I am using a piece of software that provides the aeroelastic response of a flexible wing in transonic flow. I am hoping to use this software to determine the flutter speed index values for a range of mach numbers in order to identify the transonic dip.

The inputs to the solver are the mach number, density and pressure far upstream of the airfoil. The structural parameters are also input to the solver.

I am having difficulty determining the upstream aerodynamic properties. If I prescribe the mach number and flow velocity then how do I determine the corresponding values of pressure and density?

My knowledge of atmospheric properties (not to mention aeroelastic analysis) is rather limited. If I select an altitude then I assume that this provides a value of air density; does this value have to be changed, however, to account for the fact that the upstream flow is near-sonic? (as the air becomes compressible).

I am also slightly confused as to how to specify the upstream air pressure. I know that in subsonic flows that the total pressure is given by: P_total = P_static + 0.5*density*U^2

Does this equation have to be changed when dealing with transonic flows due to the effects of compressibility?

I'd really appreciate it if someone could please shed some light on any of these issues .

Frankie

Examine the aeroelastic solver documentation to determine whether the upstream inputs are (a) Mach number, density and static pressure, or (b) Mach number, density and total pressure. The (static) density and static pressure are unaffected by the relative motion of wing and air.

Then, for example, if the required inputs are Mach number, density and static pressure, pick values for all three as desired (they are among the independent parameters of your study). Then calculate the speed of the sound from the density and static pressure (for example, for an ideal gas, a^2 = gamma*R*p/rho). Then, from the assumed Mach number and the calculated speed of sound, calculate the relative speed of wing and air, U. If needed, you can calculate the total pressure by the formula you listed.

Hi Ananda, According to the notes I've been given I need the upstream mach number, total pressure and density. I am assuming that the flow is isentropic and an ideal gas.

I prescribe the mach number and select a value of airspeed. I have to determine the values of pressure and density that will give me a local speed of sound which, when combined with my selected airspeed, provides an appropriate mach number.

I therefore have 2 unknowns: the total pressure and density far upstream. I don't know how these should be calculated whilst taking into account the fact that the upstream flow is near-sonic. I'm not sure whether compressibility effects should be included and, if so, how to include them.

Cheers, Frankie

Sorry guys I was just checking to see if anyone had replied and realised I'd signed in last time using Mike's machine (as if the question wasn't complicated enough I had to log in as someone else! )

I have been working through this problem for a while now and have come up with a solution (I think). I wonder if someone could please tell me whether I'm going about this the right way.

My understanding of pressure etc is that at a particular height above the earth the air pressure for static air has a fixed value 'Ptotal'. This value changes with height above the earth; the pressure is the 'total pressure'. As the airflow accelerates (at the prescribed height) the total pressure remains constant but bernoulli's equation tells us that Ptotal = Pstatic + 0.5*rho*U^2 is a constant. The increase of velocity is therefore linked to a decrease in static pressure - this causes lift over an airfoil.

In my problem I have to specify a pressure upstream of the airfoil. If I take the airfoil to be at a certain height then this fixes the total pressure. The problem I'm trying to solve requires a fixed mach number and a prescribed velocity value: I am proposing to use the following equation:

M^2 = (U/a)^2 = (gamma*rho/Ptotal)*U^2

where 'M' is the upstream mach number and 'a' is the upstream speed of sound. rho, Ptotal and U are the upstream values of density, pressure and velocity respectively; gamma is the thermodynamic constant(1.4).

If my logic is correct - then this equation has only one unknown, the density upstream. This can be easily calculated but an increase of velocity leads to a decrease in density. This isn't what I expected intuitively. The upstream flow is near-sonic so I'm not sure whether there are some compression effects I've forgotten to include.

If someone could point me in the right direction it'd be a big help!!

Frankie (not Mike!)

Sorry, but your method has a couple of errors. According to the notes you have been provided with, the solver needs the upstream Mach number (M), total pressure (p_t) and density (r). Properly speaking, at a given altitude, the static pressure (p_s) and density r of the air should be known. These will be fixed at that altitude, independent of the motion of the airfoil. Keep in mind that the solver will generally be set up in a reference coordinate frame in which the airfoil is stationary and the upstream air is moving at velocity U, where -U (negative vector U) is the velocity with which the wing is moving in a different reference frame fixed to the upstream air (which appears stationary in the latter frame). When the velocity of the airfoil is set to different values (as a study parameter), the Mach number and total pressure of the upstream air in the solver reference frame will differ from the previous values, but the static pressure and density (and thus speed of sound) of the air will not change. Incidentally, your expression for M is incorrect. It should be U^2 = M^2*a^2 = M^2 * (k*p_s/r), where k is the ratio of specific heats. The speed of sound (a) in the ideal gas depends only on the static temperature (and thus on static pressure and density, not total pressure), since the speed is measured relative to the air. Also, since your flow is compressible, you should not use the incompressible flow Bernouilli equation, but rather the compressible flow equivalent isentropic stagnation relation: p_t / p_s = [1+0.5*(k-1)*M^2]^[k/(k-1)].

However, the requirements of your problem are a little peculiar. You prescribe both M and U (thus fixing 'a'), and therefore you cannot truly prescribe height above sea-level which would fix a contradictory value of 'a'. Hence, roughly copying your suggested method, you might proceed as follows. Pick the upstream M and prescribed U. These two yield the speed of sound, 'a'. Next, pick a static pressure p_s, presumably based on average atmosphere at a selected height h above sea-level. 'a' and 'p_s' together determine the density r from a^2 = k*p_s/r. (Note that selecting h would more realistically determine all three of p_s, r, and thus 'a'. But then you would in general be unable to find an 'h' that would give you the 'a' required by an arbitrary choice of M and U.) Next, calculate p_t from the stagnation relation above, and you would have all the upstream inputs you need. Incidentally, as you mention, for a given M, the local total pressure is fixed along inviscid streamlines, while the local static pressure drops from its upstream value as the local fluid particle speed v increases near the airfoil, but this has no bearing on the upstream properties.

Thanks Ananda - your post has given me a few ideas what to try next. I really appreciate your help on this matter - it's been driving me crazy and I've been working on it for so long now I feel that I can't see the wood for the trees! )

Many thanks, Frankie

You are welcome. The best way to see the wood without being mired in the details of the trees is to nondimensionalize all your equations. This is not difficult, especially if you are dealing with inviscid flow and therefore do not need to consider the Reynolds number. Nondimensionalization would clarify what the minimum number of (independent) study parameters is. However, the nondimensionalization is a habit that takes time to cultivate and use fluently.