Three-dimensional Disturbances Research Articles

Temperature Factor Effect on the Disturbance Propagation in Hypersonic Flow past a Yawed Flat Plate

The flow in the three-dimensional laminar boundary layer on a yawed flat plate of finite length is studied in the regime of strong viscous-inviscid interaction with a hypersonic flow. In the vicinity of the leading edge the flow functions are expanded in series under the assumption that the base pressure dependent on the transverse coordinate is given at the trailing edge of the plate. It is established that the expansions obtained include an indefinite function and its derivative with respect to the transverse coordinate. The corresponding boundary value problems are formulated and numerically solved, the eigenvalues are found, and it is shown that the exponent in the third term of the expansion differs from that in the second term only by one. The plate surface temperature effect on the flow parameters and the initiation of three-dimensional disturbances is investigated.

Effect of trailing-edge boundary conditions on acoustic feedback loops in high-pressure turbines

The occurrence and sensitivity of acoustic feedback loops was investigated for a linear high-pressure turbine (HPT) vane cascade at realistic operating conditions, i.e. isentropic exit conditions of Re = 570, 000 and M = 0.92. Forced compressible Navier–Stokes simulations were conducted to study the temporal response of various base flows to an initial small-amplitude pulse. One of the objectives was to assess the impact the quality of the base flow has on the instability behaviour, i.e. whether base flows obtained from computationally more affordable approaches such as Reynolds-averaged Navier–Stokes (RANS) produce similar results as those obtained from highly resolved large-eddy simulations (LES). Using base flows obtained from both LES and RANS of a high-pressure turbine vane with no-slip trailing-edge boundary conditions, the temporal pulse response revealed the presence of an unstable acoustic feedback loop, i.e. with amplitude increasing quickly over time. It was also found that the base flows were globally unstable with respect to three-dimensional instabilities, regardless of the perturbation location. However, the stability analysis performed on the RANS generated base flow exhibited much increased growth rates due to the recirculation region downstream of the blade trailing edge featuring a higher reverse velocity.In order to understand the sensitivity of the acoustic feedback loop to modifications of the trailing-edge boundary conditions, additional stability analyses were conducted on base flows obtained from simulations of the same HPT vane configuration with trailing-edge ejection at two different rates. For the base flows with added trailing-edge ejection, the pulse responses did not exhibit pressure waves originating from the trailing edge impinging on the suction side of the adjacent blade. More importantly, although an acoustic feedback loop was also present in these cases, the amplitude decayed over time, thus the flow was stable with regards to two- and three-dimensional disturbances. The results suggest that it is primarily the reduction in reverse flow velocity, or even the full removal of the trailing-edge recirculation region when applying trailing-edge ejection, that is responsible for the suppression of the acoustic feedback loop and the growth of three-dimensional instabilities.

FULLY 3D RAYLEIGH–TAYLOR INSTABILITY IN A BOUSSINESQ FLUID

Rayleigh–Taylor instability occurs when a heavier fluid overlies a lighter fluid, and the two seek to exchange positions under the effect of gravity. We present a linearized theory for arbitrary three-dimensional (3D) initial disturbances that grow in time, and calculate the evolution of the interface for early times. A new spectral method is introduced for the fully 3D nonlinear problem in a Boussinesq fluid, where the interface between the light and heavy fluids is approximated with a smooth but rapid density change in the fluid. The results of large-scale numerical calculation are presented in fully 3D geometry, and compared and contrasted with the early-time linearized theory.

On the Orientation of Convective Rolls in an Inclined Rectangular Channel

An analysis of the linear stability of convective flow in a channel of rectangular cross-section inclined to the horizon is presented. The behavior of three-dimensional monotonic disturbances is considered for different values of the channel width and fluid properties. The main flow is obtained in an analytical form. Two angles of inclination, at which the convective roll changes, are determined. A strong dependence of the smaller inclination angle on the channel width and its weak dependence on the medium characteristics and a weak dependence of the greater inclination angle on the channel width are established.

Self-sustaining dual critical layer states in plane Poiseuille–Couette flow at large Reynolds number

The nonlinear stability of plane Poiseuille–Couette flow subjected to three-dimensional disturbances is studied asymptotically at large Reynolds number R . By analysing the nature of the instability for increasing disturbance size Δ, the scaling Δ = O ( R −1/3 ) is identified at which a strongly nonlinear neutral wave structure emerges, involving the interaction of two inviscid critical layers. The striking feature of this structure is that the travelling wave disturbances have both streamwise and spanwise wavelengths comparable to the channel width, with an associated phase speed of O (1). An alternative method to the classical balancing of phase shifts is proposed, involving vorticity jumps, that uses a global property of the flow-field and enables the amplitude-dependence of the neutral modes to be determined in terms of the wavenumbers and the properties of the basic flow. Numerical computation of the Rayleigh equation which governs the flow outside of the critical layers shows that neutral solutions exist for non-dimensional wall sliding speeds in the range 0 ≤ V < 2. It transpires that the critical layers merge and the asymptotic structure referred to above breaks down both in the large-amplitude limit and the limit V → 2 when the maximum of the basic flow becomes located at the upper wall.

