Fluid Analysis Overview

This section details the various approaches available for analyzing a general fluid flow problem. It also provides pros and cons for each of these approaches.

Analysis of any physical problem, in general, is governed by a set of basic governing equations. The basic equations governing the motion of fluids are the Navier-Stokes equations, which were developed more than 150 years ago.

Continuity Equation: (1)
$\frac{\partial ρ}{\partial t} + \nabla \cdot (ρ \vec{u}) = 0$
Momentum Equation: (2)
$ρ \frac{\partial \vec{u}}{\partial t} + (ρ \vec{u} \cdot \nabla) \vec{u} = - \nabla p + ρ b + \nabla \cdot τ$
Energy Equation: (3)
$ρ \frac{\partial h}{\partial t} + (ρ \vec{u} \cdot \nabla) h = \nabla \cdot (k \nabla T) + \nabla \vec{u} : τ + \frac{D p}{D t} + S$

These equations are a set of coupled partial differential equations which describe the relation among the flow properties like pressure, temperature and velocity. The solution to the above equations is in general complex because of the non-linear terms and coupled nature of the equations. However, in general the following three solution approaches are available:

Analytical Approach: An analytical solution can be obtained by integrating the above boundary value problem, resulting in an algebraic equation for the dependent variables as a function of independent variables. Because of the complexity of the Navier-Stokes equations a general solution for all physical problems is an insurmountable task. However, analytical solution is possible in some simple cases with certain approximations in geometry, dimensionality and compressibility of the flow. A few examples of such flows include Couette flow, Hagen-Poiseuille flow and Parelle flows through straight channels. An obvious limitation of this approach is the practical applications of these flows.
Experimental Approach: This approach uses the physical experiments on the scale down models of the problem at hand and extrapolates the results based on similarity laws. For example, to investigate the flow around an actual aircraft a Reynolds number based miniature model is placed in the wind tunnel and the results are extrapolated to that of the actual aircraft. Although a variety of experimental techniques are available, the choice of the particular type to use depends on capturing the parameter of interest. For example, Particle Image Velocimetry (PIV), Laser Doppler Velocimetry (LDV) and Hot-wire anemometry are used to measure flow velocity. PIV is used to produce 2D or 3D vector fields. Hot-wire anemometry and LDV are used when measuring the rapidly varying velocities at a point with good spatial and time resolution is important. A few limitations of this approach include the necessity of conducting a large number of experiments for the complete description of flow, cost and the complexity of conducting them.
Numerical Approach (Computational Fluid Dynamics-CFD): The third approach available for investigating fluid flow is CFD. In this approach, the fluid region of interest is divided into finite regions and the complex governing equations are discretized to obtain a set of algebraic equations that are advanced in time and space using computers. The end result of the CFD analysis is a complete description of the fluid flow within the region of interest.