IUTAM ABCM Symposium on Laminar Turbulent Transition … · 2014-09-25 · In Gaster’s studies8...
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Available online at www.sciencedirect.com Procedia IUTAM 00 (2014) 000–000 www.elsevier.com/locate/procedia IUTAM ABCM Symposium on Laminar Turbulent Transition Wavepackets in boundary layers close to transonic speeds Andres G. Martinez a , Elmer M. Gennaro a , Marcello A. F. Medeiros a a Universidade de S˜ ao Paulo, Av. Trabalhador S˜ ao Carlense, 400, S ˜ ao Carlos, 13566-590 - SP - Brazil Transition in boundary layers is still an open field of research. Substantial progress was made in the nonlinear regime of monochromatic Tollmien-Schlichting (TS) waves, but transition in practical situations involves modulated TS waves. Wavepackets were found to be good models for transition occurring naturally and have received some attention, in particular for the incompressible regime 1,2,3,4,5 . Since airplanes always fly in the compressible regime it is important to investigate the effect of Mach number, which is a lot less researched, in particular close to sonic speeds. As the Mach number increases, oblique waves become progressively more unstable relative to two dimensional waves and the spanwise spectrum of the packet broadens. Above Mach=0.7 oblique waves become more unstable than two dimensional ones. These aspects may enhance different types of mode interaction and affect the packet behavior. The problem was studied via Direct Numerical Simulations. The code was developed in a nonconservative for- mulation, using the velocity components, the density and the internal energy as the directly time integrated variables. The fluid was considered an ideal gas. The equations were discretized by compact finite-differences in all directions. A three diagonal computacional stencil that provided 4 th -order spectral-like accuracy was developed for the code following 6 . Time integration used a 4 th Runge Kutta scheme. Grid stretching was implemented for the wall normal direction. The code was paralelized using the so called 2D pencil and 1D slab decomposition which is particularly suitable for compact schemes. It was successfully tested by comparison with results of comprehensively documented flows, such as the lid-driven cavity flow. The boundary layer studied initiated from an uniform incoming flow that reached the leading edge of a flat plate. The boundary conditions at the inflow were uniform Dirichlet for velocity and temperature, while the pressure was extrapolated from the internal grid points assuming a zero first derivative in the streamwise direction. For the outflow, the pressure was fixed, while velocity and temperature were extrapolated. At the outerflow pressure was fixed and a homogeneous Neuman condition was applied for the other variables. At the wall, Dirichlet conditions were imposed for velocity and temperature, and for pressure a compatibility condition was used 7 . At the outflow a buffer zone was implemented via streamwise grid stretching to prevent non-physical wave reflexions. The disturbances were introduced by a localized suctions and blowing circular region on the plate. The reference length used for nondimensionalization of the parameters was the displacement thickness at the excitation location. The velocity scale was the free-stream velocity. * Corresponding author. Tel.: +55-016-3373-8285. E-mail address: [email protected], [email protected],[email protected] 2210-9838 c 2014 The Authors. Published by Elsevier B.V. Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering).
Transcript of IUTAM ABCM Symposium on Laminar Turbulent Transition … · 2014-09-25 · In Gaster’s studies8...
Available online at www.sciencedirect.com
Procedia IUTAM 00 (2014) 000–000www.elsevier.com/locate/procedia
IUTAM ABCM Symposium on Laminar Turbulent Transition
Wavepackets in boundary layers close to transonic speedsAndres G. Martineza, Elmer M. Gennaroa, Marcello A. F. Medeirosa
aUniversidade de Sao Paulo, Av. Trabalhador Sao Carlense, 400, Sao Carlos, 13566-590 - SP - Brazil
Transition in boundary layers is still an open field of research. Substantial progress was made in the nonlinearregime of monochromatic Tollmien-Schlichting (TS) waves, but transition in practical situations involves modulatedTS waves. Wavepackets were found to be good models for transition occurring naturally and have received someattention, in particular for the incompressible regime1,2,3,4,5. Since airplanes always fly in the compressible regime itis important to investigate the effect of Mach number, which is a lot less researched, in particular close to sonic speeds.As the Mach number increases, oblique waves become progressively more unstable relative to two dimensional wavesand the spanwise spectrum of the packet broadens. Above Mach=0.7 oblique waves become more unstable than twodimensional ones. These aspects may enhance different types of mode interaction and affect the packet behavior.
The problem was studied via Direct Numerical Simulations. The code was developed in a nonconservative for-mulation, using the velocity components, the density and the internal energy as the directly time integrated variables.The fluid was considered an ideal gas. The equations were discretized by compact finite-differences in all directions.A three diagonal computacional stencil that provided 4th-order spectral-like accuracy was developed for the codefollowing6. Time integration used a 4th Runge Kutta scheme. Grid stretching was implemented for the wall normaldirection. The code was paralelized using the so called 2D pencil and 1D slab decomposition which is particularlysuitable for compact schemes. It was successfully tested by comparison with results of comprehensively documentedflows, such as the lid-driven cavity flow.
