Results for “conservation form” — Page 15

NASA NTRS1994

An implicit numerical scheme for the simulation of internal viscous flows on unstructured grids

The Navier-Stokes equations are solved numerically for two-dimensional steady viscous laminar flows. The grids are generated based on the method of Delaunay triangulation. A finite-volume approach is used to discretize the conservation law form of the compressible flow equations written in terms of primitive variables. A preconditioning matrix is added to the equations so that low Mach number flows can be solved economically. The equations are time marched using either an implicit Gauss-Seidel iterative procedure or a solver based on a conjugate gradient like method. A four color scheme is employed to vectorize the block Gauss-Seidel relaxation procedure. This increases the memory requirements minimally and decreases the computer time spent solving the resulting system of equations substantially. A factor of 7.6 speed up in the matrix solver is typical for the viscous equations. Numerical results are obtained for inviscid flow over a bump in a channel at subsonic and transonic conditions for validation with structured solvers. Viscous results are computed for developing flow in a channel, a symmetric sudden expansion, periodic tandem cylinders in a cross-flow, and a four-port valve. Comparisons are made with available results obtained by other investigators.

Jorgenson, Philip C. E.↗

NASA NTRS1994

Modeling jets in cross flow

Various approaches to the modeling of jets in cross flow are reviewed. These are grouped into four classes, namely: empirical models, integral models, perturbation models, and numerical models. Empirical models depend largely on the correlation of experimental data and are mostly useful for first-order estimates of global properties such as jet trajectory and velocity and temperature decay rates. Integral models are based on some ordinary-differential form of the conservation laws, but require substantial empirical calibration. They allow more details of the flow field to be obtained; simpler versions have to assume similarity of velocity and temperature profiles, but more sophisticated ones can actually calculate these profiles. Perturbation models require little empirical input, but the need for small parameters to ensure convergent expansions limits their application to either the near-field or the far-field. Therefore, they are mostly useful for the study of flow physics. Numerical models are based on conservation laws in partial-differential form. They require little empirical input and have the widest range of applicability. They also require the most computational resources. Although many qualitative and quantitative features of jets in cross flow have been predicted with numerical models, many issues affecting accuracy such as grid resolution and turbulence model are not completely resolved.

Demuren, A. O.↗

NASA NTRS1993

Modeling of Non-Spherical Droplet Dynamics

A two-dimensional time-dependent computer code based on the modified Arbitrary Lagrangian Eulerian (ALE) technique, has been developed to simulate non-spherical droplet dynamics and evaporation under convective flows at real rocket combustion chamber conditions. The equations of mass, momentum, energy and species are simultaneously solved for both liquid and gas phases with an accurate dynamic interface tracking. The jump boundary conditions across the deforming droplet surface are obtained by applying the integral forms of conservation of mass, momentum, and energy. At each time step, the interface geometry and flow properties at the droplet surface are implicitly solved by satisfying the interface boundary conditions. A Lagrangian technique was developed to track the arbitrarily moving interface between the liquid droplet and the external gas. An elliptic grid generator is adopted to dynamically reconstruct grids both inside and outside the droplet surface. This code has been used to study droplet oscillation, droplet deformation/breakup, nonspherical droplet evaporation in both low and high pressure convective flows. This presentation briefly describes the numerical algorithm for modeling of the nonspherical droplet dynamics and demonstrates the representative simulation results of nonspherical droplet evaporation at low and high pressure convective flows. Potential applications of this code to rocket combustor design and performance predictions are discussed.

Deng, Zheng-Tao↗

NASA NTRS1996

Computation Of Flow About A Helicopter Rotor

Report describes computations of flow about hovering helicopter rotor. Thin-layer Navier-Stokes equations in conservation-law form solved numerically on embedded grids.

