7 Dynamic Similitude
254
chosen as a base. The blade speed may be chosen, leading to Re = ΩD
2
/ν. The last
expression is even commonly favoured.
7.1.8 Strong and Weak Similitude Conditions
Not all mathematically determined similitude conditions have an equal practical
relevance. This is demonstrated by the friction factor in circular ducts, as shown in
Fig. 2.17 of Chap. 2. With a given relative roughness, the friction factor becomes
independent of the Reynolds number for a sufficiently high value of the Reynolds
number. The physical reason is that, at a high Reynolds number, the (laminar) sublayer at the wall becomes so thin that the roughness peaks of the wall directly affect
the core flow. Friction is then determined only by roughness and not by viscosity.
This example demonstrates that, in case of a high Reynolds number, this number is
not a practical condition for similitude anymore. In turbomachines with low viscosity fluids such as water (ν ≈ 10
−6
m
2
/s) or air (ν ≈ 15 · 10
−6
m
2
/s), the Reynolds number
is mostly so high (order of magnitude of some hundred thousands) that is does not
constitute a practical condition for similitude in a first approach. A limited correction for the influence of the Reynolds number is sometimes necessary (see below).
Analogous behaviour occurs with the Mach number for flows of compressible
fluids. The Mach number is the ratio of the flow velocity to the speed of sound.
The finite propagation velocity of sound waves results from the compressibility of
the fluid. In an incompressible fluid, this velocity (theoretically) is infinitely high.
This demonstrates that, with a sufficiently low Mach number, this number does not
constitute a practical similitude condition. We refer to Chap. 4, where it has been
demonstrated that, with Mach numbers until about 0.3, the Bernoulli and Saint Venant equations give the same results with a good approximation.
7.2 Characteristic Numbers of Turbomachines
7.2.1 Definition of a Characteristic Number
A characteristic turbomachine number is a dimensionless combination of parameters taken from flows with optimum efficiency. Such a number is unique for the
machine and for geometrically similar machines. It characterises the shape of the
machine. We should first reason the relevance of such a definition. Therefore we
consider the Q-H curve of a driven turbomachine with a constant density fluid
(pump or fan), as sketched in Fig. 7.2. From the foregoing we know that, with neglect of the influence of the Reynolds number, the only similitude condition is the
constancy of the flow factor Φ = Q/ΩD
3
. The head factor Ψ = ∆E m /Ω
2
D
2
is then a
dependent parameter. With a change in rotational speed, all flows meeting constant
Φ are similar with each other. For these flows, Ψ is constant as well. The flow rate
254
chosen as a base. The blade speed may be chosen, leading to Re = ΩD
2
/ν. The last
expression is even commonly favoured.
7.1.8 Strong and Weak Similitude Conditions
Not all mathematically determined similitude conditions have an equal practical
relevance. This is demonstrated by the friction factor in circular ducts, as shown in
Fig. 2.17 of Chap. 2. With a given relative roughness, the friction factor becomes
independent of the Reynolds number for a sufficiently high value of the Reynolds
number. The physical reason is that, at a high Reynolds number, the (laminar) sublayer at the wall becomes so thin that the roughness peaks of the wall directly affect
the core flow. Friction is then determined only by roughness and not by viscosity.
This example demonstrates that, in case of a high Reynolds number, this number is
not a practical condition for similitude anymore. In turbomachines with low viscosity fluids such as water (ν ≈ 10
−6
m
2
/s) or air (ν ≈ 15 · 10
−6
m
2
/s), the Reynolds number
is mostly so high (order of magnitude of some hundred thousands) that is does not
constitute a practical condition for similitude in a first approach. A limited correction for the influence of the Reynolds number is sometimes necessary (see below).
Analogous behaviour occurs with the Mach number for flows of compressible
fluids. The Mach number is the ratio of the flow velocity to the speed of sound.
The finite propagation velocity of sound waves results from the compressibility of
the fluid. In an incompressible fluid, this velocity (theoretically) is infinitely high.
This demonstrates that, with a sufficiently low Mach number, this number does not
constitute a practical similitude condition. We refer to Chap. 4, where it has been
demonstrated that, with Mach numbers until about 0.3, the Bernoulli and Saint Venant equations give the same results with a good approximation.
7.2 Characteristic Numbers of Turbomachines
7.2.1 Definition of a Characteristic Number
A characteristic turbomachine number is a dimensionless combination of parameters taken from flows with optimum efficiency. Such a number is unique for the
machine and for geometrically similar machines. It characterises the shape of the
machine. We should first reason the relevance of such a definition. Therefore we
consider the Q-H curve of a driven turbomachine with a constant density fluid
(pump or fan), as sketched in Fig. 7.2. From the foregoing we know that, with neglect of the influence of the Reynolds number, the only similitude condition is the
constancy of the flow factor Φ = Q/ΩD
3
. The head factor Ψ = ∆E m /Ω
2
D
2
is then a
dependent parameter. With a change in rotational speed, all flows meeting constant
Φ are similar with each other. For these flows, Ψ is constant as well. The flow rate