Key Dimensionless Groups Research Articles

The role of coherent structures in bubble transport by turbulent shear flows

Using Auton's force law for the unsteady motion of a spherical bubble in inhomogeneous unsteady flow, two key dimensionless groups are deduced which determine whether isolated vortices or shear-layer vortices can trap bubbles. These groups represent the ratio of inertial to buoyancy forces as a relaxation parameter [tcy ] = ΔU2/2gx and a trapping parameter [Gcy ] = ΔU/VT where ΔU is the velocity difference across the vortex or the shear layer, x is streamwise distance measured from the effective origin of the mixing layer and VT is the terminal slip speed of the bubble or particle. It is shown here that whilst buoyancy and drag forces can lead to bubbles moving in closed orbits in the vortex flows (either free or forced), only inertial forces result in convergent trajectories. Bubbles converge on the downflow side of the vortex at a location that depends on the inertial and lift forces. It is important to note that the latter have been omitted from many earlier studies.A discrete-vortex model is used to simulate the large-scale unsteady flows within horizontal and vertical mixing layers between streams with velocity difference ΔU. Trajectories of non-interacting small bubbles are computed using the general force law. In the horizontal mixing layer it is found that Γ needs to have a value of about 3 to trap about 50% of the bubbles if Π is about 0.5 and greater if Π is less. The pairing of vortices actually enhances their trapping of bubbles. In the vertical mixing layer bubbles are trapped mainly within the growing vortices but bubbles are concentrated on the downflow side of the vortices as Γ and Π increase. In a companion paper we show that lateral dispersion of bubbles can be approximately described by an advective diffusion equation with the diffusivity about equal to the eddy viscosity, i.e. rather less than the diffusivity of heat or other passive scalars.

The theory of shell structures Aims and methods

This paper discusses in a general way some aspects of the theory of shell structures, and the nature of the theory itself. It emphasises the importance of the interaction between bending and stretching effects in shells, and shows that the “two surface” model of shells provides a useful way of investigating this interaction at both the conceptual and the computational level. A cylindrical shell under various simple kinds of loading demonstrates how the key dimensionless groups emerge from this type of analysis. Some fresh light is thrown on the famous controversy between Love and Rayleigh on the vibration of simple shells. Various problems which arise in the numerical computation of shell structures are discussed. The paper concludes with some general remarks.

Calculation of nonlinear circuits for surge-voltage protection

The paper gives some general solutions of 2-element circuits involving nonlinear resistances combined with ohmic resistance, transmission lines, inductance or capacitance. Solutions are given so that many practical problems of using surge arresters for the protection of transmission lines can be readily solved for realistic surge-voltage waveforms. The inductive- and capacitative-circuit solutions are designed for the easy practical solution of overvoltage-protection problems, particularly those arising in the protection against lightning of semiconductor-rectifier installations feeding d.c. electric-railway systems. A novel way of handling the equations is developed which leads to the development of key dimensionless groups of constants, epitomising the whole nature of the circuit under investigation, and also gives the correct meaning to the nonlinear equivalents of time constants and damping factors.