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The basic equations for the stability analysis of flow over three dimensional swept wings are developed and numerical methods for their solution are surveyed. The equations for nonlinear stability analysis of three dimensional disturbances in compressible, three dimensional, nonparallel flows are given. Efficient and accurate numerical methods for the solution of the equations of stability theory were surveyed and analyzed.
Fractional and tempered fractional partial differential equations (PDEs) are effective models of long-range interactions, anomalous diffusion, and non-local effects. Traditional numerical methods for these problems are mesh-based, thus struggling with the curse of dimensionality (CoD). Physics-informed neural networks (PINNs) offer a promising solution due to their universal approximation, generalization ability, and mesh-free training. In principle, Monte Carlo fractional PINN (MC-fPINN) estimates fractional derivatives using Monte Carlo methods and thus could lift CoD. However, this may cause significant variance and errors, hence affecting convergence; in addition, MC-fPINN is sensitive to hyperparameters. In general, numerical methods and specifically PINNs for tempered fractional PDEs are under-developed. Herein, we extend MC-fPINN to tempered fractional PDEs to address these issues, resulting in the Monte Carlo tempered fractional PINN (MC-tfPINN). To reduce possible high variance and errors from Monte Carlo sampling, we replace the one-dimensional (1D) Monte Carlo with 1D Gaussian quadrature, applicable to both MC-fPINN and MC-tfPINN. We validate our methods on various forward and inverse problems of fractional and tempered fractional PDEs, scaling up to 100,000 dimensions. Our improved MC-fPINN/MC-tfPINN using quadrature consistently outperforms the original versions in accuracy and convergence speed in very high dimensions.
The unsteady incompressible Navier-Stokes equations are formulated in terms of vorticity and stream function in generalized curvilinear orthogonal coordinates to facilitiate analysis of flow configurations with general geometries. The numerical method developed solves the conservative form of the transport equation using the alternating-direction implicit method, whereas the stream-function equation is solved by direct block Gaussian elimination. The method is applied to a model problem of flow over a back-step in a doubly infinite channel, using clustered conformal coordinates. One-dimensional stretching functions, dependent on the Reynolds number and the asymptotic behavior of the flow, are used to provide suitable grid distribution in the separation and reattachment regions, as well as in the inflow and outflow regions. The optimum grid distribution selected attempts to honor the multiple length scales of the separated-flow model problem. The asymptotic behavior of the finite-differenced transport equation near infinity is examined and the numerical method is carefully developed so as to lead to spatially second-order accurate wiggle-free solutions, i.e., with minimum dispersive error. Results have been obtained in the entire laminar range for the backstep channel and are in good agreement with the available experimental data for this flow problem.
The article begins by examining the fundamentals of traditional deterministic design philosophy. The initial section outlines the concepts of failure criteria and limit state functions two traditional notions that are embedded in deterministic design philosophy. This is followed by a discussion regarding safety factors (a possible limit state function) and the common utilization of statistical concepts in deterministic engineering design approaches. Next the fundamental aspects of a probabilistic failure analysis are explored and it is shown that deterministic design concepts mentioned in the initial portion of the article are embedded in probabilistic design methods. For components fabricated from ceramic materials (and other similarly brittle materials) the probabilistic design approach yields the widely used Weibull analysis after suitable assumptions are incorporated. The authors point out that Weibull analysis provides the rare instance where closed form solutions are available for a probabilistic failure analysis. Since numerical methods are usually required to evaluate component reliabilities, a section on Monte Carlo methods is included to introduce the concept. The article concludes with a presentation of the technical aspects that support the numerical method known as fast probability integration (FPI). This includes a discussion of the Hasofer-Lind and Rackwitz-Fiessler approximations.
The two decades old high order central differencing via entropy splitting and summation-by-parts (SBP) difference boundary closure of Ols- son & Oliger (1994), Gerritsen & Olsson (1996), and Yee et al. (2000) is revisited. The entropy splitting is a form of skew-symmetric splitting of the nonlinear Euler flux derivatives. Central differencing applied to the entropy splitting form of the Euler flux derivatives together with SBP difference operators will, hereafter, be referred to as entropy split schemes. This study is prompted by the recent growing interest in numerical methods for which a discrete entropy conservation law holds, a discrete global entropy conservation can be proved and/or the numerical method possesses a stable entropy in the framework of SBP difference operators and L2-energy norm estimate. The objective of this paper is to recast the entropy split scheme as the re- cent definition of an entropy stable method for central differencing with SBP operators for both periodic and non-periodic boundary conditions for non- linear Euler equations. Standard high order spatial central differencing as well as high order central spatial DRP (dispersion relation preserving) spatial differencing is part of the entropy stable methodology framework. Long time integration of 2D and 3D test cases is included to show the comparison of this efficient entropy stable method with the Tadmor-type of entropy conservative methods. Studies also include the comparison among the three skew-symmetric splittings on their nonlinear stability and accuracy performance without added numerical dissipations for smooth flows. These are, namely, entropy splitting, Ducros et al. splitting and the Kennedy & Grub- ber splitting.
