Models for the Shock Cell Structure
From the model described in the previous section, it is clear that an accurate representation of the shock cell structure as well as the large scale turbulence structures is necessary for the prediction of broadband shock noise. When a supersonic jet is operated at an under-expanded condition, Pe/Po > 1, where pe and po are the exit and ambient pressures respectively, expansion waves occur at the nozzle exit so that the pressures inside and outside the jet can come into balance. In the overexpanded mode, oblique shock waves are set up in the plume. Tam and Tanna (1982) derived an expression for the fully expanded jet diameter Dj, which is larger than the physical nozzle diameter for the under-expanded case and smaller for the over-expanded case. Pack (1950), following Prandtl (1904), first provided a complete shock cell solution using a vortex sheet approximation to represent the jet shear layer. This solution is valid only for slightly imperfectly expanded jets, and only close to the nozzle exit where the shear layer is thin. However, experimental evidence shows that the region of importance for shock noise generation is close to the end of the potential core, where the shear layer is no longer thin. Further, the effect of turbulence in reducing the shock strength and smoothing sharp discontinuities must be taken into account. Tam et al. (1985) extended Pack’s linear solution to account for the slowly diverging mean flow using the method of multiple scales. The effect of turbulence was simulated through the inclusion of eddy viscosity terms. The most suitable value for the turbulent Reynolds number was determined by comparison of predictions with experimental data. They evaluated the contributions of the higher order terms to the non-parallel correction and concluded that only the first order correction term was significant. Thus, the simpler locally parallel assumption was shown to be adequate for the calculation of the shock cell structure.
Tam et al. (1985) demonstrated very good agreement between their predictions and measured data for a variety of jet operating conditions. Both the axial and radial variations of the pressure field, in terms of shock cell spacing and shock amplitude were well predicted for both over – and under-expanded jets. Morris et al. (1989) extended Tam’s vortex sheet shock cell model for jets of arbitrary geometry using a boundary element technique. Examples for circular, elliptic and rectangular jets were given. Bhat et al. (1990) included the effects of finite jet shear layer thickness and the dissipative effects of the fine-scale turbulence in a shock cell model for elliptic jets. They also concluded that the fundamental waveguide mode could be used as a good approximation for the shock spacing at the end of the potential core and that the higher order modes only contributed to the fine structure of the shock cells near the jet nozzle exit.
The linear shock cell model is valid only for weakly imperfectly expanded jets, with Mj – Mj < 1. Extensive plume surveys carried out at NASA Langley Research Center and reported by Norum and Seiner (1982a) indicated that as the degree of mismatch between the design and fully expanded Mach numbers increased, there was a dramatic change in the shock cell structure. At highly off-design conditions, the strength of the first shock cell increased tremendously while the rest of the shock cells remained regular and quasi-periodic. That is, downstream of the first shock, the shock cell structure for the strongly off-design conditions resembled that of the slightly imperfectly expanded jet. Based on this observation and the fact that the first shock plays only a negligible role in noise generation as noted by Seiner and Norum (1980), Tam (1990) suggested that the linear solution, suitably modified, could be used to model the shock cell structure of even moderately imperfectly expanded jets. Tam (1990) also developed a semi-empirical formula to estimate the initial amplitude of the linear shock cells for this situation.