28
CHAPTER 2. DIMENSIONAL ANALYSIS
added or subtracted have different dimensions, then a mistake has been
made. Because this principle may be applied to differential equations and
integral equations, as well as to algebraic equations, it is a powerful method
for checking derivations, algebraic manipulations, or empirical formulations.
Dimensionally homogeneous equations are the basis for application of
dimensional analysis to physical problems. In dimensional analysis it is
assumed that “... the solution of the problem is expressible by means of a
dimensionally homogeneous equation in terms of specified variables (Langhaar 1951). This hypothesis is justified on the fact that the fundamental
equations of physics (motion) are dimensionally homogeneous; therefore,
all relations derived from these equations must also be dimensionally homogeneous.
2.2.2 Preparing to Conduct Dimensional Analysis
The first step in preparing to conduct a dimensional analysis of a practical
problem is to decide what variables enter into the physics of the problem.
Selection of variables is very important and also difficult because it requires
that enough must be understood about the problem to recognize which
variables are important and must be included, and which variables are not
needed and would unnecessarily complicate the analysis.
Identification of important variables and parameters should not be taken
lightly because it requires considerable insight into the problem and the governing physical laws. Including too many unnecessary variables increases
the number of expensive, time-consuming experiments that must be conducted to establish empirical coefficients or to eliminate unimportant variables. Conversely, neglecting important variables will likely result in no
conclusion, or worse, incorrect conclusions or empirical design formulas.
Often it will be necessary to include variables that could be considered
constant (such as gravity or kinematic viscosity) so they can be combined
with other variables to form dimensionless products.
Munson, et al. (1990) stated that pertinent variables for most engineering problems can be classified into three general categories:
• Geometry. Geometry plays an important role in the response of most of hydrodynamic systems, and a sufficient
number of geometric variables must be included. Geometric variables pertain to characteristics such as length, area,
volume, and angle; and they are usually easily identified.
• Material Properties. Externally-applied forces produce
a response in a system that depends on the material properties of the system. For the case of fluid mechanics the
CHAPTER 2. DIMENSIONAL ANALYSIS
added or subtracted have different dimensions, then a mistake has been
made. Because this principle may be applied to differential equations and
integral equations, as well as to algebraic equations, it is a powerful method
for checking derivations, algebraic manipulations, or empirical formulations.
Dimensionally homogeneous equations are the basis for application of
dimensional analysis to physical problems. In dimensional analysis it is
assumed that “... the solution of the problem is expressible by means of a
dimensionally homogeneous equation in terms of specified variables (Langhaar 1951). This hypothesis is justified on the fact that the fundamental
equations of physics (motion) are dimensionally homogeneous; therefore,
all relations derived from these equations must also be dimensionally homogeneous.
2.2.2 Preparing to Conduct Dimensional Analysis
The first step in preparing to conduct a dimensional analysis of a practical
problem is to decide what variables enter into the physics of the problem.
Selection of variables is very important and also difficult because it requires
that enough must be understood about the problem to recognize which
variables are important and must be included, and which variables are not
needed and would unnecessarily complicate the analysis.
Identification of important variables and parameters should not be taken
lightly because it requires considerable insight into the problem and the governing physical laws. Including too many unnecessary variables increases
the number of expensive, time-consuming experiments that must be conducted to establish empirical coefficients or to eliminate unimportant variables. Conversely, neglecting important variables will likely result in no
conclusion, or worse, incorrect conclusions or empirical design formulas.
Often it will be necessary to include variables that could be considered
constant (such as gravity or kinematic viscosity) so they can be combined
with other variables to form dimensionless products.
Munson, et al. (1990) stated that pertinent variables for most engineering problems can be classified into three general categories:
• Geometry. Geometry plays an important role in the response of most of hydrodynamic systems, and a sufficient
number of geometric variables must be included. Geometric variables pertain to characteristics such as length, area,
volume, and angle; and they are usually easily identified.
• Material Properties. Externally-applied forces produce
a response in a system that depends on the material properties of the system. For the case of fluid mechanics the