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DOE OSTI2023

Consequences and benefits of utilizing continuous vibro-impact representations in constrained pipeline conveying fluid systems

Here, the effectiveness of continuous vibro-impact forcing representations for the cantilevered pipe that conveys fluid is explored and analyzed. The previously accepted forcing model utilizing a smoothened trilinear spring is estimated using three continuous forcing representations, namely, polynomial, rational polynomial, and hyperbolic tangent. The accuracy of the estimated forcing functions is investigated and analyzed by calculating the root mean square error, and bifurcation diagrams are generated and compared to the nominal system. Additionally, the dynamic response of the system is further characterized using Poincare maps, power spectra, and basins of attraction. Once all continuous forcing representations are analyzed and compared to the nominal system, the computational cost of each method is examined, and further limitations of the hyperbolic tangent method are discovered. It is proved that the hyperbolic tangent forcing representation most accurately captures the dynamic response of the pipeline, and the least accurate representation is the rational polynomial representation. Additionally, considerable computational cost is saved when employing the hyperbolic tangent representation compared to the discontinuous representation.

42 ENGINEERING↗

NASA NTRS1972

Planetary mission handbook. Appendix A: Graphical trajectory data

Graphical data is presented for Type 1 ballistic transfers to Jupiter (1974 to 1986), Saturn (1976 to 1986) and Uranus (1985). There are four graphs for each launch opportunity, namely the hyperbolic excess speed at earth departure, the hyperbolic excess speed at planet arrival, the declination of the hyperbolic departure asymptote, and the declination of the hyperbolic approach asymptote.

Source record↗

NASA NTRS1994

Preliminary estimates of Gulf Stream characteristics from TOPEX data and a precise gravimetric geoid

TOPEX sea surface height data has been used, with a gravimetric geoid, to calculate sea surface topography across the Gulf Stream. This topography was initially computed for nine tracks on cycles 21 to 29. Due to inaccurate geoid undulations on one track, results for eight tracks are reported. The sea surface topography estimates were used to calculate parameters that describe Gulf Stream characteristics from two models of the Gulf Stream. One model was based on a Gaussian representation of the velocity while the other was a hyperbolic representation of velocity or the sea surface topography. The parameters of the Gaussian velocity model fit were a width parameter, a maximum velocity value, and the location of the maximum velocity. The parameters of the hyperbolic sea surface topography model were the width, the height jump, position, and sea surface topography at the center of the stream. Both models were used for the eight tracks and nine cycles studied. Comparisons were made between the width parameters, the maximum velocities, and the height jumps. Some of the parameter estimates were found to be highly (0.9) correlated when the hyperbolic sea surface topography fit was carried out, but such correlations were reduced for either the Gaussian velocity fits or the hyperbolic velocity model fit. A comparison of the parameters derived from 1-year TOPEX data showed good agreement with values derived by Kelly (1991) using 2.5 years of Geosat data near 38 deg N, 66 deg W longitude. Accuracy of the geoid undulations used in the calculations was of order of +/- 16 cm with the accuracy of a geoid undulation difference equal to +/- 15 cm over a 100-km line in areas with good terrestrial data coverage. This paper demonstrates that our knowledge or geoid undulations and undulation differences, in a portion of the Gulf Stream region, is sufficiently accurate to determine characteristics of the jet when used with TOPEX altimeter data. The method used here has not been shown to be more accurate than methods that average altimeter data to form a reference surface used in analysis to obtain the Gulf Stream characteristics. However, the results show the geoid approach may be used in areas where lack of current meandering reduces the accuracy of the average surface procedure.