A technique is described for converting a constrained optimization problem into an unconstrained problem. The technique transforms one of more objective functions into reduced objective functions, which are analogous to goal constraints used in the goal programming method. These reduced objective functions are appended to the set of constraints and an envelope of the entire function set is computed using the Kreisselmeir-Steinhauser function. This envelope function is then searched for an unconstrained minimum. The technique may be categorized as a SUMT algorithm. Advantages of this approach are the use of unconstrained optimization methods to find a constrained minimum without the draw down factor typical of penalty function methods, and that the technique may be started from the feasible or infeasible design space. In multiobjective applications, the approach has the advantage of locating a compromise minimum design without the need to optimize for each individual objective function separately.
An analysis is presented of the equations of motion for steady MHD shock waves proopagating in interstellar clouds, for boundary conditions that preclude C shocks. In addition to J shocks, in which the neutral fluid component becomes subsonic at an adiabatic jump front, the equations admit a new class of solutions, called C-asterisk shocks, in which the transition to subsonic flow occurs continuously at a sonic point. Numerical methods are developed for computing the structure of J and C-asterisk shocks propagating in diffuse interstellar clouds. The effects of chemical, ionization, and recombination processes are included in this treatment. An alternative numerical method, which uses artificial viscosity to facilitate integration through sonic points, is analyzed and shown to be invalid. A set of exemplary solutions, computed for realistic shock parameters, shows that C-asterisk shocks occur for a broad range of conditions relevant to diffuse interstellar clouds.
A numerical method is proposed for solving the time-dependent compressible Navier-Stokes equations in two dimensions. The equations are time-split into a hyperbolic part and a parabolic part. This paper describes an explicit numerical method for solving the hyperbolic (inviscid) part. The hyperbolic operator is explicit, conservative, uses characteristic relations to predict convection and pressure fields, and is stable under a specified condition.
The calculation of unsteady flows and the development of numerical methods for solving unsteady boundary layer equations and their application to the flows around important configurations such as oscillating airfoils are presented. A brief review of recent work is provided with emphasis on the need for numerical methods which can overcome possible problems associated with flow reversal and separation. The zig-zag and characteristic box schemes are described in this context, and when embodied in a method which permits interaction between solutions of inviscid and viscous equations, the characteristic box scheme is shown to avoid the singularity associated with boundary layer equations and prescribed pressure gradient. Calculations were performed for a cylinder started impulsively from rest and oscillating airfoils. The results are presented and discussed. It is conlcuded that turbulence models based on an algebraic specification of eddy viscosity can be adequate, that location of translation is important to the calculation of the location of flow separation and, therefore, to the overall lift of an oscillating airfoil.
Computational Fluid Dynamics (CFD) has played a major role in the resurgence of hypersonic flight, on the premise that numerical methods will allow performance of simulations at conditions for which no ground test capability exists. Validation of CFD methods is being established using the experimental data base available, which is below Mach 8. It is important, however, to realize the limitations involved in the extrapolation process as well as the deficiencies that exist in numerical methods at the present time. Current features of CFD codes are examined for application to propulsion system components. The shortcomings in simulation and modeling are identified and discussed.
Computational Fluid Dynamics (CFD) has played a major role in the resurgence of hypersonic flight, on the premise that numerical methods will allow performance of simulations at conditions for which no ground test capability exists. Validation of CFD methods is being established using the experimental data base available, which is below Mach 8. It is important, however, to realize the limitations involved in the extrapolation process as well as the deficiencies that exist in numerical methods at the present time. Current features of CFD codes are examined for application to propulsion system components. The shortcomings in simulation and modeling are identified and discussed.
Recognizing the considerable advances that have been made in computational fluid dynamics, the Internal Fluid Mechanics Division of NASA Lewis Research Center sponsored this symposium with the objective of providing a forum for exchanging information regarding recent developments in numerical methods, physical and chemical modeling, and applications. This conference publication is a compilation of 4 invited and 34 contributed papers presented in six sessions: algorithms one and two, turbomachinery, turbulence, components application, and combustors. Topics include numerical methods, grid generation, chemically reacting flows, turbulence modeling, inlets, nozzles, and unsteady flows.
CFD has played a major role in the resurgence of hypersonic flight, on the premise that numerical methods will allow us to perform simulations at conditions for which no ground test capability exists. Validation of CFD methods is being established using the experimental data base available, which is below Mach 8. It is important, however, to realize the limitations involved in the extrapolation process as well as the deficiencies that exist in numerical methods at the present time. Current features of CFD codes are examined for application to propulsion system components. The shortcomings in simulation and modeling are identified and discussed.
