicles, highly simplified models are typically used to represent the vehicle dynamics and the operating environment. The goal of this work is to perform real-time, but realistic flight simulations and trajectory planning for quad-copters in low altitude (, Preprint submitted to AIAA Journal

Simulation of multiscale problems remains a challenge due to the disparate range of spatial and temporal scales and the complex interaction between the resolved and unresolved scales. This work develops a coarse-grained modeling approach for the Continuous Galerkin discretizations by combining the Variational Multiscale decomposition and the Mori-Zwanzig (M-Z) formalism. An appeal of the M-Z formalism is that - akin to Greens functions for linear problems - the impact of unresolved dynamics on resolved scales can be formally represented as a convolution (or memory) integral in a non-linear setting. To ensure tractable and efficient models, Markovian closures are developed for the M-Z memory integral. The resulting sub-scale model has some similarities to adjoint stabilization and orthogonal subscale models. The model is made parameter free by adaptively determining the memory length during the simulation. To illustrate the generalizablity of this model, it is employed in coarse-grained simulations for the one-dimensional Burgers equation and in incompressible turbulence problems., Comment: 37 pages, 18 figures

The impact of chemical reactions on the robustness and accuracy of projection-based Reduced-Order Models (ROMs) of fluid flows is investigated. Both Galerkin and Least-Squares Petrov Galerkin ROMs are shown to be less robust in reacting flows as compared to non-reacting flows. In particular, reacting flow ROMs show a strong sensitivity to \st{the} resolution and are often unstable. To identify the main underlying causes, a representative problem that contains the essential physics encountered in typical combustion dynamics problems is chosen. Comparisons with non-reacting solutions are used to assess the impact of reactions. Investigations are focused on three potential areas of significance: 1) preservation of conservation laws; 2) loss of dissipation; and 3) existence of unphysical local phenomena. Results indicate that conservation is relatively well-controlled and the global dissipation in the ROMs is actually larger than that in the underlying CFD solutions. Spurious local phenomena are, however, highly deleterious. Specifically, the steep temperature gradients that characterize combustion can lead to oscillations in local temperatures even in the absence of reactions. Representative calculations with physics-based temperature constraints verify that eliminating such excursions results in considerable improvement in both stability and future-state prediction capability.

An approximation model based on convolutional neural networks (CNNs) is proposed for flow field predictions. The CNN is used to predict the velocity and pressure field in unseen flow conditions and geometries given the pixelated shape of the object. In particular, we consider Reynolds Averaged Navier–Stokes (RANS) flow solutions over airfoil shapes as training data. The CNN can automatically detect essential features with minimal human supervision and is shown to effectively estimate the velocity and pressure field orders of magnitude faster than the RANS solver, making it possible to study the impact of the airfoil shape and operating conditions on the aerodynamic forces and the flow field in near-real time. The use of specific convolution operations, parameter sharing, and gradient sharpening are shown to enhance the predictive capabilities of the CNN. We explore the network architecture and its effectiveness in predicting the flow field for different airfoil shapes, angles of attack, and Reynolds numbers.

The Algebraic Structure-based Model (ASBM) is an engineering model of turbulence that incorporates information about the morphology of turbulent structures, and thus characterizes additional turbulence information beyond what is provided solely by the Reynolds stresses. In this work, a new formulation of the ASBM is developed that exhibits three defining characteristics. The first is a near-wall correction segregated from the homogeneous component of the model, which allows for a new paradigm for model development and comparison. The second characteristic is a set of fully-explicit equations that replace the original implicit formulation and make evident the highly nonlinear nature of the model. Finally, the last defining characteristic is based on the coupling of the ASBM with turbulent transport equations that provide the required turbulent scales. Using arguments of consistency, stability, and accuracy we arrive at a model that uses transport equations for k and ϵ. The resulting model is simpler to implement, more robust, and amenable to further development. The analysis of the new ASBM formulation is carried out by computing fully-developed channel, pipe and squared duct flows.