Rapp, Richard H.↗

NASA NTRS2024

Aerocapture: An Enabling Technology for Flagship-Class Uranus Orbiter and Probe Mission

Introduction: The current decadal survey published by the National Academies of Sciences has informed National Aeronautics and Space Administration (NASA) to prioritize the study of the Ice Giants, especially Uranus. To gather the required data that addresses the science questions raised in this survey, a mission to Uranus with an orbiter and atmospheric probe must be designed. The Uranus Orbiter and Probe (UOP) study, which the survey identified as the flagship mission of this decade, proposes a 2031 launch to take advantage of a Jupiter fly-by and utilizes a fully propulsive orbit insertion design with an Earth-to-Uranus transit times ranging from 13 to 15 years. This fully propulsive orbit insertion at Uranus will be very fuel expensive (wet mass percentages of around 60-70\%) thereby leaving less mass for the scientific payload and additional planetary probes. In addition, scientists are more interested in visiting Uranus before 2049, when the Spring Equinox will occur, as it allows studying Uranus seasons not seen during Voyager 2's flyby in 1987. A NASA Flagship-Class mission would require at least 10 years of lead time prior to launch thereby making the 2031 launch to take advantage of the Jupiter fly-by extremely challenging. The consequence of missing the Jupiter fly-by and launching in the late 2030s is the challenge of a fully-propulsive mission like UOP to have a feasible alternative interplanetary trajectory that reaches Uranus before 2050. As an alternative, to address the shortcomings of the fully propulsive mission, one can design a mission to Uranus using aerocapture. What is aerocapture: Aerocapture is an atmospheric maneuver that uses aerodynamic forces generated by flight within a planetary atmosphere to decelerate and achieve orbit insertion. Using aerocapture allows one to provide the change in velocity (Delta V) needed to slow down from the approach hyperbolic trajectory to achieve the desired captured orbit around the target planet using aerodynamic forces of the vehicle (lift and drag). Using aerodynamic forces instead of fully-propulsive maneuvers results in significant savings in the fuel. Furthermore, aerocapture can also allow one to consider interplanetary trajectories with higher approach hyperbolic velocities, thus reducing the mission transit times. Aerocapture as an enabling technology: To use aerocapture as an enabling technology for the Uranus exploration mission, one would require an integrated system-level design, including a Thermal Protection System (TPS), hardware needed for aerodynamic modulation, and autonomous Guidance, Navigation, and Control (GNC) systems. Aerocapture has yet to be demonstrated, despite considering it for several past missions. Recent advancements in TPS and GNC capabilities show the potential for using rigid, heritage entry vehicle configurations already flown at other planetary bodies for Uranus aerocapture. Aerocapture can be a robust technology that can deliver spacecraft to Uranus science orbits while substantially increasing on-orbit payload mass (more than 40\%) that can enable robust atmospheric entry probes. Additionally, the aerocapture maneuver would reduce the interplanetary transit time by 2–5 years (15-30\%) relative to fully-propulsive orbit insertion. Recent work has shown that one can conduct a flagship-class mission in a shorter period than fully-propulsive missions if using aerocapture. What does aerocapture bring in for a Uranus Mission: Using aerocapture for a Uranus orbiter and atmospheric probe mission can provide one with considerable propellant savings. Spacecraft in interplanetary trajectories to Uranus typically need an Delta V in orders of kilometers per second to insert into science orbit. One would require thousands of kilograms of fuel to achieve such a Delta V using a traditional fully propulsive maneuver, leaving less mass for payload during the mission launch. Aerocapture can reduce the propulsion needs by dissipating energy in the sizable atmosphere of Uranus without a significant mass increase due to the need for an aeroshell. One can use the mass savings achieved using the aerocapture to reduce the launch vehicle requirements. In addition, one can also have additional science instruments on the orbiter or create a robust instrumentation suite on an atmospheric probe that can significantly increase the science outcome of the Uranus exploration mission. Furthermore, since aerocapture performance is relatively insensitive to increases in hyperbolic excess velocity, one can design the interplanetary trajectory to arrive at Uranus faster, reducing the interplanetary transit time and operations cost. All these savings, achieved using aerocapture, could help fit a larger class mission, such as the Uranus mission within a smaller capital, e.g., a Flagship-class orbiter mission in a New Frontiers class capital. Summary: This talk will provide an overview of how aerocapture can enable the Uranus exploration mission. Specifically, this talk will discuss the latest advancements made in the Uranus aerocapture study, such as investigating interplanetary trajectories with higher hyperbolic approach velocities and their implications on the aero heating and the TPS design, incorporating FNPAG (an advanced numerical-predictor guidance) and comparison of multiple navigation approaches. In addition, this talk will focus on mechanical design that can house more than one atmospheric probe and the six degrees of freedom simulation of aerocapture at Uranus. Findings from a recent NASA Space Technology Mission Directorate (STMD)-funded activity studying the aerocapture as an enabling technology for a Uranus orbiter will be presented. Using the science payload recommended by the Decadal Survey for Uranus exploration, this work shows many improvements over the baseline fully-propulsive mission. These improvements include a shorter cruise phase, flexibility in launch opportunities late into the 2030s while reaching Uranus before the 2050 equinox for the desired science opportunities, and lower propellant mass needs. This talk will highlight how aerocapture can be utilized for Uranus science orbit insertion using a lower-risk, heritage entry vehicle configuration used extensively as a Mars entry, descent, and landing vehicle. Furthermore, this talk will explore how the demonstration of aerocapture at Earth can benefit the aerocapture-enabled Uranus mission.

