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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 145 records · Page 8
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NASA NTRS1990

On the use of Lagrangian variables in descriptions of unsteady boundary-layer separation

The Lagrangian description of unsteady boundary layer separation is reviewed from both analytical and numerical perspectives. It is explained in simple terms how particle distortion gives rise to unsteady separation, and why a theory centered on Lagrangian coordinates provides the clearest description of this phenomenon. Some of the more recent results for unsteady three dimensional compressible separation are included. The different forms of separation that can arise from symmetries are emphasized. A possible description of separation is also included when the detaching vorticity layer exits the classical boundary layer region, but still remains much closer to the surface than a typical body-lengthscale.

Cowley, Stephen J.↗
NASA NTRS1990

On the Lagrangian description of unsteady boundary-layer separation. I - General theory

Although unsteady, high-Reynolds number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady 3-D separating flows follow and depend on the symmetry properties of the flow. In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi 2-D with a displacement thickness in the form of a crescent shaped ridge. Physically the singularities can be understood in terms of the behavior of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.

Van Dommelen, Leon L.↗
NASA NTRS1990

On the Lagrangian description of unsteady boundary-layer separation. II - The spinning sphere

A theory to explain the initial stages of unsteady separation was proposed by Van Dommelen and Cowley (1989). This theory is verified for the separation process that occurs at the equatorial plane of a sphere or a spheroid which is impulsively spun around an axis of symmetry. A Lagrangian numerical scheme is developed which gives results in good agreement with Eulerian computations, but which is significantly more accurate. This increased accuracy, and a simpler structure to the solution, also allows verification of the Eulerian structure, including the presence of logarithmic terms. Further, while the Eulerian computations broke down at the first occurrence of separation, it is found that the Lagrangian computation can be continued. It is argued that this separated solution does provide useful insight into the further evolution of the separated flow. A remarkable conclusion is that an unseparated vorticity layer at the wall, a familiar feature in unsteady separation processes, disappears in finite time.

Van Dommelen, Leon L.↗
NASA NTRS1990

The augmented Lagrangian method for parameter estimation in elliptic systems

In this paper a new technique for the estimation of parameters in elliptic partial differential equations is developed. It is a hybrid method combining the output-least-squares and the equation error method. The new method is realized by an augmented Lagrangian formulation, and convergence as well as rate of convergence proofs are provided. Technically the critical step is the verification of a coercivity estimate of an appropriately defined Lagrangian functional. To obtain this coercivity estimate a seminorm regularization technique is used.

Ito, Kazufumi↗
NASA NTRS1992

A new Lagrangian method for real gases at supersonic speed

With the renewed interest in high speed flights, the real gas effect is of theoretical as well as practical importance. In the past decade, upwind splittings or Godunov-type Riemann solutions have received tremendous attention and as a result significant progress has been made both in the ideal and non-ideal gas. In this paper, we propose a new approach that is formulated using the Lagrangian description, for the calculation of supersonic/hypersonic real gas inviscid flows. This new formulation avoids the grid generation step which is automatically obtained as the solution procedure marches in the 'time-like' direction. As a result, no remapping is required and the accuracy is faithfully maintained in the Lagrangian level. In this paper, we give numerical results for a variety of real gas problems consisting of essential elements in high speed flows, such as shock waves, expansion waves, slip surfaces and their interactions. Finally, calculations for flows in a generic inlet and nozzle are presented.

Loh, C. Y.↗
NASA NTRS1993

Lagrangian solution of supersonic real gas flows

The present extention of a Lagrangian approach of the Riemann solution procedure, which was originally proposed for perfect gases, to real gases, is nontrivial and requires the development of an exact real-gas Riemann solver for the Lagrangian form of the conservation laws. Calculations including complex wave interactions of various types were conducted to test the accuracy and robustness of the approach. Attention is given to the case of 2D oblique waves' capture, where a slip line is clearly in evidence; the real gas effect is demonstrated in the case of a generic engine nozzle.

