Volume I - Section V - CFD Methodology 
Page V - 3 
The governing equations have actually been known for over 150 years. In the 19th century, two 
scientists, Navier and Stokes, described the equations for a viscous, compressible fluid, which 
are now known as the Navier-Stokes equations. These equations form a set of differential 
equations. The generic form of these relationships follow the advection diffusion equation, 5.1: 
^-(p(p) + ^'v(pV9-r p gr^cp) = 5^ (5.1) 
transient + advection - diffusion = source 
The variable phi (<p) represents any of the predicted quantities such as air velocity, temperature, 
or concentration at any point in the three-dimensional model. All subsequent terms are identified 
in section 5.6. This equation is derived by considering a small, or finite, volume of fluid. The 
left- hand side of the equation refers to the change in time of a variable within this volume added 
to that advected into it, minus the amount diffused out. This is in turn equal to the amount of the 
variable flux (i.e., momentum, mass, thermal energy) that is added or subtracted within the finite 
volume. Though deceptively simple, only the emergence of ever faster computers over the past 
two decades has made it possible to solve the real world problems governed by this equation. 
Despite their relatively old age, the Navier-Stokes equations have never been solved analytically. 
The numerical techniques used to solve these coupled mathematical equations are commonly 
known as computational fluid dynamics, or CFD. At the present time, CFD is the only means of 
generating complete solutions. 
The Navier-Stokes equations are a set of partial differential equations that represent the equations 
of motion governing a fluid continuum. The set contains five equations, mass conservation, three 
components of momentum conservation, and energy conservation. In addition, certain properties 
of the fluid being modeled, such as the equation of state, must be specified. The equations 
themselves can be classified as nonlinear, and coupled. Nonlinear, for practical purposes, means 
that solutions to the equations cannot be added together to get solutions to a different problem 
(i.e., solutions cannot be superimposed). Coupled means that each equation in the set of five 
depends upon the others; they must all be solved simultaneously. If the fluid can be treated as 
incompressible and nonbuoyant, then the conservation of energy equation can be decoupled from 
the others and a set of only four equations must be solved simultaneously, with the energy 
equation being solved separately, if required. 
The majority of fluid dynamics flows are modeled by the Navier-Stokes equations. The 
Navier-Stokes equations also describe the behavior of turbulent flows. The many scales of 
motion that turbulence contains, especially its microscales, cause the modeling of turbulent 
processes to require an extremely large number of grid points. These simulations are performed 
today, and fall into the realm of what is termed direct numerical simulations (DNS). The DNS 
are currently only able to model a very small region, in the range of one cubic foot, using 
supercomputers. Differential equations represent differences, or changes, of quantities. The 
changes can be with respect to time or spatial locations. For example, in Newton's Law of 