Magnetohydrodynamic instability of mixed convection in a differentially heated vertical channel

The present paper investigates the instability of mixed convection in a differentially heated vertical layer of an electrically conducting fluid under the influence of a uniform horizontal magnetic field. The key parameters which influence the instability characteristics of the system are the Hartman number, the Reynolds number, the Grashof number and the Prandtl number. The computational results reveal that two-dimensional disturbances are more unstable than the three-dimensional disturbances. Further, influence of the Grashof number and the Hartmann number is analyzed on the basic flow. The effect of increasing the Hartman number shows stabilizing effect on the fluid flow and dissimilar is the case with an increase in the Reynolds number and the Prandtl number. In the presence of Reynolds number, Re, stationary instability disappears altogether which is contrary to the results observed in the case of pure natural convection ( $ Re\rightarrow 0$ ). The numerical results obtained under the limiting cases are shown to be in excellent agreement with the existing ones.

Numerical simulation on the development of periodic three-dimensional disturbances in a supersonic boundary layer

The results of a numerical study of the development of periodic pulsations in a supersonic boundary layer on a flat plate are presented at a Mach number of 2.5 and a unit Reynolds number of 8×106 m–1. Using the software complex ANSYS, the complete Navier-Stokes equations were solved. Periodic mass flow disturbances with a frequency of 20 kHz were introduced into the boundary layer through a small-diameter hole on the surface of the model. Downstream the profiles of the longitudinal mass flow pulsations were recorded, and spectral analysis of the data was carried out. The main characteristics of the development of unstable disturbances in both physical and wave spaces are obtained.

Stability of flow through deformable channels and tubes: implications of consistent formulation

The present study is aimed at assessing the existing results concerning the stability of canonical shear flows in channels and tubes with deformable walls, in light of consistent formulations of the nonlinear solid constitutive model and linearised interface conditions at the fluid–solid interface. We show that a class of unstable shear-wave modes at low Reynolds number, predicted by previous studies for pressure-driven flows through neo-Hookean tubes and channels, is absent upon use of consistent interfacial conditions. Furthermore, we analyse the consequences of the change in solid model on the stability of the canonical shear flows by using both neo-Hookean and Mooney–Rivlin models. We show that the salient features of the stability of the system are adequately captured by a consistent formulation of the neo-Hookean solid model, thus precluding the need to employ more detailed solid models. The stability analysis of planar flows past a neo-Hookean solid subjected to three-dimensional disturbances showed that two-dimensional disturbances are more unstable than the corresponding three-dimensional disturbances within the consistent formulations. We show that prior inconsistent formulations of the solid constitutive equation predict a physically spurious spanwise instability in disagreement with experiments thereby demonstrating their inapplicability to predict instabilities in flow past deformable solid surfaces. Using the consistent formulation, the present work provides an accurate picture, over a range of Reynolds numbers, of the stability of canonical shear flows through deformable channels and tubes. Importantly, it is shown how inconsistencies in either the bulk constitutive relation or in the linearisation of the interface conditions can separately lead to physically spurious instabilities. The predictions of this work are relevant to experimental studies in flow through deformable tubes and channels in the low and moderate Reynolds number regime.

Spatio-temporal instability of two superposed fluids in a channel with boundary slip

Spatio-temporal stability analysis of a viscosity and density stratified two-fluid flow in a channel with hydrophobic walls, which experience a finite tangential velocity slip, is considered for a range of parameters for which Squire’s theorem is not valid. The study restricted only to temporal stability analysis of two/three-dimensional infinitesimal disturbances reveals that three-dimensional disturbances are more unstable than the corresponding two-dimensional disturbances. We examine the role of different types of slip boundary conditions at the channel walls on the temporal growth rate. The influence of velocity slip at the upper wall on the nature of instability in the spatio-temporal framework for low density ratio is investigated, and the boundary that demarcates the convective/absolute instabilities are presented for both two and three-dimensional disturbances. In a configuration with a thinner lower layer away from the upper wall with slip, we found that the flow in the entire domain is absolutely unstable, except for two strips at low and high Reynolds numbers. For the parameters considered in the present study, it is found that the absolutely unstable region shrinks with decrease in velocity slip. The absolute growth rate exhibits a non-monotonic behavior with respect to velocity slip. The role of velocity slip in promoting absolute instability in some parameter regimes suggests that by designing the walls of the channel as hydrophobic/rough/porous surfaces, which can be modelled as smooth surfaces with tangential velocity slip, it may be possible to achieve early transition to turbulence in two-layer flow systems.

