On the Behavior of Entropy-Stable Schemes in the Low Mach Regime
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Intrastep, Stage-Value Predictors for Diagonally-Implicit Runge-Kutta Methods
To better identify the necessary attributes of good stage-value predictors (SVPs), numerous SVPs are designed for an existing: ESDIRK4(3)7L[2]SA [26] and a new: ESDIRK4(3)8L[2]SA scheme.1 Both are stifflyaccurate, stage-order two, explicit, singly-diagonally implicit Runge–Kutta (ESDIRK) schemes. Tradeoffs are studied in the parameter spaces enforcing the constraints on accuracy, linear stability, nonlinear stability and coefficient size to determine which objectives correlate with effective predictors. The SVPs are tested on three challenging external aerodynamics problems [107 − 108 degrees of freedom (DoFs)], each with a different level of stiffness. The problems include two 3D airfoils simulations and one canonical turbulence simulation. All simulations use the compressible Navier-Stokes equations (CNSE). An entropy stable spectral collocation formulation is used for discretizing the spatial terms in the equations. Simulations are performed at a wide variety of temporal error tolerances. Problems that are sufficiently stiff (e.g., lax temporal error tolerances) benefit from SVPs designed with second-order accuracy and stability properties: A-stability, and L-stability, rather than high accuracy constraints. Simulations with modest stiffness (e.g., strict error tolerances) are better suited for SVPs designed using high accuracy constraints. Designing SVPs with enhanced stability properties is tedious but worthwhile. Simulation times are reduced with optimal SVPs by as much as 100% on some stages, with combined stepwise improvements of between 50 − 100% for both methods. A comparative study is performed with the two aforementioned methods as well as four other ESDIRKs. The newly designed ESDIRK4(3)8L[2]SA with γ ≈ 1/10, proves to be the most efficient of the six tested ESDIRK schemes simulating the CNSE.
Diagonally-Implicit Runge-Kutta↗
Entropy conservative finite element schemes
The question of entropy stability for discrete approximations to hyperbolic systems of conservation laws is studied. The amount of numerical viscosity present in such schemes is quantified and related to their entropy stability by means of comparison. To this end, two main ingredients are used: entropy variables and the construction of certain entropy conservative schemes in terms of piecewise-linear finite element approximations. It is then shown that conservative schemes are entropy stable, if and (for three-point schemes) only if, they contain more numerical viscosity than the abovementioned entropy conservation ones.
Tadmor, E.↗
DNS of Flow in a Low-Pressure Turbine Cascade Using a Discontinuous-Galerkin Spectral-Element Method
A new computational capability under development for accurate and efficient high-fidelity direct numerical simulation (DNS) and large eddy simulation (LES) of turbomachinery is described. This capability is based on an entropy-stable Discontinuous-Galerkin spectral-element approach that extends to arbitrarily high orders of spatial and temporal accuracy and is implemented in a computationally efficient manner on a modern high performance computer architecture. A validation study using this method to perform DNS of flow in a low-pressure turbine airfoil cascade are presented. Preliminary results indicate that the method captures the main features of the flow. Discrepancies between the predicted results and the experiments are likely due to the effects of freestream turbulence not being included in the simulation and will be addressed in the final paper.
Low-Pressure Turbine Cascade↗
DNS of Low-Pressure Turbine Cascade Flows with Elevated Inflow Turbulence Using a Discontinuous-Galerkin Spectral-Element Method
Recent progress towards developing a new computational capability for accurate and efficient high-fidelity direct numerical simulation (DNS) and large-eddy simulation (LES) of turbomachinery is described. This capability is based on an entropy- stable Discontinuous-Galerkin spectral-element approach that extends to arbitrarily high orders of spatial and temporal accuracy, and is implemented in a computationally efficient manner on a modern high performance computer architecture. An inflow turbulence generation procedure based on a linear forcing approach has been incorporated in this framework and DNS conducted to study the effect of inflow turbulence on the suction- side separation bubble in low-pressure turbine (LPT) cascades. The T106 series of airfoil cascades in both lightly (T106A) and highly loaded (T106C) configurations at exit isentropic Reynolds numbers of 60,000 and 80,000, respectively, are considered. The numerical simulations are performed using 8th-order accurate spatial and 4th-order accurate temporal discretization. The changes in separation bubble topology due to elevated inflow turbulence is captured by the present method and the physical mechanisms leading to the changes are explained. The present results are in good agreement with prior numerical simulations but some expected discrepancies with the experimental data for the T106C case are noted and discussed.
