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A method of correcting finite-difference solutions for the effect of truncation error or the use of an approximate basic equation is presented. Applications to transonic flow problems are described and examples are given.
A method of correcting finite-difference solutions for the effect of truncation error or the use of an approximate basic equation is presented. Applications to transonic flow problems are described and examples given.
An algebraic stress turbulence model is presented, characterized by the following: (1) the eddy viscosity expression is derived from the Reynolds stress turbulence model; (2) the turbulent kinetic energy dissipation rate equation is improved by including a production range time scale; and (3) the diffusion coefficients for turbulence equations are adjusted so that the kinetic energy profile extends further into the free stream region found in most experimental data. The turbulent flow equations were solved using a finite element method. Examples include: fully developed channel flow, fully developed pipe flow, flat plate boundary layer flow, plane jet exhausting into a moving stream, circular jet exhausting into a moving stream, and wall jet flow. Computational results compare favorably with experimental data for most of the examples considered. Significantly improved results were obtained for the plane jet flow, the circular jet flow, and the wall jet flow; whereas the remainder are comparable to those obtained by finite difference methods using the standard kappa-epsilon turbulence model. The latter seems to be promising with further improvement of the expression for the eddy viscosity coefficient.
Most published work relating to dynamically loaded journal bearings are directed to determining the minimum film thickness from the predicted journal trajectories. These do not give any information about the subsynchronous whirl stability of journal bearing systems since they do not consider the equations of motion. It is, however, necessary to know whether the bearing system operation is stable or not under such an operating condition. The stability characteristics of the system are analyzed. A linearized perturbation theory about the equilibrium point can predict the threshold of stability; however it does not indicate postwhirl orbit detail. The linearized method may indicate that a bearing is unstable for a given operating condition whereas the nonlinear analysis may indicate that it forms a stable limit cycle. For this reason, a nonlinear transient analysis of a rigid rotor supported on oil journal bearings under: (1) a unidirectional constant load, (2) a unidirectional periodic load, and (3) variable rotating load are performed. The hydrodynamic forces are calculated after solving the time-dependent Reynolds equation by a finite difference method with a successive overrelaxation scheme. Using these forces, equations of motion are solved by the fourth-order Runge-Kutta method to predict the transient behavior of the rotor. With the aid of a high-speed digital computer and graphics, the journal trajectories are obtained for several different operating conditions.
Nuclear reactions at intermediate beam energies are often interpreted using the eikonal model. In the analysis of complex reaction probes, where few-body reaction methods are needed, the eikonal method may be used as an efficient way for describing the fragment-target reaction process. In this work, we perform a systematic study to test the validity of the eikonal approximation for nucleon-nucleus reactions. We also quantify uncertainties due to the nucleon optical potential on reaction observables. We inspect the validity of the eikonal model and its semiclassical correction by comparing it to exact solutions (obtained from solving the optical-model equation with a finite-differences method) for a wide range of reactions. We also study the effect of relativistic corrections, both kinematic and dynamic, by effectively incorporating the relativistic effects at intermediate energies. The uncertainties from a Bayesian global optical potential (KDUQ) are propagated to the observables of interest. Our study includes neutron and proton reactions on 27 Al , 40 Ca , 90 Zr , and 208 Pb , for a wide range of energies 𝐸 lab = 0–400 MeV. We calculate neutron-total cross sections (elastic and reactions) as well as proton-absorption cross sections as a function of beam energy, using the eikonal model, the eikonal model with a semiclassical correction, and the exact solution. Here, we also compute angular distributions for the methods above. Our results show that for the proton-absorption cross section, the eikonal model can be used down to around 60 MeV and the semiclassical correction extends its use to 30 MeV. However, the validity of the eikonal model for the neutron-total cross section only goes down to ≈120 MeV, a range extended to ≈ 50 MeV when using the semiclassical correction. We find the semiclassical correction to the eikonal model to be less effective in describing the angular distributions. The 1𝜎 uncertainty intervals on the observables we studied is less than 5% for most of the energies considered, but increases rapidly for higher energies, namely energies outside the range of KDUQ (𝐸 lab > 200MeV).
