Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers
2010 (English)In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 650, 181-214 p.Article in journal (Refereed) Published
The global linear stability of the flat-plate boundary-layer flow to three-dimensional disturbances is studied by means of an optimization technique. We consider both the optimal initial condition leading to the largest growth at finite times and the optimal time-periodic forcing leading to the largest asymptotic response. Both optimization problems are solved using a Lagrange multiplier technique, where the objective function is the kinetic energy of the flow perturbations and the constraints involve the linearized Navier-Stokes equations. The approach proposed here is particularly suited to examine convectively unstable flows, where single global eigenmodes of the system do not capture the downstream growth of the disturbances. In addition, the use of matrix-free methods enables us to extend the present framework to any geometrical configuration. The optimal initial condition for spanwise wavelengths of the order of the boundary-layer thickness are finite-length streamwise vortices exploiting the lift-up mechanism to create streaks. For long spanwise wavelengths, it is the Orr mechanism combined with the amplification of oblique wave packets that is responsible for the disturbance growth. This mechanism is dominant for the long computational domain and thus for the relatively high Reynolds number considered here. Three-dimensional localized optimal initial conditions are also computed and the corresponding wave packets examined. For short optimization times, the optimal disturbances consist of streaky structures propagating and elongating in the downstream direction without significant spreading in the lateral direction. For long optimization times, we find the optimal disturbances with the largest energy amplification. These are wave packets of Tollmien-Schlichting waves with low streamwise propagation speed and faster spreading in the spanwise direction. The pseudo-spectrum of the system for real frequencies is also computed with matrix-free methods. The spatial structure of the optimal forcing is similar to that of the optimal initial condition, and the largest response to forcing is also associated with the Orr/oblique wave mechanism, however less so than in the case of the optimal initial condition. The lift-up mechanism is most efficient at zero frequency and degrades slowly for increasing frequencies. The response to localized upstream forcing is also discussed.
Place, publisher, year, edition, pages
2010. Vol. 650, 181-214 p.
Asymptotic response, Blasius, Computational domains, Disturbance growth, Eigen modes, Energy amplification, Flat plate, Flow perturbations, Geometrical configurations, High Reynolds number, Initial conditions, Lateral directions, Lift-up mechanism, Linear Stability, Linearized navier-stokes equations, matrix, Objective functions, Oblique wave, Optimal disturbances, Optimal time, Optimization problems, Optimization techniques, Periodic forcing, Propagation speed, Spatial structure, Streaky structure, Streamwise vortices, Tollmien-Schlichting waves, Unstable flows, Zero frequency, Amplification, Boundary layer flow, Boundary layers, Frequency response, Lagrange multipliers, Navier Stokes equations, Reynolds number, Three dimensional, Wave packets, Waves
Fluid Mechanics and Acoustics
IdentifiersURN: urn:nbn:se:kth:diva-10648DOI: 10.1017/S0022112009993703ISI: 000278212500007ScopusID: 2-s2.0-77952398610OAI: oai:DiVA.org:kth-10648DiVA: diva2:222640
QC 20100924. Uppdaterad från submitted till published (20100924).2009-06-092009-06-092011-05-18Bibliographically approved