Analysis of frequency-gas pressure characteristics of a quartz-crystal tuning fork with rough surfaces by turbulent flow model using fractal dimension

The frequency-gas pressure characteristics of a quartz-crystal tuning fork with rough surfaces that undergoes vibrating in flexural mode in N₂ gas, was studied experimentally and theoretically. On the assumption that gas flow on a vibrating quartz-crystal tuning fork with rough surfaces is in transition from laminar flow to a mixed fluid state that can be regarded as turbulent flow, so that this turbulent flow increases surface friction resistance. The author derived a phenomenological frequency equation by extending the mixing length theory of turbulent flow on a vibrating plate proposed by L. Prandtl (1925). In this theory, the mixing length was extended to have fractal dimension. Two empirical constants involved in this theoretical frequency equation, indicating frequency-gas pressure characteristics, are obtained from the experimental results by the least-squares method. The theoretical equation obtained here shows better agreement with the experimental results than that previously obtained by P.E. Rouse et al (1953).