We construct an efficient finite volume scheme to solve the three-dimensional vorticity–velocity form of the equations governing incompressible flows. The vorticity transport equations are recast in a form that allows for the development of an accurate upwind scheme. The resulting scheme employs a staggered arrangement of vorticity and velocity and utilizes generalized Riemann problems to determine the numerical flux at the cell faces. The resulting scheme is found to account for convection and stretching of vorticity in a stable manner without the need to perform any additional correction steps to control the divergence of vorticity. Several test cases including isolated and interacting vortices as well as turbulent flow are investigated to assess the effectiveness of the numerical method.

Canonical problems involving single, pairs, and arrays of vertical axis wind turbines (VAWTs) are investigated numerically with the objective of understanding the underlying flow structures and their implications on energy production. Experimental studies by Dabiri (J Renew Sustain Energy 3, 2011) suggest that VAWTs demand less stringent spacing requirements than their horizontal axis counterparts and additional benefits may be obtained by optimizing the placement and rotational direction of VAWTs. The flowfield of pairs of co-/counter-rotating VAWTs shows some similarities with pairs of cylinders in terms of wake structure and vortex shedding. When multiple VAWTs are placed in a column, the extent of the wake is seen to spread further downstream, irrespective of the direction of rotation of individual turbines. However, the aerodynamic interference between turbines gives rise to regions of excess momentum between the turbines which lead to significant power augmentations. Studies of VAWTs arranged in multiple columns show that the downstream columns can actually be more efficient than the leading column, a proposition that could lead to radical improvements in wind farm productivity.

Large-eddy simulations of a wing-body junction flow experimentally studied are presented. Wall junction flows are common in engineering applications, but the flow physics at the corners of the junction is not well understood. Moreover, in these types of flows, the performance of the subgrid-scale models frequently used for large-eddy simulations is not well characterized. To address these issues, large-eddy simulations of the wing–body junction are performed, with multiple levels of grid resolution. Two-dimensional turbulent profiles are generated every time step and introduced to the inlet plane of the computation domain to mimic the unsteady turbulent boundary layer. A Reynolds-averaged Navier–Stokes calculation is used as a precursor to initialize the flowfield of the large-eddy simulations. Then, large-eddy simulations with the Vreman model is performed and the simulation results are analyzed with respect to three specific goals: compare the computed and measured turbulence statistics in the junction ...

We propose a modeling paradigm, termed field inversion and machine learning (FIML), that seeks to comprehensively harness data from sources such as high-fidelity simulations and experiments to aid the creation of improved closure models for computational physics applications. In contrast to inferring model parameters, this work uses inverse modeling to obtain corrective, spatially distributed functional terms, offering a route to directly address model-form errors. Once the inference has been performed over a number of problems that are representative of the deficient physics in the closure model, machine learning techniques are used to reconstruct the model corrections in terms of variables that appear in the closure model. These reconstructed functional forms are then used to augment the closure model in a predictive computational setting. As a first demonstrative example, a scalar ordinary differential equation is considered, wherein the model equation has missing and deficient terms. Following this, the methodology is extended to the prediction of turbulent channel flow. In both of these applications, the approach is demonstrated to be able to successfully reconstruct functional corrections and yield accurate predictive solutions while providing a measure of model form uncertainties.

This work addresses fundamental issues related to the structure and conditioning of linear time-delayed models of non-linear dynamics on an attractor. While this approach has been well-studied in the asymptotic sense (e.g. for infinite number of delays), the non-asymptotic setting is not well-understood. First, we show that the minimal time-delays required for perfect signal recovery are solely determined by the sparsity in the Fourier spectrum for scalar systems. For the vector case, we provide a rank test and a geometric interpretation for the necessary and sufficient conditions for the existence of an accurate linear time delayed model. Further, we prove that the output controllability index of a linear system induced by the Fourier spectrum serves as a tight upper bound on the minimal number of time delays required. An explicit expression for the exact linear model in the spectral domain is also provided. From a numerical perspective, the effect of the sampling rate and the number of time delays on numerical conditioning is examined. An upper bound on the condition number is derived, with the implication that conditioning can be improved with additional time delays and/or decreasing sampling rates. Moreover, it is explicitly shown that the underlying dynamics can be accurately recovered using only a partial period of the attractor. Our analysis is first validated in simple periodic and quasi-periodic systems, and sensitivity to noise is also investigated. Finally, issues and practical strategies of choosing time delays in large-scale chaotic systems are discussed and demonstrated on 3D turbulent Rayleigh-B\'{e}nard convection., Comment: 30 pages

