Consider the following situation:

A radial impeller is centered in a cylindrical mixing tank (centered both axially and radially). The tank is filled completely (no free surface) with an incompressible fluid. The rotational speed of the impeller is quite low such that the flow is isothermal (negligible power input) and laminar.

Question:

Are the flow patterns in the tank governed solely by the impeller Reynold's number (Re = rho N D^2/vis)?

Note: there is no free surface in this problem, so the Froude number is not important here.

Reason for asking:

LDV measurements with different fluids have shown that the azmuthial velocity field penetrates further into the tank at higher rpm for the same Reynold's number, i.e. different fluid properties, different rpm, but same Re.

Are we missing something obvious here, or it is possible that there are two stable flow patterns at the same Re in the laminar flow regime, the transition of which depends on rotational speed.

Comments or suggestions,

(1). In a non-rotating system, there is dynamic force and there is viscous force.(convection and diffusion) (2). In a rotating system, using cylindrical coordinates , there is centrifugal force. This is because it will appear in the governing equations when the cylindrical coordinates system is used. (3). In 3-D cartesian coordinate system, the problem will be 3-D, transient, with moving boundary and you are not going to see this term explicitly (invisible). (4). So, your question is: In cylindrical coordintes system, how many different terms are there in the governing equations? The answer could be found in most fluid dynamics text book. ( the problem could be transient and the wake properties also could change ). You can also take a look at some turbomachinery textbook. They belong to the rotating machinery category. Sometimes, they cover the dimensionless groups which are useful to characterize the system in rotating coordinate system.

Thanks for the response.

Solving the governing equations in a rotating coordinate system (rotating with the impeller) introduces additional centrifugal and Coriolis forces to the problem.

When the density is constant, however, the centrifugal force is unimportant because it can be expressed as the gradient of a scalar quantity. In short, the pressure can be replaced by a modified pressure, p - 1/2 rho omega^2 r^2. This is completely analogous to the procedure of subtracting out the hydrostatic pressure to remove the effect of gravitational forces. Physically, the centrifugal force is balanced by a radial pressure gradient which is present whether or not there is any flow relative to the rotating reference frame.

This is not possible with the Coriolis force and thus the Rossby number appears, Ro = U/(omega L). However, there is only one forcing velocity scale (at least only one that I can see) here which equal U = omega L and hence the Rossby number is unity in all cases.

Any other suggestions?

(1). I can't invent new physics for you, and I think the solution must be in the governing equations. (don't have to solve these equations in order to see the dimensionless parameters), (2). Run a couple of CFD simulation to see whether you are getting the same characteristics. ( Laser can't detect the fluid velocity, it can only detect the seeding particles in the fluid. In a rotating system, the seeded particles are not likely to follow the fluid motion. The same is true when using oil dots on a rotating surface to trace the streamlines. The oil dots will move in the radial direction instead of the streamline direction.) (3). Ask yourself whether the seed particle has the same density as the testing fluid density.)

>> (1). I can't invent new physics for you, and I think the solution must be in the governing equations. (don't have to solve these equations in order to see the
>> dimensionless parameters).

I agree with you completely. Normalization of the N.S. equations shows that only the Reynold's numbers based on the impeller tip velocity governs this problem.

>> (2). Run a couple of CFD simulation to see whether you are getting the same characteristics. ( Laser can't detect the fluid velocity, it can only detect the seeding
>> particles in the fluid. In a rotating system, the seeded particles are not likely to follow the fluid motion. The same is true when using oil dots on a rotating surface to >> trace the streamlines. The oil dots will move in the radial direction instead of the streamline direction.)

The CFD simulations show that their is only one solution (i.e. no rpm dependence).

>> (3). Ask yourself whether the seed particle has the same density as the testing fluid density.)

We've thought of this. The LDV particles are very small, on the order of microns, and although they are not exactly neutrally buoyant (fairly close though), the Stoke's velocity for the particles is extremely small.

Any other suggestions, John?

Is it possible that there is some other (perhaps less stable) solutions to the N.S. eqn's that we are not seeing? or am I grasping at straws?

Dear sir,

(1)in your case ie laminar flow in a stirred vessel equipped with a radial impeller the mixing is coming from the molecular diffusion only ( for your flow regime ) and is independant from your power input P .

