2.3. DIMENSIONAL ANALYSIS METHODOLOGY
45
Munson, et al. (1990) pointed out that this result is known as Stokes law, and it
applies to very viscous fluids which have a Reynolds number much less than unity. At
higher Reynolds numbers inertial effects would become important, and consequently,
fluid density and gravity would also have to be included in the analysis. Including the
two additional variables gives
D = f(d, V,
p, g)
which are the same variables used in Example 2.5. Therefore, we should expect that
drag forces in flows with Reynolds numbers somewhat larger than Stokes flows will
be a function of both Reynolds number and Froude number.
When the important variables of a coastal flow problem can be incorporated into two dimensionless products, the relationship between the pi
terms is given as
ni = #(n2)
(2.9)
Experimental determination of the functional relationship involves conducting experiments where II2 is varied, and the corresponding values of fli are
determined via measurements. Sufficient data pairs of (III, II2) facilitate
a graphical representation of the relationship, shown schematically in Figure 2.1.
In some instances it may be possible to fit a mathematical expression to
the plotted data using standard curve fitting techniques; however, Munson,
et al. (1990) emphasize that any such empirical relationship is valid
only over the range of II2 covered by the experiments. Outside
the range of the experiments, the physical process may not follow the same
trend as the mathematical function.
For flow phenomena where the important process variables can be expressed in terms of three dimensionless products, we can assume a relationship given by
ih = #(n2, n3)
(2.io)
Determining the functional relationship between pi terms requires experiments where one pi term is held constant, one is varied, and the third
pi term is measured. Graphical presentation of the data might result in
a family of curves, as shown in Figure 2.2. It may be possible to develop
appropriate empirical expressions for the family of curves by applying more
involved curve fitting procedures.
As the number of dimensionless products increases beyond three, experimental efforts become laborious, and both graphical presentation of results
45
Munson, et al. (1990) pointed out that this result is known as Stokes law, and it
applies to very viscous fluids which have a Reynolds number much less than unity. At
higher Reynolds numbers inertial effects would become important, and consequently,
fluid density and gravity would also have to be included in the analysis. Including the
two additional variables gives
D = f(d, V,
p, g)
which are the same variables used in Example 2.5. Therefore, we should expect that
drag forces in flows with Reynolds numbers somewhat larger than Stokes flows will
be a function of both Reynolds number and Froude number.
When the important variables of a coastal flow problem can be incorporated into two dimensionless products, the relationship between the pi
terms is given as
ni = #(n2)
(2.9)
Experimental determination of the functional relationship involves conducting experiments where II2 is varied, and the corresponding values of fli are
determined via measurements. Sufficient data pairs of (III, II2) facilitate
a graphical representation of the relationship, shown schematically in Figure 2.1.
In some instances it may be possible to fit a mathematical expression to
the plotted data using standard curve fitting techniques; however, Munson,
et al. (1990) emphasize that any such empirical relationship is valid
only over the range of II2 covered by the experiments. Outside
the range of the experiments, the physical process may not follow the same
trend as the mathematical function.
For flow phenomena where the important process variables can be expressed in terms of three dimensionless products, we can assume a relationship given by
ih = #(n2, n3)
(2.io)
Determining the functional relationship between pi terms requires experiments where one pi term is held constant, one is varied, and the third
pi term is measured. Graphical presentation of the data might result in
a family of curves, as shown in Figure 2.2. It may be possible to develop
appropriate empirical expressions for the family of curves by applying more
involved curve fitting procedures.
As the number of dimensionless products increases beyond three, experimental efforts become laborious, and both graphical presentation of results