Page 26 - Introduction to Computational Fluid Dynamics

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1.3 NUMERICAL VERSUS ANALYTICAL SOLUTIONS
densely filled medium (a porous body or a shell-and-tube geometry), the resistance 12:20 5
will be a function of the porosity of the medium. Such empirical resistance laws are
terms represent viscous terms arising
often determined from experiments. The S u i
from Stokes’s stress laws that are not accounted for in the ∂ [µ eff ∂u i ] term in
∂x j ∂x j
Equation 1.3.
1.3 Numerical Versus Analytical Solutions
Analytical solutions to our transport equations are rarely possible for the following
reasons:

1. The equations are three-dimensional.
2. The equations are strongly coupled and nonlinear.
3. In practical engineering problems, the solution domains are almost always
complex.

The equations, however, can be made amenable to analytical solutions when
simplified through assumptions. In a typical undergraduate program, students de-
velop extensive familiarity with such analytical solutions that can be represented
in closed form. Thus, in a fluid mechanics course, for example, when fully devel-
oped laminar flow in a pipe is considered, a student is readily able to integrate the
simplified (one-dimensional) momentum equation to obtain a closed-form solution
for the streamwise velocity u as a function of radius r. The assumptions made are
as follows: The flow is steady and laminar, it is fully developed, it is axisymmetric,
and fluid properties are uniform. The solution is then interpreted to yield the scale-
neutral result f × Re = 16. The friction factor f is a practically useful quantity
that enables calculation of pumping power required to force fluid through a pipe.
Similarly, in a heat transfer course, a student learns to calculate reduction of heat
transfer rate when insulation of a given thickness is applied to a pipe. In this case,
the energy equation is simplified and the assumptions are as follows: Heat transfer is
radial and axi symmetric, steady state prevails, and the insulation conductivity may
be constant and there is no generation or dissipation of energy within the insulation.
In both these examples, the equations are one dimensional. They are, there-
fore, ordinary differential equations (ODEs), although the original transport equa-
tions were PDEs. In many situations, in spite of the assumptions, the governing
equations cannot be rendered one dimensional. Thus, the equations of a steady,
two-dimensional velocity boundary layer or that of one-dimensional unsteady heat
conduction are partial differential equations. It is important to recognise, how-
ever, that there are no direct solutions to partial differential equations. To obtain
solutions, the PDEs are always first converted to ODEs (usually more in number
than the original PDEs) and the latter are solved by integration. Thus, in an un-
steady conduction problem, the ODEs are formed by the method of separation of
variables, whereas, for the two-dimensional velocity boundary layer, the ODE is