Mean Flow Stability Research Articles

Experimental studies on the frequency selection in flat plate wakes: Mean-flow stability analyses and low-dimensional modeling

The global frequency selection of two-dimensional vortex shedding in the flat plate wake is investigated experimentally. By performing a local stability analysis, and low-dimensional modeling based on the time-averaged mean flow velocity profiles, we have shown that the time-averaged velocity profiles give an accurate estimate of the global shedding frequency. Furthermore, we have analyzed the interaction strengths between the mean flow and the higher harmonics thereby experimentally supporting the theoretical criterion outlined previously by Sipp and Lebedev.

Instability of forced planar liquid jets: mean field analysis and nonlinear simulation

The stability of forced planar liquid jets in a still gaseous environment is explored using nonlinear simulation and spatial linear stability analysis. Harmonic modulation of the transverse component of the inlet velocity leads to an excitation of sinuous modes in the jet. Two forcing amplitudes, 1 % and 5 %, are investigated. While for 1 % forcing, the interfacial disturbance retains a sinuous shape throughout the domain, for 5 % forcing, an increasing downstream deviation from the sinuous wave shape is found. Both forcings lead to sufficient mean flow correction to render linear stability analysis on a base flow unfeasible. Hence, an analysis on the time-averaged mean flow is performed. A correction scheme is introduced, to account for the spreading of the interface position in the mean flow. Comparison of eigenfunctions and growth rates with their counterparts from the nonlinear simulation shows an excellent agreement for 1 % forcing. For 5 % forcing, agreement of the eigenfunctions deteriorates significantly and growth rates are falsely predicted, resulting in a breakdown of the stability model. Subsequent analysis reveals a strong interaction of the fundamental wave with the second higher harmonic wave for 5 % forcing and a reversed energy flow from the coherent motion to the mean flow. These findings provide an explanation for the failure of the linear stability model for large forcing amplitudes. The study extends the applicability of mean flow stability analysis to convectively unstable planar liquid/gas jets and supports previous findings on the limits of mean flow stability, involving pronounced influence of higher harmonic modes.

Stability properties of the mean flow after a steady symmetry-breaking bifurcation and prediction of the nonlinear saturation

In this paper, it is shown that when a flow undergoes a steady bifurcation breaking one reflection symmetry, the mean flow obtained by averaging the two possible asymmetric flow fields resulting from the instability remains marginally stable in the postcritical regime. This property is demonstrated rigorously through an asymptotic analysis which closely follows that proposed in Sipp and Lebedev (J Fluid Mech 792:620–657, 2007) for a Hopf bifurcation with focus on wakes. In the case of wakes, the marginal stability of the mean flow is well known and had several consequences documented in the literature. To the authors’ knowledge, the marginal stability of mean flows after a symmetry-breaking pitchfork bifurcation is demonstrated here for the first time. As an example of possible consequences of marginal stability, the self-consistent model proposed for wakes in Mantic-Lugo et al. (Phys Rev Lett 113:084501, 2014) and relying on marginal stability is also applied here to the symmetry-breaking instability of the flow in a channel with a sudden expansion. For this specific case, the marginal stability of the mean flow is first demonstrated by dedicated direct numerical simulations; successively, it is shown that the resulting self-consistent model predicts the nonlinear saturation of the instability with remarkable accuracy.

Conditions for validity of mean flow stability analysis

This article provides theoretical conditions for the use and meaning of a stability analysis around a mean flow. As such, it may be considered as an extension of the works by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to non-parallel flows and by Turton et al. (Phys. Rev. E, vol. 91 (4), 2015, 043009) to broadband flows. Considering a Reynolds decomposition of the flow field, the spectral (or temporal Fourier) mode of the fluctuation field is found to be equal to the action on a turbulent forcing term by the resolvent operator arising from linearisation about the mean flow. The main result of the article states that if, at a particular frequency, the dominant singular value of the resolvent is much larger than all others and if the turbulent forcing at this frequency does not display any preferential direction toward one of the suboptimal forcings, then the spectral mode is directly proportional to the dominant optimal response mode of the resolvent at this frequency. Such conditions are generally met in the case of weakly non-parallel open flows exhibiting a convectively unstable mean flow. The spatial structure of the singular mode may in these cases be approximated by a local spatial stability analysis based on parabolised stability equations (PSE). We have also shown that the frequency spectrum of the flow field at any arbitrary location of the domain may be predicted from the frequency evolution of the dominant optimal response mode and the knowledge of the frequency spectrum at one or more points. Results are illustrated in the case of a high Reynolds number turbulent backward facing step flow.

