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Establishment of the scour formula
7.3.4 Approach based on the similitude and approximation theory
The analysis performed in the present paragraph served to determine parameters having an important
influence on the scour phenomenon. The similitude and approximation theory of KLINE
(1965) is based of a dimensional analysis 1 . Kline calls the dimensionless parameters (usually
formed by ratios of primary and secondary parameters with units 2 ) pi’s.
The pi-theorem makes the following statements:
Given a relation among m parameters of the form
f 1
( q 1
, q 2
,…,
q m ) = 0
(7.34)
an equivalent relation expressed in terms of n non dimensional parameters can be found of
the form
f 2
( π 1
, π 2
,…π , n )
where the number n is given by the relation
n = m – k
(7.35)
where m is the number of q’s in equation 7.34, and k is the large number of parameters
contained in the original list of parameters q 1
, q 2
,…,
q m that will not combine in any non
dimensional form. Generally k is equal to the minimum number of independent dimensions
needed to construct the dimensions of all parameters .
The parameters q should be independent. If they are not, additional conditions needs to be introduced
to compensate for the redundancy. The compensation is normally compensated in the following
manner. For each redundant dimension either a physical constant with the same
dimensions needs to be introduced, or the number of dimensions has to be decreased by one
( k – 1 ).
HUNTLEY (1953) extended the pi-theorem by restricting the combination to parameters acting in
the same direction.
The method of similitude is based on the following basic steps:
1. The forces that are supposed to be important in a given problem are enumerated, including
the dependent and all the independent forces. Each of these forces is then expressed in
terms of the parameters of the problem by physical or dimensional arguments.
2. The pertinent non dimensional groups (pi’s) are constructed by forming ratios of these
forces and including enough length ratios to ensure geometric similarity.
The number of pi’s constructed from force ratios are equal to the number of independent forces.
For convenience it is useful to introduce the dependent force only into one ratio in order to provide
an explicit rather than an implicit solution to the problem.
q i
1. also known as Pi-theorem, Buckingham’s method or Bridgman’s method.
2. An example for a primary parameter is the length, for a secondary parameter the velocity since it
is derived from a length and a time span.
EPFL Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 171