Maximum rotation rate of a turbine in a fluid with a given speed [closed]

Say a turbine (of any geometry) is placed into a fluid that flows at some constant speed and drives the turbine. Is there a maximum rate at which that turbine can spin?

For example, say wind turbines generally have a tip-speed ratio of roughly 5: assuming that stays constant, in a constant wind, the smaller the turbine gets, the faster it spins:

$$\lambda = \frac{\omega R}{v}$$

• $$\lambda$$ = tip-speed ratio (TSR)
• $$R$$ = disk radius
• $$\omega$$ = angular velocity
• $$v$$ = wind/fluid speed

The TSR is often over 1, so parts of the blade will move (much) faster than the airflow though the disk: a big turbine could have tip speeds well over 100mph even in moderate wind. So there is clearly no rule that says no part of the turbine can exceed the fluid speed (which you may expect if you only think of, say, a impulse turbine where the turbine probably wouldn't have a rim speed faster than the flow).

A turbine of half the radius spins twice as fast for the same TSR. But does this continue down indefinitely? Can TSR be maintained? Could a 1mm radius turbine spin 10000 times faster than a 10m radius turbine in the same fluid stream? Can TSR be increased indefinitely by changing the geometry and sacrificing all efficiency to spin the rotor as fast as possible?

Regardless of the efficiency of a turbine (which would have a peak at some TSR for a given fluid speed and blade geometry - just assume there's no load and no friction in the turbine mechanism), and regardless of material strength, is there a physical maximum speed that any rotor can turn in a fluid stream when driven by that stream? Either in terms of tip speed, angular velocity or some other physical property of the system such as viscosity, density, speed of sound, etc?

• "So a turbine of half the radius spins twice as fast for the same TSR" This is a bit misleading. To have the same TSR for a blade with half the radius, you will need to design entirely different blade. As the blade get smaller and smaller, achieving the same TSR, becomes more and more difficult and eventually it becomes impossible to design a blade with say 1/10 of length and same TSR.
– Ebi
Jul 29, 2021 at 7:04
• Well yes, but I don't care about the geometry. Obviously you can't just scale down a Siemens SWT-3.6-120 to 10mm tall and expect it to work very well. The point is, is there some physical limit to the speed at which any turbine of any size can spin in a given fluid stream? Or does it become impossible to increase TSR at some point due to that physical limit, even at the expense of all efficiency (which could be 0 if you don't extract any energy and put it all into accelerating your turbine). Jul 29, 2021 at 7:10
• @Inductiveload, you seem to have an implicit assumption that the tip speed of a turbine rotor shouldn't exceed the wind speed. That assumption can't be correct, as the design of the turbine blades and the length of the turbine blades will strongly impact the rotational velocity of the turbine, while the tip speed depends on the rotational velocity of the turbine blade, but also depends on the length of the turbine blade, which is a design feature that is independent of wind speed. Jul 30, 2021 at 18:02
• @DavidWhite As I said in the question, it's clear that the tip speed can (and, for a wind turbine, very often does) exceed the fluid speed through the turbine. But something presumably limits the maximum rotational speed of "some turbine" (and I imagine it's not the speed of light!) Jul 30, 2021 at 18:25
• Limits on the tip speed: 1) design of the blade; 2) the efficiency of the blade vs. the wind speed; 3) the length of the blade; 4) the electrical load on the turbine, which the design engineer is attempting to maximize for the design wind speed; 5) the mechanical strength of the blade - if the generator is decoupled from the turbine in a high wind, the turbine will spin VERY fast and will at some point self-destruct due to extreme centripetal forces on the turbine blades, as seen in YouTube videos. Jul 30, 2021 at 18:41