Garai, Anirban↗
Scale-Resolving Simulations of Low-Pressure Turbine Cascades with Wall Roughness Using A Spectral-Element Method
The accurate prediction of wall-roughness effects in turbomachinery is becoming critical as turbine designers address airfoil surface quality and degradation concerns arising from the shift to advanced ceramic matrix composite (CMC) or additively-manufactured airfoils operating in higher temperature environments. In this paper, a recently developed computational capability for accurate and efficient scale-resolving simulations of turbomachinery is extended to analyze the boundary- layer separation and transition characteristics in a rough-wall low-pressure turbine (LPT) cascade. The computational capability is based on an entropy-stable discontinuous-Galerkin spectral-element approach that extends to arbitrarily high orders of spatial and temporal accuracy, and is implemented in an efficient manner for a modern high performance computer architecture. Results from the scale-resolving simulations of both smooth and rough airfoil cascades are presented and compared to previous experiments and numerical simulations. The results show that the suction surface boundary layer undergoes laminar separation, transition, and turbulent reattachment for the smooth airfoil cascade, while in the presence of roughness the separation and transition behavior of the suction surface boundary layer is substantially modified. The differences between the smooth and rough airfoil cascades are then highlighted by a detailed analysis of their respective turbulent flow fields.
Spectral-Element↗
Scale–Resolving Simulations of a Fundamental Trailing–Edge Cooling Slot Using a Discontinuous–Galerkin Spectral–Element Method
The accurate prediction of turbulent mixing in high-pressure turbines that incorporate various airfoil surface-cooling strategies is becoming increasing critical to the design of modern gas turbine engines where the quest for improved efficiency is driving compressor overall pressure ratios and turbine inlet temperatures to much higher levels than ever before. In the present paper, a recently developed computational capability for accurate and efficient scale–resolving simulations of turbomachinery is extended to study the turbulent mixing mechanism of a simplified abstraction of an airfoil trailing-edge cooling slot - a plane wall jet with finite lip thickness discharging into an ambient flow. The computational capability is based on an entropy– stable, discontinuous–Galerkin approach that extends to arbitrarily high orders of spatial and temporal accuracy. The numerical results show that the present simulations capture the trends observed in the experiments. Discrepancies between the simulations and experiments are believed to be due to differences in the inflow profiles and tunnel side–wall effects. The thick lip configuration leads to a thicker wake and higher unsteadiness in the wall jet compared to the thin lip. A detailed comparison of the turbulent flowfields is presented to highlight differences arising due to lip thickness variations.
Scale Resolving↗
Intrastep, Stage-Value Predictors for Diagonally-Implicit Runge-Kutta Methods
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Diagonally-Implicit Runge-Kutta↗
A New Approach for a Wider Class of Entropy Split Methods for Compressible Gas Dynamics and MHD
The high order entropy split methods of Sjögreen & Yee [1, 2] by entropy splitting of the compressible Euler (inviscid) flux derivatives for a thermally-perfect gas are based on Harten’s entropy function [3, 4, 5]. Their derivation takes advantage of the homogeneity property of Euler flux, symmetrizable Euler flux derivatives and energy-norm stability in conjunction with high order classical spatial central, DRP (dispersion relation-preserving) [6, 7, 8] or Padé (compact) spatial discretizations [9] with summation-by-parts (SBP) operators [10]. Our entropy split methods have been proven entropy conserving and stable [1, 11, 12]. Our proofs do not rely on a two-point numerical flux, but rather only a linear difference operator is required to derive these methods. To extend the entropy split method for the MHD, we used the Godunov symmetrizable non-conservative MHD form [12, 13, 14]. These high order entropy split methods not only preserve certain physical properties of the chosen governing equations but are also known to either improve numerical stability, and/or minimize aliasing errors in long time integration of turbulent flow computations without the aid of added numerical dissipation. In our previous published work, extensive error norm comparison with grid refinement was performed to show the high accuracy performance of these methods. These studies also showed how well the entropy split methods conserve the entropy, momentum and mass, and preserve the kinetic energy for long time integration of the various flows [1, 2, 12, 13, 14]. The objective of the present work is to use a new approach to obtain a wider class of entropy split methods consisting of a two-point numerical flux portion and a non-conservative portion in such a way that the homogeneity property of the compressible Euler flux is not required. For high order classical spatial central, DRP (dispersion relation-preserving) or Padé (compact) spatial discretizations, this new approach can be proven to be entropy conservative with conservative spatial dsicretizations while at the same time allowing a wider class of symmetrizable inviscid flux derivatives. We also use this generalization to derive an entropy split scheme that is entropy conserving for the equations of MHD without the homogeneity property using the Godunov symmetrizable ideal MHD formulation [15].
