Surendra N. Tiwari

Models for Infrared Atmospheric Radiation

Publisher Summary This chapter reviews different line and band models for infrared spectral absorption and compares their absorptances and transmittances. It also indicates their limitations and establishes their usefulness for atmospheric applications. The chapter discusses absorption by spectral lines wherein it describes radiative transmittance by spectral lines, absorption of an isolated spectral line in an infinite spectral interval, absorption of a spectral line in a finite spectral interval, and absorption of an overlapping line in a finite spectral interval. It then describes band absorption and explains limiting forms of the total band absorptance, narrow band models, wide band models, band absorptance correlations, comparison of wide band absorptance results, and band emissivity (total emissivity). Theoretical formulations and evaluations of atmospheric transmittance and integrated absorptance of selected infrared bands are presented and transmittance of selected infrared bands, and integrated absorptance of selected infrared bands are discussed. Finally, the chapter illustrates basic equations for calculating the upwelling atmospheric radiance.

Grid sensitivity and aerodynamic optimization of generic airfoils

An algorithm is developed to obtain the grid sensitivity with respect to design parameters for aerodynamic optimization. The procedure is advocating a novel (geometrical) parameterization using spline functions such as NURBS (Non-Uniform Rational B- Splines) for defining the airfoil geometry. An interactive algebraic grid generation technique is employed to generate C-type grids around airfoils. The grid sensitivity of the domain with respect to geometric design parameters has been obtained by direct differentiation of the grid equations. A hybrid approach is proposed for more geometrically complex configurations such as a wing or fuselage. The aerodynamic sensitivity coefficients are obtained by direct differentiation of the compressible two-dimensional thin-layer Navier-Stokes equations. An optimization package has been introduced into the algorithm in order to optimize the airfoil surface. Results demonstrate a substantially improved design due to maximized lift/drag ratio of the airfoil.

First- and second-order sensitivity analysis of finite element equations via automatic differentiation

Sensitivity analysis is a technique for determining derivatives of system responses with respect to design parameters. Among many methods available for sensitivity analysis, automatic differentiation has been proven through many applications in fluid dynamics and structural mechanics to be an accurate and easy method for obtaining derivatives. Nevertheless, the method can be computational ly expensive and can require a high memory space. This paper will apply an automatic differentiation tool, ADIFOR, to a /^-version and an ^-version finite element code to obtain first- and second-order thermal and structural derivatives, respectively. The focus of the study is on the implementation process and the performance of the ADIFOR-enhanced codes for sensitivity analysis in terms of memory requirement, computational efficiency, and accuracy.

An analytical approach to grid sensitivity analysis

Sensitivity analysis in Computational Fluid Dynamics with emphasis on grids and surface parameterization is described. An interactive algebraic grid-generation technique is employed to generate C-type grids around NACA four-digit wing sections. An analytical procedure is developed for calculating grid sensitivity with respect to design parameters of a wing section. A comparison of the sensitivity with that obtained using a finite-difference approach is made. Grid sensitivity with respect to grid parameters, such as grid-stretching coefficients, are also investigated. Using the resultant grid sensitivity, aerodynamic sensitivity is obtained using the compressible two-dimensional thin-layer Navier-Stokes equations.

Application of Lagrangian Blending Functions for Grid Generation Around Airplane Geometries

A simple procedure has been developed and applied for the grid generation around an airplane geometry. This approach is based on a transfinite interpolation with Lagrangian interpolation for the blending functions. By using a Lagrangian interpolation function, it is possible to enforce the grid continuity across the block interfaces without the derivative information. Monotonic rational quadratic spline interpolation has been employed for the grid distributions on the boundaries. This allows any arbitrary grid spacing without overlapping of the grid lines. An efficient computer program has been developed to generate a multiblock grid around a generic airplane geometry. This procedure has proven to be very simple and effective.