Duque, Earl P.↗

NASA NTRS1976

Split Space-Marching Finite-Volume Method for Chemically Reacting Supersonic Flow

A space-marching finite-volume method employing a nonorthogonal coordinate system and using a split differencing scheme for calculating steady supersonic flow over aerodynamic shapes is presented. It is a second-order-accurate mixed explicit-implicit procedure that solves the inviscid adiabatic and nondiffusive equations for chemically reacting flow in integral conservation-law form. The relationship between the finite-volume and differential forms of the equations is examined and the relative merits of each discussed. The method admits initial Cauchy data situated on any arbitrary surface and integrates them forward along a general curvilinear coordinate, distorting and deforming the surface as it advances. The chemical kinetics term is split from the convective terms which are themselves dimensionally split, thereby freeing the fluid operators from the restricted step size imposed by the chemical reactions and increasing the computational efficiency. The accuracy of this splitting technique is analyzed, a sufficient stability criterion is established, a representative flow computation is discussed, and some comparisons are made with another method.

Rizzi, Arthur W.↗

NASA NTRS2002

Computing Axisymmetric Jet Screech Tones Using Unstructured Grids

The space-time conservation element and solution element (CE/SE) method is used to solve the conservation law form of the compressible axisymmetric Navier-Stokes equations. The equations are time marched to predict the unsteady flow and the near-field screech tone noise issuing from an underexpanded circular jet. The CE/SE method uses an unstructured grid based data structure. The unstructured grids for these calculations are generated based on the method of Delaunay triangulation. The purpose of this paper is to show that an acoustics solution with a feedback loop can be obtained using truly unstructured grid technology. Numerical results are presented for two different nozzle geometries. The first is considered to have a thin nozzle lip and the second has a thick nozzle lip. Comparisons with available experimental data are shown for flows corresponding to several different jet Mach numbers. Generally good agreement is obtained in terms of flow physics, screech tone frequency, and sound pressure level.

Jorgenson, Philip C. E.↗

NASA NTRS2009

Adiabatic Shock Capturing in Perfect Gas Hypersonic Flows

This paper considers the streamline-upwind Petrov/Galerkin (SUPG) method applied to the compressible Euler and Navier-Stokes equations in conservation-variable form. The spatial discretization, including a modified approach for interpolating the inviscid flux terms in the SUPG finite element formulation, is briefly reviewed. Of particular interest is the behavior of the shock capturing operator, which is required to regularize the scheme in the presence of strong, shock-induced gradients. A standard shock capturing operator which has been widely used in previous studies by several authors is presented and discussed. Specific modifications are then made to this standard operator which are designed to produce a more physically consistent discretization in the presence of strong shock waves. The actual implementation of the term in a finite dimensional approximation is also discussed. The behavior of the standard and modified scheme is then compared for several supersonic/hypersonic flows. The modified shock capturing operator is found to preserve enthalpy in the inviscid portion of the flowfield substantially better than the standard operator.

Kirk, Benjamin S.↗

NASA NTRS2013

Studies of Inviscid Flux Schemes for Acoustics and Turbulence Problems

Five different central difference schemes, based on a conservative differencing form of the Kennedy and Gruber skew-symmetric scheme, were compared with six different upwind schemes based on primitive variable reconstruction and the Roe flux. These eleven schemes were tested on a one-dimensional acoustic standing wave problem, the Taylor-Green vortex problem and a turbulent channel flow problem. The central schemes were generally very accurate and stable, provided the grid stretching rate was kept below 10%. As near-DNS grid resolutions, the results were comparable to reference DNS calculations. At coarser grid resolutions, the need for an LES SGS model became apparent. There was a noticeable improvement moving from CD-2 to CD-4, and higher-order schemes appear to yield clear benefits on coarser grids. The UB-7 and CU-5 upwind schemes also performed very well at near-DNS grid resolutions. The UB-5 upwind scheme does not do as well, but does appear to be suitable for well-resolved DNS. The UF-2 and UB-3 upwind schemes, which have significant dissipation over a wide spectral range, appear to be poorly suited for DNS or LES.

