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 Li Yang September 25, 2002 18:43

N-S equations in rotating frame

Dear all,

I have got a questions on on the Navier-Stokes equations in rotating frame. As we know, the N-S equations can be written in rotating frame in two forms using relative velocity or absolute velocity as variables.

My question is: when using an absolute velocity formulation in an attached blade frame, what velocity should be used in those diffusive terms in the Navier-Stokes equations and in the Baldwin- Lomax turbulence model i.e. relative velocity or absolute velocity ?

As the viscosity coefficient mu is a function of the coordinates x, y and z, it seems that the diffusive temrs will not be identitcal when using relative or absolute velocity. For example: one of the viscous term d/dx(mu*du/dy) will not be identical when using relative velocity u_r or absolute velocity u:

d/dx(mu*du/dy)=d/dx(mu*du_r/dy)-d/dx(mu*omega)

since d(mu*omega)/dx is not equal to zero.

I would really appreciate it, if you could answer my question and share your experience with me.

With best wishes

Li

 kaiser September 25, 2002 22:32

Re: N-S equations in rotating frame

absolute velocity

 Li Yang September 26, 2002 07:29

Re: N-S equations in rotating frame

An absolute velocity formulation in an attached blade frame is intrinsically still on a non-inertial system, right ? Although the source terms are not the centrifiggal force and Coriolis force anymore, it will still generate a source term due to the transformation of the variables from relative velocity to absolute velocity. For a non-inertial system, the viscous terms should also be in terms of relative velocity components. Now, we want to transfer these relative velocity to absolute velocity as well. Your conclusion must be right, however, I'm just not convinced that why these two would be identical, since u=u_r+omega*y, then

d/dx(mu*du/dy)=d/dx(mu*du_r/dy)+d/dx(mu*omega)

and d(mu*omega)/dx is not equal to zero.

Could you give me some explanations or direct me to any book please ?

Regards

Li

 Tom September 26, 2002 10:38

Re: N-S equations in rotating frame

The turbulence closure you are using is not frame indifferent and so only makes sense in a frame moving with the blade (where the boundary is stationary and the flow is in equilibrium).

 Rami September 26, 2002 10:57

Re: N-S equations in rotating frame

Hi Li,

I cannot offer a digested answer. Rather, I suggest you start by writing down explicitly all the model equations (including the turbulence model) in a stationary frame of reference (probably cylindrical coordinates would be most convenient). Then just carry out carefully the transformation to the rotational system. If you do it, you'll end up with the correct formulation, which you may put in either the absolute or the relative velocity components. I guess the algebra may be non-trivial. If you have access to some computerized algebra system it might save you some efforts.

Good luck!

 Li Yang September 26, 2002 11:23

Re: N-S equations in rotating frame

Dear Tom,

I am now dealing with a helicopter rotor in hovering flight. In an attached blade system, the flow becomes steady. I have to use an absolute velocity formulation in an attached blade frame. A complete relative velocity formulation will cause problem in the farfield boundary since the relative velocity in the farfield is too big. i.g. Suppose rotor tip Mach number is 0.8, and the farfield boundary is placed at somewhere twice radius of the rotor, then mach number in the farfield would be more than 1.6. That's certainly not ideal.

I think the turbulence model would need some changes if we use relative velocity in the Baldwin- Lomax model since the vorticity in the farfield will be 2*omega rather than zero.

Regards

Li

 John YL September 27, 2002 04:44

Re: N-S equations in rotating frame

I am wondering which coordinate system you are using, but the diffusion term is coordinate free- that is it is the same whether relative or absolute velocities are used. Let's say the cartesian coordinate is used:

u = u_r + omega * x v = v_r - omega * y

so the omega contribution is zero (that is your formula of u = u_r + omega * y is not right).

 Tom September 27, 2002 04:48

Re: N-S equations in rotating frame

I think the problem you've got is a bit tricky. If you have to use the absolute velocity with Baldwin-Lomax I'd suggest you write the equations down in the frame of reference moving with the blade, so that all velocities are relative and the turbulent stresses take their usual form, and then reverse the transformation to get everything in terms of absolute velocities. (This is not ideal but is probably unavoidable)

Alternatively you could see if you can find a turbulence closure that makes sense in both frames?

You may be interested in the book

Turbulent flow (models and physics) by Jean Piquet

(I personally don't like what I've read in the book but you may find it useful).

Since you're using Baldwin-Lomax I assume you're solving the full equations - have you considered using a strong viscous/inviscid interaction scheme? In such a method you can solve the inviscid part (farfield) for the absolute velocities and the boundary layer in terms of relative velocities. (This also partly fix some of the weird behaviour of the turbulence model since it would only be being applied within the boundary layer).

Hope some of the above is useful,

Tom.

 Li Yang September 27, 2002 08:13

Re: N-S equations in rotating frame

Dear John,

Thank you very much indeed. I don't know why I made this silly mistake. Now d/dx(mu*du/dy) is indeed equals to d/dx(mu*du_r/dy).

However, for the normal stresses, there are some terms like d/dx(mu*du/dx) and d/dy(mu*dv/dy), hence, the contribution of omega seems still there..

It is good to hear that the diffusion term is coordinate free and I guess this conlusion must be right. I Just would like to know why it is so. Any further comments ?

Thank you again !

Li

 Li Yang September 28, 2002 05:50

Re: N-S equations in rotating frame

Dear John,

I have checked the relation between the absolute velocity and the relative velocity again. Suppose the attached frame rotates around z axis with angular velocity omega, the relations should be:

u=u_r-omega*y and v=v_r+omega*x

Regards

Li

 Li Yang September 28, 2002 05:56

Re: N-S equations in rotating frame

Dear Tom,

Thank you very much indeed.

One more question: Are you sure you can apply the Baldwin-Lomax model without any changes in a non-inertial frame with relative velocity as variables?

Regards

Li

 Tom September 28, 2002 09:25

Re: N-S equations in rotating frame

It's been a while since I looked at the Baldwin-Lomax model (around 10 years) but as I recall it's is an attempt to generalize the Cebeci-Smith boundary-layer model to the full Navier-Stokes equations. This means that it should be thought of in a refernece frame with the body fixed. This is because the Outer part of the eddy-viscosity model takes the form

(boundary layer depth)X(velocity at edge of b.l.)

(I can't remember how these are estimated in Bladwin Lomax). It's this form of the eddy-diffusivity that's the problem - the choice of velocity at the edge of the boundary layer depends on your frame of reference.

I would argue, possibly incorrectly, that if such a formulation is to make sense it must be in the frame of refernece moving with the body. This is definitely the case in flow past an aerofoil (where the Cebeci-Smith model originates).

I don't think this answers your question, since I don't really know the answer, but in my opinion I don't think the Baldwin-Lomax can be used in arbitrary frames of reference,

Tom.

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