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Chapter 8
Conclusions and comments
8.1 Overview of obtained results
As wrote in introduction section, today there is a difficulty to predict buffet onset with both CFD analyses and analytical methods. This challenge results harder for conditions whose Mach number is near to 0.80 and above. Many authors tried to trace a complete buffet boundary curve by use of analytical and numerical models like that of Ref. [7]-[8]-[10]-[11]-[12], but for them all, critical zone about 𝑀𝑎𝑐ℎ = 0.80 remained unsatisfactory. Even if their results were in quite good accordance with experimental data and between themselves, obviously far enough from previous condition, appeared a large gap at highest Mach numbers for which flux becomes unstable for little incidence angles. Moreover, these last conditions are common during cruise of aircrafts flying at transonic speed.
In this work was used an URANS approach, i.e. were employed the unsteady averaged Navier-Stokes equations. The idea was that pressure oscillations, due to buffet phenomenon, involve frequencies consistently smaller than that associated to turbulence in the shear layer, thus the averaging operation does not affect results linked to buffet. This hypothesis was then confirmed by literature, in particular by Ref. [8].
Incapability of models found in literature of locate phenomenon about 𝑀𝑎𝑐ℎ = 0.80 and beyond conducted some authors to think that pioneers of buffet studies, like Mc Devitt John B. and Okuno Arthur F., could have been wrong in their experiments made in the Ames High Reynolds Number Facility; this thesis demonstrates the contrary, showing a very good accordance with their experimental data. Reasons of this fact may be due to a finer grid than others used in literature or by software advances.
Conducted analyses demonstrated capability of software ANSYS® Workbench, used
to build mesh around model and to simulate flow dynamics using an URANS approach and a ‘realizable k-ε’ method for turbulence specification, of capture transonic buffet onset for a NACA0012 airfoil. Next Figure 8.1 clearly shows buffet boundary and results obtained in this (named CFD) and other works like Ref. [5]-[8] and [12] named respectively:
• ‘Ref. [5] NASA experimental interpolated curve’ accompanied by points defining the last. named ‘NASA (PROBE 0.5*c)’ and ‘NASA (PROBE 0.8*c)’ representing experimental points with probe position.
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• ‘Ref. [8] ROM’ that represents analytical results obtained by using Reduced Order Models (ROM), ‘Ref. [8] CFD (steady)’ and ‘Ref. [8] CFD (unsteady)’ representing points obtained by use of CFD resulting in steady and unsteady solution respectively.
• ‘Ref. [12] CFD’ representing points obtained by CFD analyses.
Figure 8.1 Buffet boundary with thesis and literature results.
As can be noted, accordance between experimental and computed data is remarkable. Frequencies involved in each located buffet onset point are in the order of magnitude of that found in literature and reported in Table 8.1, moreover Figure 8.2 illustrates their classical behaviour of reducing in value while Mach number increases.
0 0,25 0,5 0,75 1 1,25 1,5 1,75 2 2,25 2,5 2,75 3 3,25 3,5 3,75 4 4,25 4,5 4,75 5 5,25 5,5 0,7 0,71 0,72 0,73 0,74 0,75 0,76 0,77 0,78 0,79 0,8 0,81 0,82 0,83 0,84 0,85 0,86 0,87 0,88 0,89 0,9 α , i n ci d e n ce [d e g] Mach number
Buffet boundary
Ref. [5] NASA experimental interpolated curve
Ref. [8] ROM
Ref. [5] NASA (PROBE 0,5*c) Ref. [5] NASA (PROBE 0,8*c) CFD
Ref. [8] CFD (steady) Ref. [8] CFD (unsteady) Ref. [12] CFD
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Buffet onset frequency and reduced frequency
Mach incidence [deg] frequency [Hz] reduced frequency
0,76 3,025 181,73 1,035737045
0,775 2,525 98,19 0,548784748
0,8 0,975 89,89 0,530751182
0,816 0,1 84,15 0,446684042
0,825 0 54,79 0,287662882
Table 8.1 Frequency and reduced frequency involved in buffet onset.
Figure 8.2 Reduced frequencies as a function of Mach number.
Last values in Table 8.1 are in red because are estimations based on performed analyses; it was not possible indeed to extract exact data from a Fast Fourier Transform analysis and would be necessary to use other methods to inspect these spectra like Wavelet method. Furthermore, Figure 8.1 shows the sudden drop at 𝑀𝑎𝑐ℎ = 0.86 due to shock wave oscillation accompanied by large amplitudes and low frequencies.