We formulate a new projection-based reduced-order modeling technique for non-linear dynamical systems. The proposed technique, which we refer to as the Adjoint Petrov–Galerkin (APG) method, is derived by decomposing the generalized coordinates of a dynamical system into a resolved coarse-scale set and an unresolved fine-scale set. A Markovian finite memory assumption within the Mori–Zwanzig formalism is then used to develop a reduced-order representation of the coarse scales. This procedure leads to a closed reduced-order model that displays commonalities with the adjoint stabilization method used in finite elements. The formulation is shown to be equivalent to a Petrov–Galerkin method with a non-linear, time-varying test basis, thus sharing some similarities with the Least-Squares Petrov–Galerkin method. Theoretical analysis examining a priori error bounds and computational cost is presented. Numerical experiments on the compressible Navier–Stokes equations demonstrate that the proposed method can lead to improvements in numerical accuracy, robustness, and computational efficiency over the Galerkin method on problems of practical interest. Improvements in numerical accuracy and computational efficiency over the Least-Squares Petrov–Galerkin method are observed in most cases.

A multi-fidelity framework is established and demonstrated for prediction of combustion instabilities in rocket engines. The major idea is to adapt appropriate fidelity modeling approaches for different components in a rocket engine to ensure accurate and efficient predictions. Specifically, the proposed framework integrates projection-based Reduced Order Models (ROMs) that are developed using bases generated on truncated domain simulations. The ROM training is performed on truncated domains, and thus does not require full order model solutions on the full rocket geometry, thus demonstrating the potential to greatly reduce training cost. Geometry-specific training is replaced by the response generated by perturbing the characteristics at the boundary of the truncated domain. This training method is shown to enhance predictive capabilities and robustness of the resulting ROMs, including at conditions outside the training range. Numerical tests are conducted on a quasi-1D model of a single-element rocket combustor and the present framework is compared to traditional ROM development approaches.

We study the use of feedforward neural networks (FNN) to develop models of nonlinear dynamical systems from data. Emphasis is placed on predictions at long times, with limited data availability. Inspired by global stability analysis, and the observation of the strong correlation between the local error and the maximum singular value of the Jacobian of the ANN, we introduce Jacobian regularization in the loss function. This regularization suppresses the sensitivity of the prediction to the local error and is shown to improve accuracy and robustness. Comparison between the proposed approach and sparse polynomial regression is presented in numerical examples ranging from simple ODE systems to nonlinear PDE systems including vortex shedding behind a cylinder, and instability-driven buoyant mixing flow. Furthermore, limitations of feedforward neural networks are highlighted, especially when the training data does not include a low dimensional attractor. Strategies of data augmentation are presented as remedies to address these issues to a certain extent., Comment: 30 pages. Complexity, 2018

Data from experiments and direct simulations of turbulence have historically been used to calibrate simple engineering models such as those based on the Reynolds-averaged Navier–Stokes (RANS) equations. In the past few years, with the availability of large and diverse data sets, researchers have begun to explore methods to systematically inform turbulence models with data, with the goal of quantifying and reducing model uncertainties. This review surveys recent developments in bounding uncertainties in RANS models via physical constraints, in adopting statistical inference to characterize model coefficients and estimate discrepancy, and in using machine learning to improve turbulence models. Key principles, achievements, and challenges are discussed. A central perspective advocated in this review is that by exploiting foundational knowledge in turbulence modeling and physical constraints, researchers can use data-driven approaches to yield useful predictive models.

