Unsteady Aerodynamics: Introduction Flashcards
What is the aerodynamic interface operator?
From the structural analysis, we have obtained the structures generalized coordinates, therefore the position, speed, acceleration of each node is known at every time instant.
On the other hand, the aerodynamic models may be geometrically incompatible with the structural ones, due to different models, grids, etc.
The aerodynamic operator is:
*A differential operator
*Unsteady
*Effective (accurate but computationally effective).
Therefore, a interface operator is needed. This interface has the symbol I(N). If db = Nq, then dA = I(dB) = I(N)q=N’q, where A the aerodynamic grid and B the structural grid.
Why isn’t CFD always preferrable?
Using CFD (or WT tests) it is in principle possible to consider arbitrary complex geometry, any flow condition, arbitraly large structural motions. But
*Availability of a detailed model (not often available, especially at early-stage design)
*Simulation time is very high: limited capability to perform parametric analysis (mass, geometry, flight conditions)
*Often the type of analysis is limited to time marching integration, so in order to go in frequency domain and examine eigenvalues, random analysis etc required a lot of post-processing.
*The amount of information obtained is very large and it is difficult to identify the important features that dominate the phenomena.
What are some simplified physical models that can be adopted?
First of all, these simplified physical models are very effective to understand the important features of the phenomenon under investigation and they give the possibility to perform massive parametric analysis (design, certification, flight test preparation).
*In many cases, the model could be inviscid (exluded Buzz, buffet, stall, flutter, which are phenomena related to separation and CFD is a must).
*For transonic flows, it is possible to use Euler instead of N-S equations (when viscosity is not relevant)
*For subsonic or high supersonic flows the linearized potential flow are good approximations (away from transonic conditions).
*For low speed the incompressible flow models are very effective
*Geometry is often simplified (lifting surface, lifting line, non-lifting bodies neglected) especially when the study of perturbations with respect to a trim state is sought.
What are the basic governing equations? What is the hierarchy of the models?
We start from the Navier-Stokes equations (which re the most general model to represent the behaviour of the dynamics of a fluid). Full N-S are never used.
From N-S we can move to:
*Incompressible Reynolds Average N-S (RANS)
*Compressible RANS
*Euler Equations (neglect viscosity)
Euler equations are widely used and can be separated to:
*Incompressible
*Full Potential Flow
The Full Potential Flow equations can be linearized, reaching to the linearized potential.
What are the governing equations?
Starting from Reynold’s Transport Theorem:
1)Mass Conservation (continuity equation)
2)Balance of Momentum (Newton’s 2nd law of dynamics, rate of change of momentum = total body forces + sum of surface forces per unit area on the boundary) + the Cauchy-Poisson constitutive law for a Newtonian fluid, where T is related only to thermodynamic pressure p, strain rate D and dilatation. Assuming that the fluid is invisicid (Euler eqs), only the thermodynamic pressure remains for the stresses and so the surface forces per area unit depend on the pressure and the normal vector.
3)Energy balance, which is the 1st principle of thermodynamics. It states that the variation of the total energy is the work done by external forces + heat exchange.
It is necessary to add the equation of state of the ideal gas for internal energy (p=ρRT, e=CvT, h=CpT=e+p/ρ).
How can the rate of change of kinetic energy or Bernoulli equation be obtained?
If we take the balance of momentum and multiply by the speed, we can also see there is the balance of mechanical energy or Bernoulli equation.
For invisicid flows, the rate of change of kinetic energy is equal to the work done by pressure forces.
When is the flow isentropic?
Using the fundamental differential equation of thermodynamics for entropy, and assuming that the fluid is inviscid, we can obtain that ds/dt = 0, or the rate of entropy in the flux is null i.e. the derivative of entropy for each particle along its path is null.
If there is no gradient of entropy at the initial instant in time the flow remains isentropic.
How is vorticity defined?
Vorticity is the curl of velocity.
What is circulation?
Circulation is the integral of the vorticity normal to a surface over the surface.
What is Kelvin’s theorem and how is it related to a sudden start of a 2D airfoil?
If we consider a barotropic fluid (a fluid which p=p(ρ), it’s pressure depends only on the density, then the rate of circulation is equal to zero. An isentropic fluid is barotropic.
This means that if the vorticity is null at the initial instant in time (or the circulation is null), then the flow will remain with null vorticity, i.e. irrotational for all times.
Only intense strong shock waves, or intense heating an spoil the hypothesis of isoentropic and so barotropic flow.
For a sudden start of 2D airfoil:
At the initial instant of time, there is a vortex that starts on the airfoil and another opposite and equal vortex that appears at the trailing edge. This vortex will start travelling at the asymptotic speed of the wing. From the POV of the observer, the airfoil moves, and the wake stays where it was generated. If there is constant lift, there is no vorticity in the wake. For this reason, in a steady 2D flow, the wake is not taken into account. The wake exists only if there is a variation of circulation, which means a variation of lift or in a 3D wing.
What is the velocity potential?
