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9 Experimental Methods in Fluid Mechanics
9.4.2 Dimensional Analysis
Dimensional analysis is a rational method for combining physical variables
into a dimensionless product, thereby reducing the number of variables that
need to be considered (Hughes, 1993). This analysis gives qualitative, rather
than quantitative relationships and after performing this analysis, theoretical or
experimental work is needed to establish the functional relationship between the
dimensionless variables. Following Hughes, we identify three steps for applying
dimensional analysis to a given problem:
1. identification of important independent variables of the process,
2. determination of how many independent dimensionless products can be
formed from the variables,
3. reduction of the system's variables to the proper number of independent
dimensionless variables.
The above three steps of methodology are based on the theorem known as the
Buckingham IT Theorem, which states that in a dimensionally homogeneous
equation involving In' variables, the number of dimensionless products that can
be formed from In' variables is In - r' where 'r'is the number of fundamental
dimensions encompassed by the variables (Hughes, 1993). Thus, if
(9.9)
is the equation involving n variables, then according to the Buckingham IT
theorem, this equation can be rearranged into a new equation expressed in
terms of dimensionless products:
(9.10)
The number of terms in the equation is now less than the number of original
variables by the number r. If the original n variables are independent, then
the n - r dimensionless products will also be independent when all original
variables appear at least once in one of the dimensionless products.
The most important step in the dimensional analysis is an identification of
important variables which should be included in the analysis and is guided by
a sound understanding of the underlying physical processes. We have to make
judgement as to which variables strongly influence the result and must be included, and which do not affect the result and can be omitted. We demonstrate
the dimensional methodology by three examples involving the predominant influence of viscosity, gravity and buoyancy.
Sphere Moving in Viscous Fluid. The first example demonstrates an application of dimensional analysis for the determination of the drag force for a
small particle moving in a fluid at low Reynolds number. We have considered
this problem in Appendix C.3.2 using the Navier-Stokes equation. Now we will
9 Experimental Methods in Fluid Mechanics
9.4.2 Dimensional Analysis
Dimensional analysis is a rational method for combining physical variables
into a dimensionless product, thereby reducing the number of variables that
need to be considered (Hughes, 1993). This analysis gives qualitative, rather
than quantitative relationships and after performing this analysis, theoretical or
experimental work is needed to establish the functional relationship between the
dimensionless variables. Following Hughes, we identify three steps for applying
dimensional analysis to a given problem:
1. identification of important independent variables of the process,
2. determination of how many independent dimensionless products can be
formed from the variables,
3. reduction of the system's variables to the proper number of independent
dimensionless variables.
The above three steps of methodology are based on the theorem known as the
Buckingham IT Theorem, which states that in a dimensionally homogeneous
equation involving In' variables, the number of dimensionless products that can
be formed from In' variables is In - r' where 'r'is the number of fundamental
dimensions encompassed by the variables (Hughes, 1993). Thus, if
(9.9)
is the equation involving n variables, then according to the Buckingham IT
theorem, this equation can be rearranged into a new equation expressed in
terms of dimensionless products:
(9.10)
The number of terms in the equation is now less than the number of original
variables by the number r. If the original n variables are independent, then
the n - r dimensionless products will also be independent when all original
variables appear at least once in one of the dimensionless products.
The most important step in the dimensional analysis is an identification of
important variables which should be included in the analysis and is guided by
a sound understanding of the underlying physical processes. We have to make
judgement as to which variables strongly influence the result and must be included, and which do not affect the result and can be omitted. We demonstrate
the dimensional methodology by three examples involving the predominant influence of viscosity, gravity and buoyancy.
Sphere Moving in Viscous Fluid. The first example demonstrates an application of dimensional analysis for the determination of the drag force for a
small particle moving in a fluid at low Reynolds number. We have considered
this problem in Appendix C.3.2 using the Navier-Stokes equation. Now we will