Gravitational and Shear Instabilities in Compound and Composite Channels

Linear analysis of gravitational instabilities in the presence of a shear layer and shear instabilities in the presence of a free surface is performed. This study is relevant to shallow mixing layers, such as flow in compound and composite channels and inflows at channel junctions. The variations of the channel bed, velocity profile, Froude number, and friction coefficients with the transverse (lateral) coordinate are considered. It is found that there is a threshold Froude number above which the flow is unstable with respect to gravity waves and below which the flow is unstable with respect to shear waves for a certain range of the bed friction number. For values of Froude number larger than the threshold value, the influence of the shear layer and channel walls on the characteristics of the gravitational instability is strong when the channel and the shear layer are of comparable width. This influence reduces as the channel becomes wider and disappears in the limit when the channel width becomes infinite. When the Froude number is below the threshold value, free surface deformation in the form of gravitational waves exerts a strong stabilizing influence on the shear instability. In particular, the value of the critical bed friction number decreases when either the Froude number of the fast stream (main channel) or the slow stream (flood plain) increases. That is, shallow mixing layers become more stable as the Froude number increases. Comparisons of the linear stability calculations with experimental data show reasonable agreement.     

Eddy current probe impedance due to a flaw: The influence of phase-shifted fields

An approximate formula for the change in impedance of an eddy current two-coil probe situated above a conducting medium with a volumetric flaw is derived in this paper. The primary field is generated by the superposition of two phase-shifted alternating currents of the same frequency. The Lorentz reciprocity theorem and a layer approximation are used to obtain the change in impedance. Numerical results for different values of the phase difference are presented. These results can be used for developing more selective eddy current testing methods.     

Interaction of phase-shifted fields of two single-turn coils situated above a conducting medium

An analytical solution is obtained for the problem of the interaction of two phase-shifted electromagnetic fields generated by two coaxial single-turn coils carrying alternating currents of the same frequency, but of different amplitudes and nonzero phase difference, . Two cases are considered: coils situated above a conducting half-space and above a two-layer medium. Numerical results show that is the most important parameter. If the values of and of the other parameters are chosen properly, then the curve representing the change in impedance can lie in any quadrant of the complex plane. These results can be used for developing more sensitive and more selective eddy current testing methods.     

Linear Stability Analysis of Lateral Motions in Compound Open Channels

A linear stability analysis of lateral motions in shallow flow open channels with a free surface is presented. Unlike previous studies, this work does not employ the rigid-lid assumption. Computations revealed that the rigid-lid assumption works well for weak shear flows and∕or small Froude numbers. However, the validity of the rigid-lid assumption is reduced as the Froude number increases. Also, when Reynolds numbers are larger than 1,000, the stability domain is insensitive to the Reynolds number for all Froude numbers. The size of the stability domain is found strongly affected by the velocity and length scales of base velocity and the shape of the velocity profile. The analysis shows that the stability domains obtained by the Manning formula are larger than those obtained by the Chézy formula for the same Froude number. The analysis shows that a compound channel, whose floodplain has larger flow resistance than the main channel does, has larger lateral transport of flow mass, momentum, and energy.     

Range of Validity of the Transient Damping Leakage Detection Method

In a recent paper, an elegant, efficient, and easy to apply transient-based leakage detection method was proposed. The method exploits the fact that friction and leakage damp the modes of transient waves in a different manner. The method involves six major assumptions. These are: (1) the periodic motion in time of each mode is linearly independent of all other modes; (2) the amplitude of the induced transient is small; (3) the magnitude of the leak is small in comparison with the flow rate; (4) the wall friction can be represented by the Darcy–Weisbach equation; (5) the transient is initiated by an instantaneous small amplitude disturbance; and (6) the pipe system is a simple reservoir–pipe–valve type system or reservoir–pipe–reservoir type system. These six assumptions are relaxed and the validity of the transient damping method is assessed. The analysis shows that the first four assumptions do not pose any serious restriction to the applicability of the damping rate method provided that the mathematical model, used to generate the transient head trace in the leak-free pipe, accurately represents the frictional damping in the system. On the other hand, Assumptions (5) and (6) restrict the applicability of the method to systems that do not involve internal boundary conditions, such as junctions and pumps, and to transients triggered by impulses whose duration is smaller than the wave travel time. Extension of this method to complex pipe systems requires that the linearized waterhammer equations are solved under more general initial and boundary conditions. In addition, more investigation in relation to the frequency content of the input signal and its importance in leakage detection is warranted. The general framework used to derive the damping rate method has led to an efficient and direct algorithm for identifying leaks and future research should seek ways to adapt this framework to more complex pipe systems.     

Inhomogeneity correction in calibration of electrical conductivity standards

The method for inhomogeneity investigation of measurement standards for electrical conductivity has been studied in the conductivity range from 2 MS/m to 14 MS/m. For this purpose a planar spiral coil has been developed and characterized. The impedance values of the coil are measured in the air and with a conducting plate above the coil at discrete frequencies from 2 kHz to 70 kHz. By selecting the corresponding operating frequency, relative changes in the conductivity value throughout the sample thicknesses have been measured. It was revealed that inhomogeneity of measurement standards has an effect on further measurements with calibrated conductivity meters. The relative conductivity changes of 0.1% due to inhomogeneity can be detected with the measurement uncertainty (k = 2) of less than 0.3% of the measured value.     

Series solution of an eddy-current problem for a sphere withvarying conductivity and permeability profiles

Two-parameter families of analytical solutions are found for a single-turn coil symmetrically located above a two-layer sphere. Several cases are considered where the conductivity and relative magnetic permeability of the outer spherical layer are functions of the distance ρ from the sphere's center. The change in impedance of the coil is obtained in terms of a series containing Bessel functions. Computational impedance results are presented for different values of the parameters of the problem and the exact conductivity and permeability profiles are given diagrammatically for comparison with the impedance results     

Application of a perturbation method to the solution of eddycurrent testing problems

This paper presents an analytical solution for the change of the impedance of a double conductor line due to eddy currents induced in presence of an infinitely long horizontal cylindrical flaw with vertically symmetric cross-section. The solution, found by a perturbation method, is applicable in the case the conductivities of the flaw and of the surrounding material do not differ by much. Numerical results for flaws of circular and elliptic cross-sections are presented