Complementary Method for Deriving Concentration of Diffusing Substance in a Medium for Multi-Dimensional Diffusion

ABSTRACTConcentration of a diffusing substance in a medium was derived in various cases of uni-dimensional diffusion, including in a semi-infinite medium and a plate-shaped medium. Multi-dimensional diffusion involves boundary conditions in each coordinate direction. The algorithm dealing with uni-dimensional case becomes very complicated in multi-dimensional cases. This study proposes an algorithm, which is called the complementary method, that combines complementary functions of the normalized solution in uni-dimensional diffusion case by multiplication to solve those in various multi-dimensional diffusion cases with dramatically simplified mathematics. Besides, the complementary method is used to solve various kinds of boundary conditions for multi-dimensional diffusion.

Turbulent Boundary-Layer Shear Flows

Turbulent shear flows next to solid boundaries are one of the most important types of flow in geology. In such flows, turbulence is generated primarily by boundary effects; vorticity originates near a boundary in association with the velocity gradients that arise from the no-slip condition at the boundary. Such gradients provide a ready source of vorticity for eddies and eddy-like structures to develop in response to the destabilizing effects of inertial forces, and then move outward into the adjacent flow. Eddies are also generated within the wakes of bumps that comprise boundary roughness, for example, sediment particles on the bed of a stream channel (Example Problem 11.4.2). As we have seen in Chapter 14, the fluctuating motions of turbulence involve, over any elementary area, fluxes of fluid momentum that are manifest as apparent (Reynolds) stresses. In addition, the complex motions of eddies and eddy-like structures efficiently advect heat and solutes from one place to another within a turbulent flow, and thereby facilitate mixing of heat and solutes throughout the fluid. For similar reasons, turbulent motions are responsible for lofting fine sediment into the fluid column of a stream channel and in the atmosphere. We will concentrate in this chapter on steady unidirectional flows where the mean streamwise velocity varies only in the coordinate direction normal to a boundary and the mean velocity normal to the boundary is zero. We also will adopt a classic treatment of turbulent boundary flow in developing the idea of L. Prandtl’s mixing-length hypothesis, from which we will obtain the logarithmic velocity law, a function that describes how the mean streamwise velocity varies in the coordinate direction normal to a boundary. In developing Prandtl’s hypothesis, we will see that the presence of apparent stresses associated with fluctuating motions leads to the idea of an eddy viscosity or apparent viscosity. Unlike the Newtonian viscosity, the eddy viscosity is a function of the mean velocity, and therefore coordinate position. This means that the eddy viscosity cannot, in general, be removed from stress terms involving spatial derivatives, as we previously did with the Newtonian viscosity in simplifying the Navier–Stokes equations.

Extension of the Wall-Driven Enclosure Flow Problem to Toroidally Shaped Geometries of Square Cross-Section

The two-dimensional wall-driven flow in an enclosure has been a numerical paradigm of long-standing interest and value to the fluid mechanics community. In this paradigm the enclosure is infinitely long in the x-coordinate direction and of square cross-section (d × d) in the y-z plane. Fluid motion is induced in all y-z planes by a wall (here the top wall) sliding normal to the x-coordinate direction. This classical numerical paradigm can be extended by taking a length L of the geometry in the x-coordinate direction and joining the resulting end faces at x = 0 and x = L to form a toroid of square cross-section (d × d) and radius of curvature Rc. In the curved geometry, axisymmetric fluid motion (now in the r-z planes) is induced by sliding the top flat wall of the toroid with an imposed radial velocity, ulid, generally directed from the convex wall towards the concave wall of the toroid. Numerical calculations of this flow configuration are performed for values of the Reynolds number (Re = ulidd/ν) equal to 2400, 3200, and 4000 and for values of the curvature ratio (δ = d/Rc) ranging from 5.0 · 10−6 to 1.0. For δ ≤ 0.05 the steady two-dimensional flow pattern typical of the classical (straight) enclosure is faithfully reproduced. This consists of a large primary vortex occupying most of the enclosure and three much smaller secondary eddies located in the two lower corners and the upper upstream (convex wall) corner of the enclosure. As δ increases for a fixed value of Re, a critical value, δcr, is found above which the primary center vortex spontaneously migrates to and concentrates in the upper downstream (concave wall) corner. While the sense of rotation originally present in this vortex is preserved, that of the slower moving fluid below it and now occupying the bulk of the enclosure cross-section is reversed. The relation marking the transition between these two stable steady flow patterns is predicted to be δcr1/4 = 3.58 Re-1/5 (δ ± 0.005).

Diffractographic Techniques for Vibration Measurement

The single slit diffraction phenomenon is shown to be useful as a noncontacting vibration measurement technique. The diffracting slit is formed between the vibrating object and stationary reference point. Little or no mass need be added to the object. Several readout techniques are described as well as a method for determining vibratory data along a line, in which case vibratory amplitude may be determined at all points in one coordinate direction.