含摩擦效应的三维直管中定常可压缩亚音速Euler流

Subsonic flows passing a duct for three-dimensional steady compressible Euler systems with friction

本文研究由带摩擦效应的三维非等熵可压缩Euler方程组描述的管道内气体的定常流动.这种流动在工程中被称为Fanno流.本文在等方截面平直管道中分别构造非平凡的亚音流、超音流和跨音激波.由于对亚音流,三维定常可压缩Euler方程组是典型的拟线性双曲-椭圆复合型方程组,尚无一般理论,本文提出一个源于跨音激波的边值问题,通过证明上述特殊的亚音流关于进出口边界条件的高维扰动的稳定性,说明该边值问题提法的合理性.本文的证明基于对Euler方程组中双曲部分和椭圆部分的主部的分离,以及设计恰当的非线性迭代格式.特别地,由于摩擦效应, Euler方程组中双曲部分和椭圆部分出现了较强的相互作用,诱导出一类含积分非局部项的二阶线性椭圆型方程混合边值问题.本文用Fourier分析方法和二阶椭圆型方程正则性理论等研究了该非局部问题的适定性.

This paper studies steady motion of gas in a rectilinear duct with square cross-sections, governed by the three-dimensional(3-d) non-isentropic compressible Euler equations with a friction term. Such flows are called Fanno flows in engineering. We construct respectively special subsonic flows, supersonic flows and transonic shocks in the duct. Since the 3-d steady compressible Euler equations are of quasi-linear hyperbolicelliptic composite type for subsonic flows, and there is no general theory up to now, we formulate a boundary value problem arising from studies of transonic shocks, and prove the well-posedness of this problem by showing that the special subsonic flows constructed above are stable under small multi-dimensional perturbations. The proof depends on separation of the elliptic and hyperbolic parts in the Euler equations, and designation of a suitable nonlinear iteration scheme. Particularly, there are strong interactions between the elliptic part and the hyperbolic part due to the appearance of friction, and we deduce a linear mixed boundary value problem of a second-order elliptic equation with an integral-type nonlocal term. Its well-posedness is established by applying methods of Fourier analysis and regularity theory of second-order elliptic equations.