Kind Bessel Function Research Articles

On Turbulent Flow

Problemson turbulent flow were mostfy discussed using the equations of fluid motion in stationary state expressed in terms of mean velocity, because it is difficult to solve the equations of fluid motion in unstationary state. The author found that we can solve the equations of fluid motion in unstationary state, if we modify and generalize the Navier-Stoke's equations as given in the author's paper read at the meeting of applied mechanics last autumn.Chief results of the solution for the case of turbulent flow and the transition from laminar to turbulent flow are as followings.1. The ratio of frequency of harmonics of the variation of velocity in a turbulent flow to that of fundamental one is not integer but a ratio of the zero of the 2nd kind Bessel's function of zero order to the smallest one.2. The frequency increases to a fixed value from the point of transition and the decreases as the velocity increases.3. The thickness of boundary layer decreases from laminar flow to the point of transition and then increases as the velocity increases. After the boundary layer becomes completely turbulent, the thickness decreases.4. The velocity distribution takes the form of mixed type of lamniar and turbulent flow between the point of transition and turbulent flow.5. The instability of the point of transition relates to the increment of the y-component of velocity v and the limit of this increment determines the limit of the variation of the point of transition. This change of v seems the origin of turbulent flow.

Integrated Intensities of the Diffracted and Transmitted X-rays due to ideally Perfect Crystal (Laue Case)

The analytical expressions of the integrated reflecting power and transmitting power of X-rays for absorbing perfect crystals are obtained as follows: The diffracted wave: \begin{aligned} {R_{H}}^{y}{=}e^{-\mu_{0}t}{\cdot}\frac{\pi}{2}\left[2\sum\limits_{n{=}0}^{\infty}J_{2n+1}(2A)+I_{0}(h)-1\right]. \end{aligned} The transmitted wave: \begin{aligned} {R_{T}}^{y}{=}e^{-\mu_{0}t}{\cdot}(-2\pi)\left[\sum\limits_{m{=}1}^{\infty}m\cos m\beta I_{m}(h)\right]-{R_{H}}^{y}. \end{aligned} In these formulae \begin{aligned} h{=}2A\sqrt{k^{2}+g^{2}}\quad\text{and}\quad \beta{=}\tan^{-1}k/g. \end{aligned} J m is the first kind Bessel function of m -th order and I m is the modified Bessel function of m -th order, and the other notations are the same as those of Zachariasen's text-book on X-ray diffraction (1944). A practical method of obtaining the numerical values of R H y and R T y , and calculated results are shown.