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NASA NTRS1992

Runge-Kutta methods combined with compact difference schemes for the unsteady Euler equations

Recent development using compact difference schemes to solve the Navier-Stokes equations show spectral-like accuracy. A study was made of the numerical characteristics of various combinations of the Runge-Kutta (RK) methods and compact difference schemes to calculate the unsteady Euler equations. The accuracy of finite difference schemes is assessed based on the evaluations of dissipative error. The objectives are reducing the numerical damping and, at the same time, preserving numerical stability. While this approach has tremendous success solving steady flows, numerical characteristics of unsteady calculations remain largely unclear. For unsteady flows, in addition to the dissipative errors, phase velocity and harmonic content of the numerical results are of concern. As a result of the discretization procedure, the simulated unsteady flow motions actually propagate in a dispersive numerical medium. Consequently, the dispersion characteristics of the numerical schemes which relate the phase velocity and wave number may greatly impact the numerical accuracy. The aim is to assess the numerical accuracy of the simulated results. To this end, the Fourier analysis is to provide the dispersive correlations of various numerical schemes. First, a detailed investigation of the existing RK methods is carried out. A generalized form of an N-step RK method is derived. With this generalized form, the criteria are derived for the three and four-step RK methods to be third and fourth-order time accurate for the non-linear equations, e.g., flow equations. These criteria are then applied to commonly used RK methods such as Jameson's 3-step and 4-step schemes and Wray's algorithm to identify the accuracy of the methods. For the spatial discretization, compact difference schemes are presented. The schemes are formulated in the operator-type to render themselves suitable for the Fourier analyses. The performance of the numerical methods is shown by numerical examples. These examples are detailed. described. The third case is a two-dimensional simulation of a Lamb vortex in an uniform flow. This calculation provides a realistic assessment of various finite difference schemes in terms of the conservation of the vortex strength and the harmonic content after travelling a substantial distance. The numerical implementation of Giles' non-refelctive equations coupled with the characteristic equations as the boundary condition is discussed in detail. Finally, the single vortex calculation is extended to simulate vortex pairing. For the distance between two vortices less than a threshold value, numerical results show crisp resolution of the vortex merging.

Yu, Sheng-Tao↗

NASA NTRS1991

Calculations of separated 3-D flows with a pressure-staggered Navier-Stokes equations solver

A Navier-Stokes equations solver based on a pressure correction method with a pressure-staggered mesh and calculations of separated three-dimensional flows are presented. It is shown that the velocity pressure decoupling, which occurs when various pressure correction algorithms are used for pressure-staggered meshes, is caused by the ill-conditioned discrete pressure correction equation. The use of a partial differential equation for the incremental pressure eliminates the velocity pressure decoupling mechanism by itself and yields accurate numerical results. Example flows considered are a three-dimensional lid driven cavity flow and a laminar flow through a 90 degree bend square duct. For the lid driven cavity flow, the present numerical results compare more favorably with the measured data than those obtained using a formally third order accurate quadratic upwind interpolation scheme. For the curved duct flow, the present numerical method yields a grid independent solution with a very small number of grid points. The calculated velocity profiles are in good agreement with the measured data.

Kim, S.-W.↗

NASA NTRS1991

Calculations of separated 3-D flows with a pressure-staggered Navier-Stokes equations solver

Kim, S.-W.↗

NASA NTRS1975

Theoretical study of lift-generated vortex wakes designed to avoid rollup

Two hypothetical vortex wakes are introduced and studied theoretically to explore whether the rollup of lift-generated vortex sheets can be suppressed. The circulation distribution across each wake is specified such that one rotates and the other translates as a unit due to their self-induced velocities. Several span loadings are constructed from these solutions and the resulting inviscid wake structure is computed for several span lengths behind the generating wing by use of the discrete vortex method wherein the vortex wake is represented by an array of vortices. The final distribution of vortices is then used to estimate the rolling moment on an encountering wing. It is found that, even though the initial specified motions are not sustained, substantial reductions in rolling moment are predicted for certain ranges of the ratio of the span of the generating wing to the following wing.

Rossow, V. J.↗

NASA NTRS1980

Steady, Oscillatory, and Unsteady Subsonic and Supersonic Aerodynamics, production version (SOUSSA-P 1.1). Volume 1: Theoretical manual

Recent developments of the Green's function method and the computer program SOUSSA (Steady, Oscillatory, and Unsteady Subsonic and Supersonic Aerodynamics) are reviewed and summarized. Applying the Green's function method to the fully unsteady (transient) potential equation yields an integro-differential-delay equation. With spatial discretization by the finite-element method, this equation is approximated by a set of differential-delay equations in time. Time solution by Laplace transform yields a matrix relating the velocity potential to the normal wash. Premultiplying and postmultiplying by the matrices relating generalized forces to the potential and the normal wash to the generalized coordinates one obtains the matrix of the generalized aerodynamic forces. The frequency and mode-shape dependence of this matrix makes the program SOUSSA useful for multiple frequency and repeated mode-shape evaluations.

