Numerical modeling of the laminar-turbulent transition control using a dielectric barrier discharge

The control of laminar-turbulent transition driven by Tollmien–Schlichting waves is studied. The control is realized by means of accelerating the boundary layer flow using a dielectric barrier discharge. As distinct from the previous studies based on the solution of the boundary layer equations, the discharge effect on the main flow and unstable disturbances are described by the Navier–Stokes equations.     

Is Helmholtz Resonance a Problem for Micro-jet Actuators?

A theoretical analysis is described that determines the conditions for Helmholtz resonance for a popular class of self-contained microjet actuator used in both synthetic- and pressure-jump (pulse-jet) mode. It was previously shown that the conditions for Helmholtz resonance are identical to those for optimizing actuator performance for maximum mass flux. The methodology is described for numerical-simulation studies on how Helmholtz resonance affects the interaction of active and nominally inactive micro-jet actuators with a laminar boundary layer. Two sets of numerical simulations were carried out. The first set models the interaction of an active actuator with the boundary layer. These simulations confirm that our criterion for Helmholtz resonance is broadly correct. When it is satisfied we find that the actuator cannot be treated as a predetermined wall boundary condition because the interaction with the boundary layer changes the pressure difference across the exit orifice thereby affecting the outflow from the actuator. We further show that strong inflow cannot be avoided even when the actuator is used in pressure-jump mode. In the second set of simulations two-dimensional Tollmien–Schlichting waves, with frequency comparable with, but not particularly close to, the Helmholtz resonant frequency, are incident on a nominally inactive micro-jet actuator. The simulations show that under these circumstances the actuators act as strong sources of 3D Tollmien–Schlichting waves. It is surmised that in the real-life aeronautical applications with turbulent boundary layers broadband disturbances of the pressure field, including acoustic waves, would cause nominally inactive actuators, possibly including pulsed jets, to act as strong disturbance sources. Should this be true it would probably be disastrous for engineering applications of such massless microjet actuators for flow control.     

Nonlinear Instability in the Region of Laminar-Turbulent Transition in Supersonic Three-Dimensional Flow over a Flat Plate

Direct numerical simulation is applied to obtain laminar-turbulent transition in supersonic flow over a flat plate. It is shown that, due to the nonlinear instability, Tollmien–Schlichting waves generated in the boundary layer lead to the formation of oblique disturbances in the flow. These represent a combination of compression and expansion waves, whose intensities can be two orders higher than that of external harmonic disturbances. The patterns of the three-dimensional flow over the plate are presented and the structures of the turbulent flat-plate boundary layers are described for the freestream Mach numbers M = 2 and 4.     

The advantages of using high‐order finite differences schemes in laminar–turbulent transition studies

This paper presents various finite difference schemes and compare their ability to simulate instability waves in a given flow field. The governing equations for two‐dimensional, incompressible flows were solved in vorticity–velocity formulation. Four different space discretization schemes were tested, namely, a second‐order central differences, a fourth‐order central differences, a fourth‐order compact scheme and a sixth‐order compact scheme. A classic fourth‐order Runge–Kutta scheme was used in time. The influence of grid refinement in the streamwise and wall normal directions were evaluated. The results were compared with linear stability theory for the evolution of small‐amplitude Tollmien–Schlichting waves in a plane Poiseuille flow. Both the amplification rate and the wavenumber were considered as verification parameters, showing the degree of dissipation and dispersion introduced by the different numerical schemes. The results confirmed that high‐order schemes are necessary for studying hydrodynamic instability problems by direct numerical simulation. Copyright © 2005 John Wiley & Sons, Ltd.     

