The Froude number deals with the relationship between gravity and inertial forces, whilst the Reynolds number deals with the relationship between frictional and inertial forces. For various reasons the similarity requirements posed by the Froude and Reynolds numbers can typically not be satisfied simultaneously. It is then necessary to decide which are the dominant force according to which the scaling must be done. It may then be necessary to compensate for the effect of the other force. Typically free surface flows are governed by gravity forces, although affected by frictional forces as well.

The Froude number deals with the relationship between gravity and inertial forces, whilst the Reynolds number deals with the relationship between frictional and inertial forces. For various reasons the similarity requirements posed by the Froude and Reynolds numbers can typically not be satisfied simultaneously. It is then necessary to decide which are the dominant force according to which the scaling must be done. It may then be necessary to compensate for the effect of the other force. Typically free surface flows are governed by gravity forces, although affected by frictional forces as well.

Thanks Charl. This makes sense. I decided to go for Froude similarity in the end, but in what way could I compensate for the effect of the viscous force? Thanks! :)

Froude number is the ratio of flow speed to the characteristic speed of shallow water waves in the channel. As such, it is similar to a Mach number: when greater than one no wave-bourne information can travel upstream against the flow; shock-waves or hydraulic jumps can form with corresponding jumps in Froude number.

This is, of course very different from Reynolds number which is more closely related to viscous drag, particularly in laminar flow. It is likely that the Reynolds number in your flow is well above the critical value for transition to turbulence and that the flow is turbulent with relatively weak effects of drag on the process compared with the effects of gravitational free-surface waves.

So, in your case, the phenomenon is most likely to scale well with Froude number.

Reynolds number (viscous drag) effects are most likely to become more significant at lower Reynolds numbers, which could be arrived at by reducing both flow velocity and fluid depth in order to keep Froude number constant. At some stage scaling based on Reynolds number would then become more appropriate. But it seems you are well away from this point.

May I suggest that you read some text of Fluids Mechanics written by Milne &Thomson or Lamb. Another good reference is by V.T Chow....you will get a good definition and explainaton on Re or Fr. 

Now in your exp...if the flow have free surface then Fr is the  factor

As Charl Gabriel Du Toit suggested first you find out which one is the dominant force. Generally for a model involving flow of fluid with gravity as the dominant force; Froude number is used for the dynamic similarity purpose.

Thank you very much for your explanation. This makes more sense now :)

As you said yourself, you don't actually have a choice between Reynolds or Froude scaling. It has to be Froude since you are not going to do your tests with low viscosity fluids I guess (like liquid Nitrogen or the like :) ).

The implications that I could think of are these (not stating anything about the importance of either since I would not know that). A low Reynolds number would mean a different boundary layer profile. This might have some implications on the flowrate or flume limit. Furthermore it would have an implication on the friction. Low Reynolds numbers usually lead to higher relative friction on model scale, however low Reynolds numbers could inhibit transition actually leading to lower frictional resistance. Furthermore, if you have bends and corners and the like you could have more separation than on full scale. Lower Reynolds number flows, especially laminar flow, separates easier than high Reynolds number flows.

The beauty of CFD is however that you can test both; the scale of the experiment and full scale. That would show you the relative influence of all these issues. That is, if you can model this problem with CFD accurately enough.

I illustrate the effect of Froude and Reynolds by the calculation of the drag of a ship.

The Froude hypothesis says that the non-dimensional drag is the sum of two terms, the wave drag dependant on the Froude number and the friction drag dependant on the Reynolds number. You should use the Froude mumber for the similarity between model and full scale. Knowing the desired full scale boat speed, the similariry imposes the model speed. You measure the total drag of the model for that speed. Using formulas, you then calculate the friction drag for the model, and by difference the wave drag for the model. Using similarity, you obtain the wave drag for the full scale ship, then calculate using the formulas again the friction drag for full scale, and by adding the two you predict the total drag for full scale. 

This sort of ratio is sought to weigh the influence of gravity against the influence of friction. One may then decide to ignore or not to ignore any of them. Use of Fr and Re, rather the actual forces ensures generality and the convenience of limiting the need for multiple tests.