Morino, L.↗

NASA NTRS1995

Sensitivity analysis for the dynamic response of viscoplastic shells of revolution

A computational procedure is presented for evaluating the sensitivity coefficients of the dynamic axisymmetric response of viscoplastic shells of revolution. The analytical formulation is based on Reissner's large deformation shell theory with the effects of transverse shear deformation, rotatory inertia and moments turning around the normal to the middle surface included. The material model is chosen to be isothermal viscoplasticity, and an associated flow rule is used with a von Mises effective stress. A mixed formulation is used with the fundamental unknowns consisting of six stress resultants, three generalized displacements and three velocity components. Spatial discretization is performed using finite elements, with discontinuous stress resultants across element interfaces. The temporal integration is performed by using an explicit central difference scheme (leap-frog method) with an implicit constitutive update. The sensitivity coefficients are evaluated using a direct differentiation approach. Numerical results are presented for a spherical cap subjected to step loading, and a circular plate subjected to impulsive loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to thickness, mass density, Young's modulus, and two of the material parameters characterizing the viscoplastic response. Time histories of the response and sensitivity coefficients are presented, along with spatial distributions of these quantities at selected times.

Kulkarni, Makarand↗

DOE OSTI2023

A Conservative Finite-Volume Based Interface-Tracking Algorithm Using the Signed Distance Function

Methods for tracking an interface between two fluid phases are developed to ensure desired fluid properties, conservation, and stability are preserved in a finitevolume (FV) discretization. Here, the interface is tracked using a level set method where the signed distance function implicitly defines the interface. Marching methods are used to evaluate the value of the signed distance function, including a novel initialization method to initialize any implicit function to the signed distance function around sharp corners in the level set. Global conservation and consistency with a set of governing equations is enforced by a compression coefficient that measures the volumetric compression or expansion due to inaccuracies in the level set evaluation. A redistribution method is integrated into the volume correction to eliminate the small-cell instability while maintaining global conservation. This suite of methods is implemented and tested using static uniform velocity, and potential flow cases with multiple interface geometries. Results show these methods achieve up to second order accuracy, and are conservative. The application for these methods is intended to track the interface of a 3D printing filament in a finite-volume discretization of the all-speed Navier-Stokes equations.

42 ENGINEERING↗

DOE OSTI2021

Discrete stochastic model of point defect-dislocation interaction for simulating dislocation climb

Dislocation climb is an important high temperature process in metals plasticity, responsible for the phenomena such as creep, swelling, or hardening. Climb is defined by the ability of dislocations to leave their original glide plane by interacting with point defects. As such, dislocation climb is controlled by point defect diffusion/absorption/emission, all of which involve thermal activation. The existing thermodynamically consistent models for climb are generally formulated in a continuum framework, through the definition of effective defect fluxes and climb propensities in response to thermodynamic driving forces. However, the point-wise discrete nature of vacancies (and/or self-interstitials) confers a highly discrete nature to the climb dynamics, which is also strongly affected by elastic forces. The combination of discreteness, thermal activation, and elasticity makes this process too challenging for direct atomistic methods such as molecular dynamics. Here we develop a kinetic Monte Carlo model that captures vacancy generation and transport kinetics acting in conjuction with the evolving elastic fields provided by discrete dislocation dynamics simulations. The two models are coupled via the applied stresses and stress gradients generated by dislocation structures at vacancy locations. Our simulations reveal two surprising results. First that climb is dominated by vacancy emission even when the background vacancy concentration is much higher than the equilibrium one. And, second, that climb velocities might be much faster than otherwise believed when one uses the classical theories of climb. These effects are due to the locality of vacancy-dislocation processes, which are not captured in classical treatments that assume smooth vacancy fluxes and homogeneous concentrations. We apply the method to study elementary climb processes in body-centered cubic iron and furnish climb mobility functions to be used in parametric dislocation dynamics and/or crystal plasticity simulations. Here, we apply the technique to study non-conservative plastic bypass of spherical precipitates by edge dislocations and point out the differences between our discrete approach and existing continuum formulations.