Found similarity with literature were values of buffet onset up to Mach numbers about 0.775. In these conditions was highlighted that buffet phenomenon develops on upper airfoil surface with a frequency content similar to that indicated in cited references, furthermore, were analysed related Limit Cycle Oscillations. Charts of the last have shown an interesting similitude with multiple degrees of freedom systems led to the hypothesis that both would react at the same manner to a disturbance; this is related to presence in LCO charts of more than one ring, more or less circular, testifying presence of multiple frequencies governing physical phenomenon. Moreover, was found the presence of very
0 0,2 0,4 0,6 0,8 1 1,2 0,75 0,76 0,77 0,78 0,79 0,8 0,81 0,82 0,83 0,84 0,85 0,86 0,87 re d u ced fr e q u e n cy Mach number
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high frequencies in spectrum probably associated to other phenomena than transonic buffet. A further analysis with POD methods (Proper Orthogonal Decomposition) would help to define the shape of involved modes in the dynamic response of the physical system starting from data sets of conducted analyses giving an insight on what kinds of phenomena are present together with transonic buffet.
Although the remarkable similitudes with results of other works, were found some interesting news made reliable by good results obtained in comparison with experimental data not only for previous cited lower Mach numbers, but for higher ones too.
First of all, when buffet starts at all Mach numbers simulated, it seems to appear as a phenomenon concentrated in little zones of fluid highlighted by RMSE of static pressure analyses despite of popular opinion. For Mach number near and beyond 0.80 it was found that unsteadiness starts on the lower surface of the airfoil differently from other conditions. As highlighted in paragraph 7.1.3 also in case of null incidence it was found an asymmetrical solution with most of disturbances concentrated on the lower surface; even if this fact would seems to be strange, as happens in elastic theory of equilibrium when a stable configuration of equilibrium becomes unstable, the system tends to reach a new stable configuration if subjected to a perturbation even if very small. In present case the symmetrical solution became unstable and the system reached a new stable asymmetrical configuration like that found.
Moreover, were not found, when instability starts, macroscopic oscillations of shock wave. Variations of pressure field involved are of very little entity and the derived unsteadiness does not seems to be due to shock oscillation mechanism. It was noted indeed that, as Mach or incidence angle increases, buffet phenomenon grows up to the famous condition of shock wave self-excited oscillation only in a condition quite far from the onset one i.e. in a case of well-developed instability. Actually the derived hypothesis is that flux bothered by some kind of perturbation, like presence of a body (an airfoil in present case), reacts like an elastic body showing proper frequencies dependant by nature of fluid itself and by its conditions in terms of pressure and temperature, causing perturbations of pressure field that turn out in pressure fluctuations, (thus lift, drag and moment fluctuations around a body immersed in the flux); these oscillations seems to be the most probable cause of shock wave oscillation and in this condition only was observed a global instability of flow field. Practically flow field shown a behaviour typical of multiple degrees of freedom system testified by LCO and PSD charts.
Global instability analysis based on a smoothing of shock wave (see Ref. [7]) although capable of giving results near buffet boundary for lowest Mach number considered in Figure 8.1 shows great troubles in prediction at higher velocities. This fact may be due to smoothing operation: theory is based on hypothesis of global instability behaviour of flow field, but present thesis shown that, at buffet onset conditions at least, unsteadiness is very concentrated, thus, a smoothing operation probably cuts phenomena involved hiding repetitive and structured little oscillations. A sort of proof of this fact is that in this work was possible to trace whole buffet boundary with a good accuracy.
