Application of parallel-flow linear stability theory to wind turbine wakes with implications for active wake control.
Abstract not provided.
Brown, Kenneth↗
SEARCH · Engineering Papers
Results for “stability theory”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Second mode growth in a high-velocity boundary layer using stability theory and DNS.
Abstract not provided.
Harris, Shaun↗
Enabling "Fuel" Switching through On-Demand Wind Control
This work demonstrates that classical shear-flow stability theory can be successfully applied to modify wind turbine wakes and also explains the success of several emerging, empirically-arrived control methods (i.e., dynamic induction and helix control). Linear stability theory predictions are extended to include the effects of non-axisymmetric inflow profiles, such as wind shear, which is shown to not strongly affect the primary forcing frequency. The predictions, as well as idealized large-eddy simulations using actuator-line representation of the turbine blades, agree that the n = 0 and ±1 modes have faster initial growth rates than higher-order modes, suggesting the lower-order modes are more appropriate for wake control. Exciting the lower-order modes with periodic pitching of the blades produces higher entrainment into the wake and consequently faster wake recovery.
17 WIND ENERGY↗
Boundary-layer receptivity to oblique freestream vorticity waves for a high-enthalpy hypersonic flow
The receptivity of a Mach 15 straight-cone boundary layer to oblique freestream vorticity waves is investigated using direct numerical simulation (DNS) alongside linear stability theory and the linear parabolized stability equations. A thermochemical nonequilibrium gas model is used. Oblique freestream vorticity waves at frequencies of 400, 800, and 1200 kHz are considered, with incident angles ranging from 0° to 29.4° at 400 kHz, 0° to 15.7° at 800 kHz, and 0° to 20.6° at 1200 kHz. The 400 kHz case is of primary interest due to the strong second-mode amplification at this frequency. Although the underlying base flow is axisymmetric, the oblique vorticity waves lead to a fully three-dimensional boundary-layer disturbance whose characteristics vary depending on the azimuthal ray relative to the freestream wave. Moving downstream within the second-mode instability region, some clear trends emerge in terms of the boundary-layer disturbance amplitudes; that is, disturbance amplitudes are highest at the leeward ray (relative to the freestream wave), but weakest about halfway between the windward and leeward rays. Increasing the incident angle causes the amplitudes to increase on the leeward ray and decrease on the windward ray. Moreover, the boundary-layer disturbance throughout contains a wide spectrum of azimuthal wavenumbers in which the disturbance energy falls off at higher wavenumbers. Increasing the incident angle causes the azimuthal spectrum of the boundary-layer disturbance to broaden overall. Qualitatively similar results are found for the two higher frequencies leading up to the peak-amplitude locations corresponding to the second-mode instability.
42 ENGINEERING↗
On closures for reduced order models - A spectrum of first-principle to machine-learned avenues
For over a century, reduced order models (ROMs) have been a fundamental discipline of theoretical fluid mechanics. Early examples include Galerkin models inspired by the Orr–Sommerfeld stability equation and numerous vortex models, of which the von Karman vortex street is one of the most prominent. Subsequent ROMs typically relied on first principles, like mathematical Galerkin models, weakly nonlinear stability theory, and two- and three-dimensional vortex models pioneered the data-driven proper orthogonal decomposition (POD) modeling. In early POD modeling, available data were used to build an optimal basis, which was then utilized in a classical Galerkin procedure to construct the ROM, but data have made a profound impact on ROMs beyond the Galerkin expansion. In this paper, we take a modest step and illustrate the impact of data-driven modeling on one significant ROM area. Specifically, we focus on ROM closures, which are correction terms that are added to the classical ROMs in order to model the effect of the discarded ROM modes in under-resolved simulations. Through simple examples, we illustrate the main modeling principles used to construct the classical ROMs, motivate and introduce modern ROM closures, and show how data-driven modeling, artificial intelligence, and machine learning have changed the standard ROM methodology over the last two decades. Finally, we outline our vision on how the state-of-the-art data-driven modeling can continue to reshape the field of reduced order modeling.