NUMERICAL SIMULATION OF LINEAR AND NONLINEAR DISTURBANCE EVOLUTION IN A BOUNDARY LAYER WITH COMPLIANT WALLS

A computational method capable of simulating the spatial evolution of disturbances in a boundary-layer flow with compliant coatings has been developed. The flow geometry being an unknown of the problem, this difficulty is overcome by use of a mapping, the domain being fixed in the computational coordinates. The model takes into account the nonlinear fluid-structure interaction all over the flow field, as well as nonparallel effects due to the wall displacement and to the boundary-layer growth. First, the numerical solution procedure is tested by focusing on the linear and nonlinear spatial disturbance evolution for a spring-backed elastic plate which is only unstable with respect to Tollmien–Schlichting-type travelling waves. The numerical procedure is then used to study the influence of the initial disturbance amplitude on the disturbance development for a tensioned membrane. Finally, to simulate a true physical experiment, a spring-backed elastic plate of finite length is considered. It is shown that the numerical model is capable of detecting the interaction between Tollmien–Schlichting instabilities and flow-induced surface instabilities at the interface.     

On the wave-cancelling nature of boundary layer flow control

This work deals with the feedforward active control of Tollmien–Schlichting instability waves over incompressible 2D and 3D boundary layers. Through an extensive numerical study, two strategies are evaluated; the optimal linear–quadratic–Gaussian (LQG) controller, designed using the Eigensystem realization algorithm, is compared to a wave-cancellation scheme, which is obtained using the direct inversion of frequency-domain transfer functions of the system. For the evaluated cases, it is shown that LQG leads to a similar control law and presents a comparable performance to the simpler, wave-cancellation scheme, indicating that the former acts via a destructive interference of the incoming wavepacket downstream of actuation. The results allow further insight into the physics behind flow control of convectively unstable flows permitting, for instance, the optimization of the transverse position for actuation. Using concepts of linear stability theory and the derived transfer function, a more efficient actuation for flow control is chosen, leading to similar attenuation of Tollmien–Schlichting waves with only about 10% of the actuation power in the baseline case.     

Global two-dimensional stability measures of the flat plate boundary-layer flow

The stability of the two-dimensional flat plate boundary-layer is studied by means of global eigenmodes. These eigenmodes depend both on the streamwise and wall-normal coordinate, hence there are no assumptions on the streamwise length scales of the disturbances. Expanding the perturbation velocity field in the basis of eigenmodes yields a reduced order model from which the stability characteristics of the flow, i.e. the initial condition and forcing function leading to the largest energy growth, are extracted by means of non-modal analysis. In this paper we show that, even when performing stability analysis using global eigenmodes, it is not sufficient to consider only a few of the least damped seemingly relevant eigenmodes. Instead it is the task of the optimization procedure, inherent in the non-modal analysis, to decide which eigenmodes are relevant. We show that both the optimal initial condition and the optimal forcing structure have the form of upstream tilted structures. Time integration reveals that these structures gain energy through the so called Orr mechanism, where the instabilities extract energy from the mean shear. This provides the optimal way of initiating Tollmien–Schlichting waves in the boundary layer. The optimal initial condition results in a localized Tollmien–Schlichting wavepacket that propagates downstream, whereas the optimal forcing results in a persistent Tollmien–Schlichting wave train.     

Boundary layer sensitivity and receptivitySensibilité et réceptivité d'une couche limite

The relation between the receptivity and the sensitivity of the incompressible flow in the boundary layer over a flat plate to harmonic perturbations is determined. Receptivity describes the birth of a disturbance, whereas sensitivity is a concept of larger breath, describing the modification incurred by the state of a system as a response to parametric variations. The governing equations ruling the system's state are the non-local stability equations. Receptivity and sensitivity functions can be obtained from the solution of the adjoint system of equations. An application to the case of Tollmien–Schlichting waves spatially developing in a flat plate boundary layer is studied. To cite this article: C. Airiau et al., C. R. Mecanique 330 (2002) 259–265.     