Analytical and empirical studies of a finite difference method for the solution of the transonic flow about harmonically oscillating wings and airfoils are presented. The procedure is based on separating the velocity potential into steady and unsteady parts and linearizing the resulting unsteady equations for small disturbances. The steady velocity potential is obtained first from the well-known nonlinear equation for steady transonic flow. The unsteady velocity potential is then obtained from a linear differential equation in complex form with spatially varying coefficients. Since sinusoidal motion is assumed, the unsteady equation is independent of time. An out-of-core direct solution procedure was developed and applied to two-dimensional sections. Results are presented for a section of vanishing thickness in subsonic flow and an NACA 64A006 airfoil in supersonic flow. Good correlation is obtained in the first case at values of Mach number and reduced frequency of direct interest in flutter analyses. Reasonable results are obtained in the second case. Comparisons of two-dimensional finite difference solutions with exact analytic solutions indicate that the accuracy of the difference solution is dependent on the boundary conditions used on the outer boundaries. Homogeneous boundary conditions on the mesh edges that yield complex eigenvalues give the most accurate finite difference solutions. The plane outgoing wave boundary conditions meet these requirements.
Recent imaging of supermassive black holes by the Event Horizon Telescope has relied on exhaustive parameter-space searches, matching observations to large, precomputed libraries of theoretical models. As observational data become increasingly precise, the limitations of this computationally expensive approach grow more acute, creating a pressing need for more efficient methods. In this work, we present Jipole, an automatically differentiable (AD), ipole-based code for radiative transfer in curved spacetimes, designed to compute image gradients with respect to underlying model parameters. These gradients quantify how parameter changes—such as the black hole’s spin or the observer’s inclination—affect the image, enabling more efficient parameter estimation and reducing the number of required images. We validate Jipole against ipole in two analytical tests and then compare pixelwise intensity derivatives from AD with those from finite-difference methods. We then demonstrate the utility of these gradients by performing parameter recovery for an analytical model in three increasingly complex cases for the injected image: ideal, blurred, and blurred with added noise. In most cases, high-accuracy fits are obtained in only a few optimization steps, failing only in cases with extremely low signal-to-noise ratios. These results highlight the potential of AD-based methods to accelerate robust, high-fidelity model-data comparisons in current and future black hole imaging efforts.
The applications of high-order, compact finite difference methods in shock calculations are discussed. The main concern is to define a local mean which will serve as a reference for introducing a local nonlinear limiting to control spurious numerical oscillations while maintaining the formal accuracy of the scheme. For scalar conservation laws, the resulting schemes can be proven total-variation stable in one space dimension and maximum-norm stable in multiple space dimensions. Numerical examples are shown to verify accuracy and stability of such schemes for problems containing shocks. These ideas can also be applied to other implicit schemes such as the continuous Galerkin finite element methods.
A class of nonlinearly stable Runge-Kutta local projection discontinuous Galerkin (RKDG) finite element methods for conservation laws is investigated. Two dimensional Euler equations for gas dynamics are solved using P1 elements. The generalization of the local projections, which for scalar nonlinear conservation laws was designed to satisfy a local maximum principle, to systems of conservation laws such as the Euler equations of gas dynamics using local characteristic decompositions is discussed. Numerical examples include the standard regular shock reflection problem, the forward facing step problem, and the double Mach reflection problem. These preliminary numerical examples are chosen to show the capacity of the approach to obtain nonlinearly stable results comparable with the modern nonoscillatory finite difference methods.
The growth of InP by metalorganic chemical vapor deposition (MOCVD) in a horizontal reactor is being modeled with a commercially available computational fluid dynamics modeling code. The mathematical treatment of the MOCVD process has four primary areas of concern: 1) transport phenomena, 2) chemistry, 3) boundary conditions, and 4) numerical solution methods. The transport processes involved in CVD are described by conservation of total mass, momentum, energy, and atomic species. Momentum conservation is described by a generalized form of the Navier-Stokes equation for a Newtonian fluid and laminar flow. The effect of Soret diffusion on the transport of particular chemical species and on the predicted deposition rate is examined. Both gas-phase and surface chemical reactions are employed in the model. Boundary conditions are specified at the inlet and walls of the reactor for temperature, fluid flow and chemical species. The coupled set of equations described above is solved by a finite difference method over a nonuniform rectilinear grid in both two and three dimensions. The results of the 2-D computational model is presented for gravity levels of zero- and one-g. The predicted growth rates at one-g are compared to measured growth rates on fused silica substrates.
