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This paper presents (1) a brief overview of mathematical models used in the numerical study of turbulent flows; (2) a k-epsilon model of turbulence; and (3) extensions of the k-epsilon model. These account for some of the effects of compressibility, low Reynolds number, streamline curvature, and preferential stress dissipation.
A robust, sufficiently accurate and practical hydrodynamic simulation toolset is required as a key component of the modeling and simulation of air-gap electrostatic discharge events. This work was performed to complement these ongoing efforts. In particular, hydrodynamic simulations must be vetted to ensure they are robust and sufficiently accurate over relevant characteristic scales. Verification models were generated in order to cultivate the technical knowledge and expertise needed to properly create, implement and execute numerical simulations. Furthermore, this effort was utilized extensively to educate students on the mathematical and numerical principles underlying hydrodynamic simulations. This education opportunity, provided in a holistic and rigorous manner, has greatly benefited developing scientists and engineers with the necessary understandings and toolsets required to excel at accomplishing the task at hand, and, more generally, it has enabled them to generate key programmatic deliverables. This report articulates several subtilties; specifically, how perturbations, nonlinear behavior, and dissipative mechanisms influence numerical stability, how to properly structure mathematical and numerical solutions, and how to properly generate error estimation/assignment. A more rigorous discussion of the consequences of such topics can be found in the body of this report in Chapters 2 and 3 with qualitative findings discussed in Chapter 4.
Mathematical and structural analysis for optimal control of aeroelasticity in unswept wings
Mathematical model and computerized simulation of flexible space vehicle response to docking impact
This paper describes a new separation cascade model that allows for arbitrary cascade connectivity and includes the effects of mixing, material losses and generalized separation element performance. The model also accommodates mixtures with an arbitrary number of component species. The mathematical and numerical framework of this model includes both time-dependent and steady state cascade simulation. Robust numerical approximations of the continuous mathematical model are accomplished using the Sundials suite of nonlinear algebraic and differential-algebraic equation solvers. Verification of the model is accomplished using manufactured solutions while validation is demonstrated by comparison to published experimental results and benchmarks.
The processes and mechanisms involved in the rotation and alignment of interstellar dust grains have been of great interest in astrophysics ever since the surprising discovery of the polarization of starlight more than half a century ago. Numerous theories, detailed mathematical models and numerical studies of grain rotation and alignment with respect to the Galactic magnetic field have been presented in the literature. In particular, the subject of grain rotation and alignment by radiative torques has been shown to be of particular interest in recent years. However, despite many investigations, a satisfactory theoretical understanding of the processes involved in grain rotation and alignment has not been achieved. As there appears to be no experimental data available on this subject, we have carried out some unique experiments to illuminate the processes involved in rotation of dust grains in the interstellar medium. In this paper we present the results of some preliminary laboratory experiments on the rotation of individual micron/submicron size nonspherical dust grains levitated in an electrodynamic balance evacuated to pressures of approximately 10(exp -3) to 10(exp -5) torr. The particles are illuminated by laser light at 5320 Angstroms, and the grain rotation rates are obtained by analyzing the low frequency (approximately 0-100 kHz) signal of the scattered light detected by a photodiode detector. The rotation rates are compared with simple theoretical models to retrieve some basic rotational parameters. The results are examined in the light of the current theories of alignment.
The processes and mechanisms involved in the rotation and alignment of interstellar dust grains have been of great interest in astrophysics ever since the surprising discovery of the polarization of starlight more than half a century ago. Numerous theories, detailed mathematical models and numerical studies of grain rotation and alignment along the Galactic magnetic field have been presented in the literature. In particular, the subject of grain rotation and alignment by radiative torques has been shown to be of particular interest in recent years. However, despite many investigations, a satisfactory theoretical understanding of the processes involved in grain rotation and alignment has not been achieved. As there appears to be no experimental data available on this subject, we have carried out some unique experiments to illuminate the processes involved in rotation of dust grains in the interstellar medium. In this paper we present the results of some preliminary laboratory experiments on the rotation of individual micron/submicron size nonspherical dust grains levitated in an electrodynamic balance evacuated to pressures of approx. 10(exp -3) to 10(exp -5) torr. The particles are illuminated by laser light at 5320 A, and the grain rotation rates are obtained by analyzing the low frequency (approx. 0-100 kHz) signal of the scattered light detected by a photodiode detector. The rotation rates are compared with simple theoretical models to retrieve some basic rotational parameters. The results are examined in the light of the current theories of alignment.
