# Interdisciplinary Applied Mathematics

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The nozzles fabricated by (Bayt and Breuer, 2000) were planar (see Fig-

Cn

FIGURE 6.34. Mass discharge coefficient: comparisons between DSMC results (circles) and experimental results in (Bayt and Breuer, 2000). (Courtesy of M. Ivanov.)

ure 6.31), while the ones presented by (Grisnik et al., 1987) were axisym-metric, and the one by (Janson and Helvajian, 1997) had a square throat. In all cases a high Reynolds number is achieved, and the chamber pressure is relatively high. This, in turn, implies that the rarefaction effects are not dominant, with the possible exception in the results of (Grisnik et al., 1987) corresponding to low stagnation pressure. However, in these experiments the characteristic length scale is large, which implies reduced values of Knudsen number as well.

Despite the continuum operating regime, both DSMC and Navier-Stokes simulations overpredict significantly the thrust efficiency as well as the mass discharge coefficient. This is shown in Figures 6.34 and 6.35 for a nozzle with 34    p,m    throat    and 7:1    expansion    ratio    fabricated    by    (Bayt,    1999)

and (Bayt and Breuer, 2000). The computational results in (Markelov and Ivanov, 2001), utilized both two-dimensional and three-dimensional simulations with the program SMILE, which is described below. Independent two-dimensional simulations based on compressible Navier-Stokes equations performed in (Bayt, 1999) gave similar results.

More specifically, a comparison between two- and three-dimensional results by (Markelov and Ivanov, 2001) shows a decrease of the discharge coefficient from 99.6% to 89.3%, and of thrust efficiency from 87.1% to 80.6%. However, in addition to these adverse three-dimensionality effects there are other physical factors and experimental uncertainties that contribute to this large overprediciton by simulation. The following is a list of possible effects investigated systematically in (Bayt, 1999): 9

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