Two-dimensional Tollmien-Schlichting Waves Research Articles

A CFD-Compatible Amplification Factor Transport Equation for Oblique Tollmien-Schlichting Waves in Supersonic Boundary Layers

Boundary layer transition is a hot research topic in fluid mechanics and aerospace engineering. In low-speed flows, two-dimensional Tollmien-Schlichting (T-S) waves always dominate the flow instability, which has been modeled by Coder and Maughmer from 2013. However, in supersonic flows, three-dimensional oblique Tollmien-Schlichting waves become dominant in flow instability. Inspired by Coder and Maughmer’s NTS amplification factor transport equation for two-dimensional Tollmien-Schlichting waves in low-speed flows and Kroo and Sturdza’s linear stability theory (LST) analysis results for oblique Tollmien-Schlichting waves in supersonic flows, a new amplification factor transport equation for oblique Tollmien-Schlichting waves has been developed based on LST. The compressible Falkner-Skan similarity equations are introduced to build the relationships between nonlocal variables and local variables so that all the variables used in the present model can be calculated using local variables. Applications of this new transport equation to the flows over supersonic flat plate, 3% thick biconvex airfoil, and one modified supersonic laminar airfoil show promising results compared with the standard LST analysis results.

Acoustic receptivity and transition modeling of Tollmien-Schlichting disturbances induced by distributed surface roughness

Acoustic receptivity to Tollmien-Schlichting waves in the presence of surface roughness is investigated for a flat plate boundary layer using the time-harmonic incompressible linearized Navier-Stokes equations. It is shown to be an accurate and efficient means of predicting receptivity amplitudes and, therefore, to be more suitable for parametric investigations than other approaches with direct-numerical-simulation-like accuracy. Comparison with the literature provides strong evidence of the correctness of the approach, including the ability to quantify non-parallel flow effects. These effects are found to be small for the efficiency function over a wide range of frequencies and local Reynolds numbers. In the presence of a two-dimensional wavy-wall, non-parallel flow effects are quite significant, producing both wavenumber detuning and an increase in maximum amplitude. However, a smaller influence is observed when considering an oblique Tollmien-Schlichting wave. This is explained by considering the non-parallel effects on receptivity and on linear growth which may, under certain conditions, cancel each other out. Ultimately, we undertake a Monte Carlo type uncertainty quantification analysis with two-dimensional distributed random roughness. Its power spectral density (PSD) is assumed to follow a power law with an associated uncertainty following a probabilistic Gaussian distribution. The effects of the acoustic frequency over the mean amplitude of the generated two-dimensional Tollmien-Schlichting waves are studied. A strong dependence on the mean PSD shape is observed and discussed according to the basic resonance mechanisms leading to receptivity. The growth of Tollmien-Schlichting waves is predicted with non-linear parabolized stability equations computations to assess the effects of stochasticity in transition location.

Influence of localised smooth steps on the instability of a boundary layer

We consider a smooth, spanwise-uniform forward-facing step defined by a Gauss error function of height 4 %–30 % and four times the width of the local boundary layer thickness $\unicode[STIX]{x1D6FF}_{99}$. The boundary layer flow over a smooth forward-facing stepped plate is studied with particular emphasis on stabilisation and destabilisation of the two-dimensional Tollmien–Schlichting (TS) waves and subsequently on three-dimensional disturbances at transition. The interaction between TS waves at a range of frequencies and a base flow over a single or two forward-facing smooth steps is conducted by linear analysis. The results indicate that for a TS wave with a frequency ${\mathcal{F}}\in [140,160]$ (${\mathcal{F}}=\unicode[STIX]{x1D714}\unicode[STIX]{x1D708}/U_{\infty }^{2}\times 10^{6}$, where $\unicode[STIX]{x1D714}$ and $U_{\infty }$ denote the perturbation angle frequency and free-stream velocity magnitude, respectively, and $\unicode[STIX]{x1D708}$ denotes kinematic viscosity), the amplitude of the TS wave is attenuated in the unstable regime of the neutral stability curve corresponding to a flat plate boundary layer. Furthermore, it is observed that two smooth forward-facing steps lead to a more acute reduction of the amplitude of the TS wave. When the height of a step is increased to more than 20 % of the local boundary layer thickness for a fixed width parameter, the TS wave is amplified, and thereby a destabilisation effect is introduced. Therefore, the stabilisation or destabilisation effect of a smooth step is typically dependent on its shape parameters. To validate the results of the linear stability analysis, where a TS wave is damped by the forward-facing smooth steps direct numerical simulation (DNS) is performed. The results of the DNS correlate favourably with the linear analysis and show that for the investigated frequency of the TS wave, the K-type transition process is altered whereas the onset of the H-type transition is delayed. The results of the DNS suggest that for the perturbation with the non-dimensional frequency parameter ${\mathcal{F}}=150$ and in the absence of other external perturbations, two forward-facing smooth steps of height 5 % and 12 % of the boundary layer thickness delayed the H-type transition scenario and completely suppressed for the K-type transition. By considering Gaussian white noise with both fixed and random phase shifts, it is demonstrated by DNS that transition is postponed in time and space by two forward-facing smooth steps.

