7 Dynamic Similitude
250
So: Re = inertia force/viscosity force, Fr
2
= inertia force/gravity force. If the pressure unit had not been chosen a priori as ρV
2
because of the consistency of the unit
system, a third similitude condition would become
For instance, we could have chosen the pressure difference over the channel as pressure unit. This would constitute a principal error. This means that the Euler number
is not a similitude condition. But it is a dimensionless group. Consequently, for all
flows that are similar with each other (= family of similar flows), there must be an
unambiguous relation of the form:
All other possible dimensionless groups are also functions of Re and Fr.
The above example implies that similitude conditions are created with a general
flow problem because of the requirement that forces of different origins must have
a fixed ratio. In the same way, velocities of different origins must have a fixed ratio. With turbomachines, there are always two velocities of different origins: the
through-flow velocity and the blade speed. In a general flow problem, there are
dynamic and kinematic similitude conditions.
7.1.4 Purpose of Similitude Analysis
Similitude analysis demonstrates that a dependent parameter cannot depend individually on the independent parameters of a problem. The relation must necessarily apply between a dimensionless group containing the dependent parameter and
dimensionless groups of independent parameters. Similitude analysis thus reduces
the number of degrees of freedom of the relations. The result demonstrates that a
physical relation cannot depend on the unit system chosen to measure the variables.
Therefore, it is advantageous to determine the dimensionless groups uniting the
independent parameters when performing a flow analysis, either experimentally or
numerically.
Similitude is mostly applied with keeping geometry constant, i.e. the geometric
factor constitutes the unit. The degrees of freedom are then only kinematic and
dynamic. Formulating a problem as in Fig. 7.1 in a dimensionless form generates
the solution for an arbitrary inflow velocity (kinematic) and an arbitrary density
(dynamic) in one effort. So, ∞
2
problems are analysed in one effort. Strictly, even
∞
3
are, as the solution also applies to each geometrically similar problem, but the
extension to geometrically similar problems mostly has no practical relevance (except with reduced scale models, see Sect. 7.4.5).
/
(Euler) pressure force / inertia force.
2
p
Eu e
V
r
=
=
( , ).
2
p
Eu
f Re Fr
V
D
r
=
=
250
So: Re = inertia force/viscosity force, Fr
2
= inertia force/gravity force. If the pressure unit had not been chosen a priori as ρV
2
because of the consistency of the unit
system, a third similitude condition would become
For instance, we could have chosen the pressure difference over the channel as pressure unit. This would constitute a principal error. This means that the Euler number
is not a similitude condition. But it is a dimensionless group. Consequently, for all
flows that are similar with each other (= family of similar flows), there must be an
unambiguous relation of the form:
All other possible dimensionless groups are also functions of Re and Fr.
The above example implies that similitude conditions are created with a general
flow problem because of the requirement that forces of different origins must have
a fixed ratio. In the same way, velocities of different origins must have a fixed ratio. With turbomachines, there are always two velocities of different origins: the
through-flow velocity and the blade speed. In a general flow problem, there are
dynamic and kinematic similitude conditions.
7.1.4 Purpose of Similitude Analysis
Similitude analysis demonstrates that a dependent parameter cannot depend individually on the independent parameters of a problem. The relation must necessarily apply between a dimensionless group containing the dependent parameter and
dimensionless groups of independent parameters. Similitude analysis thus reduces
the number of degrees of freedom of the relations. The result demonstrates that a
physical relation cannot depend on the unit system chosen to measure the variables.
Therefore, it is advantageous to determine the dimensionless groups uniting the
independent parameters when performing a flow analysis, either experimentally or
numerically.
Similitude is mostly applied with keeping geometry constant, i.e. the geometric
factor constitutes the unit. The degrees of freedom are then only kinematic and
dynamic. Formulating a problem as in Fig. 7.1 in a dimensionless form generates
the solution for an arbitrary inflow velocity (kinematic) and an arbitrary density
(dynamic) in one effort. So, ∞
2
problems are analysed in one effort. Strictly, even
∞
3
are, as the solution also applies to each geometrically similar problem, but the
extension to geometrically similar problems mostly has no practical relevance (except with reduced scale models, see Sect. 7.4.5).
/
(Euler) pressure force / inertia force.
2
p
Eu e
V
r
=
=
( , ).
2
p
Eu
f Re Fr
V
D
r
=
=