Loh, Ching-Yuen↗
NASA NTRS1993

Variational data assimilation with a semi-Lagrangian semi-implicit global shallow-water equation model and its adjoint

An adjoint model is developed for variational data assimilation using the 2D semi-Lagrangian semi-implicit (SLSI) shallow-water equation global model of Bates et al. with special attention being paid to the linearization of the interpolation routines. It is demonstrated that with larger time steps the limit of the validity of the tangent linear model will be curtailed due to the interpolations, especially in regions where sharp gradients in the interpolated variables coupled with strong advective wind occur, a synoptic situation common in the high latitudes. This effect is particularly evident near the pole in the Northern Hemisphere during the winter season. Variational data assimilation experiments of 'identical twin' type with observations available only at the end of the assimilation period perform well with this adjoint model. It is confirmed that the computational efficiency of the semi-Lagrangian scheme is preserved during the minimization process, related to the variational data assimilation procedure.

Li, Y.↗
NASA NTRS1993

An extended Lagrangian method

A unique formulation of describing fluid motion is presented. The method, referred to as 'extended Lagrangian method', is interesting from both theoretical and numerical points of view. The formulation offers accuracy in numerical solution by avoiding numerical diffusion resulting from mixing of fluxes in the Eulerian description. Meanwhile, it also avoids the inaccuracy incurred due to geometry and variable interpolations used by the previous Lagrangian methods. The present method is general and capable of treating subsonic flows as well as supersonic flows. The method proposed in this paper is robust and stable. It automatically adapts to flow features without resorting to clustering, thereby maintaining rather uniform grid spacing throughout and large time step. Moreover, the method is shown to resolve multidimensional discontinuities with a high level of accuracy, similar to that found in 1D problems.

Liou, Meng-Sing↗
NASA NTRS1946

The Lagrangian multiplier method of finding upper and lower limits to critical stresses of clamped plates

The theory of Lagrangian multipliers is applied to the problem of finding both upper and lower limits to the true compressive buckling stress of a clamped rectangular plate. The upper and lower limits thus bracket the true stress, which cannot be exactly found by the differential-equation approach. The procedure for obtaining the upper limit, which is believed to be new, presents certain advantages over the classical Rayleigh-Ritz method of finding upper limits. The theory of the lower-limit procedure has been given by Trefftz, but, in the present application, the method differs from that of Trefftz in a way that makes it inherently more quickly convergent. It is expected that in other buckling problems and in some vibration problems the Lagrangian multiplier method of finding upper and lower limits may be advantageously applied to the calculation of buckling stresses and natural frequencies.

Budiansky, Bernard↗
NASA NTRS1993

A new Lagrangian method for three-dimensional steady supersonic flows

In this report, the new Lagrangian method introduced by Loh and Hui is extended for three-dimensional, steady supersonic flow computation. The derivation of the conservation form and the solution of the local Riemann solver using the Godunov and the high-resolution TVD (total variation diminished) scheme is presented. This new approach is accurate and robust, capable of handling complicated geometry and interactions between discontinuous waves. Test problems show that the extended Lagrangian method retains all the advantages of the two-dimensional method (e.g., crisp resolution of a slip-surface (contact discontinuity) and automatic grid generation). In this report, we also suggest a novel three dimensional Riemann problem in which interesting and intricate flow features are present.

Loh, Ching-Yuen↗
NASA NTRS1993

The piecewise-linear predictor-corrector code - A Lagrangian-remap method for astrophysical flows

We describe a time-explicit finite-difference algorithm for solving the nonlinear fluid equations. The method is similar to existing Eulerian schemes in its use of operator-splitting and artificial viscosity, except that we solve the Lagrangian equations of motion with a predictor-corrector and then remap onto a fixed Eulerian grid. The remap is formulated to eliminate errors associated with coordinate singularities, with a general prescription for remaps of arbitrary order. We perform a comprehensive series of tests on standard problems. Self-convergence tests show that the code has a second-order rate of convergence in smooth, two-dimensional flow, with pressure forces, gravity, and curvilinear geometry included. While not as accurate on idealized problems as high-order Riemann-solving schemes, the predictor-corrector Lagrangian-remap code has great flexibility for application to a variety of astrophysical problems.