Passive boundary layer control of oblique disturbances by finite-amplitude streaks

AbstractRecent experimental results on the attenuation of two-dimensional Tollmien–Schlichting wave (TSW) disturbances by means of passive miniature vortex generators (MVGs) have shed new light on the possibility of delaying transition to turbulence and hence accomplishing skin-friction drag reduction. A recurrent concern has been whether this passive flow control strategy would work for other types of disturbances than plane TSWs in an experimental configuration where the incoming disturbance is allowed to fully interact with the MVG array. In the present experimental investigation we show that not only TSW disturbances are attenuated, but also three-dimensional single oblique wave (SOW) and pair of oblique waves (POW) disturbances are quenched in the presence of MVGs, and that transition delay can be obtained successfully. For the SOW disturbance an unusual interaction between the wave and the MVGs occurs, leading to a split of the wave with one part travelling with a ‘mirrored’ phase angle with respect to the spanwise direction on one side of the MVG centreline. This gives rise to $\Lambda $-vortices on the centreline, which force a low-speed streak on the centreline, strong enough to overcome the high-speed streak generated by the MVGs themselves. Both these streaky boundary layers seem to act stabilizing on unsteady perturbations. The challenge in a passive control method making use of a non-modal type of disturbances to attenuate modal disturbances lies in generating stable streamwise streaks which do not themselves break down to turbulence.

Turbulence, instabilities and passive scalars in rotating channel flow

Fully developed channel flow with a passive scalar rotating about the spanwise axis is studied by direct numerical simulations. The Reynolds number based on the bulk mean velocity Reb is up to 30000, substantially higher than in previous studies, and the rotation rates cover a broad range. Turbulence on the stable channel side is less strongly damped at moderate rotation rates than in channel flow at lower Reb. At high rotation rates and sufficiently high Reb, intermittent strong instabilities occur on the stable side caused by rapidly growing modes resembling two-dimensional Tollmien-Schlichting waves which at some instant become unstable and break down into intense turbulence. The turbulence decays and after some time the waves form again and the process is repeated in a cyclic manner.Rotation also strongly affects the mean passive scalar profiles and turbulent scalar fluxes. Large scalar fluctuations are observed on the border between the stable and unstable channel sides. While in non-rotating channel flow the turbulent Prandtl number of the passive scalar is about one like in other shear flows, it is much smaller in the rotating cases.

Numerical Simulation of Riblet Controlled Spatial Transition in a Zero-Pressure-Gradient Boundary Layer

To analyze the fundamental physical mechanism which determines the damping effect of a riblet surface on three-dimensional transition several numerical simulations of spatial transition in a flat plate zero-pressure-gradient boundary layer above a riblet wall are performed in this study. Two types of forced transition scenarios are investigated. The first type of transition is defined by K-type transition induced by a dominant two-dimensional Tollmien–Schlichting (TS) wave and a weak spanwise disturbance. The second type of transition is purely excited by two oblique waves. By a qualitative analysis of the occurring maximum wall-normal and spanwise velocity components and the Fourier modes of the disturbances the two-dimensional TS waves are found to be amplified by riblets, whereas three-dimensional structures, i.e., Λ-, hairpin, and streamwisely aligned vortices, are damped. At oblique transition the breakdown to turbulence is delayed by the riblets compared to transition on a clean surface. The investigation of the near wall flow structure reveals secondary flows induced by the riblets and reduced wall normal ejections as well as a reduced downwash.

