绕钝前缘机翼跨音速势流的松弛计算及其稳定性与收敛性讨论

本文在机翼钝前缘处用精确速势方程和精确的边界条件,其他地方用纵向大扰动而横向小扰动的速势方程和相应的边界条件,联立求解。数值算例1为矩形机翼,展弦比λ=12,翼剖面为NACA0012,自由流的马赫数M_∞=0.63,迎角α=2°,翼根剖面压力分布的计算结果与二元亚音速精确数值解（Sells,1968）接近。算例2为NACA RM A51G31实验的机翼,垂直于1/4弦线的翼剖面为NACA64A010,其后退角χ_1/4=45°,λ=3,根梢比η=2,M_∞=0.4,0.8,0.9,α=2°。计算与实验接近。 本文建立跨音速定常小扰动速势差分方程的线松弛改进迭代在局部线化假设下的稳定性条件和松弛解收敛到原来的微分方程解的条件。这些条件大多数与数值实验相符。

RELAXATION COMPUTATION OF TRANSONIC FLOWS AROUND WINGS WITH BLUNT LEADING-EDGE AND DISCUSSION ON ITS STABILITY AND CONVERGENCE

In this paper, the blunt leading-edge of a wing is taken as mesh points, and there the exact velocity potential equation with central difference scheme and the exact boundary condition are used, while in the other places, the approximate velocity potential equation, which assumes small perturbation in the transverse plane but allows large perturbation in the longitudinal direction, and the corresponding boundary condition are employed. Two numerical examples are following:（1） A rectangular wing having airfoil NACA0012, aspect ratio λ = 12, angle of attack α=2°, free stream Mach number M∞ = 0.63. The computed pressure distribution of the root section agrees with the exact numerical subsonic solution given by Sells （1968）.（ 2 ） The sweepback wing tested by NACA RM A51G31 having airfoil NA-CA64A010 which is perpendicular to 1/4 chord line with sweepback angle X1/4 = 45°, λ = 3 and taper ratio η = 2 ,α=2°, M∞=0.4, 0.8 and 0.9. The computed pressure distributions agree well with those obtained by tests.Under the assumption of local linearization, the stability of the difference equation in line relaxation with Seidel iteration is studied by the von Neumann method and the convergence of the solution of the differential equation equivalent to the above difference equation to the solution of the original differen- tial equation is discussed by the method of separation of variables.The following conclusions are obtained:（ 1 ） The stability condition for the line relaxation with Seidel iteration is 0<ω≤2 at locally subsonic points, where ω is the relaxation factor.（ 2 ） At locally supersonic points, the relaxation is always unstable. The convergence conditions are as follows. Let the steps Δx （chordwise） and Δz （spanwise） perpendicular to the relaxation line.（3） 0<ω<2, at locally subsonic points.（4） 0 <ω<1+ , at locally supersonic points.The numerical experiences agree with the conclusions （ 1 ）, （ 3 ） and （ 4 ）, but do not agree with the conclusion （ 2 ）.