42 ENGINEERING↗
Impact of Vibrational Nonequilibrium on the Supersonic Mode in Hypersonic Boundary Layers
The supersonic mode in hypersonic boundary layers has recently been shown to achieve a larger peak growth rate than the traditional second mode in particular circumstances relevant to free flight. In this environment, high levels of aerodynamic heating are encountered such that incorporation of chemical and vibrational nonequilibrium effects can become critical. The impact of thermal nonequilibrium (TNE) on the supersonic mode in particular has not been thoroughly and systematically investigated. This work uses thermochemical nonequilibrium (TCNE) direct numerical simulation (DNS) and TCNE linear stability theory (LST) to obtain a more complete investigation of the supersonic mode using both nonequilibrium and frozen gas models. The simulation is Mach 5 flow over a 1-mm-nose-radius axisymmetric cone 1 m in length, representative of experimental conditions in ground test facilities. Both LST and DNS results indicate that TNE effects are destabilizing compared with frozen flow and can lead to an N-factor difference of approximately +1 at the end of the cone. LST results also indicate that TNE effects are stabilizing to the second and supersonic modes in comparison with thermal equilibrium. The supersonic mode appeared to be slightly more affected by TNE than the second mode. In conclusion, collectively, the findings in this work suggest that even in ground test facilities where TCNE effects are relatively small compared with free flight, incorporating nonequilibrium models in both the mean flow and stability solvers can have a practical effect on hypersonic boundary-layer transition prediction analyses.
42 ENGINEERING↗
Fluid Dynamics Effects on Microstructure Prediction in Single-Laser Tracks for Additive Manufacturing of IN625
Single-track laser fusion were simulated using a heat-transfer-solidification-only (HTS) model and its extension with fluid dynamics (HTS_FD) model using a parallel open-source code, which included laminar fluid dynamics, flat-free surface of the molten alloy, heat transfer, phase-change, evaporation, and surface tension phenomena. The results illustrate that the fluid dynamics affects the solidification and ensuing microstructure. For the HTS_FD simulations, thermal gradient, G was found to exhibit a maximum at the extremity of the solidified pool ( i.e. , at the free surface), while for HTS simulations, G exhibited a maximum around the entire edge of the solidified pool. HTS_FD simulations predicted a wider range of cooling rates than the HTS simulations, exhibited an increased spread in the solidification speed, V variation within the melt-pool with respect to the HTS model results. Primary dendrite arm spacing (PDAS) were evaluated based on power law correlations and marginal stability theory models using the ( G , V ) from HTS and HTS_FD simulations to quantify the effect of the fluid dynamics on the microstructure. At low-laser powers and low-scan speeds, the PDAS obtained with the fluid dynamics model (HTS_FD) was larger by more than 30 pct with respect to the PDAS calculated with the simple HTS model. A new PDAS correlation, i.e. , \( \lambda_{1} \left[ {\mu {\text{m}}} \right] = 832\;G\left[ {\text{K/m}} \right]^{ - 0.5} V\left[ {\text{m/s}} \right]^{ - 0.25} \) , which uses the ( G , V ) results from the HTS_FD model was developed and validated against experimental results.