Reduced Order Models (ROMs) of complex, nonlinear dynamical systems often require closure, which is the process of representing the contribution of the unresolved physics on the resolved physics. The Mori-Zwanzig (M-Z) procedure allows one to write down formally closed evolution equations for the resolved physics. In these equations, the unclosed terms are recast as a memory integral involving the past history of the resolved variables, and a "noise" term. While the M-Z procedure does not directly reduce the complexity of the original system, these equations can serve as a mathematically consistent starting point to develop closures based on approximations of the memory. In this scenario, a priori knowledge of the memory kernel, which is not explicitly known for nonlinear systems, is of paramount importance to assess the validity of a memory approximation. Unraveling the memory kernel requires the determination of the orthogonal dynamics which is a projected high-dimensional partial differential equation that is not tractable in general. A method to estimate the memory kernel a priori, using full-order solution snapshots, is proposed. The main idea is to solve a pseudo orthogonal dynamics equation, that has a convenient Liouville form, instead of the original one. This ersatz arises from the assumption that the semi-group of the orthogonal dynamics operator is a composition operator, akin to semi-groups of Liouville operators, for one observable. The method is exact in the linear case where the kernel is known explicitly. Results for under-resolved simulations of the Burgers and Kuramoto-Sivashinsky equations demonstrate that the proposed technique can accurately reconstruct the transfer of information between the resolved and unresolved dynamics through memory, and provide valuable information about the kernel.

Reynolds Averaged Navier Stokes (RANS) closures assume that the state of the turbulent flow field can be uniquely described using a finite set of tensors. However, these tensors are averaged statistics, and thus, varied turbulent flow fields with different internal structuring can correspond to the same values for this finite tensor basis. This causes the modeled initial value problem governing turbulence evolution to be ill-posed. This non-unique specification of the initial turbulent flow field may lead to non-unique evolution for the real turbulent flow at this level of description, introducing epistemic uncertainty in the RANS modeling problem. In this investigation, we show that for homogeneous turbulence, there is a range of possible evolution of a real turbulent flow field if only a finite set of local tensors are specified. Furthermore, this range can include diametrically opposite behavior of the flow. The dependence of this uncertainty is analyzed and quantified for different mean velocity gradients and at different flow parameters. It is exhibited that this uncertainty is engendered by the linear mechanisms in turbulence physics. The magnitude of sensitivity of flow evolution on flow history is explained, using the spectral space structure of the linear instabilities manifested for different mean flows.

Starting from one value of the Reynolds stress, the variability of its evolution is measured in different mean flows by ensembles of 5000 simulations of 20,000 Fourier modes.

A data–informed approach is presented with the objective of quantifying errors and uncertainties in the functional forms of turbulence closure models. The approach creates modeling information from higher-fidelity simulations and experimental data. Specifically, a Bayesian formalism is adopted to infer discrepancies in the source terms of transport equations. A key enabling idea is the transformation of the functional inversion procedure (which is inherently infinite-dimensional) into a finite-dimensional problem in which the distribution of the unknown function is estimated at discrete mesh locations in the computational domain. This allows for the use of an efficient adjoint-driven inversion procedure. The output of the inversion is a full-field of discrepancy that provides hitherto inaccessible modeling information. The utility of the approach is demonstrated by applying it to a number of problems including channel flow, shock-boundary layer interactions, and flows with curvature and separation. In all these cases, the posterior model correlates well with the data. Furthermore, it is shown that even if limited data (such as surface pressures) are used, the accuracy of the inferred solution is improved over the entire computational domain. The results suggest that, by directly addressing the connection between physical data and model discrepancies, the field inversion approach materially enhances the value of computational and experimental data for model improvement. The resulting information can be used by the modeler as a guiding tool to design more accurate model forms, or serve as input to machine learning algorithms to directly replace deficient modeling terms.