Floquet analysis of secondary instability of boundary layers distorted by Klebanoff streaks and Tollmien–Schlichting waves

Previous studies of the interaction between boundary layer streaks and Tollmien–Schlichting (TS) waves have shown puzzling effects. Streaks were shown to reduce the growth rate of primary TS waves and, thereby, to delay transition; however, they can also promote transition by inducing a secondary instability. The outcome of the interaction depends on the spanwise wavelength and intensity of the streaks as well as on the amplitude of the TS waves. A Floquet analysis of secondary instability is able to explain many of these features. The base state is periodic in two directions: it is an Ansatz composed of a saturated TS wave (periodic in x) and steady streaks (periodic in z). Secondary instability analysis is extended to account for the doubly periodic base flow. Growth rate computations show that, indeed, the streak can either enhance or diminish the overall stability of the boundary layer. The stabilizing effect is a reduction in the growth rate of the primary two-dimensional TS wave; the destabilizing effect is a secondary instability. Secondary instability falls into two categories, depending on the spanwise spacing of the streaks. The response of one category to perturbations is dominated by fundamental and subharmonic instability; the response of the other is a detuned instability.

Turbulence production in an APG-boundary-layer transition induced by randomized perturbations

This paper is devoted to an experimental investigation of formation and development of coherent vortical structures at late stages of a laminar-turbulent transition initiated by a harmonic, almost two-dimensional Tollmien-Schlichting (TS) wave perturbed by weak (initially) broadband disturbances. The initial base flow represented a self-similar boundary layer with an adverse pressure gradient (APG) with Hartree parameter βH = −0.115. Experiments were performed at controlled disturbance conditions with the help of the ‘deterministic noise’ method and a universal disturbance source of instability waves. The main measurements were carried out by means of a hot-wire anemometer in a broad spatial region of the flow starting with stages of quasi-sinusoidal small-amplitude instability wave and ending with final stages of transition characterized by formation of concentrated localized vortical structures. The excited perturbations were partly random (within 20 TS-wave fundamental periods) but periodical at very large time scales during which the flow passes the model several times. The detailed measurements and the experimental data processing gave us the possibility to obtain instantaneous velocity and vorticity fields in the (x, y, z, t)-space and to perform computer-aided ‘visualization’ of the instantaneous flow structure. Specific features of the turbulence production mechanism occurring at late stages of transition are studied and compared with previously reported data obtained at sinusoidal excitation. A qualitative similarity is found between essentially nonlinear stages of transition observed in the present (randomized) case and those studied previously in cases of transition initiated by a harmonic TS wave or by a TS wave packet. It is found that interaction of primary wave with a broadband ‘noise’ of 3D TS waves leads at late stages of transition to formation of Λ-vortices, intensive Λ -shaped high-shear (HS) layers, Ω -shaped vortices, ring-like vortices and associated ‘spikes’ in time-traces, and other phenomena. These structures are found to be disposed in a random order in time and space and have somewhat distorted shapes (in contrast to the periodic-excitation case studied earlier). However, their general properties are qualitatively similar. The obtained instantaneous pictures of velocity and vorticity fields seem to correspond qualitatively to those present in the near-wall region of the developed turbulent boundary layer.

A methodology for optimal laminar flow control: Application to the damping of Tollmien–Schlichting waves in a boundary layer

A methodology for determining the optimal steady suction distribution for the delay of transition in a boundary layer is presented. The flow state is obtained from the coupled system of boundary layer equations and parabolized stability equations (PSE), to account for the spatially developing nature of the flow. The wall suction is defined by an optimal control procedure based on the iterative solution of the equations for the state and the dual state; the latter is available from the adjoint boundary layer equations and the adjoint PSE. The technique is applied to the control of two-dimensional Tollmien–Schlichting (TS) waves. Results show that the onset of the instability can be significantly postponed and/or the growth rate considerably reduced by applying an appropriate suction through the whole wall length, in a wide frequency band. Control over panels of finite length completes the study and brings useful, preliminary information on the practicality of the approach in view of implementation. Finally, a simplified methodology which does not rely on the PSE is discussed, based on the minimization of the shape factor. Satisfactory results are achieved with this simpler approach which might, thus, constitute a method of choice when results are needed rapidly, i.e., during on-line control of TS waves.

Linear and Nonlinear PSE for Stability Analysis of the Blasius Boundary Layer Using Compact Scheme

A highly accurate finite-difference PSE code has been developed to investigate the stability analysis of incompressible boundary layers over a flat plate. The PSE equations are derived in terms of primitive variables and are solved numerically by using compact method. In these formulations, both nonparallel as well as nonlinear effects are accounted for. The validity of present numerical scheme is demonstrated using spatial simulations of two cases; two-dimensional (linear and nonlinear) Tollmien-Schlichting wave propagation and three-dimensional subharmonic instability breakdown. The PSE solutions have been compared with previous numerical investigations and experimental results and show good agreement.