36 MATERIALS SCIENCE↗
Experimental Observation of Magnetic Island Heteroclinic Bifurcation in Tokamaks
We report empirical observations of magnetic island heteroclinic bifurcation for the first time. This behavior is observed in interacting coupled 2/1 tearing modes in the core of a DIII-D tokamak plasma. Poincaré maps constrained by measured magnetic amplitudes and phasing show bifurcation from heteroclinic to homoclinic topology in the 2/1 island as the 4/2 relative amplitude (R 4/2 ) decreases. Initially, the local electron temperature peak in the 2/1 island splits, consistent with two O points. As R 4/2 decreases a single peak forms, consistent with one O point. Lastly, these call for developing tearing stability theory and control solutions for heteroclinic islands in tokamaks.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Model Reference Adaptive Control (MRAC) for Additive Manufacturing (AM)
Model Reference Adaptive Control (MRAC) is based on the fundamental concept that the process under investigation is to be controlled to follow or “track” a reference system (model) characterized by a state/input/output model employing an adaptive optimization algorithm to adjust the controller parameters in real-time. The generic structure of the MRAC is shown in Fig. 1 consisting of the following primary components: Reference model, Process (system) model, controller and the adaption algorithm. The basic structure of the controller is specified by a linear construct with the corresponding real-time adaption algorithms given by a gradient-type (so-called MIT rule) or based on stability theory (Lyapunov, hyperstability). This approach to adaptive control is termed “direct”, since the controller (parameters) are adjusted based on the component models/algorithm in contrast to the “indirect” approach that adjusts the process model parameters applying real-time system identification techniques.
42 ENGINEERING↗
A Convex Data-Driven Approach for Nonlinear Control Synthesis
We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems. Unlike Lyapunov-based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently using the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory can be understood through the lens of Perron–Frobenius and Koopman operators. This allows us to use data-driven methods to approximate these operators and combine them with the SOS techniques to establish a convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples.
97 MATHEMATICS AND COMPUTING↗
Data-driven modeling and control of dynamical systems using Koopman and Perron-Frobenius operators
This dissertation studies the data-driven modeling and control problem of nonlinear systems by exploiting the linear operator theoretic framework involving Koopman and Perro-Frobenius operator. A systematic linear-operator based controller design procedure has been established, which can be used to solve a variety of nonlinear control problems, including feedback stabilization using control Lyapunov functions, optimal quadratic regulation using Koopman eigenfunctions and convex optimization formulation of optimal control problem using P-F and Koopman operator approximation. As the core of data-driven modeling, we first propose a new algorithm for the finite-dimensional approximation of the linear transfer Koopman and Perron-Frobenius operator from time-series data. We argue that the existing approach for the finite-dimensional approximation of these transfer operators such as Dynamic Mode Decomposition (DMD) and Extended Dynamic Mode Decomposition (EDMD) do not capture two important properties of these operators, namely positivity and Markov property. The algorithm we propose preserves these two properties. We call the proposed algorithm as naturally structured DMD (NSDMD) since it retains the inherent properties of these operators. Naturally structured DMD algorithm leads to a better approximation of the steady-state dynamics of the system regarding computing Koopman and Perron- Frobenius operator eigenfunctions and eigenvalues. However, preserving positivity property is critical for capturing the real transient dynamics of the system. This positivity property of the transfer operators and it's finite-dimensional approximation play an important role for controller and estimator design of nonlinear systems. To solve the feedback stabilization problem for nonlinear control systems, we tried to take advantage of the Koopman operator framework. The Koopman operator approach provides a linear representation for a nonlinear dynamical system and a bilinear representation for a nonlinear control system. The problem of feedback stabilization of a nonlinear control system is then transformed to the stabilization of a bilinear control system. We propose a control Lyapunov function (CLF)-based approach for the design of stabilizing feedback controllers for the bilinear system. The search for finding a CLF for the bilinear control system is formulated as a convex optimization problem. This leads to a schematic procedure for designing CLF-based stabilizing feedback controllers for the bilinear system and hence the original nonlinear system. Another advantage of the proposed controller design approach outlined in this dissertation is that it does not require explicit knowledge of system dynamics. In particular, the bilinear representation of a nonlinear control system in the Koopman eigenfunction space can be obtained from time-series data. Next, we study the optimal quadratic regulation problem for nonlinear systems. The linear operator theoretic framework involving the Koopman operator is used to lift the dynamics of nonlinear control system to an infinite-dimensional bilinear system. The optimal quadratic regulation problem for nonlinear system is formulated in terms of the finite-dimensional approximation of the bilinear system. A convex optimization-based approach is proposed for solving the quadratic regulator problem for bilinear system. We applied a variety of examples and compared the simulation results between our framework and conventional LQR control using linearized model. For more general optimal control problems, we provide a density-function based convex formulation for the optimal control problem of the nonlinear system. The convex formulation relies on the duality result in the stability theory of a dynamical system involving density function and Perron-Frobenius operator. The optimal control problem is formulated as an infinite-dimensional convex optimization program. The finite-dimensional approximation of the optimization problem relies on the recent advances made in the data-driven computation of the Koopman operator, which is dual to the Perron-Frobenius operator. Simulation results are presented to demonstrate the application of the developed framework.
