Tollmien-Schlichting Waves Research Articles

Analyses of Reynolds and Mach number effects on Tollmien–Schlichting wave–bump interaction in subsonic flows

We investigated the effects of two-dimensional sharp-edged rectangular bumps on Tollmien–Schlichting (TS) wave evolution using direct numerical simulation. The bump height, $h$ , ranged from 5 % to 40 % of the local displacement thickness, $\delta ^*$ . Behind the bump, a recirculating flow region could be formed whose length increased nonlinearly with $h$ . The bump height effect on the TS wave, which was the dominant, scaled super-exponentially with $h$ . We also showed a substantial effect of the $\delta ^*$ -based Reynolds number, ${\textit {Re}} _{\delta ^*}$ . Firstly, the bump wake extended with ${\textit {Re}} _{\delta ^*}$ , promoting larger TS wave growth rates. The second effect is related to proximity to the upper branch of the instability loop, accounting for the influence of the TS frequency, as well. It dictates the bump impact increases as it gets closer to transition, either by the bump moving downstream or the transition moving upstream. For a 40 % high bump, for example, changing the ${\textit {Re}} _{\delta ^*}$ at the bump location from 1500 to 2000 increased $\Delta N$ by a factor of 2 ( $\Delta N$ represents a measure of a surface irregularity effect on the smooth plate N-factor). We also found that $(\Delta N)_{max}$ increases linearly with ${\textit {Re}} _{hh}$ . Results in the subsonic regime showed that the bump impact attenuates with Mach number up to 0.7 but above it, stabilisation is surpassed by the destabilising effect caused by the recirculation lengthening. This is mostly associated with the bump wake that extends with the pressure gradient which increases substantially towards the sonic speed. This is enhanced if the surface is adiabatic rather than isothermal.

Breakdown of laminar regime in flat plate boundary layer seen in numerical solutions of the Navier–Stokes equations obtained with 16th-order scheme

Numerical solutions of the Navier–Stokes equations describing the onset and the development of the boundary layer instability of a subsonic flow about a finite width flat plate are described. They were obtained via exciters-free calculations with the 16th-order multioperators-based scheme optimized to resolve very small solutions scales. Some relevant details of the scheme are briefly outlined. The occurrence of the Tollmien–Schlichting waves near the leading edge, their downstream propagation and breakdowns some distance away are displayed. The role of the oblique waves seen in the solutions is emphasized. The nonlinear stage in the form of structured vortices followed by the solutions randomization are described.

Subcritical transitional flow in two-dimensional plane Poiseuille flow

Recently, subcritical transition to turbulence in the quasi-two-dimensional (quasi-2-D) shear flow with strong linear friction (Camobreco et al., J. Fluid Mech., vol. 963, 2023, R2) has been demonstrated by the 2-D mechanism at $Re = 71\,211$ , and the nonlinear Tollmien–Schlichting (TS) waves related to the edge state were approached independently of initial optimal disturbances. For 2-D plane Poiseuille flow, transition to the fully developed turbulence requires that the Reynolds number is several times larger than the critical Reynolds number $Re_c$ (Markeviciute & Kerswell, J. Fluid Mech., vol. 917, 2021, A57). In this paper, we observed the subcritical transitional flow in 2-D plane Poiseuille flow driven by the nonlinear TS waves by both linear and nonlinear optimal disturbances ( $Re < Re_c$ ) with different quantitative edge states. The nonlinear optimal disturbances could trigger the sustained subcritical transitional flow for $Re \geqslant 2400$ . The initial energy for nonlinear optimal disturbance is more efficient than the linear optimal disturbance in reaching the subcritical transitional flow for $2400 \leqslant Re \leqslant 5000$ . Moreover, the initial energy of linear optimal disturbance is larger than the energy of its edge state. The nonlinear TS waves along the edge state are formed by the nonlinear optimal disturbances to trigger transitional flow, which agrees well with the main conclusions of Camobreco et al. (J. Fluid Mech., vol. 963, 2023, R2), while the required $Re$ of 2-D plane Poiseuille flow is much smaller.

