Inflectional Velocity Profiles Research Articles

The onset of turbulence in decelerating diverging channel flows

This work investigates the stability and transition to turbulence in a diverging channel subjected to a time-varying trapezoidal-shaped inflow boundary condition. Numerical simulations are performed for different deceleration rates and Reynolds numbers while maintaining a constant acceleration rate. The flow transition begins with two-dimensional primary instability with the formation of inflectional velocity profiles, followed by local separation and the emergence of an array of shear layer vortices. We divide simulation cases systematically into three categories based on the onset of secondary instability and the generation of streamwise vorticity. At low and medium Reynolds numbers (type I), the spanwise vortex rolls formed by inflectional instability remain two-dimensional and diffuse at the channel centre without exhibiting further instabilities. At high Reynolds numbers and deceleration rates (type II), the rolled shear layer exhibits secondary instability during the zero mean inflow phase, followed by local incipient turbulent structure formation. The streamwise vorticity that develops over the shear layer structures causes oscillations with a spanwise wavelength similar to those associated with the elliptic instability in a counter-rotating vortex pair. Using the Lamb–Oseen approximation of vortices in conjunction with the dynamic mode decomposition algorithm of the three-dimensional flow field, we captured successfully the characteristics of the secondary instability. The third category (type III) is characterized by periodic unsteady separation, secondary instability, and merging of shear layer vortices, which occurs when Reynolds numbers are high and deceleration rates are low.

Spatiotemporal characterization of turbulent channel flow with a hyperelastic compliant wall

Direct numerical simulations of turbulent flow in a channel with one rigid and one viscoelastic wall are performed. An Eulerian–Eulerian model is adopted with a level-set approach to identify the fluid–compliant material interface. Focus is placed on the propagation of Rayleigh waves in the compliant material, whose speed depends on the shear modulus of elasticity and whose dominant wavelength depends on the thickness of the viscoelastic layer. These parameters are selected to ensure coupling between the compliant surface and turbulence. When the phase speed of Rayleigh waves is commensurate with the advection velocity of near-wall pressure fluctuations, sheets of vorticity are lifted up and detached near the critical layer and lead to a local pressure minimum. These events are caused by the inflectional velocity profile near the troughs, and are controlled by the net vorticity flux at the elastic surface. This phenomenon is central to understanding the statistical characteristics of the flow, including the surface deformation–pressure correlation and enhanced stochastic Reynolds shear stresses. Finally, we discuss the influence of three-dimensionality of the surface topography on the generation of streamwise vorticity, secondary motions and lateral turbulent transport.

A simplified model for transition prediction applicable to wind-turbine rotors

Abstract. This work aims to develop a simple framework for transition prediction over wind-turbine blades, including effects of the blade rotation and spanwise velocity without requiring fully three-dimensional simulations. The framework is based on a set of boundary-layer equations (BLEs) and parabolized stability equations (PSEs), including rotation effects. An important element of the developed BL method is the modeling of the spanwise velocity at the boundary-layer edge. The two analyzed wind-turbine geometries correspond to a constant airfoil and the DTU 10-MW Reference Wind Turbine blades. The BL model allows an accurate prediction of the chordwise velocity profiles. Further, for regions not too close to the stagnation point and root of the blade, profiles of the spanwise velocity agree with those from Reynolds-averaged Navier–Stokes (RANS) simulations. The model also allows predicting inflectional velocity profiles for lower radial positions, which may allow crossflow transition. Transition prediction is performed at several radial positions through an “envelope-of-envelopes” methodology. The results are compared with the eN method of Drela and Giles, implemented in the EllipSys3D RANS code. The RANS transition locations closely agree with those from the PSE analysis of a 2D mean flow without rotation. These results also agree with those from the developed model for cases with low 3D and rotation effects, such as at higher radial positions and geometries with strong adverse pressure gradients where 2D Tollmien–Schlichting (TS) waves are dominant. However, the RANS and PSE 2D models predict a later transition in the regions where 3D and rotation effects are non-negligible. The developed method, which accounts for these effects, predicted earlier transition onsets in this region (e.g., 19 % earlier than RANS at 26 % of the radius for the constant-airfoil geometry) and shows that transition may occur via highly oblique modes. These modes differ from 2D TS waves and appear in locations with inflectional spanwise velocity. However, except close to the root of the blade, crossflow transition is unlikely since the crossflow velocity is too low. At higher radial positions, where 3D and rotation effects are weaker and the adverse pressure gradient is more significant, modes with small wave angles (close to 2D) are found to be dominant. Finally, it is observed that an increase in the rotation speed modifies the spanwise velocity and increases the Coriolis and centrifugal forces, shifting the transition location closer to the leading edge. This work highlights the importance of considering the blade rotation and the three-dimensional flow generated by that in transition prediction, especially in the inner part of the blade.