Huang, Bowen↗
The Computational Analysis of a Generic Hypersonic Store Separation
Many current hypersonic vehicles involve a main vehicle from which components separate during flight and presents simulation challenges such as shock-shock interactions, highly nonlinear interactions, and shock wave boundary layer interactions. These aerodynamic phenomena influence the attitude and trajectory of each object, which needs to be understood before large-scale experiments can be run. To gain a generic understanding of the process, high-fidelity Reynolds-averaged Navier-Stokes solutions are performed on a conical vehicle geometry passing through an oblique shock wave that is representative of different configurations and scenarios. Trajectory and applied forces are tracked and show the vehicle’s dynamics are predominantly the result of differential flow incidence angles causing a strong shock; this leads to a large pressure increase over a fraction of the vehicle, which influences pitch. The vehicle appears to follow conventional stability theory with detached eddy simulation and small disturbances in initial attitude shown to have minimal influence on the scenario.
Savery, Ryan↗
A review of high order strong stability preserving two-derivative explicit, implicit, and IMEX methods
High order strong stability preserving time discretizations ensure the nonlinear non-inner-product strong stability properties of spatial discretizations suited for the stable simulation of hyperbolic PDEs in a wide variety of application areas including fluid dynamics, magnetohydrodynamics, semiconductor devices, electromagnetics, and astrophysics. Over the past decade multiderivative time-stepping have been increasingly used for the time-evolution hyperbolic PDEs, so that the strong stability properties of these methods have become important. In this work we review sufficient conditions for a two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and different conditions on the second derivative. In particular we present the strong stability preserving theory for explicit and implicit two-derivative Runge–Kutta schemes, including a special condition on the second derivative under which these implicit methods may be unconditionally strong stability preserving. This special condition is natural for the stiff component of wide range of plasma physics problems, and can be useful in the context of strong stability preserving implicit-explicit multi-derivative Runge–Kutta schemes, where the time-step restriction is then independent of the stiff term. Lastly, we present the strong stability preserving theory for implicit-explicit multi-derivative general linear methods, and some novel second and third order methods where the time-step restriction is independent of the stiff term.