High Order Physical Preserving Methods↗
Crystallographic Stability of Metastable Phase Formed by Containerless Processing in REFeO3 (RE: Rare-Earth Element)
Undercooling a melt often facilitates a metastable phase to nucleate preferentially. Although the classical nucleation theory shows that the most critical factor for forming a metastable phase is the interface free energy, the crystallographic stability is also indispensable for the phase to be frozen at ambient temperature. In compound materials such as oxides, authors have suggested that the decisive factors for forming a critical nucleus are not only the free energy difference but also the difference of the entropy of fusion between stable and metastable phases. In the present study, using REFeO3 (RE: rare-earth element) as a model material, we investigate the formation of a metastable phase from undercooled melts with respect to the competitive nucleation and crystallographical stabilities of both phases.
Kuribayashi, Kazuhiko↗
The Impact of Stable Layers on Downdrafts in Tropical Convection
While the idea that downdraft vertical velocity is influenced by thermodynamic stability may seem obvious, there remains a gap in the scientific literature surrounding the interaction between downdrafts and stable layers. A number of questions remain unanswered. Can a stable layer stop the downward progression of a downdraft or merely slow its velocity? Related to this, can a stable layer stop the injection of low entropy air into the boundary layer? Do downdrafts modify the environment in a way that removes pre-existing stable layers (limiting the influence of the stable layer)? Is it possible for evaporative cooling to generate and maintain a stable layer that is sufficiently strong to stop the downward progression of downdrafts?The present study aims to answer these questions using a combination of high-altitude airborne radar and dropsonde observations, both collected within four tropical cyclones (Gaston, Hermine, Karl, and Matthew) during the 2016 NOAA SHOUT field campaign, as well as idealized simulations of tropical deep convection. This presentation will focus on the potential for a stable layer to halt the downward progression of a downdraft (and the associated low entropy air) with particular attention given to conditions and processes that might enhance or mitigate the influence of the stable layer.
Helms, Charles N.↗
Entropy Splitting for High Order Numerical Simulation of Compressible Turbulence
A stable high-order numerical scheme for direct numerical simulation (DNS) of shock-free compressible turbulence is presented. The method is applicable to general geometries. It contains no upwinding, artificial dissipation, or filtering. Instead the method relies on the stabilizing mechanisms of an appropriate conditioning of the governing equations and the use of compatible spatial difference operators for the interior points (interior scheme) as well as the boundary points (boundary scheme). An entropy-splitting approach splits the inviscid flux derivatives into conservative and nonconservative portions. The spatial difference operators satisfy a summation-by-parts condition, leading to a stable scheme (combined interior and boundary schemes) for the initial boundary value problem using a generalized energy estimate. A Laplacian formulation of the viscous and heat conduction terms on the right hand side of the Navier–Stokes equations is used to ensure that any tendency to odd–even decoupling associated with central schemes can be countered by the fluid viscosity. The resulting methods are able to minimize the spurious high-frequency oscillations associated with pure central schemes, especially for long time integration applications such as DNS. For validation purposes, the methods are tested in a DNS of compressible turbulent plane channel flow at low values of friction Mach number, where reference turbulence data bases exist. It is demonstrated that the methods are robust in terms of grid resolution, and in good agreement with published channel data. Accurate turbulence statistics can be obtained with moderate grid sizes. Stability limits on the range of the splitting parameter are determined from numerical tests.