Direct numerical simulation of vortex-induced instability for a zero-pressure-gradient boundary layer.

Vortex-induced instability caused by a free-stream vortical excitation is explored here quantitatively with the help of controlled computational results. First, the computed results are compared with experimental results in Lim etal. [Exp. Fluids 37, 47 (2004)10.1007/s00348-004-0783-5] for the purpose of validation of the three-dimensional (3D) computations. Thereafter, the computed results are explained using methods developed to study nonlinear and spatiotemporal aspects of receptivity and instability for incompressible flows. Here a zero-pressure-gradient (ZPG) boundary layer is perturbed by a constant strength vortex traveling at a fixed height, moving with constant speed, as in the cited experiment. The vortex is created by a translating and rotating circular cylinder in the experiment, with absolute control of the physical parameters. The sign of the translating vortex is fixed by the direction of rotation of the translating cylinder. A high accuracy computing method is employed to solve the 3D Navier-Stokes equation (NSE) for different translation speeds and signs of the free-stream vortex. A nonlinear disturbance enstrophy transport equation (DETE) for incompressible flows is used to explain the vortex-induced instability. This equation is exact and explains the instabilities, as governed by the NSE. The DETE approach has been successfully developed to explain two-dimensional (2D) vortex-induced instability in Sengupta etal. [Phys. Fluids 30, 054106 (2018)10.1063/1.5029560], to trace the linear and nonlinear stages of disturbance growth. Apart from quantification of vortex-induced instability, another major goal is to show how the disturbance evolves from an initial 2D to a 3D stage. While the sign of the translating vortex is important in creating the response field, we additionally highlight the distinct differences caused by increased translation speed and strength of the free-stream vortex on the overall instability. These explain creation of small-scale vortices via the instability of an equilibrium flow, even though the excitation is 2D only. For some cases, this causes 3D bypass transition. We also show a case which demonstrates strong unsteady separation with inflectional velocity profiles, yet the disturbance flow remains essentially 2D, which can be termed a bypass transition.

Silent inflow condition for turbulent boundary layers

A method is proposed to trigger inflow turbulence while minimizing spurious noise in order to study the noise radiated directly by a turbulent boundary layer. The strategy is based on a controlled transition by entering a resonant triad of unstable modes based on an inflectional velocity profile.

On stability of channel flow of thermoviscous fluid

The paper presents key results on a large-scale entrainment of thermoviscous liquid layers with different temperatures and their further mixing observed in the plane-parallel flow with an inflectional velocity profile. We show that the instability development in the channel is more intensive at the inflection point vicinity and is not related directly to vorticity generation in the near-wall region. The considered flow being unstable relative to the finite-amplitude harmonic disturbances possesses several resonant frequencies initiating the most intense entrainment. Temperature fields are analyzed based on the time-averaged entrainment layer thickness and temperature isoline displacement. We discuss the spectral properties of flow enstrophy, vorticity, and kinetic energy in terms of asymptotics of cascades observed and coherent structures. Okubo-Weiss criterion is used for mapping of four flow zones wherein an active filamentation of the turbulent veil or long-term existence of vortex structures is possible.