Stability of the flow over superhydrophobic micro roughnesses: The influence of the interface

Superhydrophobic surfaces are known for their drag reduction properties. However, the interface between the lubricant and the overlying flow may easily become unstable, leading to the depletion of the superhydrophobic layer and to a consequent drag increase. In this paper, we investigate the modal and non-modal instability of the flow over longitudinal trapezoidal superhydrophobic riblets, including, for the first time, the gas/liquid interface dynamics in the stability analyses. A two-dimensional stability problem, obtained with a domain transform technique and interface modelling using a linearised Young–Laplace equation, is coupled with the n−periodic stability framework introduced by Schmid et al. (2017). The latter technique, using a Bloch wave formalism, allows the computation of the stability of an array of n riblet units of given periodicity at a reasonable numerical cost. For small periodicities, the most unstable mode is a fundamental instability stemming from the three-dimensionalisation of a Tollmien–Schlichting wave. Conversely, in the case of large riblet periodicities, a subharmonic mode linked to capillarity effects becomes the most unstable. Nonmodal transient growth analysis shows that the superhydrophobic riblets have a weak effect on the overall growth. However, riblets having small periodicities induce a slight stabilisation of the flow, while large ones induce an increase of the energy growth on detuned perturbations. The resulting energy growth mechanism induces interface deformations encompassing more than one subunit.

DeHNSSo: The Delft Harmonic Navier-Stokes Solver for nonlinear stability problems with complex geometric features

A nonlinear Harmonic Navier-Stokes (HNS) framework is introduced for simulating instabilities in laminar spanwise-invariant shear layers, featuring sharp and smooth wall surface protuberances. While such cases play a critical role in the process of laminar-to-turbulent transition, classical stability theory analyses such as parabolized or local stability methods fail to provide (accurate) results, due to their underlying assumptions. The generalized incompressible Navier-Stokes (NS) equations are expanded in perturbed form, using a spanwise and temporal Fourier ansatz for flow perturbations. The resulting equations are discretized using spectral collocation in the wall-normal direction and finite-difference methods in the streamwise direction. The equations are then solved using a direct sparse-matrix solver. The nonlinear mode interaction terms are converged iteratively. The solution implementation makes use of a generalized domain transformation to account for geometrical smooth surface features, such as humps. No-slip conditions can be embedded in the interior domain to account for the presence of sharp surface features such as forward- or backward-facing steps. Common difficulties with Navier-Stokes solvers, such as the treatment of the outflow boundary and convergence of nonlinear terms, are considered in detail. The performance of the developed solver is evaluated against several cases of representative boundary layer instability growth, including linear and nonlinear growth of Tollmien-Schlichting waves in a Blasius boundary layer and stationary crossflow instabilities in a swept flat-plate boundary layer. The latter problem is also treated in the presence of a geometrical smooth hump and a sharp forward-facing step at the wall. HNS simulation results, such as perturbation amplitudes, growth rates, and shape functions, are compared to benchmark flow stability analysis methods such as Parabolized Stability Equations (PSE), Adaptive Harmonic Linearized Navier-Stokes (AHLNS), or Direct Numerical Simulations (DNS). Good agreement is observed in all cases. The HNS solver is subjected to a grid convergence study and a simple performance benchmark, namely memory usage and computational cost. The computational cost is found to be considerably lower than high-fidelity DNS at comparable grid resolutions. Program summaryProgram Title: DeHNSSoCPC Library link to program files:https://doi.org/10.17632/9bnms99kk2.1Developer's repository link:https://github.com/SvenWesterbeek/DeHNSSoLicensing provisions: GPLv3Programming language: MatlabSupplementary material: The supplementary material contains the code as well as a user manual.Nature of problem: Fluid flows are subject to laminar-to-turbulent transition following the growth of instabilities. To avoid computationally demanding Direct Numerical Simulations (DNS), perturbation theory is often applied to their analysis. However, classical stability methods based on the Orr-Sommerfeld equation or the Parabolized Stability Equations neglect the influence of streamwise gradients in varying degrees. The validity of these assumptions is difficult to estimate a priori.Solution method: The Delft Harmonic Navier-Stokes Solver (DeHNSSo) solves the harmonic Navier-Stokes equations nonlinearly on domains featuring sharp and smooth spanwise invariant surface features using a generalized grid approach in combination with an embedded boundary method. This allows the user to include the effects of streamwise gradients on flow at a fraction of the cost of DNS.Additional comments including restrictions and unusual features: In DeHNSSo, the equations are solved using direct matrix solvers. As such, memory is treated as a scarce resource. The problem is formulated in a mode-independent manner such that left-hand side matrices need only be computed and stored once despite incorporating nonlinear terms. Additionally, DeHNSSo offers the user the possibility to prescribe inhomogeneous boundary conditions to introduce and solve the receptivity problem or manipulate instabilities. Due to the double Fourier expansion, the solver is restricted to spanwise and temporally periodic problems.