The processes and mechanisms involved in the rotation and alignment of interstellar dust grains have been of great interest in astrophysics ever since the surprising discovery of the polarization of starlight more than half a century ago. Numerous theories, detailed mathematical models, and numerical studies of grain rotation and alignment with respect to the Galactic magnetic field have been presented in the literature. In particular, the subject of grain rotation and alignment by radiative torques has been shown to be of particular interest in recent years. However, despite many investigations, a satisfactory theoretical understanding of the processes involved in subject, we have carried out some unique experiments to illuminate the processes involved in the rotation of dust grains in the interstellar medium. In this paper we present the results of some preliminary laboratory experiments on the rotation of individual micron/submicron-sized, nonspherical dust grains levitated in an electrodynamic balance evacuated to pressures of approximately 10(exp -3) to 10(exp -5) torr. The particles are illuminated by laser light at 5320 A, and the grain rotation rates are obtained by analyzing the low-frequency (approximately 0 - 100 kHz) signal of the scattered light detected by a photodiode detector. The rotation rates are compared with simple theoretical models to retrieve some basic rotational parameters. The results are examined in light of the current theories of alignment.
The history and advances of neutronics calculations at Los Alamos during the Manhattan Project through the present are reviewed. Substantial improvements to neutron diffusion methods and the invention of both the Monte Carlo neutron transport methods in 1947 and deterministic discrete ordinates Sn in 1953 were all made at Los Alamos just after the Manhattan Project. We briefly summarize early simpler and more approximate neutronics methods and then describe the need to better predict neutronics behavior through consideration of theoretical equations, models and algorithms, experimental measurements, and available computing capabilities and their limitations. This paper briefly covers key advances in deterministic methods during the Manhattan Project. These capabilities, coupled with increasing postwar defense needs and the invention of electronic computing with the Electronic Numeric Integrator and Computer, known as ENIAC, and the Mathematical Analyzer Numerical Integrator and Automatic Computer Model, known as MANIAC, led to the creation of Monte Carlo and deterministic discrete ordinates neutronics transport methods. We note the important role that the scientific comradery between the Los Alamos scientists played in the process. This paper briefly covers the early methods, algorithms, computers, and electronic and women pioneers that enabled Monte Carlo to spread to all areas of science. We focus heavily on these early developments and the subsequent creation of the MCNP® code, advances in its associated nuclear data, and its applications to problems of national defense at Los Alamos.
Poorer than expected surface reflectivity was observed in an early Tracking and Data Relay Satellite System antenna utilizing a tricot mesh weave. This poor reflectivity was determined to be caused by inadequate electrical contact at wire crossover points. A proper mathematical and numerical approach to assess the impact of wire junctions on reflectivity performance is developed. A mathematical method is presented for computing the surface reflectivity of complex mesh configurations like those on unfurlable-type spacecraft antennas. The method is based on the Floquet mode expansion to establish an integral equation for mesh wire currents. The equation is solved using the method of moments with triangular basis functions. It is observed that it is necessary to give special attention to the junction treatment among different branches of the mesh configurations. A vector junction current approach that resulted in satisfactory solutions for the current is described. The results of numerical simulations are compared against measured data and excellent agreement is observed.