Effect of perforation on flow past a conic cylinder at $${\varvec{Re}}~=~100$$ Re = 100 : wavy vortex and sign laws

In order to find the intrinsic physical mechanism of the original Karman vortex wavily distorted across the span due to the introduction of three-dimensional (3-D) geometric disturbances, a flow past a peak-perforated conic shroud is numerically simulated at a Reynolds number of 100. Based on previous work by Meiburg and Lasheras (1988), the streamwise and vertical interactions with spanwise vortices are introduced and analyzed. Then vortex-shedding patterns in the near wake for different flow regimes are reinspected and illustrated from the view of these two interactions. Generally, in regime I, spanwise vortices are a little distorted due to the weak interaction. Then in regime II, spanwise vortices, even though curved obviously, are still shed synchronously with moderate streamwise and vertical interactions. But in regime III, violently wavy spanwise vortices in some vortex-shedding patterns, typically an $$\Omega $$ -type vortex, are mainly attributed to the strong vertical interactions, while other cases, such as multiple vortex-shedding patterns in sub-regime III-D, are resulted from complex streamwise and vertical interactions. A special phenomenon, spacial distribution of streamwise and vertical components of vorticity with specific signs in the near wake, is analyzed based on two models of streamwise and vertical vortices in explaining physical reasons of top and bottom shear layers wavily varied across the span. Then these two models and above two interactions are unified. Finally two sign laws are summarized: the first sign law for streamwise and vertical components of vorticity is positive in the upper shear layer, but negative in the lower shear layer, while the second sign law for three vorticity components is always negative in the wake.

Synchronization of second-mode instability waves for high-enthalpy hypersonic boundary layers

The stability of a hypersonic boundary layer for a $5^{\circ }$ half-angle cone at the Caltech T5 high-enthalpy flow conditions was investigated using direct numerical simulations. For the ‘linear’ stability investigations, the boundary layer was perturbed by small axisymmetric disturbances with very small amplitudes, and for the nonlinear regime, three-dimensional pulse disturbances with larger amplitudes were introduced. The surprising result from these investigations was that the 3D wave packet undergoes strong spatial modulations, which we have not observed for other experimental conditions (e.g. the Purdue BAM6QT). This modulation was found to be directly due to the synchronization between second-mode wave components and vorticity/entropy modes. Furthermore, it was found that a synchronization with slow acoustic waves leads to a sudden and strong emission of acoustic waves deep into the free stream, which was observed for both a linear wave train and a 3D nonlinear wave packet. Therefore, it can be concluded that this is a linear mechanism that is not suppressed by nonlinear effects.

Generation of Three-Dimensional Disturbances in a Boundary Layer on Strong Interaction with a Hypersonic Flow

Laminar boundary layer flow over an infinite-span, finite-length flat plate is investigated in the regime of strong interaction with a hypersonic gas flow. Under the assumption that an additional condition dependent on the transverse coordinate can be imposed on the trailing edge of the plate the flow functions are expanded in power series in the vicinity of the leading edge. It is shown that these expansions include an indefinite function dependent on the transverse coordinate. The corresponding boundary value problems are formulated and solved and the eigenvalues are determined. It is established that in this case the two-dimensional boundary layer can rearrange itself into a three-dimensional boundary layer.

Auto-oscillations in supersonic boundary layer

The production of periodic oscillations in a supersonic boundary layer at the moderate and high Mach numbers (M = 2 and 5.35) is investigated within the framework of the weakly nonlinear stability theory of the second order in nonlinearity. The model includes the effects of self-action, such as the generation of stationary secondary harmonics and the disturbances of double frequencies. It is shown that for two-dimensional vortex disturbances, the character of the excitation of vortex disturbances changes from the mild one to the stiff one with the increasing Mach number, which leads to a reduction of the critical Reynolds number Rec. For three-dimensional disturbances of low azimuthal wave numbers, a supercritical auto-oscillatory regime sets in. A complex regime realizes for two-dimensional acoustic disturbances at M = 5.35 with a stiff excitation in the region of Rec.