36 MATERIALS SCIENCE↗

NASA NTRS1997

Computations of Complex Three-Dimensional Turbulent Free Jets

Three-dimensional, incompressible turbulent jets with rectangular and elliptical cross-sections are simulated with a finite-difference numerical method. The full Navier- Stokes equations are solved at low Reynolds numbers, whereas at high Reynolds numbers filtered forms of the equations are solved along with a sub-grid scale model to approximate the effects of the unresolved scales. A 2-N storage, third-order Runge-Kutta scheme is used for temporary discretization and a fourth-order compact scheme is used for spatial discretization. Although such methods are widely used in the simulation of compressible flows, the lack of an evolution equation for pressure or density presents particular difficulty in incompressible flows. The pressure-velocity coupling must be established indirectly. It is achieved, in this study, through a Poisson equation which is solved by a compact scheme of the same order of accuracy. The numerical formulation is validated and the dispersion and dissipation errors are documented by the solution of a wide range of benchmark problems. Three-dimensional computations are performed for different inlet conditions which model the naturally developing and forced jets. The experimentally observed phenomenon of axis-switching is captured in the numerical simulation, and it is confirmed through flow visualization that this is based on self-induction of the vorticity field. Statistical quantities such as mean velocity, mean pressure, two-point velocity spatial correlations and Reynolds stresses are presented. Detailed budgets of the mean momentum and Reynolds stresses are presented. Detailed budgets of the mean momentum and Reynolds stress equations are presented to aid in the turbulence modeling of complex jets. Simulations of circular jets are used to quantify the effect of the non-uniform curvature of the non-circular jets.

Wilson, Robert V.↗

NASA NTRS1996

Large-eddy simulation of a backward facing step flow using a least-squares spectral element method

We report preliminary results obtained from the large eddy simulation of a backward facing step at a Reynolds number of 5100. The numerical platform is based on a high order Legendre spectral element spatial discretization and a least squares time integration scheme. A non-reflective outflow boundary condition is in place to minimize the effect of downstream influence. Smagorinsky model with Van Driest near wall damping is used for sub-grid scale modeling. Comparisons of mean velocity profiles and wall pressure show good agreement with benchmark data. More studies are needed to evaluate the sensitivity of this method on numerical parameters before it is applied to complex engineering problems.

Chan, Daniel C.↗

DOE OSTI2025

Additive manufacturing of high explosives with inert dilution for wave shaping and detonation velocity grading

The performance of a high explosive charge is highly influenced by the detonation velocity and any inherent internal structure. Additively manufactured (AM) high explosive charges allow for a precisely engineered internal structure and the selective placement of discrete or graded volumes of a tailored material, which are unachievable by conventional means. The dilution of a solid explosive by an inert additive for detonation velocity reduction is herein evaluated for a method of material extrusion called a direct-ink-write AM, where an 1,3,5,7-tetranitro-1,3,5,7-tetrazoctane-based ultraviolet -curable energetic-binder paste explosive was diluted by the inert density mock 5-iodo-2′-deoxyuridine for use in the manufacture of multi-material explosive charges. The explosive ink was characterized for printing by rheological and particle size measurements, and the detonation velocities for 0–40 wt. % inert diluent were found to have a continuous linear reduction in detonation velocity down to 14.3% difference. A two-component 3D plane wave explosive lens with 8.038 km/s “fast” and 7.491 km/s “slow” diluted ink in an internal cone demonstrated a reduced time of breakout at the output face from 361 to 53 ns. A linearly graded “fast–slow–fast” cylinder was successfully printed and fired, showing engineered control along the length of the charge as the detonation velocity dropped from 7.851 to 6.700 km/s and then recovered to 7.241 km/s. The extreme control over the internal composition of a high explosive charge via a dual-material extruder/mixer challenges conventional manufacturing approaches and opens new avenues for engineered detonation wavefronts, high explosive charge performance, and new applications.

36 MATERIALS SCIENCE↗

NASA NTRS1977

A numerical method for the exact calculation of airloads associated with impulsively started wings

A numerical method is developed to calculate three-dimensional potential flows due to the steady and impulsive motion of isolated wing and wing-wing interaction problems. The velocity potential is represented by a discrete set of constant-doublet quadrilaterals on wing and wake surfaces. The exact surface boundary condition is enforced, and the solution is obtained in a step-by-step fashion, configurations being impulsively started from rest. Free-wake geometries are generated for each time step with Rankine or Lamb viscous vortex segments used in wake-velocity calculations. Sample results include calculated performance to steady state for a thick wing and indicial lift of a wing-wing interaction problem.

Summa, J. M.↗

NASA NTRS1986

On the free vibrations of spinning paraboloids

The dynamic behavior of a spinning linear-elastic paraboloid subject to nonaxisymmetric deformation is investigated analytically, applying the Rayleigh-Ritz procedure described by Utku et al. (1983). Energy-density, strain-displacement, and velocity-displacement expressions are generated; expressions for the generalized strain and position vector are derived; and the discretized dynamic equations are obtained. Numerical results obtained with a computer-program implementation of the method are presented in extensive tables and graphs. The effects of spin rate and bending rigidity and results for the special case of a spinning disk are included.