Lufkin, Eric A.↗
NASA NTRS1993

Testing higher-order Lagrangian perturbation theory against numerical simulation. 1: Pancake models

We present results showing an improvement of the accuracy of perturbation theory as applied to cosmological structure formation for a useful range of quasi-linear scales. The Lagrangian theory of gravitational instability of an Einstein-de Sitter dust cosmogony investigated and solved up to the third order is compared with numerical simulations. In this paper we study the dynamics of pancake models as a first step. In previous work the accuracy of several analytical approximations for the modeling of large-scale structure in the mildly non-linear regime was analyzed in the same way, allowing for direct comparison of the accuracy of various approximations. In particular, the Zel'dovich approximation (hereafter ZA) as a subclass of the first-order Lagrangian perturbation solutions was found to provide an excellent approximation to the density field in the mildly non-linear regime (i.e. up to a linear r.m.s. density contrast of sigma is approximately 2). The performance of ZA in hierarchical clustering models can be greatly improved by truncating the initial power spectrum (smoothing the initial data). We here explore whether this approximation can be further improved with higher-order corrections in the displacement mapping from homogeneity. We study a single pancake model (truncated power-spectrum with power-spectrum with power-index n = -1) using cross-correlation statistics employed in previous work. We found that for all statistical methods used the higher-order corrections improve the results obtained for the first-order solution up to the stage when sigma (linear theory) is approximately 1. While this improvement can be seen for all spatial scales, later stages retain this feature only above a certain scale which is increasing with time. However, third-order is not much improvement over second-order at any stage. The total breakdown of the perturbation approach is observed at the stage, where sigma (linear theory) is approximately 2, which corresponds to the onset of hierarchical clustering. This success is found at a considerable higher non-linearity than is usual for perturbation theory. Whether a truncation of the initial power-spectrum in hierarchical models retains this improvement will be analyzed in a forthcoming work.

Buchert, T.↗
NASA NTRS1994

A Lagrangian dynamic subgrid-scale model turbulence

A new formulation of the dynamic subgrid-scale model is tested in which the error associated with the Germano identity is minimized over flow pathlines rather than over directions of statistical homogeneity. This procedure allows the application of the dynamic model with averaging to flows in complex geometries that do not possess homogeneous directions. The characteristic Lagrangian time scale over which the averaging is performed is chosen such that the model is purely dissipative, guaranteeing numerical stability when coupled with the Smagorinsky model. The formulation is tested successfully in forced and decaying isotropic turbulence and in fully developed and transitional channel flow. In homogeneous flows, the results are similar to those of the volume-averaged dynamic model, while in channel flow, the predictions are superior to those of the plane-averaged dynamic model. The relationship between the averaged terms in the model and vortical structures (worms) that appear in the LES is investigated. Computational overhead is kept small (about 10 percent above the CPU requirements of the volume or plane-averaged dynamic model) by using an approximate scheme to advance the Lagrangian tracking through first-order Euler time integration and linear interpolation in space.

Meneveau, C.↗
NASA NTRS1995

Lagrangian mixed layer modeling of the western equatorial Pacific

Processes that control the upper ocean thermohaline structure in the western equatorial Pacific are examined using a Lagrangian mixed layer model. The one-dimensional bulk mixed layer model of Garwood (1977) is integrated along the trajectories derived from a nonlinear 1 1/2 layer reduced gravity model forced with actual wind fields. The Global Precipitation Climatology Project (GPCP) data are used to estimate surface freshwater fluxes for the mixed layer model. The wind stress data which forced the 1 1/2 layer model are used for the mixed layer model. The model was run for the period 1987-1988. This simple model is able to simulate the isothermal layer below the mixed layer in the western Pacific warm pool and its variation. The subduction mechanism hypothesized by Lukas and Lindstrom (1991) is evident in the model results. During periods of strong South Equatorial Current, the warm and salty mixed layer waters in the central Pacific are subducted below the fresh shallow mixed layer in the western Pacific. However, this subduction mechanism is not evident when upwelling Rossby waves reach the western equatorial Pacific or when a prominent deepening of the mixed layer occurs in the western equatorial Pacific or when a prominent deepening of the mixed layer occurs in the western equatorial Pacific due to episodes of strong wind and light precipitation associated with the El Nino-Southern Oscillation. Comparison of the results between the Lagrangian mixed layer model and a locally forced Eulerian mixed layer model indicated that horizontal advection of salty waters from the central Pacific strongly affects the upper ocean salinity variation in the western Pacific, and that this advection is necessary to maintain the upper ocean thermohaline structure in this region.