Considerable progress has been made in recent years toward advancing the state-of-the-art in solid mechanics boundary element technology. In the present work, much of this new technology is applied in the development of a general-purpose boundary element method (BEM) for steady-state heat conduction. In particular, the BEM implementation involves the use of higher-order conforming elements, self-adaptive integration and multi-region capability. Two- and three-dimensional, as well as axisymmetric analysis, are incorporated within a unified framework. In addition, techniques are introduced for the calculation of boundary flux, and for the inclusion of thermal resistance across interfaces. As a final extension, an efficient formulation is developed for the analysis of solid three-dimensional bodies with embedded holes. For this last class of problems, the new BEM formulation is particularly attractive, since use of the alternatives (i.e. finite element or finite difference methods) is not practical. A number of detailed examples illustrate the suitability and robustness of the present approach for steady-state heat conduction.
The paper outlines a comparative study designed to assess and compare the accuracy of the finite element method (FEM) for linear and quadratic elements as applied to problems in duct acoustics. The acoustic disturbances are assumed to be irrotational and isentropic so that the problem can be formulated in terms of the acoustic velocity potential. It is shown that for the case of plane wave propagation in a hard-walled annular cylinder, the accuracy of the FEM solution can be increased at higher frequencies by using quadratic triangular elements instead of linear triangular elements. Evidence is presented to enhance the confidence in applying the developed FEM by comparing results with those obtained by other independently developed numerical approaches such as an integral equation technique and a finite difference method.
In this paper (a third in a series) the construction and the analysis of essentially non-oscillatory shock capturing methods for the approximation of hyperbolic conservation laws are presented. Also presented is a hierarchy of high order accurate schemes which generalizes Godunov's scheme and its second order accurate MUSCL extension to arbitrary order of accuracy. The design involves an essentially non-oscillatory piecewise polynomial reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell. The reconstruction algorithm is derived from a new interpolation technique that when applied to piecewise smooth data gives high-order accuracy whenever the function is smooth but avoids a Gibbs phenomenon at discontinuities. Unlike standard finite difference methods this procedure uses an adaptive stencil of grid points and consequently the resulting schemes are highly nonlinear.
A numerical computing procedure was developed for calculating vaporization histories of evaporating drops in a combustor in which travelling transverse oscillations occurred. The liquid drop was assumed to have a finite thermal conductivity. The system of equations was solved by using a finite difference method programmed for solution on a high speed digital computer. Oscillations in the ratio of vaporization of an array of repetitivity injected drops in the combustor were obtained from summation of individual drop histories. A nonlinear in-phase frequency response factor for the entire vaporization process to oscillations in pressure was evaluated. A nonlinear out-of-phase response factor, in-phase and out-of-phase harmonic response factors, and a Princeton type 'n' and 'tau' were determined. The resulting data was correlated and is presented in graphical format. Qualitative agreement with the open literature is obtained in the behavior of the in-phase response factor. Quantitatively the results of the present finite conductivity spray analysis do not correlate with the results of a single drop model.
Recent developments which have improved the understanding of how finite difference methods resolve discontinuous solutions to hyperbolic partial differential equations are discussed. As a result of this understanding improved shock capturing methods are currently being developed and tested. Some of these methods are described and numerical results are presented showing their performance on problems containing shocks in one and two dimensions. A conservative difference scheme is defined. Conservation implies that, except in very special circumstances, shocks must be spread over at least two grid intervals. These two interval shocks are actually attained in one dimension if the shock is steady and an upwind scheme is used. By analyzing this case, the reason for this excellent shock resolution can be determined. This result is used to provide a mechanism for improving the resolution of two dimensional steady shocks. Unfortunately, this same analysis shows that these results cannot be extended to shocks which move relative to the computing grid. Total variation diminishing (TVD) finite difference schemes and flux limiters are introduced to deal with money shocks and contact discontinuities.
