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Today high-performance computing systems and new analytical and numerical techniques enable engineers to explore the use of advanced materials for shell design. This paper reviews some of the historical developments of shell buckling analysis and design. The paper concludes by identifying key research directions for reliable and robust methods development in shell stability analysis and design.
A numerical method for solving the compressible form of the unsteady Navier-Stokes equations is described. This method was originally presented in 1970 and has since been modified during the development of computer programs at Ames for implementing models that account for the effects of turbulence in shock-induced separated flows. Although this paper does not describe the turbulence models themselves, a complete description of the basic numerical method is given with emphasis on the choice of a computational mesh for high Reynolds number flows, finite-difference approximations for mixed partial derivatives, extension of the Courant-Friedrichs-Lewy stability condition for viscous flows, mesh boundary conditions, and numerical smoothing for strong shock-wave calculations.
In this paper we consider an isothermal finite liquid bridge under zero-gravity. A sinusoidal interfacial shear stress drives a nonparallel flow in the bridge. The linear stability of this flow to three-dimensional disturbances (which may deform the cylindrical gas-liquid interface) is determined numerically by solving an elliptic eigenvalue problem. Previous results on shear stabilization of capillary break-up of axially unbounded cylindrical interfaces containing a parallel flow are hereby extended to a nonparallel flow. The turning flow regions influence the stability significantly. However, for the particular cases considered, a small area in parameter space remains where the capillary instability is suppressed through interfacial shear. Second, non-axisymmetric oscillatory instabilities are found which originate from an interaction of the interface deformation and the basic flow. These instabilities may be the isothermal limit of the oscillatory instabilities observed in float-zone crystal growth.
We propose a neural entropy-stable conservative flux form neural network (NESCFN) for learning hyperbolic conservation laws and their associated entropy functions directly from solution trajectories, without requiring any predefined numerical discretization. While recent neural network architectures have successfully integrated classical numerical principles into learned models, most rely on prior knowledge of the governing equations or assume a fixed discretization. Our approach removes this dependency by embedding entropy-stable design principles into the learning process itself, enabling the discovery of physically consistent dynamics in a fully data-driven setting. By jointly learning both the flux function and a corresponding entropy, NESCFN promotes conservation and entropy dissipation, which is critical for long-term stability and fidelity in the system of hyperbolic conservation laws. Furthermore, numerical results demonstrate that the method achieves stability and conservation over extended time horizons and accurately captures shock propagation speeds, even without oracle access to future-time solution profiles in the training data.
The stability of a decelerating boundary-layer flow is investigated experimentally and numerically. Experimentally, a flat plate having a Blasius boundary layer is decelerated in an 18 m towing tank. The boundary layer becomes unstable to two-dimensional waves, which break down into three-dimensional patterns, hairpin vortices, and finally turbulent bursts when the vortices lift off the wall. The unsteady boundary-layer equations are solved numerically to generate instantaneous velocity profiles for a range of boundary and initial conditions. A quasi-steady approximation is invoked and the stability of local velocity profiles is determined by solving the Orr-Sommerfeld equation using Chebyshev matrix methods. Comparisons are made between the numerical predictions and the experimentally observed instabilities.
A numerical study was performed to investigate boundary conditions for bleed openings that are needed in the numerical simulation of critical flow through turbojet inlets with bleed. Nine different boundary conditions (BCs) were devised and examined. Of the BCs examined, three were able to stabilize the terminal shock near the inlet's throat. This numerical study was based on the ensemble-averaged, unsteady, three-dimensional, compressible Navier-Stokes equations closed by the Baldwin-Lomax algebraic turbulence model. Solutions to the Navier-Stokes equations were obtained by using a hybrid implicit-explicit method (Beam and Warming/MacCormack). The multiblock grid system used in the solution procedure was generated by solving a set of elliptic partial differential equations. Solutions obtained were compared with available experimental data.