Prediction of effect of small local surface irregularities on natural transition to turbulence based on Global Stability Analysis

Reliable prediction of the effect of surface irregularities on laminar-turbulent transition (LTT) is of great importance for many industrial applications. At low levels of free-stream turbulence typical of external aerodynamics, the primary mechanism of LTT is associated with the so-called convective instability of the Tollmien-Schlichting waves. In this paper, a non-empiric methodology based on Global Stability Analysis, which has been developed and validated previously to predict the evolution of these waves and the location of the LTT on a smooth flat plate, is applied to evaluation of this effect for backward- and forward- facing steps of different heights and locations. The results obtained are analyzed and compared with results of 2D time-accurate unsteady Navier-Stokes simulations, as well as with available experimental data and with the semi-empiric “variable N-factor” method.

Tollmien–Schlichting wave manipulation by a multi-input multi-output phononic subsurface

Recently, there has been a reinvigorated effort to identify passive control mechanisms to reduce drag on vehicles moving through a fluid medium. In 2015, Hussein et al. engineered a phononic crystal subsurface (PSub) to interact with Tollmien–Schlichting waves of a channel flow through an interaction surface, which caused a localized reduction in the kinetic energy of the flow. Since then, arrayed PSubs, Helmholtz resonators, and other embedded structures have been studied for their ability to produce similar effects. In this talk, an implementation of a multi-input multi-output (MIMO) PSub is presented and its ability to delay the onset of turbulence is shown to depend on simultaneously satisfying single-input single-output (SISO) PSub phase requirements at both interaction surfaces of the MIMO PSub. The MIMO architecture provides a method to place an arbitrary phasing between the T-S wave induced forcing and the displacement response of the interaction surface and can be optimized for the flow conditions of the environment.

Simulating the propagation of boundary-layer disturbances by solving boundary-value and initial-value problems

Abstract The article deals with the downstream propagation of small–amplitude disturbances of viscous incompressible laminar boundary layers, using the linearized equations for disturbance amplitudes. Two different methods are proposed. The first one solves a two-dimensional boundary-value problem, using a buffer-domain technique to mimic the outflow boundary condition. The second one solves a streamwise initial-value problem, using a spectral parabolization at each integration step. Both methods show good performance in simulating the propagation of the Tollmien–Schlichting waves and the Görtler vortices and can be applied to compute the spatial optimal disturbances.

Turbulent transition in a channel with superhydrophobic walls: anisotropic slip and shear misalignment effects

Superhydrophobic surfaces dramatically reduce skin friction of overlying liquid flows. These surfaces are complex and numerical simulations usually rely on models to reduce this complexity. One of the simplest consists of finding an equivalent boundary condition through a homogenisation procedure, which in the case of channel flow over oriented riblets, leads to the presence of a small spanwise component in the homogenised base flow velocity. This work aims at investigating the influence of such a three-dimensionality of the base flow on stability and transition in a channel with walls covered by oriented riblets. Linear stability for this base flow is investigated: a new instability region, linked to cross-flow effects, is observed. Tollmien–Schlichting waves are also retrieved but the most unstable are three-dimensional. Transient growth is also affected as oblique streaks with non-zero streamwise wavenumber become the most amplified perturbations. When transition is induced by Tollmien–Schlichting waves, after an initial exponential growth regime, streaky structures with large spanwise wavenumber rapidly arise. Modal mechanisms appear to play a leading role in the development of these structures and a secondary stability analysis is performed to retrieve successfully some of their characteristics. The second scenario, initiated with cross-flow vortices, displays a strong influence of nonlinearities. The flow develops into large quasi-spanwise-invariant structures before breaking down to turbulence. Secondary stability on the saturated cross-flow vortices sheds light on this stage of transition. In both cases, cross-flow effects dominate the flow dynamics, suggesting the need to consider the anisotropicity of the wall condition when modelling superhydrophobic surfaces.

