Instability Theory of Shock Wave in a Channel

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.