Aerocapture↗

DOE OSTI2025

The tensor-train stochastic finite volume method for uncertainty quantification

The stochastic finite volume method offers an efficient one-pass approach for assessing uncertainty in hyperbolic conservation laws. Still, it struggles with the curse of dimensionality when dealing with multiple stochastic variables. Here, we introduce the stochastic finite volume method within the tensor-train framework to counteract this limitation. This integration, however, comes with its own set of difficulties, mainly due to the propensity for shock formation in hyperbolic systems. To overcome these issues, we have developed a tensor-train-adapted stochastic finite volume method that employs a global WENO reconstruction, making it suitable for such complex systems. This approach represents the first step in designing tensor-train techniques for hyperbolic systems and conservation laws involving shocks.

97 MATHEMATICS AND COMPUTING↗

NASA NTRSDate not supplied

Finite-Volume Diffusion Schemes for Svard's Eulerian Governing Equations

This paper, contains discussion on the implementation of new diffusion schemes for Svard’s Eulerian flow (EF) governing equations. It contains an alpha-damping type scheme and a new hyperbolic Eulerian flow (HEF) discretization paralleling previous work on hyperbolic Navier-Stokes (HNS). It shows these equations have simplified hyperbolic discretizations due to the simplicity of the new stress tensor and compares these schemes to established Navier-Stokes (NS) discretizations to verify the predictive utilities of such diffusion schemes.

Computational Fluid Dynamics↗

NASA NTRS2024

Finite-Volume Diffusion Schemes for Svard's Eulerian Governing Equations

Computational Fluid Dynamics↗

DOE OSTI2023

Causal explicit algorithm for heat conduction in a plasma

Hyperbolic heat conduction extends standard Spitzer-Harm heat conduction by including a term proportional to the time derivative of the heat flux. The new term arises from a kinetic derivation of the heat flux that includes higher order corrections. Here we present a causal explicit numerical algorithm for solving the nonlinear hyperbolic heat conduction equation in an unmagnetized plasma. The maximum stable timestep for the causal explicit algorithm scales linearly with the cell size, owing to the hyperbolic nature of the problem. This is in contrast to the quadratic scaling of the maximum stable timestep with the cell size for the parabolic forward time centered space algorithm. The favorable scaling of the timestep with the cell size enables a practical explicit implementation of heat conduction in high-performance massively parallel plasma codes. In particular, we have implemented the causal explicit algorithm in the laser plasma interaction code pF3D. We verify the CE algorithm and analyze its convergence rate by simulating a harmonic mode, which has an analytic solution within the context of the HHC model. We also compare simulations using the CE algorithm to those using the forward time centered space algorithm on a pair of test problems: evolution in time of a Gaussian temperature perturbation in a uniform plasma and heat transport in the presence of inverse bremsstrahlung heating by a Gaussian laser speckle.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2020

Overset meshes for incompressible flows: On preserving accuracy of underlying discretizations