This paper describes and discusses the textbook, Fundamentals of Computational Fluid Dynamics by Lomax, Pulliam, and Zingg, which is intended for a graduate level first course in computational fluid dynamics. This textbook emphasizes fundamental concepts in developing, analyzing, and understanding numerical methods for the partial differential equations governing the physics of fluid flow. Its underlying philosophy is that the theory of linear algebra and the attendant eigenanalysis of linear systems provides a mathematical framework to describe and unify most numerical methods in common use in the field of fluid dynamics. Two linear model equations, the linear convection and diffusion equations, are used to illustrate concepts throughout. Emphasis is on the semi-discrete approach, in which the governing partial differential equations (PDE's) are reduced to systems of ordinary differential equations (ODE's) through a discretization of the spatial derivatives. The ordinary differential equations are then reduced to ordinary difference equations (O(Delta)E's) using a time-marching method. This methodology, using the progression from PDE through ODE's to O(Delta)E's, together with the use of the eigensystems of tridiagonal matrices and the theory of O(Delta)E's, gives the book its distinctiveness and provides a sound basis for a deep understanding of fundamental concepts in computational fluid dynamics.
The objective was to develop and implement a numerical method capable of solving the nonlinear partial differential equations governing heat, mass, and momentum transfer in a 3-D cylindrical geometry in order to examine the character of convection in an asymmetrically heated cylindrical ampoule. The details of the numerical method, including verification tests involving comparison with results obtained from other methods, are presented. The results of the study of 3-D convection in an asymmetrically heated cylinder are described.
A three-dimensional unsteady, viscous aerodynamic analysis has been developed for the flow inside a transonic, high-through-flow, single stage compressor. The compressor stage is comprised of a low-aspect-ratio rotor and a closely coupled stator. The analysis is based on a numerical method for solving the three-dimensional Navier-Stokes equation for unsteady viscous flow through multiple turbomachinery blade rows. The method solves the fully three-dimensional Navier-Stokes equation with an implicit scheme. A two-equation turbulence model with a low-Reynolds-number modification is applied for the turbulence closure. A third-order accurate upwinding scheme is used to approximate convection terms while a second-order accurate central difference scheme is used for the discretization of the viscous terms. A second-order accurate scheme is employed for the temporal discretization. The numerical method is applied to study the unsteady flow field inside a transonic, high-through-flow, axial compressor stage. The numerical results are compared with available experimental data.
This report summarizes research done over the past two years as part of NASA Grant NCC 2-729. This research has been aimed at validating numerical methods for computing the flow about the complete F-18 HARV at alpha = 30 deg and alpha = 45 deg. At 30 deg angle of attack, the flow about the F-18 is dominated by the formation, and subsequent breakdown, of strong vortices over the wing leading-edge extensions (LEX). As the angle of attack is increased to alpha = 45 deg, the fuselage forebody of the F-18 contains significant laminar and transitional regions which are not present at alpha = 30 deg. Further, the flow over the LEX at alpha = 45 deg is dominated by an unsteady shedding in time, rather than strong coherent vortices. This complex physics, combined with the complex geometry of a full aircraft configuration, provides a challenge for current computational fluid dynamics (CFD) techniques. The following sections present the numerical method and grid generation scheme that was used, a review of prior research done to numerically model the F-18 HARV, and a discussion of the current research. The current research is broken into two main topics: the effect of engine-inlet mass-flow rate on the F-18 vortex breakdown position, and the results using a refined F-18 computational model to compute the flow at alpha = 30 deg and alpha = 45 deg.
This research has been aimed at validating numerical methods for computing the flow about the complete F-18 HARV at alpha = 30 deg and alpha = 45 deg. At 30 deg angle of attack, the flow about the F-18 is dominated by the formation, and subsequent breakdown, of strong vortices over the wing leading-edge extensions (LEX). As the angle of attack is increased to alpha = 45 deg, the fuselage forebody of the F-18 contains significant laminar and transitional regions which are not present at alpha = 30 deg. Further, the flow over the LEX at alpha = 45 deg is dominated by an unsteady shedding in time, rather than strong coherent vortices. This complex physics, combined with the complex geometry of a full-aircraft configuration, provides a challenge for current computational fluid dynamics (CFD) techniques. The following sections present the numerical method and grid generation scheme that was used, a review of prior research done to numerically model the F-18 HARV, and a discussion of the current research. The current research is broken into three main topics; the effect of engine-inlet mass-flow rate on the F-18 vortex breakdown position, the results using a refined F-18 computational model to compute the flow at alpha = 30 deg and alpha = 45 deg, and research done using the simplified geometry of an ogive-cylinder configuration to investigate the physics of unsteady shear-layer shedding. The last section briefly summarizes the discussion.