Shinoda, Toshiaki↗
NASA NTRS1946

The Lagrangian Multiplier Method of Finding Upper and Lower Limits to Critical Stresses of Clamped Plates

The theory of Lagrangian multipliers is applied to the problem of finding both upper and lower limits to the true compressive buckling stress of a clamped rectangular plate. The upper and lower limits thus bracket the truss, which cannot be exactly found by the differential-equation approach. The procedure for obtaining the upper limit, which is believed to be new, presents certain advantages over the classical Raleigh-Rite method of finding upper limits. The theory of the lower-limit procedure has been given by Trefftz but, in the present application, the method differs from that of Trefftz in a way that makes it inherently more quickly convergent. It is expected that in other buckling problems and in some vibration problems problems the Lagrangian multiplier method finding upper and lower limits may be advantageously applied to the calculation of buckling stresses and natural frequencies.

Budiansky, Bernard↗
NASA NTRS1998

A Globally Convergent Augmented Lagrangian Pattern Search Algorithm for Optimization with General Constraints and Simple Bounds

We give a pattern search adaptation of an augmented Lagrangian method due to Conn, Gould, and Toint. The algorithm proceeds by successive bound constrained minimization of an augmented Lagrangian. In the pattern search adaptation we solve this subproblem approximately using a bound constrained pattern search method. The stopping criterion proposed by Conn, Gould, and Toint for the solution of this subproblem requires explicit knowledge of derivatives. Such information is presumed absent in pattern search methods; however, we show how we can replace this with a stopping criterion based on the pattern size in a way that preserves the convergence properties of the original algorithm. In this way we proceed by successive, inexact, bound constrained minimization without knowing exactly how inexact the minimization is. So far as we know, this is the first provably convergent direct search method for general nonlinear programming.

Lewis, Robert Michael↗
NASA NTRS1994

Eulerian-Lagrangian Simulations of Transonic Flutter Instabilities

This paper presents an overview of recent applications of Eulerian-Lagrangian computational schemes in simulating transonic flutter instabilities. This approach, the fluid-structure system is treated as a single continuum dynamics problem, by switching from an Eulerian to a Lagrangian formulation at the fluid-structure boundary. This computational approach effectively eliminates the phase integration errors associated with previous methods, where the fluid and structure are integrated sequentially using different schemes. The formulation is based on Hamilton's Principle in mixed coordinates, and both finite volume and finite element discretization schemes are considered. Results from numerical simulations of transonic flutter instabilities are presented for isolated wings, thin panels, and turbomachinery blades. The results suggest that the method is capable of reproducing the energy exchange between the fluid and the structure with significantly less error than existing methods. Localized flutter modes and panel flutter modes involving traveling waves can also be simulated effectively with no a priori knowledge of the type of instability involved.

Bendiksen, Oddvar O.↗
NASA NTRS1998

Effects of Helicity on Lagrangian and Eulerian Time Correlations in Turbulence

Taylor series expansions of turbulent time correlation functions are applied to show that helicity influences Eulerian time correlations more strongly than Lagrangian time correlations: to second order in time, the helicity effect on Lagrangian time correlations vanishes, but the helicity effect on Eulerian time correlations is nonzero. Fourier analysis shows that the helicity effect on Eulerian time correlations is confined to the largest inertial range scales. Some implications for sound radiation by swirling flows are discussed.

Rubinstein, Robert↗