A systematic study is presented to guide the selection of a numerical solution strategy for URANS computation of a subsonic transport configuration undergoing simulated forced oscillation about its pitch axis. Forced oscillation is central to the prevalent wind tunnel methodology for quantifying aircraft dynamic stability derivatives from force and moment coefficients, which is the ultimate goal for the computational simulations. Extensive computations are performed that lead in key insights of the critical numerical parameters affecting solution convergence. A preliminary linear harmonic analysis is included to demonstrate the potential of extracting dynamic stability derivatives from computational solutions.
We consider the transition to turbulence in three-dimensional reconnection of a magnetic neutral sheet. We find that the transition can occur via a three-step process. First, the sheet undergoes the usual tearing instability. Second, the tearing mode saturates to form a two-dimensional quasi-steady state. Third, this secondary equilibrium is itself unstable when it is perturbed by three-dimensional disturbances. Most of this paper is devoted to the analysis and simulation of the three-dimensional linear stability properties of the two-dimensional saturated tearing layer. The numerical simulations are performed with a semi-implicit, pseudospectral-Fourier collocation algorithm. We identify a three-dimensional secondary linear stability which grows on the ideal timescale. An examination of the modal energetics reveals that the largest energy transfer is from the mean field to the three-dimensional field, with the two-dimensional field acting as a catalyst.
A finite element of Galerkin type semidiscrete method is proposed for numerical solution of a linear hyperbolic partial differential equation. The question of stability is reduced to the stability of a system of ordinary differential equations for which Dahlquist theory applied. Results of separating the part of numerical solution which causes the spurious oscillation near shock-like response of semidiscrete scheme to a step function initial condition are presented. In general all methods produce such oscillatory overshoots on either side of shocks. This overshoot pathology, which displays a behavior similar to Gibb's phenomena of Fourier series, is explained on the basis of dispersion of separated Fourier components which relies on linearized theory to be satisfactory. Expository results represented.
A new, accurate, and nondiffusive method for three-dimensional advection of trace species is presented. The method preserves tracer structures by conserving the second-order moments of the spatial distribution of tracer during advection. Upstream transport and second-order tracer distribution are described, and the moments of the tracer distribution about the center of a grid box are formally defined and related to the polynomial distribution. Formulas are presented which describe how the moments of a grid box are decomposed into a unique set of moments centered about each subbox and how they are reassembled into new grid boxes. A one-dimensional example if tracer transport is given, and limits necessary to maintain positive tracer concentrations are derived. The accuracy and stability of this method are analytically examined, and numerical experiments testing the effective resolution are presented. The method is compared with other methods for numerical advection of tracers.
Various tools for the aeroelastic stability and response analysis of rotor blades in hover and forward flight were developed and incorporated in a comprehensive package capable of performing aeroelastic tailoring of rotor blades in forward flight. The results indicate that substantial vibration reductions, of order 15-40%, in the vibratory hub shears can be achieved by relatively small modifications of the initial design. Furthermore the optimized blade can be up to 20% lighter than the original design. Accomplishments are reported for the following tasks: (1) finite element modeling of rotary-wing aeroelastic problems in hover and forward flight; (2) development of numerical methods for calculating the aeroelastic response and stability of rotor blades in forward fight; (3) formulation of the helicopter air resonance problem in hover with active controls; and (4) optimum design of rotor blades for vibration reduction in forward flight.
In an attempt to increase computational efficiency in the numerical solution of highly nonlinear problems in solid and fluid mechanics, underintegrated finite element methods have been employed by many analysts. Underintegration refers to the use of a rule of an order lower than that required to integrate polynomial integrands exactly. The main drawback of this technique is related to the production of rank-deficient stiffness matrices, or equivalently an expanded kernel of the governing linear momentum operators. Such a development can introduce numerical instabilities. In order to overcome this difficulty, artificial stiffness or viscosity methods, or other stabilization methods have been proposed. One approach involves the elimination of spurious modes in a postprocessing operation. The present study is concerned with this a posteriori elimination method, taking into account the results which can be expected from it, and some of its possible extensions.