When a suspension of bacterial cells of the species Bacillus subtilis is placed in a chamber with its upper surface open to the atmosphere complex bioconvection patterns are observed. These arise because the cells: (1) are denser than water; and (2) usually swim upwards, so that the density of an initially uniform suspension becomes greater at the top than the bottom. When the vertical density gradient becomes large enough, an overturning instability occurs which ultimately evolves into the observed patterns. The reason that the cells swim upwards is that they are aerotactic, i.e., they swim up gradients of oxygen, and they consume oxygen. These properties are incorporated in conservation equations for the cell (N) and oxygen (C) concentrations, and these are solved in the pre-instability phase of development when N and C depend only on the vertical coordinate and time. Numerical results are obtained for both shallow- and deep-layer chambers, which are intrinsically different and require different mathematical and numerical treatments. It is found that, for both shallow and deep chambers, a thin boundary layer, densely packed with cells, forms near the surface. Beneath this layer the suspension becomes severely depleted of cells. Furthermore, in the deep chamber cases, a discontinuity in the cell concentration arises between this cell-depleted region and a cell-rich region further below, where no significant oxygen concentration gradients develop before the oxygen is fully consumed. The results obtained from the model are in good qualitative agreement with the experimental observations.
The phenomenon of crack propagation is among the predominant modes of failure in many natural and engineering structures, often leading to severe loss of structural integrity and catastrophic failure. Thus, the ability to understand and a priori simulate the evolution of this failure mode has been one of the cornerstones of applied mechanics and structural engineering and is broadly referred to as "fracture mechanics." The work reported herein focuses on extending this understanding, in the context of through-thickness crack propagation in cohesive materials, through the development of a continuum-level multiscale numerical framework, which represents cracks as displacement discontinuities across a surface of zero measure. This report presents the relevant theory, mathematical framework, numerical modeling, and experimental investigations of through-thickness crack propagation in fiber-reinforced composites using the Variational Multiscale Cohesive Method (VMCM) developed by the authors.
We consider the inverse source problem in the parabolic equation, where the unknown source possesses the semi-discrete formulation. Theoretically, we prove that the flux data from any nonempty open subset of the boundary can uniquely determine the semi-discrete source. This means the observed area can be extremely small, and that is the reason we call it sparse boundary data. For the numerical reconstruction, we formulate the problem from the Bayesian sequential prediction perspective and conduct the numerical examples which estimate the space-time-dependent source state by state. To better demonstrate the method’s performance, we solve two common multiscale problems from two models with a long source sequence. The numerical results illustrate that the inversion is accurate and efficient.
Mathematical models and numerical methods are presented for computer modeling of aeroengine afterburners. A computer code GEMCHIP is described briefly. The algorithms SIMPLER, for gas flow predictions, and DROPLET, for droplet flow calculations, are incorporated in this code. The block correction technique is adopted to facilitate convergence. The method of handling irregular shapes of combustors and flameholders is described. The predicted results for a low-bypass-ratio turbofan afterburner in the cases of gaseous combustion and multiphase spray combustion are provided and analyzed, and engineering guides for afterburner optimization are presented.
Two-dimensional convective flows in shallow cavities with adiabatic.
Details are presented of the implementation of the new formulation into NASTRAN including descriptions of the DMAP statements required for conversion of the program and details pertaining to problem definition and bulk data considerations. Details of the current 1/8-scale space shuttle external tank mathematical model, numerical results and analysis/test comparisons are also presented. The appendices include a description and listing of a FORTRAN program used to develop harmonic transformation bulk data (multipoint constraint statements) and sample bulk data information for a number of hydroelastic problems.
The measurement of plasma electron velocity distribution functions as a problem in imaging and image reconstruction is considered. A model instrument that measures the integral of the distribution function along lines in velocity space is presented. This allows the use of the powerful mathematical and numerical methods that have recently been so successful in other areas of imaging. It is found that this approach leads to classes of instruments that are qualitatively different from contemporary designs. An investigation of different methods of reconstruction of the distribution function from integral measurements reveals that the mathematical tools appropriate to one particular imaging problem may be very different from those required to deal with another.
The study will develop mathematical and numerical models of the turbulent ignition process that comprises plasma species interaction with turbulence and principles of ignition kernel growth or extinction based on a flamelet modeling approach.