On the nonlinear growth of two-dimensional Tollmien–Schlichting waves in a flat-plate boundary layer

We study the nonlinear development of two-dimensional Tollmien-Schlichting waves in an incompressible flat-plate boundary layer at asymptotically large values of the Reynolds number. Attention is restricted to the far-downstream lower-branch' regime where a multiple-scales analysis is possible. It is supposed that to leading-order the waves are inviscid and neutral, and governed by the [Davis-Acrivos-]Benjamin-Ono equation. This has a three-parameter family of periodic solutions, the large-amplitude (soliton) limit of which bears a qualitative resemblance to the 'spikes' observed in certain K-type'transition experiments. The variation of the parameters over slow length- and timescales is controlled by a viscous sublayer. For the case of a purely temporal evolution, it is shown that a solution for this sublayer ceases to exist when the amplitude reaches a certain finite value. For a purely spatial evolution, it appears that an initially linear disturbance does not evolve to a fully nonlinear stage of the envisaged form. The implications of these results for the soliton theory of spike formation are discussed

Interaction between Görtler vortices and two-dimensional Tollmien–Schlichting waves

The nonlinear interaction between Görtler vortices and two-dimensional Tollmien–Schlichting (TS) waves is studied with a spatial, nonparallel model based on the parabolized stability equations. The effect of the TS waves on the development of the vortices is accounted for, showing that TS wave amplitudes of the same order of magnitude as the vortices result in significant nonlinear interaction. The range of governing parameters that has been studied so far is extended and the main effects of Görtler number, spanwise wave number, and initial amplitudes are identified. The study shows that the relative growth rates and initial amplitudes are the two most significant parameters.

Effect of the second harmonic on the fundamental parametric resonance in boundary layers

The effect of the second harmonic of a two-dimensional Tollmien-Schlichting wave on the spatial fundamental parametric resonance in boundary layers is analysed using Floquet theory. The resulting problem is numerically solved by using finite differences. The effect of this second harmonic is found to be significant. Results on this effect at different spanwise wavenumbers, primary-wave amplitudes, flow Reynolds numbers, and disturbance frequencies are presented.

FLOW STRUCTURES AND HEAT TRANSFER DURING TRANSITION IN PERTURBED MIXED-CONVECTION CHANNEL FLOWS

This research seeks to increase our understanding of fluid mechanics and heat transfer phenomena in transitional mixed-convection flows between two heated vertical plates. A direct numerical simulation by the spectral method with weak formulation is used to solve the transient three-dimensional (3-D) Navier-Stokes equations and energy equation. Initial disturbances consist of the finite amplitude two-dimensional Tollmien-Schlichting wave and two 3-D oblique waves. The Nusselt number (Nu) is mainly a function of the Rayleigh number (Ra) and also depends on the Reynolds number (Re) and the Richardson number (Ri). Nu dependence on Re increases with increase of Ra. Disturbance competitions among different modes during transition are shown for different sets of dimensionless parameters. Flow visualization by passive particles is also presented.

Spatial direct numerical simulation of boundary-layer transition mechanisms: Validation of PSE theory

A study of instabilities in incompressible boundary-layer flow on a flat plate is conducted by spatial direct numerical simulation (DNS) of the Navier-Stokes equations. Here, the DNS results are used to evaluate critically the results obtained using parabolized stability equations (PSE) theory and to study mechanisms associated with breakdown from laminar to turbulent flow. Three test cases are considered: two-dimensional Tollmien-Schlichting wave propagation, subharmonic instability breakdown, and oblique-wave breakdown. The instability modes predicted by PSE theory are in good quantitative agreement with the DNS results, except a small discrepancy is evident in the mean-flow distortion component of the two-dimensional test problem. This discrepancy is attributed to far-field boundary-condition differences. Both DNS and PSE theory results show several modal discrepancies when compared with the experiments of subharmonic breakdown. Computations that allow for a small adverse pressure gradient in the basic flow and a variation of the disturbance frequency result in better agreement with the experiments.

Multiple paths to subharmonic laminar breakdown in a boundary layer.

Numerical simulations demonstrate that laminar breakdown in a boundary layer induced by the secondary instability of two-dimensional Tollmien-Schlichting waves to three-dimensional subharmonic disturbances need not take the conventional lambda vortex/high-shear layer path.