97 MATHEMATICS AND COMPUTING↗
The theory of kinetic effects on resistive wall mode stability in tokamaks
Tokamak fusion plasmas benefit from high pressures but are then susceptible to modes of instability. These magnetohydrodynamic (MHD) modes are macroscopic distortions of the plasma, but certain collective motions of individual particles can provide stabilizing effects opposing them. The presence of a resistive wall slows the mode growth, converting a kink to a resistive wall mode (RWM). A kinetic MHD model includes Maxwell's equations, ideal MHD constraints, and kinetic effects included through the pressure tensor, calculated with the perturbed drift-kinetic distribution function of the particles. The kinetic stabilizing effects on the RWM arise through resonances between the plasma rotation and particle drift motions: precession, bounce, and transit. A match between particle motions and the mode allows efficient transfer of energy that would otherwise drive the growth of the mode, thus damping the growth. The first approach to calculating RWM stability is to write a set of equations for the complex mode frequency in terms of known quantities and then to solve the system. The “energy principle” approach, which has the advantage of clarity in distinguishing the various stabilizing and destabilizing effects, is to change the force balance equation into an equation in terms of changes of kinetic and potential energies, and then to write a dispersion relation for the mode frequency in terms of those quantities. These methods have been used in various benchmarked codes to calculate kinetic effects on RWM stability. Importantly, the theory has illuminated the important roles of plasma rotation, energetic particles, and collisions in RWM stability.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
A unified framework of stabilized finite element method for solving the Boltzmann transport equation
This paper presents a unified framework of stabilized finite element method for solving the Boltzmann transport equation. Unlike the traditional Petrov-Galerkin finite element method which modifies the test function to construct the stabilization term, we derive the stabilization methods from the standard Galerkin weak form with Sub-grid scale model. The basic idea of this method is to decompose the unknowns into its numerical solution and residual, with an approximation for the residual and embeds it in the Galerkin weak form to yield a stabilized variational formula. Different approximations of the residual lead to different stabilization methods, all the frequently used stabilized methods, including the Streamline Upwinding Petrov-Galerkin (SUPG) method, Galerkin/Least-Square (GLS) method, and Algebraic Sub-Grid Scale (ASGS) method can be obtained from this framework. The similarities and differences of the different approximations are compared in this paper. The numerical results show that the behaviors of the different methods area similar with the same stabilization parameter, and all these stabilized techniques can obtain a correct and stable solution. (authors)
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
Laser-induced fluorescence measurements of ion fluctuations in electron and ion presheaths
Electrostatic flow-driven instabilities have been predicted in the presheath region of low-temperature plasmas for both ion and electron rich sheaths. The ion-acoustic fluctuations that arise from these instabilities can influence transport through an enhanced collision rate. In this work, ion fluctuation spectra were measured using laser-induced fluorescence (LIF) in the presheath region for both sheath polarities. Furthermore, the non-invasive nature of LIF compared to that of probes is particularly valuable for these measurements because the sheath and presheath around a probe may generate their own flow-driven instabilities, which obscure the measurement. Measurements were made using a recently implemented field programmable gate array-based system able to measure ion fluctuation spectra up to 1 MHz using a two-point correlation function technique. Narrow bandwidth ion fluctuations were observed near 0:46 f pi (460 kHz) for both sheath polarities, where f pi is the ion plasma frequency. The observed fluctuations were significantly stronger in the electron sheath case and were observed over a larger volume of plasma. Fluctuations were measured in the ion sheath case at locations far enough from the sheath that linear theory predicts stability, suggesting that ion-acoustic instabilities generated in the relatively small unstable region near the boundary reflect from the sheath and perturb a much larger volume of the plasma. This is expected to affect probes with both ion and electron rich sheaths, meaning any dc biased probe may effectively act as an ion-acoustic wave antenna. These measurements are consistent with the recent theory and particle-in-cell results.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Quasicrystal stability and nucleation kinetics from density functional theory
The aperiodic order of quasicrystals bridges the amorphous and crystalline regime, so it has remained unclear whether quasicrystals are metastable or stable phases of matter. Density functional theory is often used to evaluate thermodynamic stability, but quasicrystals are long-range aperiodic and their energies cannot be calculated using conventional ab initio methods. Here, in this work, we perform first-principles calculations on quasicrystal nanoparticles of increasing size, from which we can directly extrapolate their bulk and surface energies. Using this technique, we determine with high confidence that the icosahedral quasicrystals ScZn 7.33 and YbCd 5.7 are ground-state phases, thus revealing that translational symmetry is not a necessary condition for the zero-temperature stability of inorganic solids. Although we found the ScZn 7.33 quasicrystal to be thermodynamically stable, we show on a mixed thermodynamic and kinetic phase diagram that its solidification from the melt is limited by nucleation, which illustrates why even stable materials may be kinetically challenging to grow. Our techniques broadly open the door to first-principles investigations into the structure–bonding–stability relationships of aperiodic materials.
density functional theory↗