On the odd and even secondary instabilities of Görtler vortices

Boundary layer flows over concave wall can be unstable to disturbances giving rise to streamwise counter-rotating vortices known as Gortler vortices. These vortices in its nonlinear form are responsible for a strong distortion of the streamwise velocity profiles in the wall-normal and spanwise directions. The resulting inflectional velocity profiles are unstable to unsteady disturbances. These disturbances are called secondary instabilities and can develop into horseshoe vortices or a sinuous motion of the Gortler vortices. These types of secondary instabilities are known as even (varicose) and odd (sinuous) modes, respectively. Although many studies focused this subject, it has not been stated which mode dominates the transition process. In the present study the secondary instability of Gortler flow is investigated using high-order spatial numerical simulation. Multi-frequency unsteady disturbances are introduced with the same spanwise wavelength as the Gortler vortices, but different spanwise phases. Three different spanwise phases are used and the effect on the secondary instability is analyzed. Both, even and odd secondary instabilities are observed, according to the relative spanwise position of the unsteady disturbances. The growth analysis for each secondary crossplane instability mode is made using a temporal Fourier analysis and the physics is explored with the aid of the flow structures visualization. The results introducing disturbances that give rise to odd and even modes simultaneously show that, for the spanwise wavelength analyzed, the odd modes grow first and dominate the transition process.

Secondary instability analysis of crossflow on a hypersonic yawed straight circular cone

The purpose of this paper is to provide secondary instability analysis of stationary crossflow vortices on a hypersonic yawed straight circular cone with a $7^{\circ }$ half-angle at $6^{\circ }$ angle of attack, free-stream Mach number 6 and unit Reynolds number $10.09\times 10^{6}~\text{m}^{-1}$. At an angle of attack, a three-dimensional boundary layer is developed between the windward and leeward symmetry planes. Under the action of azimuthal pressure gradients, the flow near the surface is deflected more than the flow near the edge of the boundary layer. This results in an inflectional velocity profile that can sustain the growth of crossflow vortices. The stationary crossflow instability is computed by means of the nonlinear parabolized stability equations, including a methodology to predict the stationary-crossflow marching path and variation of the spanwise number of waves in the marching direction solely from the basic state. Secondary instability analysis is performed using spatial BiGlobal equations based on two-dimensional partial differential equations. The secondary instabilities are calculated at different axial locations along two crossflow vortex trajectories selected to complement experiments conducted in the Mach 6 Quiet Tunnel at Texas A&M University and in the Boeing/AFOSR Mach 6 Quiet Tunnel at Purdue University. The secondary instability analysis captures various instability modes. Similar to observations in the low-speed regime for an infinite swept wing, secondary shear-layer instabilities are amplified as a consequence of the three-dimensional shear layer formed by crossflow vortices. Also, low-frequency travelling crossflow and high-frequency second modes coexist with the shear-layer instabilities. These results are shown to be in good agreement with the two sets of hypersonic yawed cone experiments (one with natural surface roughness and one with artificial discrete roughness) and compare well with experimental measurements of an incompressible swept wing.

Numerical analysis of laminar and transitional flow in a planar sudden expansion

Abstract The laminar and transitional flow through a planar sudden expansion with various expansion ratios is studied using two-dimensional direct numerical simulations at flow Reynolds numbers up to 5000 and expansion ratios in the range of 1.33 to 4.00. Results show the relationship between the reattachment length and the flow Reynolds number and expansion ratio. Correlations are developed for the non-dimensional reattachment length and the maximum velocity magnitude in the reverse-flow region in terms of Reynolds numbers and expansion ratios. The correlations and computed results achieve excellent agreement with published literature. Bifurcation phenomena resulting in the loss of flow symmetry downstream of the sudden expansion is observed and critical Reynolds numbers for the onset of bifurcation for various expansion ratios are identified. The initiation mechanism for bifurcation is attributed to the amplification of streamwise velocity disturbances at a spatial location that depends on the flow Reynolds number. For Reynolds numbers near the critical value for bifurcation onset, the disturbances grow from the sudden expansion, while for Reynolds numbers well above the critical value, the disturbances grow from the location of flow reattachment that occurred in the symmetric condition prior to bifurcation. Finally, the effect of the bifurcation phenomena on the development of flow unsteadiness and transition to turbulence downstream of the sudden expansion is linked to the successive formation of smaller-scale, steady recirculation zones. Transition is initiated by inviscid instability of the inflectional velocity profiles associated with these smaller-scale zones.