Grid and design variables sensitivity analyses for NACA four-digit wing-sections

Two distinct parameterization procedures are developed for investigating the grid sensitivity with respect to design parameters of a wing-section example. The first procedure is based on traditional (physical) relations defining NACA four-digit wing-sections. The second is advocating a novel (geometrical) parameterization using spline functions such as NURBS (Non-Uniform Rational B-Splines) for defining the wing-section geometry. An interactive algebraic grid generation technique, known as Hermite Cubic Interpolation, is employed to generate C-type grids around wing-sections. The grid sensitivity of the domain with respect to design and grid parameters has been obtained by direct differentiation of the grid equations. A hybrid approach is proposed for more geometrically complex configurations. A comparison of the sensitivity coefficients with those obtained using a finite-difference approach has been made to verify the feasibility of the approach. The aerodynamic sensitivity coefficients are obtained using the compressible two-dimensional thin-layer Navier-Stokes equations.

Grid adaption for hypersonic flow

The methods of grid adaption are reviewed and a method is developed with the capability of adaption to several flow variables. This method is based on a variational approach and is an algebraic method which does not require the solution of partial differential equations. Also the method has been formulated in such a way that there is no need for any matrix inversion. The method is used in conjunction with the calculation of hypersonic flow over a blunt nose body. The equations of motion are the compressible Navier-Stokes equations where all viscous terms are retained. They are solved by the MacCormack time-splitting method. A movie has been produced which shows simultaneously the transient behavior of the solution and the grid adaption.

Aerodynamic design optimization using Euler equations and variational methods

Abstract An optimization methodology which uses the conservative field variables, is developed to solve a design optimization problem in fluid dynamical distributed parameter systems. This approach which is completely based on the variational method, is employed to derive the costate partial differential equations (pdes) and their transversality (boundary) conditions from the continuous pdes of the fluid flow. The costate equations coupled with the flow field equations are solved iteratively to get the functional derivative coefficients. Then, these derivative coefficients combined with the flow field variables are used to find the boundary shape which extremizes the performance index (functional). To demonstrate the method through examples, the shape of the nozzle is optimized for the maximum thrust. For this maximization problem, inlet and outlet flow conditions that depend on the upstream and downstream Mach numbers respectively are considered. In order to build confidence in the optimization procedure, high convergences of the state and costate equations were sought and the numerical and analytical solutions (in the form of pressure distributions) of the state equations are compared. In the purely supersonic flow case, the gain in thrust is remarkably high. Even in the supersonic-inlet–subsonic-outlet and the purely subsonic cases, the improvement of the thrust is found to be substantial. As demonstrated through the cases investigated, a new achievement is that the present variational shape optimization approach is capable of resolving flows with shocks.

Numerical solutions of Navier-Stokes equations for a Butler wing

The flow field is simulated on the surface of a given delta wing (Butler wing) at zero incident in a uniform stream. The simulation is done by integrating a set of flow field equations. This set of equations governs the unsteady, viscous, compressible, heat conducting flow of an ideal gas. The equations are written in curvilinear coordinates so that the wing surface is represented accurately. These equations are solved by the finite difference method, and results obtained for high-speed freestream conditions are compared with theoretical and experimental results. In this study, the Navier-Stokes equations are solved numerically. These equations are unsteady, compressible, viscous, and three-dimensional without neglecting any terms. The time dependency of the governing equations allows the solution to progress naturally for an arbitrary initial initial guess to an asymptotic steady state, if one exists. The equations are transformed from physical coordinates to the computational coordinates, allowing the solution of the governing equations in a rectangular parallel-piped domain. The equations are solved by the MacCormack time-split technique which is vectorized and programmed to run on the CDC VPS 32 computer.

Stiffness characteristic and the applications of the method of lines on nonuniform grids

The method of lines is investigated for the numerical solution of the stream-function-and-vorticity form of the Navier-Stokes equations on nonuniform grids. Stiffness characteristics of a linear one-dimensional model equation are examined to establish the feasibility of applying the method to the vorticity equation in two dimensions. The governing equations are transformed from the physical domain with a highly variable grid to a computational domain with a uniform grid. The method of lines is used to solve only the vorticity equation, and the successive-over relaxation technique is used to solve the stream-function equation. It is observed that the transformed governing equations become stiffer with increased concentration of grid points and also as the number of grid points increases. It is also observed that the differencing technique affects the stiffness characteristics. The use of forward differencing is not feasible, and backward differencing is preferable to central differencing for high Reynolds numbers. The results of specific applications for the solution of flow in curved-wall diffusers and a driven cavity demonstrate that the method of lines under certain circumstances is feasible for the numerical solution of physical problems on domains covered with variable grids.