The stability of the laminar flow in a rectangular duct of an arbitrary aspect ratio is investigated numerically by expanding the flow fields of both the main flow and the disturbance into series of Legendre polynomials and solving the eigenvalue problem of the resulting matrix equation. The stability of the flow is found to depend upon the aspect ratio of the duct and the mode of the disturbance. The flow is unstable to two of the four possible modes of different parity and stable to the other two. With respect to the most unstable mode, the flow is stable or unstable according as the aspect ratio is below or above a critical value of 3.2 respectively, and the critical Reynolds number decreases monotonically with increasing aspect ratio towards the known value of 5772 for plane Poiseuille flow. The flow field of the disturbance shows the existence of strong vortex layers along the critical layer at which the velocity equals the phase velocity of the disturbance.

The velocity field of the Burgers one-dimensional model of turbulence at extremely large Reynolds numbers is expressed as a train of random triangular shock waves. For describing this field statistically the distributions of the intensity and the interval of the shock fronts are defined. The equations governing the distributions are derived taking into account the laws of motion of the shock fronts, and the self-preserving solutions are obtained. The number of shock fronts is found to decrease with time t as t−α, where α (0 [les ] α < 1) is the rate of collision, and consequently the mean interval increases as tα. The distribution of the intensity is shown to be the exponential distribution. The distribution of the interval varies with α, but it is proved that the maximum entropy is attained by the exponential distribution which corresponds to α = ½. For α = ½, the turbulent energy is shown to decay with time as t−1, in good agreement with the numerical result of Crow & Canavan (1970).

A method of multiple-scale expansion is applied to the theory of incompressible isotropic turbulence in order to close the infinite system of cumulant equations. The dynamical equation for the energy spectrum derived from this method is found to give positive-definite solutions at all Reynolds numbers. At large Reynolds numbers the spectrum takes the form of Kolmogorov's$-\frac{5}{3}$power spectrum in the inertial subrange, whose extent increases indefinitely with Reynolds number. The spectrum in the energy-containing range satisfies an inviscid similarity law, so that the rate of energy decay or of viscous dissipation is also independent of the viscosity. In the higher wavenumber region beyond the inertial subrange the spectrum takes a universal form which is independent of its structure at lower wavenumbers. The universal spectrum is composed of three different subspectra, which are, in order of increasing wavenumber, the$k^{-\frac{5}{3}}$spectrum, thek−1spectrum and the exp [−σk1·5] spectrum, σ being a constant. Various statistical quantities such as the energy, the skewness of the velocity derivative, the microscale and the microscale Reynolds number are calculated from the numerical data for the energy spectrum. Theoretical results are discussed in detail in comparison with experimental results.

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The binding position and the electronic properties of acidic proton in anhydrous dodecatungstophosphoric acid, H 3 PW 12 O 40 , were studied by means of 1 H, 2 H NMR, and 1 H-31 P spin-echo double resonance (SEDOR) NMR techniques. 1 H broad line NMR spectrum showed a bell-type resonance line shape with 4.7 kHz line width and MAS NMR spectrum has a single resonance peak, suggesting that the hetero-polyanion PW 12 O 40 3provides a single site to the acidic proton. 2 H NMR spectrum leads to the quadrupole coupling constant (QCC) of 210 kHz and the asymmetry parameter of the electric-field-gradient tensor (η) of 0.15. The value of QCC suggests that the deuterons are bonded to oxygen atoms in the hetero-polyanion very rigidly, and that any motion of the hetero-polyanion does not take place. A remarkable decay of the 1 H-31 P SEDOR signal was observed, due to the dipolar interaction between the central 31 P and three protons strongly bonded to the hetero-polyanion. The P-H distance was evaluated by simulating the decay behavior of the echo to be 0.50 ( 0.02 nm. Comparing this P-H distance with model structures in which protons are bonded to cornershared, edge-shared, or terminal oxygen atom of WO 6 unit in PW 12 O 40 3-, the edge-shared oxygen atom seems to be the most probable binding position for the acidic proton.