Hello Folks, I'm dealing with a problem that involves unsteady flow over a simple 2D airfoil; the airfoil has 2 degrees of freedom (pitch and plunge) and the idea is that the unsteady aerodynamic load will cause oscillations of the system. I'm trying to determine the physical parameters that cause the oscillations to grow with time (flutter).

The model aerodynamic equation I'm trying to solve is:

k*p_t + p_xx + p_yy = 0

This equation is dimensionless, p is the potential and _t represents differentiation with respect to non-dimensional time 't', x & y are my independent position variables. The 'k' term is a dimensionless parameter that involves the frequency of the system and the airspeed.

I know the structural parameters of the system including the modeshapes and the natural frequencies for the 2 modes. Calculating the response of the structure to a given aerodynamic load is easy (for this problem) and I expect an oscillating solution. If I can determine the airload then it's a simple matter to determine the system response. The difficulty comes when calculating the actual airloads - what values of frequency do I use to calculate my parameter 'k' in the aerodynamic equation?

I have 4 frequencies involved in the problem: wn1 & wn2 (natural frequencies of mode 1 & 2) and wh & wp are the uncoupled plunge & pitch frequencies respectively. The pitch & plunge motions are coupled so I am unsure how to work out the frequency of the combined motion.

It seems I have a bit of a chicken & egg problem: I can't work out airloads until I know the correct frequency term to use and I can't determine system response without knowing the airloads.

If someone could please advise me how to solve this then I'd be very greatful, Bren

The aeroelastic problem is coupled...ie you need to solve both the structural and fluid equations together in one big "matrix solution"(monolithic scheme) or iterate the solution of each within a timestep (strong coupling algorithm). Have you checked out books on aeroelasticity? Fung, Bisplinghoff, Dowell, etc?

Thanks for the advice Peter. I haven't looked through those texts - I'll have a check through the library tomorrow to see if we have them in stock.

My aerodynamic problem is 2D and is solved using a rather complex ADI scheme - there's no way (that I'm aware of) to solve the problem directly as a single matrix equation. The structural equations don't contain the frequency term explicitly. I guess that I will have to extract the frequency from the actual physical response to applied load. This is where I have a problem.

I'm very new to structural dynamics and am having a problem with the concept of 'frequency'. I understand the term completely when talking about single entities but am a bit unsure about how to deal with coupled systems. How would one define the frequency of, say, a coupled pitching & plunging airfoil? - is the frequency related to the pitching or the plunging motion?

The iterative technique sounds promising though - I'll give it a try later this evening.

Many thanks, Bren

Hi Bren, is the unsteady aerodynamic flow incompressible?

The flow can be either compressible or incompressible depending upon the parameters chosen in the solver. The aerodynamics are all dealt with by a black-box solver. The inputs to the aerodynamic solver include the frequency term and that's the difficult part. The outputs of the solver are a set of non-dimensional pressure values which provide the loads used in my structural solver.

First of all, the problem is linear, thus the lift and moment coefficients computed from the solver are linearly related to the pitch and plunge by a matrix of aerodynamic influence coefficients (AIC),i.e.

Q = [AIC]x

The AIC matrix should be a function of the frequency and maybe Mach number, and airspeed. For a given Mach number, M the equations of motion become

[M]x" + [K]x = (1/2)*(rho*U^2)*[AIC]x

Assuming a harmonic motion of frequency, w

[-(w^2)[M] + [K] - (1/2)*(rho*U^2)*[AIC]]x = 0

At the critical flutter point, the determinant of the matrix, [-(w^2)[M] + [K] - (1/2)*(rho*U^2)*[AIC]], should be zero.

Hence the values of the frequency, w and airspeed, U for which this determinant becomes zero are the critical flutter frequency and flutter speed.

The correct way to solve for this problem is to compute the AIC matrix for a range of values of the frequency, w. For each frequency, solve for the airspeed from the equation obtained by setting the determinant to zero. The solutions obtained for the airspeed should be complex numbers. However when the correct value of w is used, the solution for U is a real number. This is why you should solve for U for a range of w-values to find the values of w where the imaginary part of U changes sign.

NB: x is the vector containing the plunge and pitch motions [M] is the inertia or mass matrix, [K] is the stiffness matrix and [AIC(w,M)] is the aerodynamic influence coefficient matrix which is function of w, and M. Is k = wb/U?

Sorry Nicholas I just realised that I'd written the wrong aerodynamic equation in my original post (doh!). The aerodynamic method is actually quasilinear - I'd missed out a term in my original problem statement. The equation should be:

k*p_t + f(p_x)*p_xx + p_yy = 0 where f(p_x) is a function of p_x

The method I'm using is a time-domain method which solves the system at each timestep by using an aerodynamic and structural component. The aerodynamic solver is essentially a black box utility, I input a value of reduced frequency, the pitch & plunge & velocities and it gives the lift & moment coefficients for the prescribed airfoil.

The structural component takes the airloads and applies them to my structure to work out the response at the current timestep. The structural model gives me the new pitch, plunge & velocities which are used for the aerodynamic solver for the next timestep.

Will the method that you suggested be capable of dealing with this type of problem when the aerodynamics are non-linear? The output matrix of lift & moment coefficients is not an analytical function fo frequency so I'm unsure how to form the determinant.

