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Numerical methods and hypoexponential approximations for gamma distributed delay differential equations
Abstract Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic gamma distributed DDEs have not previously been implemented. Modellers have therefore resorted to approximating the gamma distribution with an Erlang distribution and using the linear chain technique to derive an equivalent system of ordinary differential equations (ODEs). In this work, we address the lack of appropriate numerical tools for gamma distributed DDEs in two ways. First, we develop a functional continuous Runge–Kutta (FCRK) method to numerically integrate the gamma distributed DDE without resorting to Erlang approximation. We prove the fourth-order convergence of the FCRK method and perform numerical tests to demonstrate the accuracy of the new numerical method. Nevertheless, FCRK methods for infinite delay DDEs are not widely available in existing scientific software packages. As an alternative approach to solving gamma distributed DDEs, we also derive a hypoexponential approximation of the gamma distributed DDE. This hypoexponential approach is a more accurate approximation of the true gamma distributed DDE than the common Erlang approximation but, like the Erlang approximation, can be formulated as a system of ODEs and solved numerically using standard ODE software. Using our FCRK method to provide reference solutions, we show that the common Erlang approximation may produce solutions that are qualitatively different from the underlying gamma distributed DDE. However, the proposed hypoexponential approximations do not have this limitation. Finally, we apply our hypoexponential approximations to perform statistical inference on synthetic epidemiological data to illustrate the utility of the hypoexponential approximation.
Description and planned use of a data distribution evaluation system for fiber optic data buses
A general description of a laboratory data distribution evaluation system (DDES) is given and some plans to use the system. The DDES is a microprocessor-based evaluation system consisting of three identical terminals. The DDES provides the capability for evaluating different system protocols and data word structures by making appropriate software changes, and different transmission medias and modulation schemes by making front-end hardware changes. A data multiplexing standard for fiber optic data transmission which parallels MIL-STD-1553B has proposed four different data modulation schemes. These modulation schemes will be described along with the plans to evaluate each of them. NASA-Langley has also been developing a data distribution approach which utilizes optical wavelength division multiplexing (WDM). A four-port, four-wavelength WDM system will be described along with the planned use of the DDES to evaluate its performance.
Scale resolving simulations of the NASA Juncture Flow Model using the LAVA solver
Two distinct simulation methodologies: Delayed Detached Eddy Simulation (DDES) and stress based Wall-modelled Large Eddy Simulation (WMLES) are evaluated using structured overset curvilinear grids for the NASA juncture flow model. While both methodologies are shown to mitigate the primary shortcomings of steady state Reynolds Averaged Navier Stokes(RANS) simulations, several unresolved aspects are identified. Strong sensitivity to RANS-typegrid refinement is observed in the DDES with a substantial deterioration of the solution quality with increasing spatial resolution associated with deficiencies in the shielding function. Mean profiles for attached boundary layers on the fuselage show spurious inflections suggesting modelled stress depletion on finer grids. Lower numerical dissipation in terms of spatial discretization and time step size is seen to improve the solution quality, and the advantage of using RC and QCR2000 corrections in the underlying RANS closure is demonstrated for DDES on RANS-type meshes. Equilibrium Wall-Modelled LES used grids that resolved the tripping dots over the fuselage nose and the wing leading edge, consistent with the experiment setup. These simulations resulted in a cost-competitive approach compared to DDES on RANS-type grids. Although the agreement between WMLES predictions of first and second order single point statistics with experimental measurements is promising, a fundamental shortcoming is noted in terms of an overshoot in streamwise momentum in corner regions of the wing-fuselage juncture. This excess momentum subsequently delays the onset of separation,thereby resulting in an underprediction in length of the separation bubble. Some quantitative sensitivity to numerical discretization is observed; lowering of numerical dissipation shows better agreement with the experiment.
Multiphysics Computational Analysis of a Perforated Plate Cooling Flow
A multiphysics analysis of a film cooling flow was conducted using a computational fluid dynamics (CFD) solver in conjunction with a conjugate heat transfer (CHT) solver. The film cooling problem consisted of a square nozzle producing a heated subsonic jet over a perforated plate with three sections of 45 cooling holes each. Results from the multiphysics analysis were compared to a CFD-only approach and experimental results. Reynolds-averaged Navier-Stokes (RANS) computations were employed first, with and without the CHT solver. The multiphysics approach provided much closer agreement with experimentally measured plate temperatures than the CFD-only approach. Comparisons of computed velocities and temperatures were also made with experimental data. Hybrid RANS-large eddy simulation calculations employing the Delayed Detached Eddy Simulation (DDES) approach were also conducted within the multiphysics framework. The DDES approach was unable to resolve turbulent structures in the near wall regions. Hence, the RANS part of the DDES model was used in near wall regions, and produced surface temperatures along with velocity and temperature profiles close to the wall that were very similar to the RANS results. The DDES approach did resolve structures in the outer portion of the boundary layer especially near the end of the plate where the boundary layer was thicker. The key result was the improved surface temperatures produced by the multiphysics approach employing CHT over the fluid-only approach.