Stability Analysis of Time-averaged Jet Flows: Fundamentals and Application

We report on experimental and theoretical investigations of shear flow instabilities in jet flows. Linear stability analysis is applied to the time-averaged flow taken from experiments, contrasting the ‘classic’ stability approach that is based on a stationary base flow. To some extend, mean flow stability eigenmodes may deal as a model for instability waves at their nonlinearly saturated state, which is typically encountered in experiments. The capability of mean flow stability models is first demonstrated on laminar oscillating jets where the primary interaction takes place between the mean flow and the instability wave. We then focus on turbulent swirling jets where additional interactions occur between the fine-scale turbulence and the instability waves. Swirling flows are widely used in combustion applications where the associated high turbulence levels and internal recirculation zones (vortex breakdown bubble) are exploited for flame stabilization. We demonstrate the application of mean flow stability analysis on the flow field of a industry- relevant swirl-stabilized flame. We show that the flame response to acoutstic perturbations is closely linked to the flow receptivity predicted from linear stability analysis, which suggests that the adopted theoretical framework is very useful for thermoacoustic modeling.

Mean flow stability analysis of oscillating jet experiments

AbstractLinear stability analysis (LSA) is applied to the mean flow of an oscillating round jet with the aim of investigating the robustness and accuracy of mean flow stability wave models. The jet’s axisymmetric mode is excited at the nozzle lip through a sinusoidal modulation of the flow rate at amplitudes ranging from 0.1 % to 100 %. The instantaneous flow field is measured via particle image velocimetry (PIV) and decomposed into a mean and periodic part utilizing proper orthogonal decomposition (POD). Local LSA is applied to the measured mean flow adopting a weakly non-parallel flow approach. The resulting global perturbation field is carefully compared with the measurements in terms of spatial growth rate, phase velocity, and phase and amplitude distribution. It is shown that the stability wave model accurately predicts the excited flow oscillations during their entire growth phase and during a large part of their decay phase. The stability wave model applies over a wide range of forcing amplitudes, showing no pronounced sensitivity to the strength of nonlinear saturation. The upstream displacement of the neutral point and the successive reduction of gain with increasing forcing amplitude is very well captured by the stability wave model. At very strong forcing ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{>}40\, \%$), the flow becomes essentially stable to the axisymmetric mode. For these extreme cases, the prediction deteriorates from the measurements due to an interaction of the forced wave with the geometric confinement of the nozzle. Moreover, the model fails far downstream in a region where energy is transferred from the oscillation back to the mean flow. This study supports previously conducted mean flow stability analysis of self-excited flow oscillations in the cylinder wake and in the vortex breakdown bubble and extends the methodology to externally forced convectively unstable flows. The high accuracy of mean flow stability wave models as demonstrated here is of great importance for the analysis of coherent structures in turbulent shear flows.

Shear flow instabilities in swirl-stabilized combustors and their impact on the amplitude dependent flame response: A linear stability analysis

The hydrodynamic instabilities in the flow field of a swirl-stabilized combustor are investigated theoretically. These instabilities give rise to large-scale flow structures that interact with the flame front causing unsteady heat release rate fluctuations. The streamwise growth of these coherent structures depends on the receptivity of the shear layer, which can be predicted numerically by means of linear stability analysis. This analysis is applied to the reacting flow field of a perfectly premixed swirl-stabilized combustor that is subjected to strong axial forcing mimicking thermoacoustic oscillations. The linear stability analysis reveals a clear correlation between the shear layer receptivity and the measured amplitude dependent flame transfer function. The stability analysis based on the natural flow predicts the distinctive frequency dependent flame response to low amplitude forcing. At these conditions, the flow reveals strong spatial amplification near the nozzle, causing the inlet perturbations to be significantly amplified before they reach the flame. At higher forcing amplitudes, the flow instabilities saturate, which manifests in a saturation of the flame response. The saturation of the shear layers predicted from the linear stability analysis is compared to phase-locked measurements of the forced flow field revealing good qualitative agreement. The analysis of the mean flow stability offers a powerful analytical tool to investigate the impact of shear flow instabilities on the flame describing function.

Modeling the Relations Between Flow Regime Components, Species Traits, and Spawning Success of Fishes in Warmwater Streams

Modifications to stream hydrologic regimes can have a profound influence on the dynamics of their fish populations. Using hierarchical linear models, we examined the relations between flow regime and young-of-year fish density using fish sampling and discharge data from three different warmwater streams in Illinois, Alabama, and Georgia. We used an information theoretic approach to evaluate the relative support for models describing hypothesized influences of five flow regime components representing: short-term high and low flows; short-term flow stability; and long-term mean flows and flow stability on fish reproductive success during fish spawning and rearing periods. We also evaluated the influence of ten fish species traits on fish reproductive success. Species traits included spawning duration, reproductive strategy, egg incubation rate, swimming locomotion morphology, general habitat preference, and food habits. Model selection results indicated that young-of-year fish density was positively related to short-term high flows during the spawning period and negatively related to flow variability during the rearing period. However, the effect of the flow regime components varied substantially among species, but was related to species traits. The effect of short-term high flows on the reproductive success was lower for species that broadcast their eggs during spawning. Species with cruiser swimming locomotion morphologies (e.g., Micropterus) also were more vulnerable to variable flows during the rearing period. Our models provide insight into the conditions and timing of flows that influence the reproductive success of warmwater stream fishes and may guide decisions related to stream regulation and management.

