CFD 10 : Partial Differential Equations (PDEs)

From the mathematical viewpoint, the equations of fluid flows and heat transfer are partial differential equations (PDE).

Certain properties of the equations have profound effect on the behavior of solutions and, significantly for us, on the choice
of numerical method.


Heat Equation

It expresses the energy conservation principle in the case of conduction heat transfer with constant physical properties and absent sources of internal heat generation:

where u(x, t ) is the temperature field and a^2 = κ/ρC is the temperature diffusivity coefficient. In fact, the same equation can be used to describe many other processes, such as, for example, diffusion of an admixture in a quiescent fluid or evolution of an initially sharp velocity gradient in a viscous flow. In the one-dimensional case, the equation reduces to

Wave Equation

describes wavelike phenomena such as sound propagation or oscillations of a string or membrane. In the one-dimensional case, the equation is

Linear Convection Equation

Another, even simpler, equation can be used as a representative of the equations with wavelike solutions. This is
the so-called linear convection equation

Laplace and Poisson Equations

can be considered as a version of the heat equation when ∂u/∂t = 0.An important generalization is the Poisson equation

where f is a known function of spatial coordinates. The simplest PDE form of the two-dimensional case u = u(x, y) is given below :

Burgers and Generic Transport Equations

where u = u(x, t ) and μ ≥ 0 is a constant coefficient. The equation was suggested by J. M. Burgers in 1948 as a one-dimensional model for the Navier-Stokes dynamics, and, presumably, for turbulence in fluid flows. The terms of the equation can be considered as counterparts of unsteady, convective, and viscous terms of the momentum conservation equation of the Navier-Stokes system.

A modification often considered in the literature is the one dimensional generic transport equation:

where φ is a transported and diffused scalar field (e.g., temperature) and u(x, t ) is a known function acting as a velocity-like transporting agent.

It has become clear with time that turbulence is an essentially three dimensional phenomenon and cannot be modeled by both above equations . Similarly, modified equation is not a good model for the majority of heat and mass transfer processes, which are either two- or three-dimensional. It has also become clear that the equations serve as excellent benchmarks for development and testing of CFD methods.