Using Density-Corrected DFT to Understand Density-Driven and Functional-Dependent Errors in Ab Initio Simulations of the Hydrated Electron
The hydrated electron, an excess electron in liquid water, plays a crucial role in a plethora of chemical processes, motivating extensive research efforts to characterize its structure, dynamics, and reactivity in solution. Recent theoretical approaches to understanding this intriguing object have involved ab initio simulations based on density functional theory (DFT). Although DFT allows for the study of hydrated electron reactivity and quantum mechanical behavior, it is well-known that anionic systems can suffer from significant density-driven errors (DDEs). Density-corrected DFT (DC-DFT) provides a framework to mitigate such errors; the method reduces DDEs by replacing the self-consistent (SC) density associated with a given density functional with the Hartree–Fock (HF) density. Since HF densities tend to be more localized than DFT SC densities, the DC-DFT scheme significantly improves errors in calculations where the SC density is spuriously delocalized. Here, we investigate how the use of density correction affects the calculated properties of the DFT-simulated (PBEh) hydrated electron, a particularly challenging diffuse anionic system to simulate. First, we analyze charge delocalization in a system consisting of a model octahedral hydrated electron water cluster (the so-called Kevan structure) along with a spatially separated sulfur atom. We show that the use of density correction indeed reduces DDEs in comparison to a standard DFT global hybrid functional. We then propagate molecular dynamics trajectories of the hydrated electron using DC-DFT, where we find that DC further localizes electron density in the cavity region, a signature of reduced charge delocalization. Unfortunately, the decreased radius of gyration of the spin density and corresponding tightening of the local solvation structure from density correction causes predicted observables to deviate further from experimental measurements than when density correction is not employed. Here, we argue that DC’s worse agreement with experiment results from the removal of a fortuitous cancellation of errors that is intrinsic to the PBEh functional. This indicates that the difficulties with DFT to simulate hydrated electrons are primarily due to the inherent approximations in DFT rather than to density-driven errors.
Turbulence Simulations of Transonic Flows over an NACA-0012 Airfoil
Three different simulation approaches, namely unsteady Reynolds-averaged Navier-Stokes (URANS), delayed detached-eddy simulation (DDES), and wall-modeled large-eddy simulation (WMLES) are employed to simulate transonic flow over an NACA-0012 airfoil at different angles of attack covering pre- and post-buffet-onset regimes. The freestream Mach number is 0.75, and the Reynolds number based on the chord length is 10 million. These conditions are the same as the wind-tunnel experimental conditions of McDevitt and Okuno (1985). The NASA FUN3D solver is used for the simulations, which is an unstructured, compressible flow solver. The URANS simulations are performed using the Spalart-Allmaras (SA) model with the compressibility correction, the DDES predictions are based on the SA model, and the WMLES are performed using an equilibrium wall-model. The unsteady RANS simulations, only with the compressibility correction, predict the pre- and post- buffet characteristics, which compare well with the experimental results. DDES results predicted a lower buffet onset angle compared to experiment. The predicted shock locations are upstream of the locations predicted by URANS. Using a fine grid in the spanwise direction, WMLES predictions show buffeting consistent with the experiment.
Turbulence Simulations of Transonic Flows over an NACA-0012 Airfoil
Three different simulation approaches, namely unsteady Reynolds-averaged Navier-Stokes (URANS), delayed detached-eddy simulation (DDES), and wall-modeled large-eddy simulation (WMLES) are employed to simulate transonic flow over an NACA-0012 airfoil at different angles of attack covering pre-and post-buffet-onset regimes. The freestream Mach number is 0.75,and the Reynolds number based on the chord length is 10million. These conditions are the same as the wind-tunnel experimental conditions of McDevitt and Okuno (1985). The NASA FUN3D solver is used for the simulations, which is an unstructured, compressible flow solver. The URANS simulations are performed using the Spalart-Allmaras (SA) model with the compressibility correction, the DDES predictions are based on the SA model, and the WMLES are performed using an equilibrium wall-model. The unsteady RANS simulations, only with the compressibility correction, predict the pre-and post-buffet characteristics, which compare well with the experimental results. DDES results predicted a lower buffet onset angle compared to experiment. The predicted shock locations are upstream of the locations predicted by URANS. Using a fine grid in the span wise direction, WMLES predictions show buffeting consistent with the experiment.
