32
CHAPTER 2. DIMENSIONAL ANALYSIS
were established by physical arguments before the advent of dimensional
analysis.
In this chapter, we will see how these important flow parameters can be
derived through formal dimensional analysis. In the following chapter on
similitude, these same numbers will be derived from a physical standpoint
in order to understand their importance in scale modeling.
2.2.4 Complete Sets of Dimensionless Products
Dimensional analysis offers a structured methodology for arranging variables into dimensional products; but before we can proceed with an analysis, it is first necessary to know how many dimensionless products can be
formed from a given set of variables.
Every finite set of variables describing a physical process can produce a
complete set of dimensionless products.
Complete Set. A set is “complete” if (a) each dimensionless
product in the set is independent of the other products, and (b)
all other dimensionless combinations that can be formed using
the same variables produces a product that can be alternately
expressed as powers of the original products in the set.
The abstract concept of a “complete set of dimensionless products” is
difficult to explain, so it may be best to illustrate the concept with an
example.
Example 2.3. Complete Set of Dimensionless Products
The fluid flow variables L, V, g, p, and p form a complete set2 of dimensionless
products given by the Froude number and the Reynolds number, i.e.,
It will be shown in the next section how to calculate a complete set of dimensionless
products.)
Re=^
Therefore, the dimensionless products pV3/pg and pgL2/pV must be combinations
of the Reynolds and Froude numbers. These combinations are shown below
,/3/
PVL\
V2\
.
,
PV / P9 = -----• — = Re1 Ft2
\ M J \gLJ
CHAPTER 2. DIMENSIONAL ANALYSIS
were established by physical arguments before the advent of dimensional
analysis.
In this chapter, we will see how these important flow parameters can be
derived through formal dimensional analysis. In the following chapter on
similitude, these same numbers will be derived from a physical standpoint
in order to understand their importance in scale modeling.
2.2.4 Complete Sets of Dimensionless Products
Dimensional analysis offers a structured methodology for arranging variables into dimensional products; but before we can proceed with an analysis, it is first necessary to know how many dimensionless products can be
formed from a given set of variables.
Every finite set of variables describing a physical process can produce a
complete set of dimensionless products.
Complete Set. A set is “complete” if (a) each dimensionless
product in the set is independent of the other products, and (b)
all other dimensionless combinations that can be formed using
the same variables produces a product that can be alternately
expressed as powers of the original products in the set.
The abstract concept of a “complete set of dimensionless products” is
difficult to explain, so it may be best to illustrate the concept with an
example.
Example 2.3. Complete Set of Dimensionless Products
The fluid flow variables L, V, g, p, and p form a complete set2 of dimensionless
products given by the Froude number and the Reynolds number, i.e.,
It will be shown in the next section how to calculate a complete set of dimensionless
products.)
Re=^
Therefore, the dimensionless products pV3/pg and pgL2/pV must be combinations
of the Reynolds and Froude numbers. These combinations are shown below
,/3/
PVL\
V2\
.
,
PV / P9 = -----• — = Re1 Ft2
\ M J \gLJ