Combined effects of EDL and boundary slip on mean flow and its stability in microchannels

The effects of an electrical double layer (EDL), boundary slip and their combined effects on the microchannel flow stability are investigated. Instead of applying the traditional streaming electrical current balance (ECB) mode, a newly introduced electrical current density balance (ECDB) mode is used to compute the conduction current when the effects of EDL is considered. The modified N-S equations and Poisson–Boltzmann equation together with the ECDB mode and Navier slip boundary are the theoretical basis for the present approach. The stability analysis considering the modifications of EDL and boundary slip is built up by the small perturbation method. It is found that the effect of EDL results in inflexions on the mean velocity profile near walls and destabilizes the stability of flow. On the contrary, the effect of boundary slip stabilizes the stability of flow. The effectiveness of boundary slip on the mean velocity and flow stability is influenced strongly by the effect of EDL. The effect of boundary slip can be disappeared when the Zeta potential is large enough.

Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows

This article deals with the first Hopf bifurcation of a cylinder flow, and more particularly with the properties of the unsteady periodic Kármán vortex street regime that sets in for supercritical Reynolds numbers Re > 46. Barkley (Europhys. Lett. vol.75, 2006, p. 750) has recently studied the linear properties of the associated mean flow, i.e. the flow which is obtained by a time average of this unsteady periodic flow. He observed, thanks to a global mode analysis, that the mean flow is marginally stable and that the eigenfrequencies associated with the global modes of the mean flow fit the Strouhal to Reynolds experimental function well in the range 46 < Re < 180. The aim of this article is to give a theoretical proof of this result near the bifurcation. For this, we do a global weakly nonlinear analysis valid in the vicinity of the critical Reynolds number Rec based on the small parameter ε = Rec−1 − Re−1 ≪ 1. We compute numerically the complex constants λ and μ′ which appear in the Stuart-Landau amplitude equation: dA/dt = ε λA − εμ′ A|A|2. Here A is the scalar complex amplitude of the critical global mode. By analysing carefully the nonlinear interactions yielding the term μ′, we show for the cylinder flow that the mean flow is approximately marginally stable and that the linear dynamics of the mean flow yields the frequency of the saturated Stuart-Landau limit cycle. We will finally show that these results are not general, by studying the case of the bifurcation of an open cavity flow. In particular, we show that the mean flow in this case remains strongly unstable and that the frequencies associated with the eigenmodes do not exactly match those of the nonlinear unsteady periodic cavity flow. It will be demonstrated that two precise conditions must hold for a linear stability analysis of a mean flow to be relevant and useful.

A Total Turbulent Energy Closure Model for Neutrally and Stably Stratified Atmospheric Boundary Layers

Abstract This paper presents a turbulence closure for neutral and stratified atmospheric conditions. The closure is based on the concept of the total turbulent energy. The total turbulent energy is the sum of the turbulent kinetic energy and turbulent potential energy, which is proportional to the potential temperature variance. The closure uses recent observational findings to take into account the mean flow stability. These observations indicate that turbulent transfer of heat and momentum behaves differently under very stable stratification. Whereas the turbulent heat flux tends toward zero beyond a certain stability limit, the turbulent stress stays finite. The suggested scheme avoids the problem of self-correlation. The latter is an improvement over the widely used Monin–Obukhov-based closures. Numerous large-eddy simulations, including a wide range of neutral and stably stratified cases, are used to estimate likely values of two free constants. In a benchmark case the new turbulence closure performs indistinguishably from independent large-eddy simulations.

Convective and Absolute Instability in the Incompressible Boundary Layer on a Rotating Disk in the Presence of a Uniform Magnetic Field

The stability of a conducting fluid flow over a rotating disk with a uniform magnetic field applied normal to the disk, is investigated. It is assumed that the magnetic field is unaffected by the motion of the fluid. The mean flow and linear stability equations are solved for a range of magnetic field-strength parameters and the absolute/convective nature of the stability is investigated. It is found that increasing the magnetic field parameter is in general stabilizing.