Turbulence Simulations of Transonic Flows over an NACA-0012 Airfoil
Three different simulation approaches, namely unsteady Reynolds-averaged Navier-Stokes (URANS), delayed detached-eddy simulation (DDES),and wall-modeled large-eddy simulation (WMLES)are employed to simulate transonic flow over an NACA-0012 airfoil at different angles of attack covering pre- and post-buffet-onset regimes. The freestream Mach number is 0.75,and the Reynolds number based on the chord length is 10million. These conditions are the same as the wind-tunnel experimental conditions of McDevitt and Okuno (1985).1The NASA FUN3D solver is used for the simulations, which is an unstructured, compressible flow solver. The URANS simulations are performed using the Spalart-Allmaras (SA) model with the compressibility correction, the DDES predictions are based on the SA model, and the WMLES are performed using an equilibrium wall-model.The unsteady RANS simulations,only with the compressibility correction,predict the pre- and post-buffet characteristics,which compare well with the experimental results. DDES results predicted a lower buffet onset angle compared to experiment.The predicted shock locations are upstream of the locations predicted by URANS. Using a fine grid in the spanwise direction, WMLES predictions show buffeting consistent with the experiment
Simulations of Mach 0.8 Transonic Truss-Braced Wing Aircraft Aerodynamics at High Angles of Attack
This paper presents numerical simulations of the Mach 0.8 Transonic Truss-Braced Wing (TTBW) aircraft at high angle of attack using CFD solver FUN3D. Three different approaches - steady Reynolds Averaged Navier-Stokes (RANS), Unsteady Reynolds Averaged Navier-Stokes (URANS), and Delayed Detached Eddy Simulation (DDES) are used to simulation the TTBW aircraft at Mach number 0.8 and altitude 40, 000 ft. The pitch break phenomenon is observed at this flight condition. The simulation results show that the pitch break angle of attack occurs at 2.75˚ by DDES, 2.81◦ by URANS, and 2.89˚ by RANS. The DDES unsteady pressure begins to grow at angle of attack of 3.5˚, which might indicate the buffet onset, and rises one order of magnitude larger at angle of attack of 4.0˚. However, the URANS unsteady pressure begins to grow at angle of attack of 4.0˚. Both unsteady simulation results show the pitch break occurs before buffet onset.
Turbulence Simulations of Transonic Flows over NACA-0012 and OAT15A Airfoils
Three different simulation approaches, namely unsteady Reynolds-averaged Navier-Stokes (URANS), delayed detached-eddy simulation (DDES), and wall-modeled large-eddy simulation (WMLES) are employed to simulate transonic flow over an NACA-0012, OAT15A airfoils at different angles of attack covering pre- and post-buffet-onset regimes. The freestream Mach number is 0.75, and the Reynolds number based on the chord length is 10 and 3million, respectively. The NASA FUN3D solver is used for the simulations, which is an unstructured, compressible flow solver. The URANS simulations are performed using the Spalart-Allmaras (SA) model with the compressibility correction, the DDES predictions are based on the SA model, and the WMLES are performed using an equilibrium wall-model. The unsteady RANS simulations, only with the compressibility correction, predict the pre- and post- buffet characteristics, which compare well with the experimental results. DDES results predicted a lower buffet onset angle compared to experiment. The predicted shock locations are upstream of the locations predicted by URANS. Using a fine grid in the spanwise direction, WMLES predictions show buffeting consistent with the experiment.
Turbulence Simulations of Transonic Flows over NACA-0012 and OAT15A Airfoils
Three different simulation approaches, namely unsteady Reynolds-averaged Navier-Stokes (URANS), delayed detached-eddy simulation (DDES), and wall-modeled large-eddy simulation (WMLES) are employed to simulate transonic flow over an NACA-0012, OAT15A airfoils at different angles of attack covering pre- and post-buffet-onset regimes. The freestream Mach number is 0.75, and the Reynolds number based on the chord length is 10 and 3million, respectively. The NASA FUN3D solver is used for the simulations, which is an unstructured, compressible flow solver. The URANS simulations are performed using the Spalart-Allmaras (SA) model with the compressibility correction, the DDES predictions are based on the SA model, and the WMLES are performed using an equilibrium wall-model. The unsteady RANS simulations, only with the compressibility correction, predict the pre- and post- buffet characteristics, which compare well with the experimental results. DDES results predicted a lower buffet onset angle compared to experiment. The predicted shock locations are upstream of the locations predicted by URANS. Using a fine grid in the spanwise direction, WMLES predictions show buffeting consistent with the experiment.
