Luigi Martinelli

Aerodynamic Shape Optimization of Complex Aircraft Configurations via an Adjoint Formulation

This work describes the implementation of optimization techniques based on control theory for complex aircraft configurations. Here control theory is employed to derive the adjoint differential equations, the solution of which allows for a drastic reduction in computational costs over previous design methods (13, 12, 43, 38). In our earlier studies (19, 20, 22, 23, 39, 25, 40, 41, 42) it was shown that this method could be used to devise effective optimization procedures for airfoils, wings and wing-bodies subject to either analytic or arbitrary meshes. Design formulations for both potential flows and flows governed by the Euler equations have been demonstrated, showing that such methods can be devised for various governing equations (39, 25). In our most recent works (40, 42) the method was extended to treat wing-body configurations with a large number of mesh points, verifying that significant computational savings can be gained for practical design problems. In this paper the method is extended for the Euler equations to treat complete aircraft configurations via a new multiblock implementation. New elements include a multiblock-multigrid flow solver, a multiblock-multigrid adjoint solver, and a multiblock mesh perturbation scheme. Two design examples are presented in which the new method is used for the wing redesign of a transonic business jet.

Fast multigrid method for solving incompressible hydrodynamic problems with free surfaces

We develop of a finite-volume multigrid Euler scheme for solving three-dimensional, fully nonlinear ship wave problems. The flowfield and the a priori unknown free surface location are calculated by coupling the free surface kinematic and dynamic equations with the equations of motion for the bulk flow. The evolution of the free surface boundary condition is linked to the evolution of the bulk flow via a novel iteration strategy that allows temporary leakage through the surface before the solution is converged. The method of artificial compressibility is used to enforce the incompressibility constraint for the bulk flow. A multigrid algorithm is used to accelerate convergence to a steady state

Multigrid unsteady Navier-Stokes calculations with aeroelastic applications

An implicit approach to the solution of the unsteady two-dimensional Navier-Stokes equations is presented. After spatial discretization, the resulting set of coupled implicit non-linear equations is solved iteratively. This is accomplished using well proven convergence acceleration techniques for explicit schemes such as multigrid, residual averaging, and local time-stepping in order to achieve large computational efficiency in the calculation. Calculations are performed in parallel using a domain decomposition technique with optimized communication requirements. In addition, particular care is taken to minimize the effect of numerical dissipation with flux-limited dissipation schemes. Results for the unsteady shedding flow behind a circular cylinder and for a pitching NACA 64A010 airfoil are presented with experimental comparisons, showing the feasibility of accurate, efficient, time-dependent viscous calculations. Finally, a two-dimensional structural model of the cylinder is coupled with the unsteady flow solution, and time responses of the deflections of the structure are analyzed. Nomenclature C l coefficient of lift C d coefficient of drag Cx, Cy damping coefficients in the two coordinate directions D cylinder diameter, cylinder drag E total energy (internal plus kinetic) E(wij) convective Euler fluxes f , g Euler flux vectors H total enthalpy Kx, Ky spring constants in the two coordinate directions L airfoil section lift (normal to free stream), positive up m cylinder mass M∞ free stream Mach number n frequency, 1/sec NS(wij) viscous flux residual for cell i,j p static pressure qi heat flux component R(wij) total flux residual for cell i,j R * modified residual R, S viscous flux vectors ReD Reynolds number based on the diameter St Strouhal frequency, St = nD U∞ T static temperature u, v cartesian velocity components U∞ free stream velocity Vij volume of i,j cell w vector of flow variables xt, yt mesh cartesian velocity components ∆α pitching motion forcing amplitude ∆t implicit real time step γ ratio of specific heats, γ = 1.4 ρ air density σij viscous stress tensor components ω f frequency of the forced oscillations Ω, ∂Ω cell element and boundary Introduction U NSTEADY flow solvers are becoming a necessary part of the toolkit of the computational fluid dynamicist. In order to solve problems which are naturally unsteady (such as vortex shedding flows, moving boundary problems, fluid-structure interaction flows, etc.) it is essential to develop numerical schemes which provide accurate solutions at a reasonable cost. Therefore, computational efficiency is of paramount importance for unsteady numerical solutions. As the governing equations …

Flux-Limited Schemes for the Compressible Navier-Stokes Equations

Several high-resolution schemes are formulated with the goal of improving the accuracy of solutions to the full compressible Navier-Stokes equations. Calculations of laminar boundary layers at subsonic, transonic, and supersonic speeds are carried out to validate the proposed schemes. It is concluded that these schemes, which were originally tailored for nonoscillatory shock capturing, yield accurate solutions for viscous flows. The results of this study suggest that the formulation of the limiting process is more important than the choice of a particular flux splitting technique in determining the accuracy of computed viscous flows. Symmetric limited positive and upstream limited positive schemes hold the promise of improving the accuracy of the results, especially on coarser grids. HE calculation of compressible flows at transonic, supersonic, and hypersonic Mach numbers requires the implementation of nonoscillatory discrete schemes which combine high accuracy with high resolution of shock waves and contact discontinuities. These schemes must also be formulated in such a way that they facilitate the treatment of complex geometric shapes. In the past decade numerous schemes have been developed to meet these requirements in conjunction with the solution of the Euler equations.l More recently, the application of such schemes to the Navier-Stokes equations has produced algorithms which have progressively gained acceptance as analysis tools in the aerospace industry. There remains, however, a need to understand and improve Navier-Stokes schemes beyond the current state of the art. The most compelling reason for this rests on the fact that shock capturing requires the construction of schemes which are numerically dissipative, a requirement which could affect the global accuracy of the solution of the physical viscous problem. In a recent paper2 Jameson has shown that a theory of nonoscillatory schemes can be developed for scalar conservation laws based upon the local extremum diminishing (LED) principle that maxima should not increase and minima should not decrease. Moreover, although it is equivalent to the total variation diminishing principle (TVD) for one-dimensional problems, the LED principle can be applied naturally to multidimensional problems on both structured and unstructured meshes. This recent development has shed new light on the principles underlying the construction of both high-resolution switched and flux-limited dissipation schemes. In particular, it allowed the new formulation of two families of flux-limited schemes denominated, symmetric limited positive (SLIP) and upstream limited positive (USLIP), respectively. The present work merges several dissipation schemes based on both the SLIP and USLIP construction with a well-developed cell-centered, finite-volume formulation for solving the two-dimensional Navier-Stokes equations.3 The aim is to analyze and validate these new discretizations for the solution of viscous flow problems. In Sec. II the design principles of nonoscillator y discrete approximations to a scalar convection equation are reviewed together