Numerical simulations of inviscid and viscous free‐surface flows with application to nonlinear water waves

The purpose of the present study is to establish a numerical model appropriate for solving inviscid/viscous free‐surface flows related to nonlinear water wave propagation. The viscous model presented herein is based on the Navier–Stokes equations, and the free‐surface is calculated through an arbitrary Lagrangian–Eulerian streamfunction‐vorticity formulation. The streamfunction field is governed by the Poisson equation, and the vorticity is obtained on the basis of the vorticity transport equation. For computing the inviscid flow the Laplace streamfunction equation is used. These equations together with the respective (appropriate) fully nonlinear free‐surface boundary conditions are solved using a finite difference method. To demonstrate the model feasibility, in the present study we first simulate collision processes of two solitary waves of different amplitudes, and compute the phenomenon of overtaking of such solitary waves. The developed model is subsequently applied to calculate (both inviscid and the viscous) flow field, as induced by passing of a solitary wave over submerged rectangular structures and rigid ripple beds. Our study provides a reasonably good understanding of the behavior of (inviscid/viscous) free‐surface flows, within the framework of streamfunction‐vorticity formulation. The successful simulation of the above‐mentioned test cases seems to suggest that the arbitrary Lagrangian–Eulerian/streamfunction‐vorticity formulation is a potentially powerful approach, capable of effectively solving the fully nonlinear inviscid/viscous free‐surface flow interactions. Copyright © 2009 John Wiley & Sons, Ltd.     

A Combined Experimental/Numerical Study of Unsteady Phenomena in a Laminar Separation Bubble

A laminar boundary layer separates in a region of adverse pressure gradient on a flat plate and undergoes transition. Finally the turbulent boundary layer reattaches, forming a laminar separation bubble (LSB). Laminar-turbulent transition within such a LSB is investigated by means of Laser-Doppler-Anemometry (LDA), Particle Image Velocimetry (PIV), and direct numerical simulation (DNS). The transition mechanism occurring in the flow-field under consideration is discussed in detail. Observations for the development of small disturbances are compared to predictions from viscous linear instability theory (Tollmien–Schlichting instability). Non-linear development of these disturbances and their role in final breakdown to turbulence is analyzed.     

Measuring Tollmien–Schlichting waves using phase-averaged particle image velocimetry

This article addresses the direct experimental measurement of Tollmien?CSchlichting waves on a flat plate, when the laminar boundary layer is excited by velocity perturbations; the free stream velocity was 16?m/s, the excitation frequency 250?Hz. The two-dimensional velocity field in proximity of the flat plate was captured using a conventional PIV system; however, the image recording was phase locked with the disturbance source and ensemble averaging was used to obtain characteristics of the Tollmien?CSchichting waves. In particular, after subtraction of the mean velocity, the characteristics of the excited waves in terms of streamlines were extracted, revealing that the investigated waves represented velocity deviations with an order of magnitude of 1?% of the undisturbed free stream flow. This study is a prelude to the use of the same technique to visualize the effect of dielectric barrier discharge plasma actuators on the suppression of such Tollmien?CSchlichting waves, which is difficult using other measurement techniques.     

Mach Wave Effect on Laminar-Turbulent Transition in Supersonic Flow over a Flat Plate

The effect of a Mach wave (N wave) on laminar-turbulent transition induced by the first instability mode (Tollmien–Schlichting wave) in the flat-plate boundary layer is investigated on the basis of the numerical solution of Navier–Stokes equations at the freestream Mach number of 2.5. In accordance with the experiment, the N wave is generated by a two-dimensional roughness at the computation domain boundary corresponding to the side wall of the test section of a wind tunnel. It is shown that the disturbance induced by the backward front of the N wave in the boundary layer has no effect on the beginning of transition but displaces downstream the nonlinear stage of the first mode development. The disturbance induced by the forward front of the N wave displaces the beginning of transition upstream.     