Effect of Surface Roughness on Dynamic Stall in Pitching Motion
Dynamic stall plays a critical role in determining the performance and stability of a wide range of fluid-dynamics systems in various engineering applications. This unsteady aerodynamic phenomenon is particularly significant for maneuvering aircraft wings, jet aircraft subjected to gust encounters, helicopter rotor blades, and wind turbine blades [1–3]. The prediction of the dynamic stall vortex (DSV) is challenging due to factors such as unsteady aerodynamics, three-dimensional (3-D) effects, turbulence and flow separation, incoming gust, and surface impact effects [4–6]. Hence, advanced computational techniques and modeling approaches in computational fluid dynamics (CFD) would be required to enhance the accuracy and reliability of DSV predictions in dynamic stall scenarios. In the past, Batther and Lee [7] employed delayed detached eddy simulations (DDES) to understand the flow physics associated with the onset of dynamic stall. Their approach demonstrated that DDES achieves results comparable to those obtained from large-eddy simulations at a reduced computational cost. In another study, Khalifa et al. [8] examined the 3-D aspects of dynamic stall on a NACA 0012 airfoil using DES solvers. In conclusion, the findings underlined the superiority of 3-D simulations over two-dimensional approaches, particularly in predicting the lift coefficient values and capturing dynamic stall stages more precisely.
Time-Accurate Unsteady Pressure Loads Simulated for the Space Launch System at Wind Tunnel Conditions
A transonic flow field about a Space Launch System (SLS) configuration was simulated with the Fully Unstructured Three-Dimensional (FUN3D) computational fluid dynamics (CFD) code at wind tunnel conditions. Unsteady, time-accurate computations were performed using second-order Delayed Detached Eddy Simulation (DDES) for up to 1.5 physical seconds. The surface pressure time history was collected at 619 locations, 169 of which matched locations on a 2.5 percent wind tunnel model that was tested in the 11 ft. x 11 ft. test section of the NASA Ames Research Center's Unitary Plan Wind Tunnel. Comparisons between computation and experiment showed that the peak surface pressure RMS level occurs behind the forward attach hardware, and good agreement for frequency and power was obtained in this region. Computational domain, grid resolution, and time step sensitivity studies were performed. These included an investigation of pseudo-time sub-iteration convergence. Using these sensitivity studies and experimental data comparisons, a set of best practices to date have been established for FUN3D simulations for SLS launch vehicle analysis. To the author's knowledge, this is the first time DDES has been used in a systematic approach and establish simulation time needed, to analyze unsteady pressure loads on a space launch vehicle such as the NASA SLS.
Turbulence Model Validation through Joint Experimental / Computational Studies of Separated Flow Over a Three-Dimensional Tapered Bump: Part II - Computational Investigation
This report serves as documentation of activities related to Computational Fluid Dynamics (CFD) analysis applied to the Smooth Body Separation Experiment designed to produce validation-quality experimental data from wind-tunnel tests performed at the University of Notre Dame. All CFD analysis utilized the Boeing CFD (BCFD) flow solver. BCFD was used to analyze flow over the Boeing Speed Bump with second-order accuracy in space and time. Inviscid fluxes were computed using the HLLE scheme for steady-state simulations, and the Low-Dissipation Roe scheme for time-accurate simulations. The working fluid was modeled as calorically perfect air using Sutherland's law for viscosity. Thermal diffusivity was determined by assuming a constant molecular Prandtl number of 0.72, and a turbulent Prandtl number of 0.9. Reynolds Averaged Navier-Stokes (RANS) simulations were conducted using the one-equation Spalart-Allmaras turbulence model with rotation correction3 and the quadratic constitutive relation (SA-RC-QCR), as well as Menter's two-equation, Shear-Stress Transport (SST) model. Time-accurate computations were conducted using Delayed Detached eddy Simulation (DDES) with the one-equation RANS model, and Large Eddy Stress balance (LESb) with the two-equation RANS model. Implicit dual-time stepping was used to achieve second-order temporal accuracy. The experimental investigation conducted during the course of this study is described in detail in an accompanying report.
