# Interdisciplinary Applied Mathematics

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F/F ideal

FIGURE 6.35. Thrust    coefficient    defined    as    the    ratio    of    actual    to ideal    thrust

force: comparisons between DSMC results (circles/squares) and experimental results in (Bayt and Breuer, 2000). The solid square corresponds to a three-dimensional simulation. (Courtesy of M. Ivanov.)

Transition to turbulence,

Nonequilibrium, and

Outflow conditions.

The main question regarding roughness is whether it can induce shocks. This was examined by (Bayt, 1999) by considering the Mach number based on the roughness height h, where

Mh ж Reh • Knh .

Therefore, if    the    product    Reh    •    Knh    is greater    than    one,    then    shocks may

form. In the experiments of Bayt, Reh ranges from 0.1 to 10, and Kn varies between 0.1 and 5, and thus it is possible for this product to be greater than one. However, for a shock to appear, the roughness height h has to be larger than the height of the subsonic region, which does not seem to be the case for Bayt’s nozzles, as corresponding simulations show. In addition, transition to turbulence does not seem to occur, since the critical Re^9 for supersonic boundary layers is 250 (based on momentum thickness 69), which is    much    larger    than    the    Re^9    of    the    experiments.    In    Section 15.4,

we will show the effects of roughness corresponding to different models on micronozzle performance.

The effects of nonequilibrium and associated “frozen losses” may come about because of the short length of the nozzles, which results in a short residence time for particles. There are two time scales associated with this

process. The    first    relates    to the    relaxation    time,    which is    the    time    for    gas

particles to reach equilibrium. It is defined as

tr = nAcc,

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