Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers

• 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.

Ämnesord

TEKNIK OCH TEKNOLOGIER  -- Maskinteknik -- Strömningsmekanik och akustik (hsv//swe)

ENGINEERING AND TECHNOLOGY  -- Mechanical Engineering -- Fluid Mechanics and Acoustics (hsv//eng)

Nyckelord

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

Strömningsmekanik

Publikations- och innehållstyp

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