摘要:
本文将无限大激波阵面的激波不稳定性理论[1]推广到矩形截面管道内的激波不稳定性问题.首先,给出这个问题的数学提法,包括扰动方程与三类边界条件.其次,给出扰动方程的普遍解.上游和下游的普遍解分别含有5个待定常数.再次,在一类边界条件和一个假定下,证明了激波前扰动为0,激波后两个声扰动之一为0.边界条件是,X→±∞处扰动物理量为0.假定只讨论激波不稳定性问题,从而可先设ω=iγ,γ是不稳定性增长率,为正实数.另一类边界条件是管壁上法向速度扰动为0,它使波数只能取一组离散值.最后,用扰动激波上的5个守恒方程这一边界条件来决定激波后4个待定常数和扰动激波振幅这个未知量时,导出了色散关系.结果表明,正实数γ确是存在.不稳定激波有两种模式,一种模式为γ=-W·k(W<0)它代表激波的绝对不稳定性,是新得到的模式.另一种模式与过去工作中给出的[2,3]大体相同.本文则进一步给出了这种模式的激波不稳定性增长率,并指出j2((∂V/∂P)H=1+2M为最不稳定点(即无量纲化的不稳定性增长率Г=∞).如果不假定ω是纯虚数,而是复数,其虚部为正实数Im(ω)≥0.本文也严格证明了其不稳定性判据仍有两种模式,ω仍为纯虚数.
Abstract:
The instability theory of shock wave was extended from the case with an infinite to the case of a channel with a rectangular cross section. First, the mathematical formulation of the problem was given which included a system of disturbed equations and three kinds of boundary conditions. Then, the general solutions of the equations upstream and downstream were given and each contained five constants to be determined. Thirdly, under one boundary condition and one assumption, it was proved that all of the disturbances in front of the shock front and one of the two acoustic disturbances behind the shock front should be zero. The boundary condition was that all of the disturbed physical quantities should approach to zero at infinity. The assumption was that only the unstable shock wave was concerned here. So it was reasonable to assume, ω=iY. Ywas the instability growth rate and was a positive real number. Another kind of boundary conditions was that the normal disturbed velocities should be zero at the solid wall of the channel, and it led to the result that the wave number of disturbances could only be a set of discrete values. Finally, a total of five conservation equations across the disturbed shock front was the third kind of boundary conditions which was used to determine the remained four undetermined constants downstream and an undetermined constant representing the amplitude of disturbed shock front. Then a dispersion relation was derived. The results show that a positive real γ does exist, so the assumption made above is self-consistent, and there are two modes, instead of one, for unstable shock. One mode corresponds to γ=-W·k(W<0) It is a newly discovered mode and represents an absolute instability of shock. The instability criterion derived from another mode is nearly the same as the one obtained in [2, 3], in addition, its growth rate is newly derived in this paper, and on this basis, it is further pointed out that at j2(∂v/∂p)H=1+2M the shock wave is most unstable, i.e. its nondimensional growth rate Γ=∞ If ω is assumed to be a complex number with Im(ω≥0) instead of being assumed a pure imaginary number at the beginning, it can be proved in Section V that there are still two modes for the instability criteria, besides, the roots ω of the dispersion equation are still imaginary.
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