Multi-input multi-output phononic subsurfaces for passive boundary layer transition delay

This paper documents progress towards developing a passive boundary layer transition delay strategy using a phononic subsurface (PSub). High-order implicit large-eddy simulation (ILES) is used to explore control requirements for boundary layer transition delay in the context of guiding passive flow control strategies. Positive phasing of surface displacement relative to forcing from a Tollmien-Schlichting (T-S) wave is shown to be effective at attenuating instability for the present flow conditions. Although difficult to achieve with a single input system, positive phasing can be realized with a multi-input, multi-output (MIMO) system. To this end, a MIMO PSub was developed and a reduced-order model was coupled with the flow solver. Positive phasing, and therefore a small degree of passive transition delay, was demonstrated on a flat plate boundary layer. Further optimization of subsurface properties and placement of multiple devices are likely to improve controller efficacy.

Mechanism of vorticity amplification by elastic waves in a viscoelastic channel flow

Inertia-less viscoelastic channel flow displays a supercritical nonnormal mode elastic instability due to finite-size perturbations despite its linear stability. The nonnormal mode instability is determined mainly by a direct transition from laminar to chaotic flow, in contrast to normal mode bifurcation leading to a single fastest-growing mode. At higher velocities, transitions to elastic turbulence and further drag reduction flow regimes occur accompanied by elastic waves in three flow regimes. Here, we demonstrate experimentally that the elastic waves play a key role in amplifying wall-normal vorticity fluctuations by pumping energy, withdrawn from the mean flow, into wall-normal fluctuating vortices. Indeed, the flow resistance and rotational part of the wall-normal vorticity fluctuations depend linearly on the elastic wave energy in three chaotic flow regimes. The higher (lower) the elastic wave intensity, the larger (smaller) the flow resistance and rotational vorticity fluctuations. This mechanism was suggested earlier to explain elastically driven Kelvin-Helmholtz-like instability in viscoelastic channel flow. The suggested physical mechanism of vorticity amplification by the elastic waves above the elastic instability onset recalls the Landau damping in magnetized relativistic plasma. The latter occurs due to the resonant interaction of electromagnetic waves with fast electrons in the relativistic plasma when the electron velocity approaches light speed. Moreover, the suggested mechanism could be generally relevant to flows exhibiting both transverse waves and vortices, such as Alfven waves interacting with vortices in turbulent magnetized plasma, and Tollmien-Schlichting waves amplifying vorticity in both Newtonian and elasto-inertial fluids in shear flows.

A Waveguide Model of the Developed Turbulent Boundary Layer

A study of the developed turbulent boundary layer that emerges when incompressible viscous fluid flows around a plate at a zero angle of attack and with zero longitudinal pressure gradient is presented. The waveguide approach is used for describing the turbulent boundary layer; in this approach, turbulent fluctuations are related with Tollmien–Schlichting waves that are in three-wave resonance. To study the original nonlinear system of equations, an estimate of hydrodynamic quantities is proposed that does not violate the generally accepted approach in the boundary layer but leads to the appearance of a new small parameter—the ratio of the thickness of the boundary layer momentum loss to the damping length of the least damped mode of the Tollmien–Schlichting waves. Equations for the coherent and stochastic parts of fluctuations are obtained on the basis of the method of multiple scales. The dispersion characteristics of waves of the least damped mode on the profile of the average longitudinal velocity of the developed turbulent boundary layer are determined, and the conditions for the multiple three-wave resonance of this mode of the Tollmien–Schlichting waves are analyzed. For the coherent part of the fluctuations, the fluctuation characteristics are compared with the known numerical results.

Attenuation of Tollmien–Schlichting waves using resonating surface-embedded phononic crystals

A novel method for control of convective boundary layer instabilities using metamaterial concepts is investigated. Attenuation of Tollmien–Schlichting (TS) waves with surface-embedded one-dimensional phononic crystals (PCs) is theoretically and numerically modeled, capitalizing on the inherent frequency band stop of PCs. The PC is tuned to the targeted TS wave characteristics through the use of analytical models derived from transfer matrix and interface response theories, verified using a finite elements analysis. The interaction between TS waves and a single PC is investigated using coupled two-dimensional fluid structure interaction simulations in the frequency domain. It is shown that TS waves are either amplified or attenuated depending on whether the PC free-face surface displacement and unsteady perturbation pressure at the wall are in-phase or out-of-phase, respectively. The perturbation pressure acts solely as the driver for the mechanical oscillation of the PC. The emerging hydrodynamic coupling between TS waves and the PC is found to be governed by a combination of the Orr mechanism and wall-normal velocity linear superposition near the wall. Finally, a metasurface comprised of an array of streamwise-distributed PCs is evaluated, resulting in an amplitude growth delay of 11.3% of the TS wavelength along the metasurface extent.

