ADI方法求解完全跨声速非定常小扰动方程

CALCULATION OF THE COMPLETE TRANSONIC UNSTEADY SMALL DISTURBANCE EQUATION BY ADI METHOD

本文修改了Whitlow的ADI格式,以便求解另一变形的完全跨声速小扰动方程。本文给出的NACA64A006和NACA0012翼型的非定常跨声速流动的计算结果与其它数值结果和实验结果的比较表明本文格式是合理的且便于实际应用的。

One of the most extensively used programs for transonic unsteady aerodynamics analysis is the LTRAN2 code obtained by Ball-haus and Goorjian[1]. The code used ADI algorithm to solve the low frequency approximation of the transonic small-disturbance ( TSD ) equation with steady boundary conditions satisfied at the airfoil, in the wake, and on the computational boundary.Use of the low frequency approximation and steady state aerofoil and wake conditions limits the frequency of unsteady motion that can be analyzed with LTRAN2. Houwink and van der Vooren[3] extend the range of application of LTRAN2 by adding unsteady terms to the airfoil and wake boundary conditions with a resulting code LTRAN2-NLR. Hessenius and Goorjian[4] added a time derivative term in the downstream far-field condition as well as unsteady airfoil and wake conditions with a resulting code termed LTRAN2-HI. All these codes are confined within relatively low frequency motions ( because of the use of the low frequency TSD ) and have to place the computational boundaries far enough from the airfoil due to the steady far-field boundary conditions used. Whitlow[5] developed a new code XTRAN2L which removed the low frequency limitation by solving the complete TSD equation. Also the computational domain was reduced by the implementation of nonreflecting far-field boundary conditions which were consistent with the complete TSD.In the present paper, the algorithm due to Whitlow has been modified to resolve a new nontime-linearized small disturbance equation of transonic unsteady flows derived from the complete TSD equation. The unsteady solution is divided into an unsteady part and a steady part which is just the solution of the steady transonic flow around the airfoil in mean position. Since only existing in the form of coefficients in the new equation developed, the steady solution by any numerical method is available. Although theoretically the steady solution is that of the steady transonic small disturbance equation (TSD), for practical applications the solutions of the other steady transonic equations such as full potential equation (TFP) and Euler equation can also be used. In fact the difference between the steady solution of TSD equation and that of TFP equation (or Euler equation) is evident only in the regions near the shocks for thin airfoils, with the latter providing more accurate shock position and strength. By the idea of dividing the unsteady solution into an unsteady part and a steady part, we just want to make use of the above point. It is predicted that the present algorithm can provide more accurate shock motion and strength providing the steady solution of the TFP equation (or Euler equation) is used as the steady part.As a test of the present algorithm, results of unsteady transonic flows are presented for a NACA64A006 airfoil with oscillating trailing-edge flap hinged at its 3/4 chord and a NACA0012 airfoil oscillating in pitch about its 1/4 chord. The numerical results show that the present algorithm is reasonable. Because the steady solution by any numerical method can be used in the present algorithm, we believe that the present algorithm is more convenient for engineering applications.