A comprehensive gaze stabilization controller based on cerebellar internal models

Gaze stabilization is essential for clear vision; it is the combined effect of two reflexes relying on vestibular inputs: the vestibulocollic reflex (VCR), which stabilizes the head in space and the vestibulo-ocular reflex (VOR), which stabilizes the visual axis to minimize retinal image motion. The VOR works in conjunction with the opto-kinetic reflex (OKR), which is a visual feedback mechanism that allows the eye to move at the same speed as the observed scene. Together they keep the image stationary on the retina.In this work, we implement on a humanoid robot a model of gaze stabilization based on the coordination of VCR, VOR and OKR. The model, inspired by neuroscientific cerebellar theories, is provided with learning and adaptation capabilities based on internal models. We present the results for the gaze stabilization model on three sets of experiments conducted on the SABIAN robot and on the iCub simulator, validating the robustness of the proposed control method. The first set of experiments focused on the controller response to a set of disturbance frequencies along the vertical plane. The second shows the performances of the system under three-dimensional disturbances. The last set of experiments was carried out to test the capability of the proposed model to stabilize the gaze in locomotion tasks. The results confirm that the proposed model is beneficial in all cases reducing the retinal slip (velocity of the image on the retina) and keeping the orientation of the head stable.

Direct numerical simulation of the laminar–turbulent transition at hypersonic flow speeds on a supercomputer

A method for direct numerical simulation of three-dimensional unsteady disturbances leading to a laminar–turbulent transition at hypersonic flow speeds is proposed. The simulation relies on solving the full three-dimensional unsteady Navier–Stokes equations. The computational technique is intended for multiprocessor supercomputers and is based on a fully implicit monotone approximation scheme and the Newton–Raphson method for solving systems of nonlinear difference equations. This approach is used to study the development of three-dimensional unstable disturbances in a flat-plate and compression-corner boundary layers in early laminar–turbulent transition stages at the free-stream Mach number M = 5.37. The three-dimensional disturbance field is visualized in order to reveal and discuss features of the instability development at the linear and nonlinear stages. The distribution of the skin friction coefficient is used to detect laminar and transient flow regimes and determine the onset of the laminar–turbulent transition.

Linear stability of confined flow around a 180-degree sharp bend

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

Influence of the lateral wall velocity on three-dimensional disturbance development in plane Poiseuille–Couette flow

The linear stage of three-dimensional disturbance development in the Poiseuille–Couette flow in the case when both walls can move in the lateral direction is investigated by applying the asymptotic triple-deck theory. It is shown that the lateral wall velocity has no effect on the streamwise velocity of a wave packet. The packet does not bifurcate, but drifts in the lateral direction at the speed equal to the arithmetic mean of the walls’ speeds. Characteristic “ripples” in the lateral direction are observed at the stage of packet formation.

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.

Modeling rockfall frequency and bounce height from three-dimensional simulation process models and growth disturbances in submontane broadleaved trees

The use of dynamic computational methods has become indispensable for the assessment of rockfall hazards and the quantification of uncertainties. Although a substantial number of models with various degrees of complexity has become available over the past few years, models have only rarely been parameterized against observations, especially because long-term records of rockfalls have proven to be scarce and typically incomplete. On forested slopes, tree-ring analyses may help to fill this gap, as they have been shown to provide annually resolved data on past rockfall activity over long periods. In this paper, a total of 1495 rockfall scars recorded on the stem surface of 1004 trees have been studied at a site in the Vercors massif (French Alps) to calibrate the 3D process based simulation model RockyFor3D. Uncertainties related to the choice of parameters accounting for energy dissipation and surface roughness have been investigated in detail. Because of the lack of reliable data, these parameters typically are estimated based on expert judgments, despite the fact that they have significant impacts on runout distances and bounce height. We demonstrate that slight variations in roughness can indeed strongly affect the performance of runout modeling and that the decreasing downward gradient, observed in field data, is properly reproduced only if reduced roughness (<10cm) enables blocks to reach the distal parts of the study plot. With respect to the height of impacts, our results also reveal that differences between simulations and observations can indeed be minimized if softer soil types are preferred during simulation, as they typically limit bouncing. We conclude that field-based dendrogeomorphic approaches represent an objective tool to improve rockfall simulations and to enhance our understanding of parameterization, which is of key importance for process dynamics and thus hazard zoning.

Numerical simulation of the evolution of unstable disturbances of various modes and initial stages of the laminar-turbulent transition in the boundary layer at the freestream Mach number М = 6

Direct numerical simulations of linear and nonlinear stages of the evolution of unstable disturbances of various modes and initial stages of the laminar-turbulent transition in the boundary layer on a flat plate at the freestream Mach number M = 6 are performed on the basis of full unsteady Navier–Stokes equations for a compressible gas. A considerable effect of three-dimensional unstable disturbances on initiation of the laminar-turbulent transition is demonstrated.