Adjoint-based robust optimization design of laminar flow wing under flight condition uncertainties

It is an inherent uncertainty problem that the application of laminar flow technology to the wing of large passenger aircraft is affected by flight conditions. In order to seek a more robust natural laminar flow control effect, it is necessary to develop an effective optimization design method. Meanwhile, attention must be given to the impact of crossflow (CF) instability brought on by the sweep angle. This paper constructs a robust optimization design framework based on discrete adjoint methods and non-intrusive polynomial chaos. Transition prediction is implemented by coupled Reynolds-Averaged Navier-Stokes (RANS) and simplified eN method, which can consider both Tollmien-Schlichting (TS) wave and crossflow vortex instability. We have performed gradient enhancement processing on the general Polynomial Chaos Expansion (PCE), which is advantageous to reduce the computational cost of single uncertainty propagation. This processing takes advantage of the gradient information obtained by solving the coupled adjoint equations considering transition. The statistical moment gradient solution used for the robust optimization design also uses the derivatives of coupled adjoint equations. The framework is applied to the robust design of a 25° swept wing with infinite span in transonic flow. The uncertainty quantification and sensitivity analysis on the baseline wing shows that the uncertainty quantification method in this paper has high accuracy, and qualitatively reveals the factors that dominate in different flow field regions. By the robust optimization design, the mean and standard deviation of the drag coefficient can be reduced by 29% and 45%, respectively, and compared with the deterministic optimization design results, there is less possibility of forming shock waves under flight condition uncertainties. Robust optimization results illustrate the trade-off between the transition delay and the wave drag reduction.

Attachment-line, crossflow and Tollmien-Schlichting instabilities on swept ONERA-D and Joukowski airfoils.

Linear stability analyses are performed to investigate the boundary layer instabilities developing in an incompressible flow around the whole leading-edge of swept ONERA-D and Joukowski airfoils of infinite span. The stability analyses conducted in our study are global in the chordwise direction and local in the spanwise direction. A neutral curve is drawn at a given leading-edge Reynolds number ReR and several overlapping regions, called "lobes", are identified on a physical basis. A detailed study of the marginal modes reveals the presence of attachment-line and crossflow instabilities, as well as modes whose features do not fall within the standards of a specific type. Connected crossflow/Tollmien-Schlichting modes, that show a dominant spatial structure reminiscent of Tollmien-Schlichting waves but whose destabilization is linked to a crossflow mechanism, have been identified. The comparison of several neutral curves at different ReR values reveals the greater stabilizing effect of the increase of ReR on the crossflow instability compared to the attachment-line instability. The influence of the airfoil shape is also studied by comparing the neutral curves of the ONERA-D with the neutral curves of the Joukowski airfoil. These curves reveal similar characteristics with the presence of distinct lobes and their comparison at constant sweep angle shows that, under the conditions studied, the ONERA-D airfoil is more stable than the Joukowski airfoil, even for crossflow instabilities. The absolutely or convectively unstable nature of the flow in the spanwise direction is also tackled and our results suggest that the flow is only convectively unstable.

On the stability of supersonic boundary layer in interaction with weak shock waves

Numerical simulation of the problem of interaction of the supersonic (M∞=2) boundary layer on the plate with weak shock waves was performed. The shock wave was generated using a wedge located above the plate. The problem was solved using the computational technology, combining the ANSYS Fluent gasdynamic package, which calculates the main flow, and the LOTRAN 3.0 software package, which calculates not only the stability of no-break flow, but also the laminar flow failure. The intensity of the incident shock wave was varied by varying the wedge inclination angle. The N-factor envelopes of the Tollmien-Schlichting wave instability for different wedge tilt angles, including those in the local breakaway zones, were obtained. It is demonstrated that the proposed approach using the ANSYS Fluent and LOTRAN 3.0 combination computational technology allows one to calculate the stability of flows with the shockwave, including the presence of local breakaway zones.

