Upender K. Kaul

New boundary constraints for elliptic systems used in grid generation problems

New boundary constraints for elliptic partial differential equations as used in grid generation problems in generalized curvilinear coordinate systems are proposed in this paper. These constraints, based on the principle of local conservation of thermal energy in the vicinity of the boundaries, are derived using the Greens Theorem. They uniquely determine the so called decay parameters in the inhomogeneous terms of these elliptic systems. These constraints are designed for boundary clustered grids where large gradients in physical quantities need to be resolved adequately. It is observed that the present formulation also works satisfactorily for mild clustering. Therefore, a closure for the decay parameter specification for elliptic grid generation problems has been provided resulting in a fully automated elliptic grid generation technique. Thus, there is no need for a parametric study of these decay parameters since the new constraints fix them uniquely. It is also shown that for Neumann type boundary conditions, these boundary constraints uniquely determine the solution to the internal elliptic problem thus eliminating the nonuniqueness of the solution of an internal boundary value grid generation problem with Neumann boundary conditions.

A comparative study of the parabolized Navier-Stokes code using various grid-generation techniques

Abstract The parabolized Navier-Stokes (PNS) equations are used to calculate the flow-field characteristics about the hypersonic research aircraft X-24C. A comparison of the results obtained using elliptic, hyperbolic and algebraic grid generators is presented. The outer bow shock is treated as a sharp discontinuity, and the discontinuities within the shock layer are captured. Surface pressures and heat-transfer results at angles of attack of 6° and 20°, obtained using the three grid generators, are compared. The PNS equations are marched downstream over the body in both Cartesian and cylindrical base coordinate systems, and the results are compared. A robust marching procedure is demonstrated by successfully using large marching-step size with the implicit shock fitting procedure. A correlation is found between the marching-step size. Reynolds number and the angle of attack at fixed values of smoothing and stability coefficients for the marching scheme.

Three-dimensional elliptic grid generation with fully automatic boundary constraints

A new procedure for generating smooth uniformly clustered three-dimensional structured elliptic grids is presented here which formulates three-dimensional boundary constraints by extending the two-dimensional counterpart presented by the author earlier. This fully automatic procedure obviates the need for manual specification of decay parameters over the six bounding surfaces of a given volume grid. The procedure has been demonstrated here for the Mars Science Laboratory (MSL) geometries such as aeroshell and canopy, as well as the Inflatable Aerodynamic Decelerator (IAD) geometry and a 3D analytically defined geometry. The new procedure also enables generation of single-block grids for such geometries because the automatic boundary constraints permit the decay parameters to evolve as part of the solution to the elliptic grid system of equations. These decay parameters are no longer just constants, as specified in the conventional approach, but functions of generalized coordinate variables over a given bounding surface. Since these decay functions vary over a given boundary, orthogonal grids around any arbitrary simply-connected boundary can be clustered automatically without having to break up the boundaries and the corresponding interior or exterior domains into various blocks for grid generation. The new boundary constraints are not limited to the simply-connected regions only, but can also be formulated around multiply-connected and isolated regions in the interior. The proposed method is superior to other methods of grid generation such as algebraic and hyperbolic techniques in that the grids obtained here are C^2 continuous, whereas simple elliptic smoothing of algebraic or hyperbolic grids to enforce C^2 continuity destroys the grid clustering near the boundaries.

Do large structures control their own growth in a mixing layer? - An assessment

Two different two-dimensional free shear layers, the T-layer developing in time from an initial tangential velocity discontinuity separating the two half-spaces, and the S-layer which develops downstream of the origin where two uniform streams of unequal velocity are brought into tangential contact, are compared. Calculations are performed in order to determine to what extent the perturbations induced upstream by large concentrations of vorticity found downstream hasten or retard the subharmonic instability that leads to the formations of these large structures. The results show that the elliptic influence, or the feedback, in a mixing layer is relatively small for small velocity ratios.

Machine Learning for Detecting and Locating Damage in a Rotating Gear

This paper describes a multi-disciplinary damage detection methodology that can aid in detecting and diagnosing a damage in a given structural system, not limited to the example of a rotating gear presented here. Damage detection is performed on the gear stress data corresponding to the steady state conditions. The normal and damage data are generated by a finitedifference solution of elastodynamic equations of velocity and stress in generalized coordinates 1 . The elastodynamic solution provides a knowledge of the stress distribution over the gear such as locations of stress extrema, which in turn can lead to an optimal placement of appropriate sensors over the gear to detect a potential damage. The damage detection is performed by a multi-function optimization that incorporates Tikhonov kernel regularization reinforced by an added Laplacian regularization term as used in semi-supervised machine learning. Damage is mimicked by reducing the rigidity of one of the gear teeth. Damage detection models are trained on a subset of the normal data and are then tested on the damage solution. The precision with which the damaged tooth and the extent of the damage are identified is very encouraging. The present methodology promises to lead to a significant damage detection, diagnosis and prognosis technology for structural health monitoring.