The frequency appears in the aerodynamic solver as the reduced frequency k=wb/U. The frequency appears in the aerodynamic equation as a consequence of non-dimensionalising the time derivative. A characteristic frequency of the flow is chosen for the non-dimensionalisation process. I'm not too sure which one to use though!

Thanks very much for your advice - it's much appreciated!

The equation bears strong resemblance with the non-dimensionalised form of the unsteady transonic small disturbance equation. The flow potential, p is expressed and solved for as the sum of a steady and unsteady harmonic component. The aerodynamic equation to be solved for the unsteady harmonic component should be linearised.

The recommended approach can be used, your only obstacle is the computation of AIC matrix from the aerodynamic code. In order to compute your AIC you need to know whether your aerodynamic code is computing the unsteady airload as a function of frequency or time. You can only know this by enquiring to the person who developed the code.

You say that the aero load is computed for a given time step, however you need k as an input. This seems conflicting because once a value of k is prescribed, the loads are harmonic functions of time, all you need to know from the solver are the amplitudes and phases for the aero loads.

I'm not very familiar with developing aerodynamic solvers, its something I'm very keen on delving into in the near future, but what is clear is that you need the harmonic component of the lift and moment coefficients from the solver given the pitch, plunge, velocity and k, in order to solve for the flutter point conditions using the Classical Theodorsen Approach or Hassig's p-k method.

The aerodynamic solver does indeed solve the unsteady TSD equation (when it works - it's a sensitive problem!). I left the description of the aerodynamic code rather vague as the problem I'm having isn't anything to do with the cfd side of things but rather the coupling of the cfd to the structural model. I'm also not 100% sure how the solver actually works!

The non-dimensionalisation used in deriving the TSD equation means that there is a reduced frequency term multiplying the p_xt term. The reduced frequency is usually specified as part of the scheme. The airfoil motion is then prescribed, this gives periodic boundary conditions that are input to the tsd code.

I'm trying to couple an airfoil to the tsd equation to do some simple flutter analysis. The structural model is very simple with only pitch & plunge DOF. Originally I had thought of simply specifying the reduced frequency in the tsd equation and using the generated output to oscillate my airfoil. I then realised that this would be a rather naive approach as the prescribed frequency might not have anything to do with the actual motion of the airfoil. This would defeat the point as the reduced frequency appears in the equation due to the non-dimensionalisation. That's where my chicken & egg problems started....

I will try the AIC method that you suggested. Thanks very much for your advice Bren

Hi Bren,

Sorry I was away for a couple of days. With the standard unsteady TSD equation, you will obtain linearised solutions for lift and pitch moment coefficients and therefore the AIC method will definately work, however a you require a full understanding of the way code works in order to couple the structure dynamics of the airfoil with the solver.

If it is frequency domain solver, the good news then is no coupling is required. All you have to do is prescribe harmonic motions separately for the pitch and plunge with a frequency, k. Compute the lift and moment coefficients, and then generate the AIC matrix for the given value of k.

On the other hand, if it is a time domain solver, you will also have to prescribe harmonic motions separately for the pitch and plunge with a frequency, k, however the means of generating the lift and moment coefficients from the solver with this information is different.

In either way you dont have to couple the structure and aerodynamics to compute the pitch and plunge responses at each time step.

By the way thanks for the Yoshihara reference you sent for the Euler code, I managed to download a copy. I'm currently occupied writing my thesis, but I'll have a read of it once thats out the way. I'd also like to look at some of the unsteady TSD stuff you're doing, I think it would be a more than useful exercise and a good start into the unsteady transonic aerodynamics I'd like to look at.

Cheers Nick

Hi Nick, The TSD solver I'm using is in the time domain and solves the full quasi-steady transonic small-potential equation. The way in which the TSD equation is derived, for this particular case, means that there is a reduced frequency term appearing in the equation.

The initial solver had a prescribed airfoil motion (pitch motion with a prescribed frequency). The solver integrates the TSD equation in a timestepping manner, in each timestep the position of the airfoil is updated which provides new tangential flow conditions for the next timestep.

With a flexible structure thrown into the mix things become much more complicated. The motion is no longer prescribed and so I don't have a reference frequency to work with. The aerodynamics are non-linear and expensive to compute and so the P-K method will likely be very expensive computationally. I had a meeting with my boss today and have a few new ideas how to solve the problem.

I can't send you my own codes until we're finished with them (as you're writing your thesis I'm sure you'll understand how these things work). But when everything is written (hopefully some time early next year) I'll be happy to send you what I have.

Good luck with your thesis, B

Thanks very much Bren,

I really appreciate this because it is so difficult to get hold of these codes and developing them is no where close to easy either. Anytime you're ready I'll be glad to have them. Besides we can also share ideas about aeroelasticity, I'm useless at CFD but I know a thing or two about non-linear aeroelasticity, limit cycle oscillations, and linear flutter.

Best of wishes, keep me posted and anytime you feel I may be of help, my e-mail is aeronick2000@yahoo.com.

Cheers Nick

Many thanks Nick, I'll get in touch when my research is finished. I'll be able to share the codes and results when it's all wrapped up. The stuff I'm doing is by no means cutting edge but as you're aware it's a massive pain in the ass trying to find some example codes. I'll probably make them publicly available (assuming my department has no objections) and maybe install a matlab gui for their use.

Good luck with your thesis, Bren