(2)As you know ,avoiding a free surface or including baffles in your vessel leads you avoid centrifugal vortex in the vessel ( bad for mixing)which make the powe number Np dependent of the reynolds number only, Np*Re=Ne , where Ne is the newton number related to the geometry of the tank

(3) you used different fluids ( differents viscosities) with approriate rotating speed to keep your Re number constant, But the reynolds number cannot give informations about your velocity flow field, only about your regime .

you should take care, by changing the rotating speed you change the velocity tip =(pi*n*impeller's diameter) .

for an explicit exemple take a flow aroun a cylinder if you take two different fluids with same Reynolds number you will obtain two different flow fields around the cylinder!

please contact me directly at : essemian@insa-tlse.fr , i'm sure we will find more to talk about ( we perform experimental( laser measurements) and numerical simulations in stirred vessels )

best regards

karim

(1). If you fix the rotating coordinate system on the rotor blade, then the only moving boundary is the casing wall. In this way , the problem is similar to a 3-D cavity flow with a moving wall. (there are gaps) (2). There is also differences between this model and the cavity flow problem, that is, it is a continuous closed loop. ( cavity-1, -2,-3,-4,...,cavity-1,-2...) So the flow over the cavity-1 will have impact on the flow over cavity-2, etc...(3).Even at low Reynolds number, there will be flow recirculating regions associated with each cavity ( flow region between two impellers). The flow will have to climb over the vertical fense with thin lips and flow separation will occur there. A shear layer will be formed there . So, the stability of this shear layer is one possible source of un-steadiness. (4). I am sure that the formation of this shear layer is sensitive to the exact shape of the blade lip geometry. And each blade probably has a slightly different lip geometry. So the separation point will occur at different point on the lip. (5). It may be possible that this slight difference in lip geometry can change between two stable states in a closed loop flow field. The number of blades is also a possible parameter in this closed loop system. (6). It would be interesting to know whether the flow fields inside each cavity ( between blades) are identical or not. (7). Is the flow sensitive to the number of blades? You could start with just one blade, and then add another blade to see the difference. (8). It is an interesting problem. And it is up to you to find the answer.

Yes, that could be a source of unsteadiness. Another possibility is that we may have reached a burification point for the given Reynold's number, i.e. there can be two or more solutions. The actual solution that one encounters in practice may depend on many things (initial start-up, etc.)

We will perform some more careful flow visualization studies in conjunction with additional CFD simulations and hopefully move towards an improved understanding of this problem.

Regards, Glenn

and there is always the effect due to the earth rotation. The direction of rotation could be another parameter. It is likely that the flow is not symmetrical ( periodic between blades), and the change in the speed of rotation simply shift the flow field in one direction. I am sure that results from flow visualization will give you a better picture of the whole flow field. You may be right that there are more than one stable solutions, but, without knowing the detailed conditions and design, it is hard to make any suggestions in that direction.

>>>>>>>>>>>>>>

for an explicit exemple take a flow aroun a cylinder if you take two different fluids with same Reynolds number you will obtain two different flow fields around the cylinder!
>>>>>>>>>>>>>>

Are there any publications on this effect? Thank You beforehand and with best regards

(1). Using Laser to measure velocity, you are likely to measure the point value, that is the velocity component at a point. The velocity measured will be the absolute velocity, say m/s, or ft/sec.. (2). Using the definition of the Reynolds number, the velocity value must be changed when the viscosity is changed in order to keep the Reynolds number fixed. So, when the viscosity is decreased, the velocity must be decreased to keep the Reynolds number fixed at the same number. From this point of view, the velocity values in the flow field will be will be everywhere lower for the low viscosity cases than the velocity field for the high viscosity cases. (3). But if you look at the pattern of the whole flow field, that pattern will be the same when the Reynolds numbers are the same. This comes from the governing equations, when non-dimensionalized, become a function of the Reynolds number only. ( there may be other parameters as well) (4). So, from direct Laser measurement point of view, the absolute velocity values are different. But the pattern of the flow field remains the same as long as the Reynolds numbers are the same.

Steady multiple solutions in laminar flows usually have something fairly obvious to lock onto/detach from as the Reynolds number is varied. A classic example is opening up a diffuser angle until it stalls and locks onto one of the walls. This raises a question about the impeller. Can you see the possibility of such a mechanism in your experiment? If so, steadily reducing the impeller speed from high speed (where it sits in one mode) and steadily increasing it from low speed (where it sits in the other) should establish the speed range where the two modes overlap (if you can find something reasonably sensitive to monitor). You may be able to see it by simply watching. The sudden attaching/detaching of the flow from the back of various car shapes can be clearly seen in many visualization experiments without the need to monitor anything.

Thanks for the input, Andy.

Something like this is certainly a possibility and some additional flow visualization studies using very small aluminum flakes should provide us with some more insight.

What sort of non-dimensional parameter would govern this? Some form of Strouhal number would be my best guess. Any other suggestions?

Reynolds Number. In an adverse pressure gradient, it is viscosity which enables a boundary layer to hang on for a while before separating. Changing the relative size of the viscous forces will alter the point of separation. An inviscid flow will separate immediately and treacle probably never.