This study on overset meshes for incompressible-flow simulations is motivated by accurate prediction of wind farm aerodynamics involving large motions and deformations of components with complex geometry. Specifically, using first-order hyperbolic and elliptic equation proxies for the incompressible Navier-Stokes (NS) equations, we investigate the influence of information exchange between overset meshes on numerical performance where the underlying discretization is second-order accurate. The first aspect of information exchange surrounds interpolation of solution where we examine Lagrange and point-cloud-based interpolation for creating constraint equations between overset meshes. To maintain overall second-order accuracy, higher-order interpolation is required for elliptic problems, but linear interpolation is sufficient for hyperbolic problems in first-order form. Higher-order point-cloud-based interpolation provides a pathway to maintaining accuracy in unstructured meshes, but at higher complexity. The second aspect of information exchange focuses on comparing the approaches of overset single system (OSS) and overset Additive Schwarz (OAS) for coupling the linear systems of the overlapping meshes. While the former involves a single linear system, in the latter the discrete linear systems are solved separately, and solving the global system is accomplished through outer iterations and sequential information exchange in a Jacobi fashion. For the test cases studied, accuracy for hyperbolic problems is maintained by performing two outer iterations, whereas many outer iterations are required for elliptic systems. The order-of-accuracy studies explored here are critical for verifying the overset-mesh coupling algorithms used in engineering simulations. Accuracy of these simulations themselves is, however, quantified using engineering quantities of interest such as drag, power, etc. Consequently, we conclude with numerical experiments using NS equations for incompressible flows where we show that linear interpolation and few outer iterations are sufficient for achieving asymptotic convergence of engineering quantities of interest.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

Cavity-altered superconductivity

Is it feasible to alter the ground-state properties of a material by engineering its electromagnetic environment? Inspired by theoretical predictions, experimental realizations of such cavity-controlled properties without optical excitation are beginning to emerge. Here we devised and implemented a new platform to realize cavity-altered materials. Single crystals of hyperbolic van der Waals (vdW) compounds provide a resonant electromagnetic environment with enhanced density of photonic states and prominent mode confinement. We interfaced hexagonal boron nitride (hBN) with the molecular superconductor κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Br (κ-ET). The frequencies of infrared hyperbolic modes (HMs) of hBN match the infrared-active carbon–carbon (C=C) stretching molecular resonance of κ-ET implicated in superconductivity. Nano-optical data supported by first-principles molecular Langevin dynamics simulations confirm the presence of resonant coupling between the hBN hyperbolic cavity modes and the C=C stretching mode in κ-ET. Meissner-effect measurements using magnetic force microscopy (MFM) demonstrate a strong suppression of superfluid density near the hBN/κ-ET interface. Non-resonant control heterostructures, including RuCl 3 /κ-ET and hBN/Bi 2 Sr 2 CaCu 2 O 8+x (BSCCO), do not show the pronounced superfluid suppression. These observations suggest that hBN/κ-ET realizes a cavity-altered superconducting ground state. Our work highlights the potential of dark cavities devoid of external photons for engineering electronic ground-state properties of complex quantum materials.

36 MATERIALS SCIENCE↗

DOE OSTI2023

Asymptotic vacuum solution at tokamak X-point tip

In the H-mode regime of diverted tokamaks, the presence of strong pressure gradients in the pedestal gives rise to a sizable bootstrap current, together with the Ohmic and Pfirsch–Schlueter currents, close to the separatrix. For such equilibria, the presence of finite current density close to the separatrix requires the reexamination of equilibrium properties. It is almost universally assumed that the two branches of the separatrix (the stable and unstable manifolds) are straight as they cross at the X-point. However, the opposite angles of the plasma-filled segment and vacuum one cannot be equal if the current density does not vanish at the separatrix on the plasma side. We solve this difficulty by chipping off a thin layer of plasma edge so that the sharp corner of the plasma-filled segment becomes a hyperbola. Using the conformal transformation, we found that in the assumption of a hyperbolic boundary, the X point moves beyond the plasma boundary to fall in the vacuum region. An acute angle of the plasma-filled segment leads to an obtuse opposite angle of vacuum segment and vice versa. In the case of an acute angle of the plasma-filled segment, the new X point shifts inside the X point formed by the asymptotes of a hyperbolic boundary; in the case of an obtuse angle of the plasma-filled segment, the new X point shifts outside the X point formed by the asymptotes of a hyperbolic plasma boundary. Furthermore, the results are important for understanding the X point features, which affect the tokamak edge stability and transport.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2024