Nonreflecting boundary conditions based on nonlinear multidimensional characteristics

Nonlinear characteristic boundary conditions based on nonlinear multidimensional characteristics are proposed for 2‐ and 3‐D compressible Navier–Stokes equations with/without scalar transport equations. This approach is consistent with the flow physics and transport properties. Based on the theory of characteristics, which is a rigorous mathematical technique, multidimensional flows can be decomposed into acoustic, entropy, and vorticity waves. Nonreflecting boundary conditions are derived by setting corresponding characteristic variables of incoming waves to zero and by partially damping the source terms of the incoming acoustic waves. In order to obtain the resulting optimal damping coefficient, analysis is performed for problems of pure acoustic plane wave propagation and arbitrary flows. The proposed boundary conditions are tested on two benchmark problems: cylindrical acoustic wave propagation and the wake flow behind a cylinder with strong periodic vortex convected out of the computational domain. This new approach substantially minimizes the spurious wave reflections of pressure, density, temperature, and velocity as well as vorticity from the artificial boundaries, where strong multidimensional flow effects exist. The numerical simulations yield accurate results, confirm the optimal damping coefficient obtained from analysis, and verify that the method substantially improves the 1‐D characteristics‐based nonreflecting boundary conditions for complex multidimensional flows. Copyright © 2009 John Wiley & Sons, Ltd.     

Three-wave interactions between disturbances in the hypersonic boundary layer on impermeable and porous surfaces

The interaction between disturbances in the hypersonic boundary layer on impermeable and porous surfaces is considered within the framework of weakly-nonlinear stability theory. It is established that on the impermeable surface nonlinear interactions between different waves (acoustic and vortex) occur in the parametric resonance regime. The role of pumping wave is played by a plane acoustic wave. The nonlinear interactions take place over a wide frequency range and can lead to the packet growth of Tollmien-Schlichting waves. On the porous surface the analogous interactions are fairly weak and result in a slight decay of the acoustic mode and a slight amplification of the vortex mode. This leads to the dragging out of the laminar flow regime and the regions of linear disturbance growth. In this situation the low-frequency spectrum of the vortex modes may be filled on account of the nonlinear processes occurring in the three-wave systems between the vortex components.     

Numerical simulation of fully nonlinear irregular wave tank in three dimension

A fully nonlinear irregular wave tank has been developed using a three‐dimensional higher‐order boundary element method (HOBEM) in the time domain. The Laplace equation is solved at each time step by an integral equation method. Based on image theory, a new Green function is applied in the whole fluid domain so that only the incident surface and free surface are discretized for the integral equation. The fully nonlinear free surface boundary conditions are integrated with time to update the wave profile and boundary values on it by a semi‐mixed Eulerian–Lagrangian time marching scheme. The incident waves are generated by feeding analytic forms on the input boundary and a ramp function is introduced at the start of simulation to avoid the initial transient disturbance. The outgoing waves are sufficiently dissipated by using a spatially varying artificial damping on the free surface before they reach the downstream boundary. Numerous numerical simulations of linear and nonlinear waves are performed and the simulated results are compared with the theoretical input waves. Copyright © 2006 John Wiley & Sons, Ltd.     

Stability of the laminar boundary layer flow encountering a row of roughness elements: Biglobal stability approach and DNS

The stability of the laminar boundary layer developing on a flat plate in the presence of a periodic row of roughness elements is investigated. A Direct Numerical Simulation is performed to compute the steady flow downstream of the roughness elements, which contains a pair of two counter-rotating streamwise vortices per element, which can be considered as a “pre-streaky” structure. The linear stability of this base flow is analyzed by means of the so-called “biglobal” stability approach. Three-dimensional eigenmodes are found, which are shown to be the continuation of the Tollmien–Schlichting waves present in the case of an unperturbed boundary layer. Moreover, a stabilizing effect due to the roughness-induced vortices is found. A Direct Numerical Simulation of the interaction between a two-dimensional Tollmien–Schlichting wave and the roughness array is also performed. The computed perturbation traveling downstream of the roughness elements is shown to be a linear combination of the biglobal eigenmodes.     