A method for integrating Aeroheating analysis into conceptual reusable launch vehicle (RLV) design is presented in this thesis. This process allows for faster turn-around time to converge a RLV design through the advent of designing an optimized thermal protection system (TPS). It consists of the coupling and automation of four computer software packages: MINIVER, TPSX, TCAT, and ADS. MINIVER is an Aeroheating code that produces centerline radiation equilibrium temperatures, convective heating rates, and heat loads over simplified vehicle geometries. These include flat plates and swept cylinders that model wings and leading edges, respectively. TPSX is a NASA Ames material properties database that is available on the World Wide Web. The newly developed Thermal Calculation Analysis Tool (TCAT) uses finite difference methods to carry out a transient in-depth 1-D conduction analysis over the center mold line of the vehicle. This is used along with the Automated Design Synthesis (ADS) code to correctly size the vehicle's thermal protection system (TPS). The numerical optimizer ADS uses algorithms that solve constrained and unconstrained design problems. The resulting outputs for this process are TPS material types, unit thicknesses, and acreage percentages. TCAT was developed for several purposes. First, it provides a means to calculate the transient in-depth conduction seen by the surface of the TPS material that protects a vehicle during ascent and reentry. Along with the in-depth conduction, radiation from the surface of the material is calculated along with the temperatures at the backface and interior parts of the TPS material. Secondly, TCAT contributes added speed and automation to the overall design process. Another motivation in the development of TCAT is optimization. In some vehicles, the TPS accounts for a high percentage of the overall vehicle dry weight. Optimizing the weight of the TPS will thereby lower the percentage of the dry weight accounted for by the TPS. Also, this will lower the cost of the TPS and the overall cost of the vehicle.
A method for integrating Aeroheating analysis into conceptual reusable launch vehicle RLV design is presented in this thesis. This process allows for faster turn-around time to converge a RLV design through the advent of designing an optimized thermal protection system (TPS). It consists of the coupling and automation of four computer software packages: MINIVER, TPSX, TCAT and ADS. MINIVER is an Aeroheating code that produces centerline radiation equilibrium temperatures, convective heating rates, and heat loads over simplified vehicle geometries. These include flat plates and swept cylinders that model wings and leading edges, respectively. TPSX is a NASA Ames material properties database that is available on the World Wide Web. The newly developed Thermal Calculation Analysis Tool (TCAT) uses finite difference methods to carry out a transient in-depth I-D conduction analysis over the center mold line of the vehicle. This is used along with the Automated Design Synthesis (ADS) code to correctly size the vehicle's thermal protection system JPS). The numerical optimizer ADS uses algorithms that solve constrained and unconstrained design problems. The resulting outputs for this process are TPS material types, unit thicknesses, and acreage percentages. TCAT was developed for several purposes. First, it provides a means to calculate the transient in-depth conduction seen by the surface of the TPS material that protects a vehicle during ascent and reentry. Along with the in-depth conduction, radiation from the surface of the material is calculated along with the temperatures at the backface and interior parts of the TPS material. Secondly, TCAT contributes added speed and automation to the overall design process. Another motivation in the development of TCAT is optimization.
Historically, large-eddy simulations (LES) have been restricted to simple geometries where spectral or finite difference methods have dominated due to their efficient use of structured grids. Structured grids, however, not only difficulty representing complex domains and adapting to complicated flow features, but also are rather inefficient for simulating flows at high Reynolds numbers. The lack of efficiency stems from the need to resolve the viscous sublayer, which requires very fine resolution in all three directions near the wall. Structured grids make use of a stretching to reduce the normal grid spacing but must carry the fine resolution in the streamwise and spanwise directions throughout the domain. The unnecessarily fine grid for much of the domain leads to disturbingly high grid estimates. Chapman (1979), and later Moin & Jimenez (1993), pointed out that, in order to advance the technology to airfoils at flight Reynolds numbers, structured grids must be abandoned in lieu of what are known as nested or unstructured grids. The finite element method can efficiently solve the Navier-Stokes equations on unstructured grids. Although the CPU cost per time step per element is somewhat higher than structured grid methods, this effect is more than offset by the reduction in the number of elements. The use of unstructured grids, coupled with the advances in dynamic subgrid-scale modeling such as those made by Germano et al. (1991) and Ghosal et al. (1994), make LES of an airfoil tractable. We have chosen the NACA 4412 airfoil at maximum lift as the first simulation since this flow has not been successfully simulated with the Reynolds-averaged Navier-Stokes equations.