Numerical solution of Euler equations for aerofoils in arbitrary unsteady motion

Abstract

This paper is part of a DLR research programme to develop a three-dimensional Euler code for the calculation of unsteady flow fields around helicopter rotors in forward flight. The present research provides a code for the solution of Euler equations around aerofoils in arbitrary unsteady motion. The aerofoil is considered rigid in motion, and an O-grid system fixed to the moving aerofoil is generated once for all flow cases. Jameson's finite volume method using Runge-Kutta time stepping schemes to solve Euler equations for steady flow is extended to unsteady flow. The essential steps of this paper are the determination of inviscid governing equations in integral form for the control volume varying with time in general, and its application to the case in which the control volume is rigid with motion. The implementation of an implicit residual averaging with variable coefficients allows the CFL number to be increased to about 60. The general description of the code, which includes the discussions of grid system, grid fineness, farfield distance, artificial dissipation, and CFL number, is given. Code validation is investigated by comparing results with those of other numerical methods, as well as with experimental results of an Onera two-bladed rotor in non-lifting flight. Some numerical examples other than periodic motion, such as angle-of-attack variation, Mach number variation, and development of pitching oscillation from steady state, are given in this paper.

Development of a Fluid-Structure Interaction Framework Using Unstructured Cartesian CFD Methods

“…There is a trend of underprediction that has also be seen by several other researchers. 36,[41][42][43][44][45] Now, considering the relative motion simulation, there is good agreement compared to the ALE approach, with the largest deviations occurring at the maximum pitch up and pitch down conditions. The normal force coefficient curve for the relative motion case also produces slightly more noise as a result of the computational surface mesh constantly changing as the airfoil sweeps through the stationary Cartesian mesh.…”

Study of the aerodynamics of in-plane motion

“…The second test case is that suggested by Lerat and Sides [10] for in-plane motion. Calculations were performed for a NACA 0012 aerofoil undergoing a motion described by equation (12) with r=Rˆ0:892, M tipˆ0 :60, í rˆ0 :61 and k rˆ0 :185. The angle of incidence and Reynolds number based upon chord were áˆ08 and Re c1…”

“…Habibie et al [11] investigated the aerodynamics of aerofoils subjected to harmonic variations of Mach number and ramping motions (dM=dtˆconstant). Subsequently, Pahlke et al [12,13] compared solutions of the two-and three-dimensional Euler equations at conditions representative of helicopter rotors at various forward flight speeds. These calculations suggest that, for the geometry considered, three-dimensional effects are significant.…”

Unsteady flow around helicopter rotor blade sections in forward flight

“…Qin and Richards" 8 - 19 > compared a sparse finite-difference Newton method with a sparse quasi-Newton method for the solution of the Navier-Stokes equations and concluded that, while the former can demonstrate a quadratic convergence, the latter can be more efficient as the approximate Jacobian is updated rather than calculated. Instead of calculating the Jacobian using finite difference, Badcock and Richards' 2 '" followed an approach to derive the analytical Jacobian using the chain rule to obtain the following expression _3E___3E_ d^dQ^ ... (10) 3Qi " 3Qi + 3Q, 9Q, Here 9E/3Q () and 3E/3Q, are the Jacobians of the flux vector with respect to the left and right states used in the approximate Riemann solver. The remaining terms dQ ( /3Qj and dQi/8Qj arise from the MUSCL interpolation of the primitive variables and were obtained using a symbolic manipulation package, REDUCE.…”

“…32 flux evaluations are required (this is equivalent to 16 right hand side calculations) for second order accuracy of the Jacobian terms together with a small overhead for the calculation of the terms arising from the interpola-tion procedure. Further improvements in computational efficiency can be obtained by using analytical expressions to evaluate all of the terms on the right hand side of Equation (10). The complicated nature of the numerical fluxes arising from Osher's scheme has dissuaded most authors from deriving analytical expressions for the flux Jacobians directly.…”