Linear Instability Analysis of Magnetohydrodynamic Flow in a Rectangular Duct with Side Walls of Unsymmetrical Conductivity

Temporal instability of liquid-metal flow in a square duct is investigated using a two-dimensional Chebyshev collocation method. In this study, the flow is subjected to a transverse magnetic field. The wall of the duct perpendicular to the magnetic field and the left parallel wall is perfectly conducting whereas the right parallel wall is insulating. Neutral stability curves are obtained for different Hartmann numbers. The five influencing factors of the instability are analyzed by energy analysis of perturbations. With the increase of Hartmann number, the critical Reynolds number first decreases rapidly and then increases gradually. The turning point of the variation of Rec with Ha is at Ha ≈ 20.4. When Ha < 20.4, velocity shear near the inflection point plays a dominant role in leading to the flow instability. When Ha becomes >20.4, perturbations produced by the inflectional velocity profile and Tollmien-Schlichting waves in the side layer are elongated by the nonuniform velocity in transverse direction; thus, the flow instability is caused by the combined effect at a much lower Reynolds number.

Instabilities in unsteady boundary layers with reverse flow

Instabilities arising in unsteady boundary layers with reverse flow have been investigated experimentally. Experiments are conducted in a piston driven unsteady water tunnel with a shallow angle diffuser placed in the test section. The ratio of temporal (Πt) to spatial (Πx) component of the pressure gradient can be varied by a controlled motion of the piston. In all the experiments, the piston velocity variation with time is trapezoidal consisting of three phases: constant acceleration from rest, constant velocity and constant deceleration to rest. The adverse pressure gradient (and reverse flow) are due to a combination of spatial deceleration of the free stream in the diffuser and temporal deceleration of the free stream caused by the piston deceleration. The instability is usually initiated with the formation of one or more vortices. The onset of reverse flow in the boundary layer, location and time of formation of the first vortex and the subsequent flow evolution are studied for various values of the ratio Πx/(Πx+Πt) for the bottom and the top walls. Instability is due to the inflectional velocity profiles of the unsteady boundary layer. The instability is localized and spreads to the other regions at later times. At higher Reynolds numbers growth rate of instability is higher and localized transition to turbulence is observed. Scalings have been proposed for initial vortex formation time and wavelength of the instability vortices. Initial vortex formation time scales with convective time, δ/ΔU, where δ is the boundary layer thickness and ΔU is the difference of maximum and minimum velocities in the boundary layer. Non-dimensional vortex formation time based on convective time scale for the bottom and the top walls are found to be 23 and 30 respectively. Wavelength of instability vortices scales with the time averaged boundary layer thickness.

An inviscid modal interpretation of the ‘lift-up’ effect

AbstractIn this paper, we give a modal interpretation of the lift-up effect, one of two well-known mechanisms that lead to an algebraic instability in parallel shearing flows, the other being the Orr mechanism. To this end, we first obtain the two families of continuous spectrum modes that make up the complete spectrum for a non-inflectional velocity profile. One of these families consists of modified versions of the vortex-sheet eigenmodes originally found by Case (Phys. Fluids, vol. 3, 1960, pp. 143–148) for plane Couette flow, while the second family consists of singular jet modes first found by Sazonov (Izv. Acad. Nauk SSSR Atmos. Ocean. Phys., vol. 32, 1996, pp. 21–28), again for Couette flow. The two families are used to construct the modal superposition for an arbitrary three-dimensional distribution of vorticity at the initial instant. The so-called non-modal growth that underlies the lift-up effect is associated with an initial condition consisting of rolls, aligned with the streamwise direction, and with a spanwise modulation (that is, a modulation along the vorticity direction of the base-state shearing flow). This growth is shown to arise from an appropriate superposition of the aforementioned continuous spectrum mode families. The modal superposition is then generalized to an inflectional velocity profile by including additional discrete modes associated with the inflection points. Finally, the non-trivial connection between an inviscid eigenmode and the viscous eigenmodes for large but finite Reynolds number, and the relation between the corresponding modal superpositions, is highlighted.

Study of instabilities in a quasi-2D MHD duct flow with an inflectional velocity profile

The mechanisms responsible for instabilities and a transition to turbulence in liquid metal duct flows of a fusion blanket are not understood very well, which limits predictive capabilities for heat and material transport in a blanket. In order to elucidate such mechanisms in quasi-two-dimensional (Q2D) magnetohydrodynamic flows with inflection points, an experimental and computational effort is underway to electromagnetically induce a Q2D turbulent flow through the injection of current at the Hartmann walls. In such a flow, inflectional instabilities arise at the two locations where current is supplied. In the experiments, Hartmann wall inductive velocimetry is employed as the main flow diagnostics. The electric potential field is measured using an array of small probes embedded in the wall material, and the fluctuating velocity field is reconstructed from the potential data using Ohm's law. First experimental data have been taken, which are in qualitative agreement with the pre-experimental analysis, where the flows are numerically simulated using a Q2D flow model.