Automorphic spectra and the conformal bootstrap

We describe a new method for constraining Laplacian spectra of hyperbolic surfaces and 2-orbifolds. The main ingredient is consistency of the spectral decomposition of integrals of products of four automorphic forms. Using a combination of representation theory of PSL 2 (R) and semidefinite programming, the method yields rigorous upper bounds on the Laplacian spectral gap. In several examples, the bound is nearly sharp. For instance, our bound on all genus-2 surfaces is λ 1 ≤ 3.8388976481, while the Bolza surface has λ 1 ≈ 3.838887258. The bounds also allow us to determine the set of spectral gaps attained by all hyperbolic 2-orbifolds. Our methods can be generalized to higher-dimensional hyperbolic manifolds and to yield stronger bounds in the two-dimensional case. The ideas were closely inspired by modern conformal bootstrap.

Kravchuk, Petr↗

DOE OSTI2024

Shearless effective barriers to chaotic transport induced by even twin islands in nontwist systems

For several decades now it has been known that systems with shearless invariant tori, nontwist Hamiltonian systems, possess barriers to chaotic transport. These barriers are resilient to breakage under perturbation and therefore regions where they occur are natural places to look for barriers to transport. Here we describe a kind of effective barrier that persists after the shearless torus is broken. Because phenomena are generic, for convenience we study the standard nontwist map (SNM), an area-preserving map that violates the twist condition locally in the phase space. The barrier occurs in nontwist systems when twin even period islands are present, which happens for a broad range of parameter values in the SNM. With a phase space composed of regular and irregular orbits, the movement of chaotic trajectories is hampered by the existence of shearless curves, total barriers, and a network of partial barriers formed by the stable and unstable manifolds of the hyperbolic points. Being a degenerate system, the SNM has twin islands and, consequently, twin hyperbolic points. We show that the structures formed by the manifolds intrinsically depend on period parity of the twin islands. For this even scenario the structure that we call a torus free barrier occurs because the manifolds of different hyperbolic points form an intricate chain atop a dipole configuration and the transport of chaotic trajectories through the chain becomes a rare event. This structure impacts the emergence of transport, the escape basin for chaotic trajectories, the transport mechanism, and the chaotic saddle. The case of odd periodic orbits is different: we find for this case the emergence of transport immediately after the breakup of the last invariant curve, and this leads to a scenario of higher transport, with intricate escape basin boundary and a chaotic saddle with nonuniformly distributed points.

classical mechanics↗

DOE OSTI2020

Modal analysis of the behavior of inertial particles in turbulence subjected to Stokes drag

The clustering of small heavy inertial particles subjected to Stokes drag in turbulence, which is the concern of present study, is known to be minimal at small and large Stokes number and substantial at St = $\textit{O}$(1). This nonmonotonic trend, which has been shown computationally and experimentally, is yet to be explained analytically. In this study, we obtain an analytical expression for the Lyapunov exponents that quantitatively predicts this trend. The sum of the exponents, which is the normalized rate of change of the signed volume of a small cloud of particles, is correctly predicted to be negative and positive at small and large Stokes numbers, respectively, asymptoting to $\textit{τQ}$ as $\textit{τ}$ → 0 and $τ^{–1/2}|Q|^{1/4}$ as $\textit{τ} → ∞$, where $\textit{τ}$ is the particle relaxation time and $\textit{Q(τ)}$ is the difference between the norm of the rotation- and strain-rate tensors computed along the particle trajectory, which remains negative at all $\textit{τ}$ in turbulent flows. Additionally, the trajectory crossing is predicted only in hyperbolic flows where Q < 0 for sufficiently inertial particles with a $\textit{τ}$ that scales with $|Q|^{–1/2}$. Following the onset of crossovers, a transition from clustering to dispersion is predicted correctly. We show these behaviors are not unique to three-dimensional isotropic turbulence and can be reproduced closely by a one-dimensional mono-harmonic flow, which appears as a fundamental canonical problem in the study of particle clustering. Analysis of this one-dimensional canonical flow shows that the rate of clustering, quantified as the product of the Lyapunov exponent and particle relaxation time, is bounded by –1/2, behaving with extreme nonlinearity in the hyperbolic flows and always remaining positive in the elliptic flows. Finally, this model problem also confirms that dispersion in a hyperbolic flow will occur only if particles' trajectory cross. These findings, which stem from our analysis, are corroborated by the direct numerical simulations.