Morris, Chris↗

NASA NTRS2020

Costa Rica & Panama Ecological Forecasting II: Identifying Current and Future Areas of Environmental Concern in La Amistad International Park to Inform Resource Management

Seven percent of all scientifically known life forms lie within the 202,230 square miles of Central America, making this area ecologically unique and increasing the need for environmental management. The Mesoamerican Biological Corridor forms a conservation partnership throughout Central America to establish a forested corridor of over 600 protected areas. Although conservation programs exist, deforestation still afflicts the area, putting strain on these diverse ecosystems. La Amistad International Park in southern Costa Rica and northern Panama in particular faces conflicting land use changes. Expanding agricultural development and urbanization, combined with concern over indigenous land rights, have raised questions about the implementation of sustainability goals and communication strategies within the region. To help address these issues, the NASA DEVELOP Costa Rica & Panama Ecological Forecasting II team continued a partnership with the Ministry of Environment and Energy in Costa Rica and the National Environmental Authority in Panama. The team created a Land Use Conflict Identification Strategy (LUCIS) model based on land cover maps for 2019 and 2029 that were created in term I using imagery from Landsat 8 Operational Land Imager (OLI). With partner input, the team applied weights to different objectives and combined suitability maps to identify areas of potential biodiversity conflict. Additionally, the team created a Short-term Forest Change (SFTC) Tool using Terra Moderate Resolution Imaging Spectroradiometer, Landsat 8 OLI, and Sentinel-2 Multispectral Instrument to help the partners identify areas of immediate, major forest changes. The LUCIS model indicated higher agricultural growth when compared to the ecological category.

Ecological Forecasting↗

NASA NTRS2020

Costa Rica & Panama Ecological Forecasting II: Identifying Current and Future Areas of Environmental Concern in La Amistad International Park to Inform Resource Management

Seven percent of all scientifically known life forms lie within the 202,230 square miles of Central America, making this area ecologically unique and increasing the need for environmental management. The Mesoamerican Biological Corridor forms a conservation partnership throughout Central America to establish a forested corridor of over 600 protected areas. Although conservation programs exist, deforestation still afflicts the area, putting strain on these diverse ecosystems. La Amistad International Park in southern Costa Rica and northern Panama in particular faces conflicting land use changes. Expanding agricultural development and urbanization, combined with concern over indigenous land rights, have raised questions about the implementation of sustainability goals and communication strategies within the region. To help address these issues, the NASA DEVELOP Costa Rica & Panama Ecological Forecasting II team continued a partnership with the Ministry of Environment and Energy in Costa Rica and the National Environmental Authority in Panama. The team created a Land Use Conflict Identification Strategy (LUCIS) model based on land cover maps for 2019 and 2029 that were created in term I using imagery from Landsat 8 Operational Land Imager (OLI). With partner input, the team applied weights to different objectives and combined suitability maps to identify areas of potential biodiversity conflict. Additionally, the team created a Short-term Forest Change (SFTC) Tool using Terra Moderate Resolution Imaging Spectroradiometer, Landsat 8 OLI, and Sentinel-2 Multispectral Instrument to help the partners identify areas of immediate, major forest changes. The LUCIS model indicated higher agricultural growth when compared to the ecological category.

Ecological Forecasting↗

NASA NTRS2020

Costa Rica & Panama Ecological Forecasting II: Identifying Current and Future Areas of Environmental Concern in La Amistad International Park to Inform Resource Management