Numerical investigation of quasi-periodic flow and vortex structure in a twin rectangular subchannel geometry using detached eddy simulation

The hybrid Unsteady Reynolds-Averaged Navier–Stokes (URANS)/Large Eddy Simulation (LES) methodology was used to investigate the flow dynamics and associated gap vortex structure in compound rectangular channels for isothermal flows. The specific form of the hybrid URANS/LES approach that was used is the Strelets (2001) version of the Shear Stress Transport (SST) based Detached Eddy Simulation (DES). The DES-SST model was used to study quasi-periodic flow across a gap connecting two rectangular sub-channels on which extensive experiments were conducted by Meyer and Rehme (1994). It was found that the DES-SST model was successful in predicting the characteristics of the flow field in the vicinity of the gap region. The span-wise velocity contours, velocity vector plots, and time traces of the velocity components showed the expected cross flow mixing between the sub-channels through the gap. The dynamics of the flow field were quantitatively described through temporal auto-correlations, spatial cross-correlations and power spectral functions. The numerical predictions were in general agreement with the experiments. Predictions from the model were used to identify different flow mixing patterns. As expected, the simulation predicted the formation of a gap vortex street which results in a quasi-periodic flow through the gap. Coherent structures were identified in the flow field to be comprised of eddies, shear zones and streams. Eddy structures with high vorticity and low pressure cores were found to exist near the vicinity of the gap edge region. A three dimensional vorticity field was identified and found to exist near the gap edge region. The instability mechanism and the probable cause behind the quasi-periodic fluid flow across the gap was identified and related to the inflectional stream-wise velocity profile.

Modeling Vortex Generating Jet-Induced Transition in Low-Pressure Turbines

The current design of low-pressure turbines (LPTs) with steady-blowing vortex generating jets (VGJs) uses steady computational fluid dynamics (CFD). The present work aims to support this design approach by proposing a new semiempirical transition model for injection-induced laminar-turbulent boundary layer transition. It is based on the detection of cross-flow vortices in the boundary layer which cause inflectional cross-flow velocity profiles. The model is implemented in the CFD code TRACE within the framework of the γ-Reθ transition model and is a reformulated, recalibrated, and extended version of a previously presented model. It is extensively validated by means of VGJ as well as non-VGJ test cases capturing the local transition process in a physically reasonable way. Quantitative aerodynamic design parameters of several VGJ configurations including steady and periodic-unsteady inflow conditions are predicted in good accordance with experimental values. Furthermore, the quantitative prediction of end-wall flows of LPTs is improved by detecting typical secondary flow structures. For the first time, the newly derived model allows the quantitative design and optimization of LPTs with VGJs.

On the longitudinal optimal perturbations to inviscid plane shear flow: formal solution and asymptotic approximation

AbstractWe study the longitudinal linear optimal perturbations (which maximize the energy gain up to a prescribed time $T$) to inviscid parallel shear flow, which present unbounded energy growth due to the lift-up mechanism. Using the phase invariance with respect to time, we show that for an arbitrary base flow profile and optimization time, the computation of the optimal longitudinal perturbation reduces to the resolution of a single one-dimensional eigenvalue problem valid for all times. The optimal perturbation and its amplification are then derived from the lowest eigenvalue and its associated eigenfunction, while the remainder of the infinite set of eigenfunctions provides an orthogonal base for decomposing the evolution of arbitrary perturbations. With this new formulation we obtain, asymptotically for large spanwise wavenumber ${k}_{z} , $ a prediction of the optimal gain and the localization of inviscid optimal perturbations for the two main classes of parallel flows: free shear flow with an inflectional velocity profile, and wall-bounded flow with maximum shear at the wall. We show that the inviscid optimal perturbations are localized around the point of maximum shear in a region with a width scaling like ${ k}_{z}^{- 1/ 2} $ for free shear flow, and like ${ k}_{z}^{- 2/ 3} $ for wall-bounded shear flows. This new derivation uses the stationarity of the base flow to transform the optimization of initial conditions in phase space into the optimization of a temporal phase along each trajectory, and an optimization among all trajectories labelled by their intersection with a codimension-1 subspace. The optimization of the time phase directly imposes that the initial and final energy growth rates of the optimal perturbation should be equal. This result requires only time invariance of the base flow, and is therefore valid for any linear optimal perturbation problem with stationary base flow.

