A Continuation Algorithm for Turbulent Flows Around a 2-D Airfoil

Abstract

A new code has been developed to study the parameter dependence of the nonlinear dynamics of turbulent compressible two dimensional flows. Using this code it is possible to track the change in steady state solutions, as a parameter such as angle of attack is varied, as well as changes in the stability of the solutions. Details of the development of the code and results showing both the stable and unstable steady state solutions, as the angle of attack is varied, are presented. The nonlinear nature of the Navier-Stokes equations gives rise to multiple solutions and complex dynamics as system parameters are varied. For example, hysteresis in the values of the coefficients of lift, drag and pitching moment for an airfoil with varying angle of attack is observed under certain flow conditions. To gain an overall understanding of a fluid problem, it is necessary to consider all possible solutions of the system, both stable and unstable, as the parameters on which it depends are varied. The resulting knowledge has applications in a number of areas: multivalued aerodynamic behaviour is important as the resulting widely different values of lift, and lift to drag ratio, could affect recovery from stall and/or a spin; parameter values beyond which stable steady solutions are unobtainable are important in aeroelastic calculations; attempts to produce reduced order models of complex unsteady aerodynamic stall behaviour require the dynamics of that behaviour to be first identified. There are two main numerical approaches to gaining insight into the possible solutions of a nonlinear system as the parameters on which it depends are varied. The first is to carry out time accurate simulations at a range of fixed parameters. This is the main method used in computational fluid dynamics. Such an approach is usually only able to identify stable solution branches, however Mittal and Saxena 1 conducted a numerical study using a finite element method to predict the static hysteresis around a NACA 0012 airfoil, using a incompressible solver coupled with the Baldwin-Lomax turbulence model. At higher angles of attack results were unsteady and so the time average of these was plotted as the solution. The unstable solution that separates the two branches was not calculated. The alternative approach is to discretise the problem to produce a system of nonlinear equations and then use methods from nonlinear analysis to compute solution paths. This method has the advantage that it can calculate the unstable solutions which are just as important as the stable solutions in driving the overall dynamics of the system. This latter approach has been adopted in this study and used to investigate the behaviour of the steady Navier-Stokes equations. There are several existing software packages available for carrying out the analysis of nonlinear equations, through path following, such as AUTO. However, these packages tend to be designed for low dimensional systems with dynamics dependent on multiple parameters. As a result they use direct solvers for dense matrices which make them unsuitable for analysing high dimensional systems of equations that result from the discretisation of the Navier-Stokes equations. With improvements in computational power of modern processors and increased memory capacity, with direct or iterative sparse linear solvers, it has recently become possible to extend existing nonlinear analysis methods to analyse the Navier-Stokes equations. Previous investigations into Navier-Stokes flows have mainly been on Taylor-Courette type flows 2,3 with Reynolds