On two-dimensional linear waves in Blasius boundary layer over viscoelastic layers

This work concerns the direct numerical simulation of small-amplitude two-dimensional ribbon-excited waves in Blasius boundary layer over viscoelastic compliant layers of finite length. A vorticity-streamfunction formulation is used, which assures divergence-free solutions for the evolving flow fields. Waves in the compliant panels are governed by the viscoelastic Navier's equations. The study shows that Tollmien–Schlichting (TS) waves and compliance-induced flow instability (CIFI) waves that are predicted by linear stability theory frequently coexist on viscoelastic layers of finite length. In general, the behaviour of the waves is consistent with the predictions of linear stability theory. The edges of the compliant panels, where abrupt changes in wall property occur, are an important source of waves when they are subjected to periodic excitation by the flow. The numerical results indicate that the non-parallel effect of boundary-layer growth is destabilizing on the TS instability. It is further demonstrated that viscoelastic layers with suitable properties are able to reduce the amplification of the TS waves, and that high levels of material damping are effective in controlling the propagating CIFI.     

Experiments on localized disturbances in a flat plate boundary layer. Part 1. The receptivity and evolution of a localized free stream disturbance

The receptivity of a laminar boundary layer to free stream disturbances has been experimentally investigated through the introduction of deterministic localized disturbances upstream of a flat plate mounted in a wind tunnel. Hot-wire measurements indicate that the spanwise gradient of the normal velocity component (and hence the streamwise vorticity) plays an essential role in the transfer of disturbance energy into the boundary layer. Inside the laminar boundary layer the disturbances were found to give rise to the formation of longitudinal structures of alternating high and low streamwise velocity. Similar streaky structures exist in laminar boundary layers exposed to free stream turbulence, in which the disturbance amplitude increases in linear proportion to the displacement thickness. In the present study the perturbation amplitude of the streaks was always decaying for the initial amplitudes used, in contrast to the growing fluctuations that are observed in the presence of free stream turbulence. This points out the importance of the continuous influence from the free stream turbulence along the boundary layer edge.     

Numerical simulation of vortical ideal fluid flow through curved channel

A numerical algorithm to study the boundary‐value problem in which the governing equations are the steady Euler equations and the vorticity is given on the inflow parts of the domain boundary is developed. The Euler equations are implemented in terms of the stream function and vorticity. An irregular physical domain is transformed into a rectangle in the computational domain and the Euler equations are rewritten with respect to a curvilinear co‐ordinate system. The convergence of the finite‐difference equations to the exact solution is shown experimentally for the test problems by comparing the computational results with the exact solutions on the sequence of grids. To find the pressure from the known vorticity and stream function, the Euler equations are utilized in the Gromeka–Lamb form. The numerical algorithm is illustrated with several examples of steady flow through a two‐dimensional channel with curved walls. The analysis of calculations shows strong dependence of the pressure field on the vorticity given at the inflow parts of the boundary. Plots of the flow structure and isobars, for different geometries of channel and for different values of vorticity on entrance, are also presented. Copyright © 2003 John Wiley & Sons, Ltd.     

Active flow control of laminar boundary layers for variable flow conditions

This work investigates the stability of a fxLMS controller for active wave cancelation of broad-band Tollmien–Schlichting disturbances in a flat plate boundary-layer with a single DBD plasma actuator. In particular the influence of a changing free stream velocity and the resulting off-design operation of the control algorithm is analyzed up to an unstable behavior. As the main reason for unstable controller operation in the off-design case the difference between actual and predicted phase angle of the disturbances at the position of the error sensor is identified. A method for an online adjustment of the secondary-path model to different free-stream velocities is presented. Finally a wall-bounded method based on the disturbances phase speed is developed that can cope with changes of the physical secondary path not only due to changes of the free-stream velocity but also due to changes of the pressure distribution. This method enables the extension of the stable operation range of the control system significantly.