Effect of a magnetic field on stability and transitions in liquid breeder flows in a blanket

We review previous studies on instabilities and transitions in magnetohydrodynamic flows in a special context of liquid-metal blanket applications. In the past, possible transitions in blanket flows were mostly attributed to instabilities in the Hartmann layers. More recent studies show, however, that the side layers can experience instabilities at sufficiently lower Reynolds numbers. This suggests that in the blanket flows, the appearance of turbulence can most likely be related to the side rather than Hartmann layers. Various factors that may affect stability in blanket flows have been discussed. In particular, buoyancy forces can result in potentially unstable inflectional velocity profiles. First computational results, illustrating possibility of instabilities and quasi-two-dimensional turbulence in vertical mixed-convection flows heated volumetrically are presented.

Turbulent boundary layers over permeable walls: scaling and near-wall structure

AbstractThis paper presents an experimental study devoted to investigating the effects of permeability on wall turbulence. Velocity measurements were performed by means of laser Doppler anemometry in open channel flows over walls characterized by a wide range of permeability. Previous studies proposed that the von Kármán coefficient associated with mean velocity profiles over permeable walls is significantly lower than the standard values reported for flows over smooth and rough walls. Furthermore, it was observed that turbulent flows over permeable walls do not fully respect the widely accepted paradigm of outer-layer similarity. Our data suggest that both anomalies can be explained as an effect of poor inner–outer scale separation if the depth of shear penetration within the permeable wall is considered as the representative length scale of the inner layer. We observed that with increasing permeability, the near-wall structure progressively evolves towards a more organized state until it reaches the condition of a perturbed mixing layer where the shear instability of the inflectional mean velocity profile dictates the scale of the dominant eddies. In our experiments such shear instability eddies were detected only over the wall with the highest permeability. In contrast attached eddies were present over all the other wall conditions. On the basis of these findings, we argue that the near-wall structure of turbulent flows over permeable walls is regulated by a competing mechanism between attached and shear instability eddies. We also argue that the ratio between the shear penetration depth and the boundary layer thickness quantifies the ratio between such eddy scales and, therefore, can be used as a diagnostic parameter to assess which eddy structure dominates the near-wall region for different wall permeability and flow conditions.

On the aeroacoustic tonal noise generation mechanism of a sharp-edged plate

This letter presents an experimental study on the tonal noise generated by a sharp-edged flat plate at low-to-moderate Reynolds number. Flow and far-field noise data reveal that, in this particular case, the tonal noise appears to be governed by vortex shedding processes. Also related to the existence of the tonal noise is a region of separated flow slightly upstream of the trailing edge. Hydrodynamic fluctuations at selected vortex shedding frequencies are strongly amplified by the inflectional mean velocity profile in the separated shear layer. The amplified hydrodynamic fluctuations are diffracted by the trailing edge, producing strong tonal noise.

A Rayleigh instability in a vortex-induced unsteady boundary layer

Numerical solutions of the unsteady, two-dimensional Navier–Stokes equations are considered for the flow induced by a thick-core vortex with a finite difference and spectral element code. The adverse pressure gradient imposed on the surface by the vortex leads to the development of inflectional velocity profiles and the formation of a recirculation region within the boundary layer, and the presence and nature of an instability are considered that may dominate the flow development at high Reynolds numbers. The instability ensues in the form of high-frequency oscillations in vorticity and streamwise pressure gradient along the wall, and the dominant wavenumber of the instability is O(Re1/2), consistent with a Rayleigh instability. The existence of a Rayleigh instability is further confirmed through evaluation of the Rayleigh equation for velocity profiles obtained from a boundary-layer calculation, and the dominant wavenumber of the instability in the Navier–Stokes solutions agrees very closely with that predicted by the Rayleigh solutions.