The boundary layer studied initiated from an uniform incoming flow that reached the leading edge of a flat plate.The boundary conditions at the inflow were uniform Dirichlet for velocity and temperature, while the pressure wasextrapolated from the internal grid points assuming a zero first derivative in the streamwise direction. For the outflow,the pressure was fixed, while velocity and temperature were extrapolated. At the outerflow pressure was fixed and ahomogeneous Neuman condition was applied for the other variables. At the wall, Dirichlet conditions were imposedfor velocity and temperature, and for pressure a compatibility condition was used7. At the outflow a buffer zonewas implemented via streamwise grid stretching to prevent non-physical wave reflexions. The disturbances wereintroduced by a localized suctions and blowing circular region on the plate.
The reference length used for nondimensionalization of the parameters was the displacement thickness at theexcitation location. The velocity scale was the free-stream velocity.
∗ Corresponding author. Tel.: +55-016-3373-8285.E-mail address: [email protected], [email protected],[email protected]
2210-9838 c© 2014 The Authors. Published by Elsevier B.V.Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering).
2 A. Martinez, E. Gennaro and M. Medeiros / Procedia IUTAM 00 (2014) 000–000
Prior to tests with wavepackets, results of periodic wave excitations were successfully compared with the linearstability theory. The disturbance used to excite the packet was built from a flat espectrum of Fourier modes coveringa range of frequencies substantially wider than the boundary layer unstable frequencies3. The effective domain size(excluding the buffer zones) was 4000 × 20 × 1200 units in the streamwise, wall-normal and spanwise directionsrespectively, and the grid had 1401 × 101 × 215 points. Tests ensured that the results were neither dependent on thedomain nor on the discretization used. The displacement thickness Reynolds number at the excitations was 835 whilethat at the effective end of the plate was 3140. These Reynolds numbers are within the unstable region of subsonic flatplate boundary layers.
Figure 1 (a) shows the evolution of a wavepacket at Mach=0.9. Carefull attention was given to the excitationmagnitude. A pulse excitation 3 times larger than that corresponding to figure 1 (a) triggered at the source nonlinearbehavior strong enough to remain self-sustained and govern the packet evolution. We wanted to study the nonlinearregime of a packet as it evolves from previous linear regime. The initial amplitudes shown in figure 1 are similar toamplitudes used in studies of incompressible wavepackets.
Figure 1 (a) shows that, even thought a large portion of the modes composing the packet are unstable, the amplitudeof the packet on the centerline decays along most of the domain. The packet displays the characteristic crescent shapeof the incompressible linear regime, figure 2(a). Throughout the evolution, the packet is composed essentially bythe band of linearly unstable waves, figure 2 (b). Some initial activity that could not be directly linked to TS-wavesappears, but eventually decays.
In Gaster’s studies8 the incompressible linear packet grows by less than 50% along its centerline evolution whichcovers a fraction of the Reynolds number range of the current compressible study. In the subsonic regime, as theMach increases the TS-waves become more stable. Moreover, owing to enhanced three-dimensionality, the dispersiveeffects are also more relevant. Therefore it is expected that the compressible packet would have a reduced growthrate along the centerline. As the packet evolves the modulation effects reduce and eventually the packet amplitudeshould start to grow in physical space. What was somewhat unexpected was that this would take so much longer thanfor incompressible flow. The results show that wavepackets in the Mach number range investigated are much moreresilient to nonlinear behaviour than their incompressible counterpart.
The results indicate that, prior to nonlinear activity, the compressible packet is likely to become a lot wider andlonger than incompressible ones. Therefore, in a compressible natural environment it will be more difficult for apacket to evolve in complete isolation. Interacting packet may be a probable scenario and this was here investigated.
A simulation was performed in which two packets were excitated at the same time. The excitation points were 70units distant from each other in the spanwise direction. This distance was chosen arbitrarily. In order to keep a goodbasis for comparison, the excitation amplitude of each packet was half that of the isolated packet above discussed.
Figure 1(b) shows that the evolution at the center of one of the packets is initially similar to the isolated one, withhalf the amplitude. At some stage distortions are seen. They are associated with the packet interaction, but disappearlater. Results along the centerline of the domain also do display strong distortions 1(c)
Figure 2(a) and (b) show the evolution of the interacting packet in x− z and α× β planes. The packets merge somedistance dowstream, and even at later stages display substantial differences from the isolated case. In the linear regimethe spectra displays a pattern that is linked with the symmetry of the interacting packet case. At later stage modesoutside the linearly unstable band arise, suggesting nonlinear activity. It is clear that the interacting packets are moreprone to nonlinear behavior. Even under interacting conditions the nonlinear activity is relatively weak. The mostsalient feature is a band of oblique modes that are close to a subharmonic of the dominant modes.
It is expected that simulating the evolution even further and extending the parameter space will bring more insightinto this important, and as yet unknown, scenario. These will be available for the final version of the paper. Transitionon wings of commercial airplanes often occur under adverse pressure. This scenario is also being considered.
Acknowledgements
The authors acknowledge support from AFOSR/USA under grant number FA9550-11-1-0354-P00002 and CNPq/Brazil.
A. Martinez, E. Gennaro and M. Medeiros / Procedia IUTAM 00 (2014) 000–000 3
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Fig. 1. Wave packet evolution. (a) single wavepacket at the centerline; pair of wavepackets, (b) at the center of the packet and (c) at the center ofthe domain. At plane y = 2, Mach=0.9.
a) b)
Fig. 2. Wave packet evolution on y=2 planes in physical and Fourier space at Mach=0.9. (a) single wavepacket; (b) pair of wavepackets.
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