The purpose of the last three sections is to demonstrate by representative examples that control theory can be used to formulate computationally feasible procedures for aerodynamic design. The cost of each iteration is of the same order as two flow solutions, since the adjoint equation is of comparable complexity to the flow equation, and the remaining auxiliary equations could be solved quite inexpensively. Provided, therefore, that one can afford the cost of a moderate number of flow solutions, procedures of this type can be used to derive improved designs. The approach is quite general, not limited to particular choices of the coordinate transformation or cost function, which might in fact contain measures of other criteria of performance such as lift and drag. For the sake of simplicity certain complicating factors, such as the need to include a special term in the mapping function to generate a corner at the trailing edge, have been suppressed from the present analysis. Also it remains to explore the numerical implementation of the design procedures proposed in this paper.

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Aerodynamic design via control theory

A distributed Lagrange multiplier/fictitious domain method for particulate flows

Finite element formulations based on stabilized bilinear and linear equal-order-interpolation velocity-pressure elements are presented for computation of steady and unsteady incompressible flows. The stabilization procedure involves a slightly modified Galerkin/least-squares formulation of the steady-state equations. The pressure field is interpolated by continuous functions for both the quadrilateral and triangular elements used. These elements are employed in conjunction with the one-step and multi-step time integration of the Navier-Stokes equations. The three test cases chosen for the performance evaluation of these formulations are the standing vortex problem, the lid-driven cavity flow at Reynolds number 400, and flow past a cylinder at Reynolds number 100.

Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements

This is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary Navier–Stokes equations. Second-order error estimates are proven for spatial discretization, using conforming or nonconforming elements. The results indicate a fluid-like behavior of the approximations, even in the case of large data, so long as the solution remains regular. The analysis is based on sharp a priori estimates for the solution, particularly reflecting its behavior as$t \to 0$ and as $t \to \infty $. It is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the Navier–Stokes equations, as it depends on nonlocal compatibility conditions for the data. The results which are presented here are independent of such compatibility conditions, which cannot be verified in practice.

Finite element approximation of the nonstationary Navier-Stokes problem. I : Regularity of solutions and second-order error estimates for spatial discretization

Publisher Summary This chapter discusses stabilized finite element formulations for incompressible flow computations. Finite element computation of incompressible flows involve two main sources of potential numerical instabilities associated with the Galerkin formulation of a problem. The stabilization techniques that are reviewed more extensively than others are the Galerkin/ least-squares (GLS), streamline-upwind/ Petrov–Galerkin (SUPG), and pressure-stabilizing/Petrov–Galerkin (PSPG) formulations. The SUPG stabilization for incompressible flows is achieved by adding to the Galerkin formulation a series of terms, each in the form of an integral over a different element. These integrals involve the product of the residual of the momentum equation and the advective operator acting on the test function. The natural boundary conditions are the conditions on the stress components, and these are the conditions assumed to be imposed at the remaining part of the boundary. The interpolation functions used for velocity and pressure are piecewise bilinear in space and piecewise linear in time. These computations involve no global coefficient matrices, and therefore need substantially less computer memory and time compared to noniterative solution of the fully discrete equations. It is suggested that for two-liquid flows, the solution and variational function spaces for pressure should include the functions that are discontinuous across the interface.

Stabilized Finite Element Formulations for Incompressible Flow Computations

Numerical methods for the navier-stokes equations. Applications to the simulation of compressible and incompressible viscous flows

On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods (II). Application to transonic flow simulations

Numerical simulation of compressible Navier-Stokes flows

On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods (I) least square formulations and conjugate gradient solution of the continuous problems

https://hal.inria.fr/inria-00073303/document

M. O. Bristeau

Papers

Numerical methods for the navier-stokes equations. Applications to the simulation of compressible and incompressible viscous flows

On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods (II). Application to transonic flow simulations

Numerical simulation of compressible Navier-Stokes flows

On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods (I) least square formulations and conjugate gradient solution of the continuous problems

Controllability Methods for the Computation of Time-Periodic Solutions; Application to Scattering