If the flow is irrotational it is possible to define a velocity potential:
ω = 0 -> ∇ x u = 0. If we define u = ∇Φ, with Φ being the velocity potential, the velocity will be irrotational by definition as ∇x∇(.) = 0. The advantage is that in this way there is only one uknown Φ, while before there were 3 uknowns, the components of speed.
Aftwerwards, we can substitute the potential in the balance of momentum and obtain the Unsteady Bernoulli Theorem (for irrotational flows). Ultimately, we obtain an expression that contains the potential, the value of velocity and enthalpy. If the flow is inviscid and irrotational, this quantity is constant in space. Otherwise, if the flow is compressible and unsteady, the LHS is a function of time only and is equal to F(t).
Then, by using the equations for isoentropic flows and the expession for the speed of sound, it is possible to obtain an expression with the potential, the |velocity|, the speed of sound and the asymptotic speed, asymptotic speed of sound and γ (adiabatic constant). ∇(∂Φ/∂t + u²/2 + h) = 0.
Αfterwards, the Pressure Coefficient can be calculated, both for compressible, adiabatic flow and incompressible flow (ρ = const).
Ultimately, two expressions, for the Potential and Speed of Sound can be found, which express the full potential equations for compressible and isentropic flows. This is a complex, nonlinear, scalar equation in the potential. Through the potential it is possible to identify the entire flow-field (pressure, density..), It is possible to simplify the problem by linearization.
It is possible to obtain the pressure coefficient distributions whenever the solution of the problem is known, so the knowledge of the potential at every instant of time. To solve an unsteady problem, we need to solve the potential problem.
What are the boundary conditions for the full potential equation?
If we call B(x,t) the implicit definition of the lifting surface, the normal to this surface is nB, so the BC is:
u nb = vb nb (where vb is the velocity of the body).
At the infinite boundary, the limit of u(x) at infinite is equal to the asymptotic speed.
However, with this condition it is impossible to obtain the right solution by using the infinite boundary, because the flow is irrotational and the rotation have to appear out of the airfoil boundary, caused by the viscosity of the boundary layer.
That is why the Kutta condition is enforced.
*The point from which the flow detached (trailing edge) should always be the rear stagnation point (the detachment of the flow is a consequence of the development of vorticity).
*The potential problem with Neumann BC admits a unique solution only in simply, connected regions.
*Any closed line around the airfoil (2D) cannot be contracted to a point, so the region is multiconnected and the solution is not unique.
*For 3D flows the region is simply connected but the wake is not a potential flow region, so it cannot be crossed. In this case, the potential is not uniquely defined.
A condition must be added to recover the effect of viscosity. The condition states that the flow must leave from the trailing edge smoothly and the velocity must be finite, meaning that the TE must be a stagnation point. Therefore:
ΔpTE = 0 or if we call γ the circulation per unit lenth, γTE = 0. For steady flows it is possible to verify that the flow leaves the airfoil along the bisector of the TE angle.
What is the linearized potential?
We are only interested in very small movements and we expect the peturbaton to the flow generated by those small movements to be small as well. We can linearize our problem, making the hpytothesis that all peturbations generated by the unsteady effect addef to the steady solutions are small.
α = α00 + δα, p = p00 + δp, ρ = ρ00 + δρ, u = Uooi + δu, Φ = Uoox + φ, ∇Φ = Uooi + ∇φ.
Ultimately we reach an expression for ∇²φ for compressible and incompressible ( = 0) flows, and Cp = -ρinf dφ.dt,
The reason it is equal to zero for incompressible flows is because the speed of sound goes to infinite, meaning that any peturbation is transmitted to all points of the flow instantaneously. Also, the equation is linear and not time dependent and the only elements that may introduce time dependance are the BC. The equation is also valid for steady and unsteady flows.
Afterwards, we have to linearize the boundary conditions, where we compare the boundaries on the body at two instants in time t+dt. The position of the boundary can be seen as the sum of the position of the cord + the position of the camber line + the position of the thickness + the deformation, since the potential problem is linear with linear boundary conditions so the superimposition effects coule be applied. The deformation is expressed as the product of a shape function for a vector of uknowns, which are the generalized coordinates (bud = N(x)q(t)). Ultimately:
Nq_dot/Uoo + ∂N/∂xq = w/Uoo. The first term is the kinematic change of the angle of attack and the second temr is the geometric change of the angle of attack. Nq_dot is the speed vector (speed of every point of the airfoil), so dividing it by Uinf we obtain the local change of AoA due to a movement. If N(x) is the moement in x direction, dN/dx represents the change of slope of the curve. The sum of the two contributions is giving the global change of AoA of the airfoil at every point of the structural, which is equal to the ratio of normal speed dividied by Uinf.
How can the incompressible problem be solved?
∇u = 0. u is solenoidal (null divergence) so it exists a vector potential that u is the rotor of it.
We define a vector potential Ψ and obtain the vectorial Poisson Equation which relates Ψ to vorticity. Then we can use the Biot-Savart formula that expresses the velocity of a point due to a vortex at poisition ξ.
Then we can calculate the speed indeced at a point P from either a straight vortex at distance r, or due to a line of infinite length vortices.