Numerical simulation of nonlinear structures in channel flow transition

Introduction T HE objective of this computational study is to investigate the development of nonlinear vortical structures in plane channel flow transition by the visualization of threedimensional vortex lines. It has long been recognized that transition may be viewed as a sequence of events that is triggered by two-dimensional Tollmien-Schlichting waves that become three-dimensional, leading to nonlinear wave interactions. In a recent experiment, Williams used hot film probes in a water channel boundary layer to create a data base containing three-dimensional instantaneous velocity and vorticity components documenting the evolution of the high shear layer and the formation of hairpin vortices. He then used this data base to obtain three-dimensional vortex lines. In this work, a similar technique is used to visualize the flowfield structures. In order to extract information from the computational data base, which provides the three-dimensional instantaneous velocity/vorticity field in over a quarter million grid points, vortex lines were calculated at time intervals that can be viewed sequentially. In addition, vorticity contours and velocity vector plots for the secondary flow are presented for a comparison with the findings inferred from the vortex line plots. Although flow visualization by vortex lines have been utilized in other numerical studies,' the present results effectively capture the continuous evolution of the flow, enabling the identification of various vortical structures.

Numerical simulation of boundary-layer transition and transition control

The laminar-turbulent transition process in a parallel boundary-layer with Blasius profile is simulated by numerical integration of the three-dimensional incompressible Navier-Stokes equations using a spectral method. The model of spatially periodic disturbances developing in time is used. Both the classical Klebanoff-type and the subharmonic type of transition are simulated. Maps of the three-dimensional velocity and vorticity fields and visualizations by integrated fluid markers are obtained. The numerical results are compared with experimental measurements and flow visualizations by other authors. Good qualitative and quantitative agreement is found at corresponding stages of development up to the one-spike stage. After the appearance of two-dimensional Tollmien-Schlichting waves of sufficiently large amplitude an increasing three-dimensionality is observed. In particular, a peak-valley structure of the velocity fluctuations, mean longitudinal vortices and sharp spike-like instantaneous velocity signals are formed. The flow field is dominated by a three-dimensional horseshoe vortex system connected with free high-shear layers. Visualizations by time-lines show the formation of A-structures. Our numerical results connect various observations obtained with different experimental techniques. The initial three-dimensional steps of the transition process are consistent with the linear theory of secondary instability. In the later stages nonlinear interactions of the disturbance modes and the production of higher harmonics are essential.We also study the control of transition by local two-dimensional suction and blowing at the wall. It is shown that transition can be delayed or accelerated by superposing disturbances which are out of phase or in phase with oncoming Tollmien-Schlichting instability waves, respectively. Control is only effective if applied at an early, two-dimensional stage of transition. Mean longitudinal vortices remain even after successful control of the fluctuations.

On the use of a phased heater array for the controlled excitation of arbitrary three-dimensional perturbations in a laminar boundary layer

Flush mounted resistive heating elements were used successfully by Liepmann, Brown and Nosenchuck (1981) to excite and control the development of two-dimensional Tollmien-Schlichting waves. Due to increased interest in the three dimensional nature of boundary-layer transition, this technique has recently been extended to provide a means for introducing arbitrary controlled three-dimensional perturbations into a laminar flow. This paper describes the design and operation of a programmable 32-element linear heater array capable of exciting a wide variety of flow disturbances hitherto unobtainable in the laboratory.

The effects of resonance interactions between wave perturbations in a boundary layer

The transition to turbulence in a boundary layer can be induced by perturbations of low intensity and is accompanied by a growth in their energy, the development of three-dimensional structures, and a change in the spectral composition of the field. A number of important properties of the process admit interpretation in the framework of nonlinear stability theory and can be due to a resonance interaction. Experiments [1, 2] have revealed a transition accompanied by an appreciable enhancement of pulsations whose period is twice that of the driving vibrating tape. Theoretical investigations [3–9] have revealed the existence of a resonance mechanism capable of strong excitation of three-dimensional Tollmien-Schlichting waves at the frequency of a subharmonic. It has been suggested [4] that the observed transition regime is the result of evolution of triplets of resonantly coupled oscillations forming symmetric triplets [10]. In contrast to the type of transition considered by Craik et al. [10, 11], the leading role is played by subharmonics distinguished parametrically in the background. Experimental confirmations have been obtained [12, 13] of the coupling of the resonances in symmetric triplets with the subharmonic regime. Further investigation of the resonance mechanism is an important topical problem. This paper presents a study on the formation and special characteristics of the initial stage in the nonlinear development of triplets; the collective interaction of a two-dimensional Tollmien-Schlichting wave with a packet of three-dimensional waves is examined; the behavior of the system is analyzed, taking into account the resonance coupling with the harmonic of the main wave. A comparison is made between Craik's model and experimental data.