Fluid Mechanics
In this chapter we study finite elements for incompressible fluids (i.e., most liquids and gases). We start by reviewing the governing equations of mass and momentum balance and derive the Navier-Stokes equations. Restricting attention to laminar flow we
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Fluid Mechanics
Abstract In this chapter we study finite elements for incompressible fluids (i.e., most liquids and gases). We start by reviewing the governing equations of mass and momentum balance and derive the Navier-Stokes equations. Restricting attention to laminar flow we then introduce the Stokes system and formulate a finite element method for the velocity and pressure. We discuss the inf-sup condition as a necessary requirement for existence and uniqueness of the solution. Three types of inf-sup stable finite elements are presented. Uzawa’s algorithm and the solution of saddle-point linear systems is briefly touched upon. The lid-driven cavity benchmark is studied numerically. Both a priori and a posteriori error estimates are derived using B-stability. Finally, we introduce Chorin’s classical projection method as a simple numerical method to simulate time-dependent nearly turbulent fluid flow.
12.1 Governing Equations 12.1.1 Conservation of Mass In classical physics mass can neither be destroyed nor created. This means that the mass of any small volume dx of matter (e.g., a fluid) can change over time only by flow in and out of the boundary ds. Letting u denote the flow velocity vector we immediately obtain the following mass balance equation for a fluid occupying the domain ˝. .; P 1/ C .; u n/@˝ D 0
(12.1)
Here, is the density of the fluid and n is the outward pointing unit normal on the boundary @˝. Because d m D dx is the mass of dx, the first term represents the rate of change of mass within the domain. Further, during the small time span dt a M.G. Larson and F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Texts in Computational Science and Engineering 10, DOI 10.1007/978-3-642-33287-6__12, © Springer-Verlag Berlin Heidelberg 2013
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total volume of matter of d m D u nds will flow out of the surface ds. Hence, the second term represents the rate of mass loss through the domain boundary. Now, using the divergence theorem on the surface integral we have P C r .u/ D 0
(12.2)
Assuming a constant density , this simplifies to r uD0
(12.3)
Physically, this means that the volume of any small fluid particle dx does not change under deformation. Such fluids are said to be incompressible. Most everyday fluids (e.g., water) are incompressible to a very high degree.
12.1.2 Momentum Balance Besides mass conservation a fluid also obeys conservation of momentum (i.e., Newton’s second law). Recall that the momentum of a particle with mass m and velocity u is defined as the product p D mu, and that Newton’s second law says that the rate of change of momentum equals the net force F acting on the particle, that is, pP D F . Now, the momentum dp of a small volume of fluid dx is given by dp D udx, so taking into consideration the fact that momentum can be transported in and out of the boundary @˝ of the domain ˝ we also have the following equation for momentum balance. .u; P 1/ C .u; u n/@˝ D F
(12.4)
Here, we can use our knowle
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