Fundamentals of Fluid Dynamics
Ultimately, the goal of Computational Fluid Dynamics (CFD) is to provide a numerical description of fluid flow behaviour. This is achieved through solving the governing equations that are mathematical statements of the physical conservation laws: conserva
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Fundamentals of Fluid Dynamics
5.1
Introduction
Ultimately, the goal of Computational Fluid Dynamics (CFD) is to provide a numerical description of fluid flow behaviour. This is achieved through solving the governing equations that are mathematical statements of the physical conservation laws: • conservation of mass; • balance of momentum (Newton’s second law, the rate of change of momentum equals the sum of forces acting on the fluid) and; • conservation of energy (first law of thermodynamics, the rate of change of energy equals the sum of rate of heat addition to, and the rate of work done on, the fluid). The governing equations represent not only the transport of mass, momentum, and energy but also the interaction of phenomena such as diffusion, convection, boundary layers, and turbulence. This chapter presents the derivation of the governing equations for a single fluid phase, and provides an understanding of the basic processes of fluid flow and the significance of different terms in the governing equations. Furthermore, a CFD user must understand the physical behaviour of fluid motion, as it is these phenomena that CFD analyses and predicts.
5.2 5.2.1
Fluid Dynamics and Governing Equations Mass Conservation
The conservation of mass is the basic principle which states that matter may neither be created nor destroyed. This means that a fluid in motion moves through a region of space in a way that the mass is conserved. In steady flow, the rate of mass entering the control volume equals the net rate at which mass exits the control volume J. Tu et al., Computational Fluid and Particle Dynamics in the Human Respiratory System, Biological and Medical Physics, Biomedical Engineering DOI 10.1007/978-94-007-4488-2_5, © Springer Science+Business Media Dordrecht 2013
101
102
5 Fundamentals of Fluid Dynamics
Fig. 5.1 Mass flow into and out of a 2D control volume element
(inflow = outflow). In other words, 0=
in
m ˙ −
m ˙
(5.1)
out
Recalling that the mass flow rate is given by m ˙ = ρUA, which is the product of density, average velocity, and cross-sectional area normal to the flow, then the rate at which mass enters the control volume is ρuA, where A = y · 1 for a unit depth, i.e. z = 1 (Fig. 5.1). Thus, m ˙ in = ρuAx = ρu(y · 1)
(5.2)
and the rate at which the mass leaves the surface at x + x may be expressed as ∂(ρu) m ˙ out = (ρu) + x y · 1 (5.3) ∂x Similarly in the y-direction, m ˙ in = ρv(dx · 1)
∂(ρv) y (x · 1) m ˙ out = (ρv) + ∂y
Substituting the mass in and mass out terms into Eq. (5.1) for a two-dimensional control volume we get ∂(ρu) 0 = (ρu) + x (y · 1) − ρu(d · 1) ∂x ∂(ρv) + (ρv) + y (x · 1) − ρv(x · 1) ∂y
5.2 Fluid Dynamics and Governing Equations Fig. 5.2 a Pipe. b trachea with two cross-sectional slices are shown. Cross section 1 is taken just after the larynx and has a cross-sectional area of 1.5 cm2 . Cross section 2 is taken in the main trachea region and has a cross-sectional area of 3.0 cm2
a .
m1 = u1 A1
103
b
A1 = 1.5cm2
cross-section
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