Assessment Of Boussinesq Approximation In A Fluid Mechanics Course

There is an absolute need for an in-depth coverage of certain important topics in an undergraduate engineering curriculum especially in the area of Thermodynamics and Fluid Mechanics. This need arises basically from the feedback received from the alumni and also from some members of the Industrial Advisory Board. A small group of employers has also indicated that there is a need for increasing the academic rigor in certain courses. The author is of the opinion that all undergraduate engineering students must know, in addition to various other topics, the five theorems that are normally encountered while treating the subject matter of Thermodynamics and Fluid Mechanics. The five theorems are Green’s Theorem, Gauss’ Theorem, Stokes’ Theorem, Buckingham-Pi Theorem and Boussinesq approximation. The author considers this ‘set’ as a part of accommodating academic rigor. The author has tried to meet these needs while he was teaching courses in Thermodynamics and Fluid Mechanics. In this presentation, the author describes how he has tried to incorporate the principles of Boussinesq approximation in a junior level fluid mechanics course. He has also outlined methods to assess students’ knowledge in certain specific areas. Introduction Boussinesq approximation is named after the French physicist and mathematician Joseph Valentin Boussinesq for his invaluable contributions in the area of hydraulics and fluid mechanics. Boussinesq was the professor of mechanics at the Faculty of Sciences of Paris, before retiring in 1918. There are several mathematical models to describe Boussinesq approximation and Boussinesq equations. Boussinesq approximation is normally encountered in three general areas. 1. Buoyancy: Assuming small differences in density of the fluid, one can utilize Boussinesq approximation for determining buoyancy-driven flow calculations. 2. Waves: Assuming gravitational actions, one can utilize Boussinesq approximation for analyzing the propagation of long water waves on the surface of fluid layer. 3. Viscosity: Eddy Viscosity to model Reynold’s Stresses is another area where Boussinesq approximation has helped in Turbulence modeling. P ge 15214.2 It is essential that students have an adequate background of partial differenital equations before they take a course in Thermodynamics and Fluid Mechanics. In this paper, the author presents some ideas pertaining to the utilization of those mathematical techniques in a Fluid Mechanics course. Boussinesq approximation for water waves is an important topic that majority of engineering students must know and understand. The approximation is valid for non-linear waves and fairly long waves. The students should also understand that fluid being studied is incompressible and is an assumption that is used in the theory of convection. Some instructors are of the opinion that Boussinesq Equations are normally not taught in an undergraduate curriculum. The author is in partial agreement with this idea. Regardless, the author wants to introduce the importance of Boussinesq approximation in an undergraduate curriculum, because many students may not choose to go to a graduate program in fluid mechanics. Boussinesq Equations It is important to observe that buoyancy-driven flow experiments, data collection and calculations have been carried out on horizontal turbulent buoyant jet with small density variations only. Validity of Boussinesq approximation has been verified in these instances. However, if one tries to analyze horizontal turbulent buoyant jet with relatively larger density variations, one finds inadequate data in the literature. Perhaps, there is enough experimental data, availabile only to certain specific researchers. It is possible to obtain an accurate description of wave evolution in coastal regions utilizing Boussinesq-type equations. Boussinesq models gained prominence during the past few decades because of the availability of high-speed computers and simulation software. The original equations were derived by Boussinesq in 1871, using a ‘depth-averaged’ model. However major subsequent contributions to the subject matter took place during the 1960’s, wherein ‘variable-depth’ models were introduced. Let us consider for example, the potential flow over a horizontal bed. Let us consider a three-dimensional space with co-ordinates, x, y, z. However, for this example let us consider only the two dimensional plane x and z. If h is the mean water depth, and z is the vertical coordinate, then z = – h. One can arrive at a Taylor Expansion of the velocity potential η∀(x,z,t) around the bed level, z = – h. Page 15214.3 Assume that u is the horizontal flow velocity component, w is the vertical flow velocity component, g is the acceleration due to gravity and φ∀ is free surface elevation. Then, horizontal and vertical flow velocities can be accounted for while deriving the partial differential equations. It is normal practice to solve a system of conservation equations of an integral model using a fourth order Runge-Kutta technique. The ultimate objective is to obtain a set of numerical solutions to the problem in question. Buoyancy Motion If one assumes that buoyancy can drive the motion even when the temperature variations and the density variations are very small, one can understand the basis for Boussinesq Approximation. Boussinesq Equations can be derived based on these fundamental assumptions. Supposing one considers determining the flow under certain specified conditions. 3 3 3 6 1 ] ) [( x u h u h x t b b ÷ ÷ ? − ÷ ÷ − ÷ ÷ φ φ x t u h x g x u u t u b b b b 2 3 2 2 1 ÷ ÷ ÷ ? ÷ ÷ − ÷ ÷ − ÷ ÷ φ