42 ENGINEERING↗

NASA NTRS1980

Material capture by double lunar gravity assist

The equations yielding the performance of a single lunar flyby in removing incoming hyperbolic excess velocity to capture payloads on interplanetary trajectories are briefly derived. The impossibility of using a single lunar flyby to capture a body entering the earth-moon system with a hyperbolic velocity in excess of about 1.9 km/s is discussed, and a method of using a double flyby of the moon to significantly improve this performance is developed. The equations for achieving a double lunar flyby are derived by solving the orbital equations and Lambert's problem both for the incoming trajectory in the plane of the moon's orbit and for arbitrary declination. For the in-plane case it is shown that the maximum removable hyperbolic excess velocity is 2.2687 km/s. For the inclined case, it is shown that the use of a double lunar flyby allows capture for declinations in excess of 54 degrees, and that for declinations less than 38 degrees the double lunar flyby offers better performance than the single lunar flyby.

Ross, D. J.↗

NASA NTRS1986

Transformation of two and three-dimensional regions by elliptic systems

Efforts in transferring computational work from the LRC computer to the IRIS Graphics Workstation at MSU are reported and the computation of a conservative solution of a simple hyperbolic equation on an overlapping grid is discussed. Several conclusions concerning computations on overlapping grids are apparent. Problems only occur when there is a major difference in grid spacing on the individual component grids. In the case of hyperbolic equations, it is necessary that both interpolation and extrapolation be applied at the grid boundaries. When interpolated values are used at outflow boundary points, excessive oscillations in the numerical solution may be the result. The same conclusions would be valid for more complicated systems of hyperbolic equations such as the Euler equations for inviscid flow. Some of the solution values would be extrapolated at the overlap boundary, the exact number depending on the number of characteristics pointing out of the overlap region. It is also possible that similar boundary conditions may be needed for some parabolic equations such as high Reynolds number viscous flow equations. Efforts were also expended on the development of three-dimensional conservative interpolation procedures. Finally, the investigation of grid smoothing procedures were initiated during this reporting period. It was decided that the first grid smoothing algorithms will be based on the concepts of variational grid generation.

Mastin, C. W.↗

NASA NTRS1993

Tsien's method for generating non-Keplerian trajectories. Part 2: The question of thrust to orbit a sphere and the restricted three-body problem

Tsien's method is extended to treat the orbital motion of a body undergoing accelerations and decelerations. A generalized solution is discussed for the generalized case where a body undergoes azimuthal and radial thrust and the problem is further simplified for azimuthal thrust alone. Judicious selection of thrust could generate either an elliptic or hyperbolic trajectory. This is unexpected especially when the body has only enough energy for a lower state trajectory. The methodology is extended treating the problem of vehicle thrust for orbiting a sphere and vehicle thrust within the classical restricted three-body problem. Results for the latter situation can produce hyperbolic trajectories through eigen value decomposition. Since eigen values for no-thrust can be imaginary, thrust can generate real eigen values to describe hyperbolic trajectories. Keplerian dynamics appears to represent but a small subset of a much larger non-Keplerian domain especially when thrust effects are considered. The need for high thrust long duration space-based propulsion systems for changing a trajectory's canonical form is clearly demonstrated.

Murad, P. A.↗

NASA NTRS1993

Semiclassical approximation for electron impact excitation of hydrogenic ions

The threshold behavior of the electron impact excitation cross sections for hydrogenic ions is investigated using the semiclassical approximation with the hyperbolic orbit for the projectile path, rather than the straight line path. The symmetric approximation is applied to modify the ordinary hyperbolic orbit. The modification factor due to the hyperbolic orbit approximation produces the correct energy dependence of the cross section near the excitation threshold. This result is very similar to that of the quantum mechanical case. The semiclassical enhancement factor due to this simple modification corresponds to the Coulomb focusing factor in the Born-Bethe approximation. In the high-energy limit, the semiclassical cross sections approach the Born-Bethe cross sections, with a finite cutoff in the momentum transfer for dipole transitions.

Jung, Young-Dae↗