Effects of Aeroelasticity on Buffet Onset of Mach 0.8 Transonic Truss-Braced Wing
This paper presents a buffet simulation of the Mach 0.8 Transonic Truss-Braced Wing (TTBW) aeroelastic model using CFD solver FUN3D. A jig twist optimization is performed to obtain a flight optimized jig twist for the flexible Mach 0.8 TTBW model. A series of Delayed Detached Eddy Simulation (DDES) aeroelastic simulations has been conducted at Mach number 0.8 and altitude 40,000 ft using a structural damping of 2% to compare the pitch break prediction and the buffet behavior for the TTBW aeroelastic model. The unsteady pressure begins to grow at angle of attack of 4.0 ◦ , which might indicate the buffet onset, and rises one order of magnitude larger at angle of attack of 4.5 ◦ . In contrast, the buffet onset for the TTBW 1g model appears to be at the angle of attack of about 3.5 ◦ . The delay in the buffet onset of the aeroelastic model is due to the load relief from the aeroelastic effect.
MITR & NBSR DDE Irradiations in BR2 – Fluence in LEU Cladding and Structural Materials
The BR2 nuclear reactor is a material testing reactor (MTR) located in Mol, Belgium, and operated by the Belgian Nuclear Research Centre (SCK CEN) since 1963. The reactor is highly versatile as the number and location of fuel elements and control rods can change significantly from cycle to cycle to accommodate different needs. Argonne National Laboratory (ANL or Argonne) Reactor Conversion (RC) team has collaborated with SCK CEN for over a decade on the conversion of domestic and international research reactors from highly enriched uranium (HEU, ≥20 wt.% of 235 U) to low enriched uranium (LEU, <20 wt.% of 235 U) fuel. The U.S. High-Performance Research Reactor (USHPRR) project within the M3 Reactor Conversion Program aims at converting five U.S. high performance research reactors (MITR, MURR, NBSR, HFIR, and ATR) and one critical facility (ATR-C) to LEU fuel. These USHPRRs still use and regularly refuel with HEU fuel. Each facility has a unique reactor design, operating conditions, and fuel element design to accomplish its mission. The goal of the USHPRR project is to convert the USHPRRs and the critical facility to LEU fuel while maintaining experimental performance and ensuring safe facility operation. The current technical report focuses on two reactors requiring very high-density LEU fuel: the Massachusetts Institute of Technology Reactor (MITR) and the National Bureau of Standards Reactor (NBSR). To support the conversion of these reactors, so-called design demonstration elements (DDE) are planned to be irradiated in the BR2 reactor under conditions similar to the targeted reactors and using a prototypic geometry. In support of this experiment, SCK CEN studied and modeled the DDE irradiations using MCNP6.2 to investigate the feasibility of irradiating the MITR DDE and NBSR DDE in BR2. Argonne reviewed and confirmed the conclusions of this study. Structural analysis is another step toward converting USHPRR to LEU fuel. The objective of the current report is to provide information useful to the structural analysis of the NBSR & MITR DDEs to support its irradiation in BR2. Specifically, the goal is to provide the fast neutron (E>0.1MeV) fluence in the cladding of the fuel plates in BR2 for the whole period of irradiation (8 cycles for MITR DDE and 10 cycles for NBSR DDE). Additionally, fast neutron fluences in the side plates and in the NBSR DDE’s outside plates were calculated and reported. Neutronic calculations were performed using MCNP6.2 on the RTRHPC cluster.
Route Monopolie and Optimal Nonlinear Pricing
To cope with air traffic growth and congested airports, two solutions are apparent on the supply side: 1) use larger aircraft in the hub and spoke system; or 2) develop new routes through secondary airports. An enlarged route system through secondary airports may increase the proportion of route monopolies in the air transport market.The monopoly optimal non linear pricing policy is well known in the case of one dimension (one instrument, one characteristic) but not in the case of several dimensions. This paper explores the robustness of the one dimensional screening model with respect to increasing the number of instruments and the number of characteristics. The objective of this paper is then to link and fill the gap in both literatures. One of the merits of the screening model has been to show that a great varieD" of economic questions (non linear pricing, product line choice, auction design, income taxation, regulation...) could be handled within the same framework.VCe study a case of non linear pricing (2 instruments (2 routes on which the airline pro_ddes customers with services), 2 characteristics (demand of services on these routes) and two values per characteristic (low and high demand of services on these routes)) and we show that none of the conclusions of the one dimensional analysis remain valid. In particular, upward incentive compatibility constraint may be binding at the optimum. As a consequence, they may be distortion at the top of the distribution. In addition to this, we show that the optimal solution often requires a kind of form of bundling, we explain explicitly distortions and show that it is sometimes optimal for the monopolist to only produce one good (instead of two) or to exclude some buyers from the market. Actually, this means that the monopolist cannot fully apply his monopoly power and is better off selling both goods independently.We then define all the possible solutions in the case of a quadratic cost function for a uniform distribution of agent types and explain the implications for airlines in terms of service differentiation.