A dimensionless number is always the factor that remains if you make a set of equations dimensionless. So that means that the characteristic parameters of the phenomena in this number indicate the phenmomena and then u indicates convection, \nu indicates viscous effects. The ratio of this gives the Re number and thus indicates convective versus viscous effects. In your case the gravitational acceleration, g comes into play as well. This appears in the Froude number that appears in your momentum equation when correctly non-dimensionalised, right?

Most student (not all) always look for short cut to problem solving...broad reading will help

Hi Pavel,

I have thought about this as I had an assignment a long time ago on this problem.

Now having read turbulence for a number of years the best way to investigate this is shown in Steven Pope's 2000 text book 'Turbulent Flows' shown in the graph on page 256 which plots the Velocity Derivative Kurtosis against Reynolds Number.

Clearly there is a difference between the Reynolds numbers between the model and prototype.

This the best information I can find that gives an indication of an effect of not obtaining Reynolds Number similarity. I think the physical reason for this is that the  turbulence spectrum is truncated in the model compared to theh prototpye i.e. the difference between the largests scales and the Kologomorov scale is less in the model. I would like to look at this but may not get the opportunity.

This is just my opinion and may be a short cut but may not be. But it is a very good idea to read widely to come up with your own ideas for research. This is the whole point of the literature review of a PhD.

Regards,

Rob

Hi Rob,

Thanks for your reply and for the reference - I will definitely look it up; it looks like a good reference. Now, i've got more time available to read widely around this topic. 

Regards,

Pavel 

The Froude number similarity will give you proper scaling of the surface waves. The Reynolds number has a (small) impact on the friction factor which triggers the kinematic instability, i.e., the critical condition for the formation of waves. A CFD turbulence model would allow you to scale the change in the friction factor between experiment and prototype.

Dear Sir;

Reynolds number deals with the relationship between frictional and inertial forces while Froude number deals with the relationship between gravity and inertial forces.

It may then be necessary to compensate for the effect of the other force. Typically free surface flows are governed by gravity forces, although affected by frictional forces as well.

For various reasons the similarity requirements posed by the Froude and Reynolds numbers can typically not be satisfied simultaneously. It is then necessary to decide which are the dominant force according to which the scaling must be done.

GOOD LUCK

Intesrting paper from Castro Orgaz and Hager in JHE this year

Froude number (Fr), in hydrology and fluid mechanics, dimensionless quantity used to indicate the influence of gravity on fluid motion. It is generally expressed as Fr = v/(gd)1/2, in which d is depth of flow, g is the gravitational acceleration (equal to the specific weight of the water divided by its density, in fluid mechanics), v is the celerity of a small surface (or gravity) wave, and Fr is the Froude number. When Fr is less than 1, small surface waves can move upstream; when Fr is greater than 1, they will be carried downstream; and when Fr = 1 (said to be the critical Froude number), the velocity of flow is just equal to the velocity of surface waves. The Froude number enters into formulations of the hydraulic jump (rise in water surface elevation) that occurs under certain conditions, and, together with the Reynolds number, it serves to delineate the boundary between laminar and turbulent flow conditions in open channels.

Froude no. is ratio of Kinetic force and gravitational force. Rynolds no. is ratio of Kinetic force and Viscous force.

It is impossible to fulfill similarity both Reynolds and Froude number based, if we used same fluid in the model and prototype, say water. In practical using Froude number criteria will solve the problem, as long as we can asure that the flow is in turbulence state.

The Froude is the relationship between gravity and inertial forces, whilst the Reynolds number is the relationship between frictional and inertial forces.

Fluid dynamics is usually not properly taught in colleges. Theory should be substantiated by practical experimentation and explanation. Concepts of non-dimensional numbers such as Froude Number, Richardason Number, Reynold Number, Burger Number, when properly learnt, are very useful to understand how fluids flow.

I think fluid mechanics is often taught well. I was taught (and taught) that one can find empirical formulas based on the Reynolds number for the turbulent drag coefficient and use similarity based on the Froude number for the form drag. We must suppose that total drag is the sum of the friction drag and the form drag (Froude hypothesis).