This project was concerned with the determination of conditions of guaranteed stability and instability for thermocapillary convection in a model of the float-zone crystal-growth process. This model, referred to as the half-zone, was studied extensively, both experimentally and theoretically. Our own earlier research determined, using energy-stability theory, sufficient conditions for stability to axisymmetric disturbances. Nearly all results computed were for the case of a liquid with Prandtl Number Pr = 1. Attempts to compute cases for higher Prandtl numbers to allow comparison with the experimental results of other researchers were unsuccessful, but indicated that the condition guaranteeing stability against axisymmetric disturbances would be a value of the Marangoni number (Ma), significantly higher than that at which oscillatory convection was observed experimentally. Thus, additional results were needed to round out the stability picture for this model problem. The research performed under this grant consisted of the following: (1) computation of energy-stability limits for non-axisymmetric disturbances; (2) computation of linear-stability limits for axisymmetric and non-axisymmetric disturbances; (3) numerical simulation of the basic state for half- and full-zones with a deformable free surface; and (4) incorporation of radiation heat transfer into a model energy-stability problem. Each of these is summarized briefly below.
A numerical simulation-based study to investigate the stability properties of inherently three-dimensional plasma magnetic configurations is presented. These configurations, called stellarators and heliotrons, use external helical coils to provide the twisting magnetic field that is required to confine ionized plasma particles. In contrast to the tokamak configuration, which nominally has the property of axisymmetry, electrical current does not need to be induced in stellarator and heliotron plasma. Experiments have shown that these three-dimensional configurations are remarkably robust to macroscopic instability, even when driven to relatively high levels of plasma pressure. Our understanding of how the plasma is able to withstand macroscopic instability will be enhanced by the work proposed here. The effort will generalize existing numerical modeling capability, and it will apply the new capability to model macroscopic dynamics in these three-dimensional configurations. The development work has been applied to the NIMROD simulation code (https://nimrodteam.org) that was originally based on axisymmetric problem domains. It has been generalized to accommodate the three-dimensional shaping of the plasma volume in stellarators and heliotrons. Its use of equilibrium data has also been generalized, and the implementation allows different numerical representation of the magnetic field. Analysis has been performed to inform the selection of numerical methods, and a method to accelerate the iterative solution of the necessary algebraic systems has been developed and implemented. Verification of the new NIMSTELL code on benchmark problems has also been an important part of the effort.
The global asymptotic nonlinear behavior of 1 1 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODES) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDES.
The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.
A plasma instability that strongly influences the efficiency and lifetime of electromagnetic plasma accelerators was quantitatively measured. Experimental measurements of dispersion relations (wave phase velocities), spatial growth rates, and stability boundaries are reported. The measured critical wave parameters are in excellent agreement with theoretical instability boundary predictions. The instability is current driven and affects a wide spectrum of longitudinal (electrostatic) oscillations. Current driven instabilities, which are intrinsic to the high-current-carrying magnetized plasma of the magnetoplasmadynmic (MPD) accelerator, were investigated with a kinetic theoretical model based on first principles. Analytical limits of the appropriate dispersion relation yield unstable ion acoustic waves for T(i)/T(e) much less than 1 and electron acoustic waves for T(i)/T(e) much greater than 1. The resulting set of nonlinear equations for the case of T(i)/T(e) = 1, of most interest to the MPD thruster Plasma Wave Experiment, was numerically solved to yield a multiparameter set of stability boundaries. Under certain conditions, marginally stable waves traveling almost perpendicular to the magnetic field would travel at a velocity equal to that of the electron current. Such waves were termed current waves. Unstable current waves near the upper stability boundary were observed experimentally and are in accordance with theoretical predictions. This provides unambiguous proof of the existence of such instabilites in electromagnetic plasma accelerators.
The properties of the flat spin mode of a general aviation configuration have been studied through analysis of rotary balance data, numerical simulation, and analytical study of the equilibrium state. The equilibrium state is predicted well from rotary balance data. The variations of yawing moment and pitching moment as functions of sideslip have been shown to be of great importance in obtaining accurate modeling. These dependencies are not presently available with sufficient accuracy from previous tests or theories. The stability of the flat spin mode has been examined extensively using numerical linearization, classical perturbation methods, and reduced order modeling. The stability exhibited by the time histories and the eigenvalue analyses is shown to be strongly dependent on certain static cross derivatives and more so on the dynamic derivatives. Explicit stability criteria are obtained from the reduced order models.