Curious aspect of onset of instability on lower airfoil surface should be further indagated to give a physical motivation. This behaviour indeed, seems to be not a spurious error of numerical simulation, but a constant for conditions near 𝑀𝑎𝑐ℎ = 0.80 and beyond. Coupled analyses of simulated dynamic behaviour and of pressure fluctuations in MATLAB® led to visualize origin of the instabilities and their propagation direction. From simulation it was noted that pressure oscillations start immediately downstream of shock
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wave on upper or lower surface of the airfoil, depending on Mach number as previously wrote, and that there was presence of pressure oscillations along shock wave and downstream of the last inside the shear layer towards airfoil trailing edge as well as towards leading edge even if of vanishingly intensity. Following step of colours-map creation in MATLAB® confirmed the hypothesis that origin of fluctuations, at buffet onset at least, was located near and downstream of shock wave moreover indicating that instabilities travel as waves from shock wave towards trailing and leading edge. Further inspection of previous colours-map seen in plant view and juxtaposed like in Figure 7.42, indicated a substantial condition of phase opposition of phenomenon on upper and lower surfaces even if with some difficulties to inspect buffet onset conditions due to smallness of perturbations, but markedly visible in case of full developed unsteadiness like in case of 𝑀𝑎𝑐ℎ = 0.86 and 𝛼 = 0° or 𝑀𝑎𝑐ℎ = 0.80 and 𝛼 = 4°. Moreover, were made films animating simulations indicating, as wrote in chapter 5 (par. 5.1.14), that there is presence of waves going back to the current.
Observations concerning direction of propagation of oscillations seem to be in accordance with theory of self-excited oscillation developed by Lee B.H.K., Tijdemann H. and Seebass R. (Ref. [13]-[14]), but for a complete confirmation of their theory should be made simulations acquiring data on a streamline outside of shear layer and should created colours-map of static pressure along this line to investigate direction of propagation of resulting waves considerable as acoustic waves.
At 𝑀𝑎𝑐ℎ = 0.80 was noted simulation by simulation that buffet, even if started in accordance with experimental data while increasing a little bit incidence angle, had tendency to vanish suddenly, newly appearing for slightly larger α. Using root mean square of power spectral density of lift coefficient and representing obtained values as a function of incidence angle, were noted formation of ramps whose peaks corresponded to conditions of α for which simulations showed well-organised unsteadiness accompanied by repetitive and predictable cycles as well as limit cycle oscillations with a well-defined shape and larger dimensions than in other conditions. Last considerations in association with the fact that normalised residuals of CFD analyses in these peaks remain quite large compared to others, led to the hypothesis of multiple solution existence for buffet onset on a NACA 0012 airfoil involving eigenvalues of second, third order etcetera referring to similarity found with dynamic response of a multiple degrees of freedom system. Authors like Kuzmin A. (see Ref. [15]) found existence of multiple solutions in non-viscous fluxes around airfoils in transonic regime with sudden return to stable conditions; Crouch J. D., Garbaruk A., Magidov D., Travin A. in their work cited in Ref. [8] hinted at the presence of two analytical solutions of a viscous flow around an airfoil in transonic regime, but they did not furnish a physical explanation of that result. Found result of present work in analyses at 𝑀𝑎𝑐ℎ = 0.80 seems to be correlated to Hopf subcritical bifurcations, i.e., despite of wing panels that maybe subjected to supercritical conditions with supercritical Hopf bifurcation presenting two stable branches of curve representing solution, when a system becomes unstable in a subcritical regime with a subcritical Hopf bifurcation its curve representing solution has two unstable branches and solution oscillate between the two. Situation is represented in a qualitative way in next Figure 8.3.
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Figure 8.3 Qualitative representation of subcritical Hopf bifurcation conditions.
In future investigations should be examined this multiple solution condition trying to give a physical explanation.
Further investigations should be focused on building an analytical process based on present model of numerical simulation that seems to be promising in giving good results.
Used method of this work indeed, is not suitable for exploring and predicting buffet boundaries of new configurations for two reasons:
• first it cannot predict if a condition could be an unstable one before a simulation was performed, thus needs to do a great number of simulations slightly varying parameters like Mach number or incidence angle.
• second, every analysis needs a long time to be performed starting from a steady simulation and then performing a transient one. Every analysis of this thesis took at least three days to provide a result by using six processors INTEL® I7 3770 working 24 hours each day at a clock frequency of 3.40 GHz in a machine equipped with 32 Gb of RAM, but it’s a two-dimensional body of little dimension. On the contrary simulations conducted on an aircraft would be too expensive in terms of computational time spent using a method practically based on attempts.
Due to that reasons, implementing an analytical model capable to predict at least enough reliably position of buffet boundaries would be a great goal allowing then this kind of simulation to reduce attempts to find the onset condition. Then a simulation using
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method like that employed in present work may predict oscillation amplitude of pressure, and consequently forces developed by fluid on a structure. A coupled simulation fluid-structure interaction (FSI) could eventually define dynamic response of fluid-structure to fluid perturbation giving good indications to structural designers.