Seven percent of all scientifically known life forms lie within the 202,230 square miles of Central America, making this area ecologically unique and increasing the need for environmental management. The Mesoamerican Biological Corridor forms a conservation partnership throughout Central America to establish a forested corridor of over 600 protected areas. Although conservation programs exist, deforestation still afflicts the area, putting strain on these diverse ecosystems. La Amistad International Park in southern Costa Rica and northern Panama in particular faces conflicting land use changes. Expanding agricultural development and urbanization, combined with concern over indigenous land rights, have raised questions about the implementation of sustainability goals and communication strategies within the region. To help address these issues, the NASA DEVELOP Costa Rica & Panama Ecological Forecasting II team continued a partnership with the Ministry of Environment and Energy in Costa Rica and the National Environmental Authority in Panama. The team created a Land Use Conflict Identification Strategy (LUCIS) model based on land cover maps for 2019 and 2029 that were created in term I using imagery from Landsat 8 Operational Land Imager (OLI). With partner input, the team applied weights to different objectives and combined suitability maps to identify areas of potential biodiversity conflict. Additionally, the team created a Short-term Forest Change (SFTC) Tool using Terra Moderate Resolution Imaging Spectroradiometer, Landsat 8 OLI, and Sentinel-2 Multispectral Instrument to help the partners identify areas of immediate, major forest changes. The LUCIS model indicated higher agricultural growth when compared to the ecological category.

Ecological Forecasting↗

NASA NTRS1978

Finite elements and finite differences for transonic flow calculations

The paper reviews the chief finite difference and finite element techniques used for numerical solution of nonlinear mixed elliptic-hyperbolic equations governing transonic flow. The forms of the governing equations for unsteady two-dimensional transonic flow considered are the Euler equation, the full potential equation in both conservative and nonconservative form, the transonic small-disturbance equation in both conservative and nonconservative form, and the hodograph equations for the small-disturbance case and the full-potential case. Finite difference methods considered include time-dependent methods, relaxation methods, semidirect methods, and hybrid methods. Finite element methods include finite element Lax-Wendroff schemes, implicit Galerkin method, mixed variational principles, dual iterative procedures, optimal control methods and least squares.

Hafez, M. M.↗

NASA NTRS1994

Magnetocentrifugally driven flows from young stars and disks. 2: Formulation of the dynamical problem

We formulate the dynamical problem of a cool wind centrifugally driven from the magnetic interface of a young star and an adjoining Keplerian disk. We examine the situation for mildly accreting T Tauri stars that rotate slowly as well as rapidly accreting protostars that rotate near break-up. In both cases a wind can be driven from a small X-region just outside the stellar magnetopause, where the field lines assume an open geometry and are rooted to material that rotates at an angular speed equal both to the local Keplerian value and to the stellar angular speed. Assuming axial symmetry for the ideal magnetohydrodynamic flow, which requires us to postpone asking how the (lightly ionized) gas is loaded onto field lines, we can formally integrate all the governing equations analytically except for a partial equation that describes how streamlines spread in the meridional plane. Apart from the difficulty of dealing with PDEs of mixed type, finding the functional forms of the conserved quantities along streamlines - the ratio beta of magnetic field to mass flux, the specific energy H of the fluid in the rotating frame, and the total specific angular momentum J carried in the matter and the field - constitutes a standard difficulty in this kind of (Grad-Shafranov) formalism. Fortunately, because the ratio of the thermal speed of the mass-loss regions to the Keplerian speed of rotation of the interface constitutes a small parameter epsilon, we can attack the overall problem by the method of matched asymptotic expansions. This procedure leads to a natural and systematic technique for obtaining the relevant functional dependences of beta, H, and J. Moreover, we are able to solve analytically for the properties of the flow emergent from the small transsonic region driven by gas pressure without having to specify the detailed form of any of the conserved functions, beta, H, and J. This analytical solution provides inner boundary conditions for the numerical computation in a companion paper by Najita & Shu of the larger region where the main acceleration to terminal speeds occurs.

Shu, Frank H.↗

NASA NTRS2003

A Vertically Lagrangian Finite-Volume Dynamical Core for Global Models

A finite-volume dynamical core with a terrain-following Lagrangian control-volume discretization is described. The vertically Lagrangian discretization reduces the dimensionality of the physical problem from three to two with the resulting dynamical system closely resembling that of the shallow water dynamical system. The 2D horizontal-to-Lagrangian-surface transport and dynamical processes are then discretized using the genuinely conservative flux-form semi-Lagrangian algorithm. Time marching is split- explicit, with large-time-step for scalar transport, and small fractional time step for the Lagrangian dynamics, which permits the accurate propagation of fast waves. A mass, momentum, and total energy conserving algorithm is developed for mapping the state variables periodically from the floating Lagrangian control-volume to an Eulerian terrain-following coordinate for dealing with physical parameterizations and to prevent severe distortion of the Lagrangian surfaces. Deterministic baroclinic wave growth tests and long-term integrations using the Held-Suarez forcing are presented. Impact of the monotonicity constraint is discussed.