N.D. Sandham↗
Heart rate dynamics in patients with stable angina pectoris and utility of fractal and complexity measures
Dynamic analysis techniques may uncover abnormalities in heart rate (HR) behavior that are not easily detectable with conventional statistical measures. However, the applicability of these new methods for detecting possible abnormalities in HR behavior in various cardiovascular disorders is not well established. Conventional measures of HR variability were compared with short-term (< or = 11 beats, alpha1) and long-term (> 11 beats, alpha2) fractal correlation properties and with approximate entropy of RR interval data in 38 patients with stable angina pectoris without previous myocardial infarction or cardiac medication at the time of the study and 38 age-matched healthy controls. The short- and long-term fractal scaling exponents (alpha1, alpha2) were significantly higher in the coronary patients than in the healthy controls (1.34 +/- 0.15 vs 1.11 +/- 0.12 [p <0.001] and 1.10 +/- 0.08 vs 1.04 +/- 0.06 [p <0.01], respectively), and they also had lower approximate entropy (p <0.05), standard deviation of all RR intervals (p <0.01), and high-frequency spectral component of HR variability (p <0.05). The short-term fractal scaling exponent performed better than other heart rate variability parameters in differentiating patients with coronary artery disease from healthy subjects, but it was not related to the clinical or angiographic severity of coronary artery disease or any single nonspectral or spectral measure of HR variability in this retrospective study. Patients with stable angina pectoris have altered fractal properties and reduced complexity in their RR interval dynamics relative to age-matched healthy subjects. Dynamic analysis may complement traditional analyses in detecting altered HR behavior in patients with stable angina pectoris.
NASA Discipline Cardiopulmonary↗
Microstructural and Mechanical Characterization of a Dispersion Strengthened Medium Entropy Alloy Produced Using Selective Laser Melting
High entropy alloys (HEAs) are an interesting new class of alloys which have been shown to exhibit both notable strength and ductility for a wide range of temperature and stresses. In addition, the remarkably small difference between the solvus and liquidus temperatures for many face centered cubic HEAs makes them an excellent candidate for selective laser melting fabrication. In this study, the microstructure and mechanical properties of a dispersion strengthened equiatomic NiCoCr alloy successfully produced using selective laser melting are explored. The effect laser speed, laser power, and powder recyclability have on final part density and microstructural segregation are analyzed through both x-ray diffraction and high resolution scanning electron microscopy. These results are further validated and compared to stable phase predictions produced using a commercially available high entropy alloy mobility database. Lastly, the tensile strengths resulting from different heat treatment pathways are detailed.
Smith, T. M.↗
The Impact of Stable Layers on Downdrafts in Tropical Deep Convection
Previous studies have demonstrated that one pathway by which dry air can interrupt tropical cyclone intensification is by inducing downdrafts that flood the surface inflow layer with low entropy air. While work has been performed examining the relationship between dry air and downdraft intensity, very little has been done to investigate the potential for features of the convective environment, such as stable layers, to inhibit the downward progression of downdrafts. This study aims to understand the role that stable layers have on inhibiting the downward progression of downdrafts and their associated low entropy air in the context of tropical deep convection. Specifically, this study asks two key questions: can a stable layer halt the downward progression of a downdraft or merely slow its velocity and can a stable layer stop the injection of low entropy air into the boundary layer? The present study aims to answer these questions using a combination of high-altitude airborne radar and dropsonde observations, both collected within four tropical cyclones (Gaston, Hermine, Karl, and Matthew) during the 2016 NOAA SHOUT field campaign, as well as idealized simulations of tropical deep convection. Preliminary results suggest that precipitation-free downdrafts can be blocked by a stable layer while precipitation-laden downdrafts are weakened, but not stopped, by the presence of a stable layer. This presentation will primarily focus on the precipitation-laden downdrafts that can be observed via airborne radar. Additionally, the presentation will conclude with a brief discussion of the implications of downdraft/stable-layer interactions for a tropical cyclone ingesting dry environmental air.
