Question

Dimensional analysis is a very useful tool for solving fluid mechanics problems. Indeed, it is a method for reducing the number and complexity of experimental variables that affect a given physical phenomenon. By applying the Buckingham Pi theorem, determine the relationship in dimensionless form for the following problems: (a) The period of oscillation T of a water surface wave is assumed to be a function of density rho, wavelength λ, depth h, gravity g, and surface tension Y. i. Establish the variables and their dimensions [2] ii. Write the relationship between the above variables in a dimensionless form. [2] iii. What is the result if Y is negligible? [2] (b) The period T of vibration of a beam is a function of its length L, area moment of inertia I, modulus of elasticity E, density rho, and Poisson's ratio σ. i. Establish the variables and their dimensions [2] ii. Write the relationship between the above variables in a dimensionless form. [2] iii. What further reduction can be made if E and I can occur only in the product form EI? (c) Water at 20∘C flows steadily through a reducing pipe bend of 1 kg weight and 1.5/1000 m3 internal volume, as schematically shown in Figure Q1.1. Knowing that at cross section 1 the pressure is p1​=350kPa, the pipe diameter is D1​=25 cm, and the fluid velocity is V1​=2.2 m/s; whereas, at cross section 2 the pressure is p2​=120kPa, and the pipe diameter is D2​=8 cm i. Calculate the velocity V2​ at the outlet (i.e., at point 2 ); [2] ii. Calculate the total force which must be resisted by the flange bolts in the vertical (y) direction; [3] iii. Calculate the total force which must be resisted by the flange bolts in the horizontal (x) direction.

Answer 1

Dimensional analysis is a fundamental method in engineering, particularly for fluid mechanics and beam vibration problems, allowing for conversion between units and identification of dimensionless variable relationships. When applying dimensional analysis, relationships can often be simplified when certain factors like surface tension or product forms of variables are negligible. Conservation of mass is applied to fluid flow through varying pipe cross-sections to define forces on pipe infrastructure.

Dimensional Analysis in Fluid Mechanics And Vibrating Beams:

Dimensional analysis is a cornerstone of engineering and physical sciences. It serves as a tool to convert quantities into different units and to deduce relationships between physical variables in a dimensionally consistent manner. The Buckingham Pi theorem is a key technique within dimensional analysis which allows for the reduction of complex physical scenarios into simpler, dimensionless relationships. This is especially useful when dealing with fluid mechanics problems where a variety of factors, such as density, velocity, and pressure, come into play.

When applying the Buckingham Pi theorem to the oscillation of a water surface wave, one must consider the dimensions of the period of oscillation (T), density (ρ), wavelength (λ), depth (h), gravity (g), and surface tension (Y). If the surface tension (Y) is deemed negligible, the relationship between the variables simplifies and no longer includes the effects of surface tension.

In the context of a vibrating beam, the period of oscillation (T) is a function of the beam's length (L), area moment of inertia (I), modulus of elasticity (E), density (ρ), and Poisson's ratio (σ). If the modulus of elasticity (E) and the area moment of inertia (I) are always in the product form EI, the equation simplifies further, illustrating how certain pairs of variables can be combined in the dimensional analysis.

Considering fluid flow through a pipe with changing cross-sectional area, the conservation of mass principle dictates that the velocity of the fluid increases as the area decreases at a constriction. This principle can be used to calculate the change in velocity and the resulting forces that must be resisted by the pipe infrastructure, such as flange bolts, where fluid pressures differ at varying cross-sections.