Lin, Shian-Jiann↗

NASA NTRS1990

Some Aspects of Essentially Nonoscillatory (ENO) Formulations for the Euler Equations, Part 3

An essentially nonoscillatory (ENO) formulation is described for hyperbolic systems of conservation laws. ENO approaches are based on smart interpolation to avoid spurious numerical oscillations. ENO schemes are a superset of Total Variation Diminishing (TVD) schemes. In the recent past, TVD formulations were used to construct shock capturing finite difference methods. At extremum points of the solution, TVD schemes automatically reduce to being first-order accurate discretizations locally, while away from extrema they can be constructed to be of higher order accuracy. The new framework helps construct essentially non-oscillatory finite difference methods without recourse to local reductions of accuracy to first order. Thus arbitrarily high orders of accuracy can be obtained. The basic general ideas of the new approach can be specialized in several ways and one specific implementation is described based on: (1) the integral form of the conservation laws; (2) reconstruction based on the primitive functions; (3) extension to multiple dimensions in a tensor product fashion; and (4) Runge-Kutta time integration. The resulting method is fourth-order accurate in time and space and is applicable to uniform Cartesian grids. The construction of such schemes for scalar equations and systems in one and two space dimensions is described along with several examples which illustrate interesting aspects of the new approach.

Chakravarthy, Sukumar R.↗

NASA NTRS1971

Transpiration and film cooling boundary layer computer program. Volume 2: Computer program and user's manual

A finite difference turbulent boundary layer computer program which allows for mass transfer wall cooling and equilibrium chemistry effects is presented. The program is capable of calculating laminar or turbulent boundary layer solutions for an arbitrary ideal gas or an equilibrium hydrogen oxygen system. Either two dimensional or axisymmetric geometric configurations may be considered. The equations are solved, in nondimension-alized physical coordinates, using the implicit Crank-Nicolson technique. The finite difference forms of the conservation of mass, momentum, total enthalpy and elements equations are linearized and uncoupled, thereby generating easily solvable tridiagonal sets of algebraic equations. A detailed description of the computer program, as well as a program user's manual is provided. Detailed descriptions of all boundary layer subroutines are included, as well as a section defining all program symbols of principal importance. Instructions are then given for preparing card input to the program and for interpreting the printed output. Finally, two sample cases are included to illustrate the use of the program.

Gloss, R. J.↗

NASA NTRS1973

Application of the method local potential to the analysis of turbulent shear flows

It has been found that, in general, the local potential cannot be employed to obtain approximate solutions for the various correlations of turbulent properties which appear in the time averaged form of the conservation equations. Although the method of local potential is equivalent to the Galerkin method when the self-consistent condition is applied, the local potential can also be applied as an iterative algorithm in place of using the selfconsistent condition. This procedure offers an alternative to the Galerkin method and may be useful in obtaining approximate solutions for the total turbulent velocity. In addition, for certain simple turbulent shear flows the iterative algorithm may permit approximate, but non-empirical, solutions by modeling only the mean velocity and the Reynolds stress.

Reed, T. D.↗

NASA NTRS1976

Theory of fracture mechanics based upon plasticity

A theory of fracture mechanics is formulated on the foundation of continuum mechanics. Fracture surface is introduced as an unknown quantity and is incorporated into boundary and initial conditions. Surface energy is included in the global form of energy conservation law and the dissipative mechanism is formulated into constitutive equations which indicate the thermodynamic irreversibility and the irreversibility of fracture process as well.

Lee, J. D.↗