Charles N Helms↗
The convergence rate of approximate solutions for nonlinear scalar conservation laws
The convergence rate is discussed of approximate solutions for the nonlinear scalar conservation law. The linear convergence theory is extended into a weak regime. The extension is based on the usual two ingredients of stability and consistency. On the one hand, the counterexamples show that one must strengthen the linearized L(sup 2)-stability requirement. It is assumed that the approximate solutions are Lip(sup +)-stable in the sense that they satisfy a one-sided Lipschitz condition, in agreement with Oleinik's E-condition for the entropy solution. On the other hand, the lack of smoothness requires to weaken the consistency requirement, which is measured in the Lip'-(semi)norm. It is proved for Lip(sup +)-stable approximate solutions, that their Lip'convergence rate to the entropy solution is of the same order as their Lip'-consistency. The Lip'-convergence rate is then converted into stronger L(sup p) convergence rate estimates.
Nessyahu, Haim↗
Onset of Reconnection in the near Magnetotail: PIC Simulations
Using 2.5-dimensional particle-in-cell (PIC) simulations of magnetotail dynamics, we investigate the onset of reconnection in two-dimensional tail configurations with finite Bz. Reconnection onset is preceded by a driven phase, during which magnetic flux is added to the tail at the high-latitude boundaries, followed by a relaxation phase, during which the configuration continues to respond to the driving. We found a clear distinction between stable and unstable cases, dependent on deformation amplitude and ion/electron mass ratio. The threshold appears consistent with electron tearing. The evolution prior to onset, as well as the evolution of stable cases, are largely independent of the mass ratio, governed by integral flux tube entropy conservation as imposed in MHD (magnetohydrodynamics). This suggests that ballooning instability in the tail should not be expected prior to the onset of tearing and reconnection. The onset time and other onset properties depend on the mass ratio, consistent with expectations for electron tearing. At onset,we found electron anisotropies T⊥⁄ T∥ (bottom tail divided by parallel tail) equals 1.1-1.3, raising growth rates and wavenumbers. Our simulations have provided a quantitative onset criterion that is easily evaluated in MHD simulations, provided the spatial resolution is sufficient. The evolution prior to onset and after the formation of a neutral line does not depend on the electron physics, which should permit an approximation by MHD simulations with appropriate dissipation terms.
PIC↗
Dynamical Signatures of Living Systems
One of the main challenges in modeling living systems is to distinguish a random walk of physical origin (for instance, Brownian motions) from those of biological origin and that will constitute the starting point of the proposed approach. As conjectured, the biological random walk must be nonlinear. Indeed, any stochastic Markov process can be described by linear Fokker-Planck equation (or its discretized version), only that type of process has been observed in the inanimate world. However, all such processes always converge to a stable (ergodic or periodic) state, i.e., to the states of a lower complexity and high entropy. At the same time, the evolution of living systems directed toward a higher level of complexity if complexity is associated with a number of structural variations. The simplest way to mimic such a tendency is to incorporate a nonlinearity into the random walk; then the probability evolution will attain the features of diffusion equation: the formation and dissipation of shock waves initiated by small shallow wave disturbances. As a result, the evolution never "dies:" it produces new different configurations which are accompanied by an increase or decrease of entropy (the decrease takes place during formation of shock waves, the increase-during their dissipation). In other words, the evolution can be directed "against the second law of thermodynamics" by forming patterns outside of equilibrium in the probability space. Due to that, a specie is not locked up in a certain pattern of behavior: it still can perform a variety of motions, and only the statistics of these motions is constrained by this pattern. It should be emphasized that such a "twist" is based upon the concept of reflection, i.e., the existence of the self-image (adopted from psychology). The model consists of a generator of stochastic processes which represents the motor dynamics in the form of nonlinear random walks, and a simulator of the nonlinear version of the diffusion equation which represents the mental dynamics. It has been demonstrated that coupled mental-motor dynamics can simulate emerging self-organization, prey-predator games, collaboration and competition, "collective brain," etc.
Zak, M.↗