Shoemaker, W. L.↗

NASA NTRS1990

Numerical simulation of rarefied gas flow through a slit

Two different approaches, the finite-difference method coupled with the discrete-ordinate method (FDDO), and the direct-simulation Monte Carlo (DSMC) method, are used in the analysis of the flow of a rarefied gas from one reservoir to another through a two-dimensional slit. The cases considered are for hard vacuum downstream pressure, finite pressure ratios, and isobaric pressure with thermal diffusion, which are not well established in spite of the simplicity of the flow field. In the FDDO analysis, by employing the discrete-ordinate method, the Boltzmann equation simplified by a model collision integral is transformed to a set of partial differential equations which are continuous in physical space but are point functions in molecular velocity space. The set of partial differential equations are solved by means of a finite-difference approximation. In the DSMC analysis, three kinds of collision sampling techniques, the time counter (TC) method, the null collision (NC) method, and the no time counter (NTC) method, are used.

Keith, Theo G., Jr.↗

NASA NTRS1992

FDDO and DSMC analyses of rarefied gas flow through 2D nozzles

Two different approaches, the finite-difference method coupled with the discrete-ordinate method (FDDO), and the direct-simulation Monte Carlo (DSMC) method, are used in the analysis of the flow of a rarefied gas expanding through a two-dimensional nozzle and into a surrounding low-density environment. In the FDDO analysis, by employing the discrete-ordinate method, the Boltzmann equation simplified by a model collision integral is transformed to a set of partial differential equations which are continuous in physical space but are point functions in molecular velocity space. The set of partial differential equations are solved by means of a finite-difference approximation. In the DSMC analysis, the variable hard sphere model is used as a molecular model and the no time counter method is employed as a collision sampling technique. The results of both the FDDO and the DSMC methods show good agreement. The FDDO method requires less computational effort than the DSMC method by factors of 10 to 40 in CPU time, depending on the degree of rarefaction.

Chung, Chan-Hong↗

NASA NTRS1992

Advances in tilt rotor noise prediction

The two most serious tilt rotor external noise problems, hover noise and blade-vortex interaction noise, are studied. The results of flow visualization and inflow velocity measurements document a complex, recirculating highly unsteady and turbulent flow due to the rotor-wing-body interactions characteristic of tilt rotors. The wing under the rotor is found to obstruct the inflow, causing a deficit in the inflow velocities over the inboard region of the rotor. Discrete frequency harmonic thickness and loading noise mechanisms in hover are examined by first modeling tilt rotor hover aerodynamics and then applying various noise prediction methods using the WOPWOP code. The analysis indicates that the partial ground plane created by the wing below the rotor results in a primary sound source for hover.

George, A. R.↗

NASA NTRS2011

Mesh-Based Entry Vehicle and Explosive Debris Re-Contact Probability Modeling

The risk to a crewed vehicle arising from potential re-contact with fragments from an explosive breakup of any jettisoned spacecraft segments during entry has long sought to be quantified. However, great difficulty lies in efficiently capturing the potential locations of each fragment and their collective threat to the vehicle. The method presented in this paper addresses this problem by using a stochastic approach that discretizes simulated debris pieces into volumetric cells, and then assesses strike probabilities accordingly. Combining spatial debris density and relative velocity between the debris and the entry vehicle, the strike probability can be calculated from the integral of the debris flux inside each cell over time. Using this technique it is possible to assess the risk to an entry vehicle along an entire trajectory as it separates from the jettisoned segment. By decoupling the fragment trajectories from that of the entry vehicle, multiple potential separation maneuvers can then be evaluated rapidly to provide an assessment of the best strategy to mitigate the re-contact risk.

McPherson, Mark A.↗

NASA NTRS1985

Improving the Accuracy of the Boundary Integral Method Based on the Helmholtz Integral

Several recent papers in the literature have been based on various forms of the Helmholtz integral to compute the radiation fields of vibrating bodies. The surface integral form is given. The symbols of P,R micron, rho,G,R,V, and S micron are acoustic pressure, source coordinate, angular frequency, fluid density, Green function, field coordinate, surface velocity and body surface respectively. A discretized form of the surface integral is also given. Solutions to the surface integral are complicated with the singularity of the Green function at R=R micron and with the uniqueness problem at interior eigen frequencies of the enclosed space. The use of the interior integral circumvents the singularity problem since the field points are chosen in the interior space of the vibrating body where a zero pressure condition exists. The interior integral form is given. The method to improve the accuracy is detailed. Examples of the method is presented for a variety of radiators.

Koopmann, G. H.↗