On transonic boundary-layer receptivity over a vibrating flat plate coated with a thin liquid film

In this paper we study the generation of Tollmien–Schlichting waves initiated by vibrations of a wall where the wall is coated with a thin liquid film in a transonic flow regime. Motion of fluids are described by the two-dimensional Navier–Stokes equations assuming the Reynolds number is large. To find asymptotic solutions of the transonic boundary layer, we conduct an inspection analysis on the affine transformations of the triple-deck model for a subsonic flow and the unsteady full potential equations, with the intention of obtaining the order quantity of the free-stream Mach number in the transonic flow. We construct a modified triple-deck model for the transonic flow by considering the scalings of the perturbations that lead to the viscous–inviscid interaction problem for the flow in a subsonic regime. In particular, we are interested in the region where the subsonic scalings become invalid as the flow approaches transonic regime. We assume the wall oscillates in the vertical direction to the oncoming flow and these vibrations are periodic in time. We outline the process where the flow in the boundary layer converts the wall vibration perturbations into the instability modes which are measured by the receptivity coefficient. The viscous–inviscid interaction problem describes the stability of the boundary layer on the lower branch of the neutral curve. We show that the governing equations for the air viscous sublayer and the film flow are quasi-steady. The equation describing the inviscid layer of the airflow is unsteady and its referred to as the unsteady Kàrmàn–Guderley equation. The influence of the film surface tension is expressed through normal shear stress condition at the interface. We present an analytic formula for the amplitude of the Tollmien–Schlichting waves that are formed in the boundary layer. We analysed our model with different values of surface tension, initial film thickness and Kàrmàn–Guderley parameter. Depending on the value of these parameters, the initial amplitude of the instability waves may grow or decay.

Numerical Research of the Pressure Fluctuation of the Bow of the Submarine at Different Velocities

A numerical analysis based on stress-blended eddy simulation was conducted to investigate the pressure fluctuation of the bow of a submarine at various velocities (5.93 kn, 10 kn, and 12 kn). The simulation results were compared with the experimental data to demonstrate the validity of the numerical method. Self-power spectrum and wave-number frequency spectrum were discussed from the perspective of energy. The results show that with increasing submarine velocity, the pressure fluctuation in the axial direction increases, the transition point moves forward, and the frequency of the Tollmien–Schlichting wave raises.

Excitation of instabilies in a Blasius boundary layer by surface vibration

Here we have demonstrated that small amplitude vibration can artificially excite both two dimensional (2D) and three dimensional (3D) instability modes. The 2D modes were typical of Tollmien–Schlichting (TS) waves provided that the frequency of excitation lies within the unstable region of the neutral stability predicted by modal linear stability theory. However, even if the frequency of the mechanically forced mode was within the stable bounds of the neutral curve the harmonics generated by the non-linear response of the flow could develop as instability modes. Further analysis of the streamwise and spanwise evolution of the instability modes identified from the temporal Fourier transform confirmed the presence of 3D modes excited due to the nature of the mode shape deflection of the vibrating panel which was not uniform in the spanwise direction. The effect of spanwise non-uniformity could be increased by activating the motors along the spanwise direction. However, due to the forcing from a combination of both streamwise and spanwise motors, strong interaction with the 3D mode led to a reduction in the growth rate of the TS wave in the far-field region despite higher initial perturbation generated by a larger number of motors.

Analysis of the Tollmien-Schlichting wave in the Prandtl-Hartmann regime

In this paper, we study the instability induced by the Tollmien-Schlichting wave governed by the incompressible MHD system in the Prandtl-Hartmann regime. The interaction of the inviscid mode and viscous mode that leads to the instability is analyzed by the introduction of a new decomposition of the Orr-Sommerfeld operator on the velocity and magnetic fields. The critical Gevrey index for the instability is justified by constructing the growing mode in the essential frequency and it is shown to be the same as the incompressible Navier-Stokes equations in the Prandtl regime. This result justifies rigorously the physical understanding that the transverse magnetic field to the boundary in the Prandtl-Hartmann regime has no extra stabilizing effect on the Tollmien-Schlichting wave.