A multigrid method for the Navier Stokes equations

A multigrid method for solving the compressible Navier Stokes equations is presented. The dimensionless conservation equations are discretized by a finite volume technique and time integration is performed by using a mltistage explicit algorithm. Convergence to a steady state. is enhanced by local time stepping, implicit smoothing of the residuals and the use of mltiple grids. The raethod has been implemented in two different ways: firstly a cell centered and secondly a corner point formulation ( i . e . the unknown variables are defined either at the center of a computational cell or at its vertices). laminar and turbulent two dimensional flows over airfoils. Computed results are presented for

Continuous Adjoint Method for Unstructured Grids

Adjoint-based shape optimization methods have proven to be computationally efficient for aerodynamic problems. The majority of the studies on adjoint methods have used structured grids to discretize the computational domain. Because of the potential advantages of unstructured grids for complex configurations, in this study we have developed and validated a continuous adjoint formulation for unstructured grids. The hurdles posed in the computation of the gradient for unstructured grids are resolved by using a reduced gradient formulation. The methods to impose thickness constraints on unstructured grids are also discussed. The results for two- and three-dimensional simulations of airfoils and wings in inviscid transonic flow are used to validate the design procedure. Finally, the design procedure is applied to redesign the shape of a transonic business jet configuration; we were able to reduce the inviscid drag of the aircraft from 235 to 216 counts resulting in a shock-free wing. Although the Euler equations are the focus of the study in this paper of the adjoint-based approach, the solution of the adjoint system and gradient formulation can be conceptually extended to viscous flows. The approach presented in this study has been successfully used by the first and third authors for viscous flows using structured grids. However, particular aspects of the design process, such as the robustness of the mesh deformation process for unstructured grids, need more attention for viscous flows and are therefore the subject of ongoing research.

Shock wave propagation and dispersion in glow discharge plasmas

Spark-generated shock waves were studied in glow discharges in argon and argon–nitrogen mixtures. Ultraviolet filtered Rayleigh scattering was used to measure radial profiles of gas temperature, and the laser schlieren method was used to measure shock arrival times and axial density gradients. Time accurate, inviscid, axisymmetric fluid dynamics computations were run and results compared with the experiments. Our simulation show that changes in shock structure and velocity in weakly ionized gases are explained by classical gas dynamics, with the critical role of thermal and multi-dimensional effects (transverse gradients, shock curvature, etc.). A direct proof of the thermal mechanism was obtained by pulsing the discharge. With a sub-millisecond delay between starting the discharge and shock launch, plasma parameters reach their steady-state values, but the temperature is still low, laser schlieren signals are virtually identical to those without the discharge, differing dramatically from the signals in discharges with fully established temperature profiles.

A new high resolution scheme for compressible viscous flows with shocks

A new flux splitting and limiting technique which yields one-point stationary shock capturing is presented. The technique is applied to the full NavierStokes and Reynolds Averaged Navier-Stokes equations. Calculations of laminar boundary layers at subsonic and supersonic speeds are presented together with calculations of transonic flows around airfoils. The results exhibit very good agreement with theoretical solutions and existing experimental data. It is found that. the proposed scheme improves the resolution of viscous flows while maintaining excellent one-point shock capturing characteristics. d

Mesh Refinement and Modeling Errors in Flow Simulation

AIAA, Fluid Dynamics Conference, 27th, New Orleans, LA, June 17-20, 1996 We present a perspective on verification and validation of CFD tools for analysis and design, identifying principal sources of error due to approximations in the physical model, numerical discretization, and implementation. Issues in algorithm design and trade-offs between modelling accuracy and computational costs are discussed in detail. Computational examples are drawn from the authors work in several applications areas. (Author)

Accelerating three-dimensional Navier-Stokes calculations

This paper addresses the widely observed breakdown in multigrid performance for turbulent Navier-Stokes computations on highly stretched meshes. Extending previous work in two dimensions, two alternative preconditioned multigrid methods are proposed based on an examination of the analytic expressions for the preconditioned Fourier footprints in an asymptotically stretched boundary layer cell. These methods provide for efficient multigrid performance by ensuring that all error modes are effectively damped inside the boundary layer. The schemes also strive to balance the trade-offs between operation count, storage overhead, and parallel scalability. The first of these methods is implemented for the present work and is shown to dramatically accelerate convergence for three-dimensional turbulent Navier-Stokes calculations.