Computational Evaluation of Quiet Tunnel Hypersonic Boundary-Layer Stability Experiments

Abstract : A computational evaluation of two stability experiments conducted in the NASA Langley Mach 6 axisymmetric quiet nozzle test chamber facility is conducted. Navier- Stokes analysis of the mean flow and linear stability theory analysis of boundary layer disturbances is performed in the computations. The effects of adverse pressure gradient and wall cooling are examined. Calculated pressure, temperature and boundary layer thickness distributions show very good overall agreement with experimental measurements. Computed mass flux and total temperature profiles show very good quantitative agreement with uncalibrated hot- wire measurements obtained with the hot-wire operated in high and low overheat modes respectively. Comparisons between calibrated hot-wire data and mean flow computations show excellent agreement in the early stages of the transitional flow. However, examination of the wire Reynolds number and mass flux and total temperature eigenfunction profiles suggest that when operated in high overheat mode the sensitivity of the hot-wire to total temperature is significant. Thus, while uncalibrated hot-wire measurements are useful to characterize the overall features of the flow, calibrated hot- wire measurements are necessary for quantitative comparison with stability theory. Computations show that adverse pressure gradient and wall cooling decrease the boundary layer thickness and increase the frequency and amplification rate of the unstable second mode disturbances; these findings are consistent with the experimental observations.

Real-Gas Effects on Mean Flow and Temporal Stability of Binary-Species Mixing Layers

Real-gas effects on the mean flow and inviscid stability of temporal mixing layers are examined for supercritical heptane/nitrogen and oxygen/hydrogen mixtures. The analysis is based on the compressible Navier–Stokes equations for conservation of mass, momentum, total energy, and species mass, with heat and species-mass fluxes derived from fluctuation-dissipation theory and incorporating Soret and Dufour effects. An approximate form of the equations is used to obtain a system of similarity equations for the streamwise velocity, the temperature, and the mass fraction. The similarity profiles show important real-gas nonideal-mixture effects, particularly for the temperature, in departing from the incompressible error-function similarity solution. Realistic Schmidt and Prandtl numbers were found to be important to the similarity profiles. A linear, inviscid stability analysis is then performed using the similarity profile, as well as analytical error-function profiles, as its basic flow. The stability analysis shows that the similarity profile has larger growth rates at a given wavelength and a shorter more unstable wavelength than the error-function profiles and than an incompressible flow. The similarity profile also has a larger range of unstable wavelengths than the error-function profiles.

Real gas effects on mean flow and temporal stability of binary-species mixing layers

Real gas effects on mean flow and temporal stability of binary-species mixing layers

DPIV-driven flow simulation: a new computational paradigm

We present a new approach to simulating unsteady fluid flows, with only very few degrees of freedom, by employing directly eigenmodes extracted from digital particle image velocimetry experimental data. In particular, we formulate standard Galerkin and nonlinear Galerkin approximations of the incompressible Navier–Stokes equations using hierarchical empirical eigenfunctions extracted from an ensemble of flow snapshots. We demonstrate that standard Galerkin approaches produce simulations capable of capturing the short–term dynamics of the flow, but nonlinear Galerkin projections are more effective in capturing both the short– and long–term dynamics, leading to bounded solutions. These findings are documented by applying these approaches to flow past a stationary circular cylinder at Reynolds number 610.

On the neutral stability of spanwise-periodic boundary-layer and triple-deck flows

SUMMARY The neutral stability of spanwise-periodic mean flows to three-dimensional TollmienSchlichting (TS) disturbances is considered theoretically and numerically. The flows under consideration are (i) a triple-deck flow with a simple pressure-displacement law and (ii) a boundary layer flow on the point of separation. In both cases the stability problem is formulated and a necessary condition for the existence of neutral solutions is derived. Using a numerical approach we show that this condition is not met in case (i) for any periodic spanwise dependence and in case (ii) provided spanwise variations in the mean flow are sufficiently small. This means that neutral waves are not rendered possible by a three-dimensional skin friction in case (i) and are unlikely to exist in case (ii) for physically realistic mean flows. The implications of these results with regard to the possibility of the occurrence of vortex-TS interactions in such flows is discussed.

Mean flow stability in a model of the eastern North Pacific Ocean

Datos de estratificación y corriente media de un modelo cuasigeostrófico de ocho capas de la Corriente de California son usados para resolver el problema de estabilidad lineal asociado. Se obtiene, al resolver un problema de autovalores complejos la frecuencia, rapidez de fase, velocidad de grupo, razón de crecimiento y estructura vertical. Se corren siete experimentos para estudiar la influencia de las diferentes condiciones de frontera (fondo plano o inclinado), dirección media de la corriente, amplitud, latitud y mecanismos de fricción en las características de las ondas de Rossby. Usando valores de la frecuencia de flotabilidad características del Pacífico Nororiental encontramos que la resolución vertical es crucial para determinar los efectos de la topografía y fricción de fondo en la estabilidad del flujo básico. La estructura vertical del flujo medio tiene un efecto importante en la escala de decaimiento exponencial. Esta escala de decaimiento de las diferentes áreas de la región de la Corriente de California varía entre 144 y 374 días para las ondas más inestables. La estructura vertical de nuestra solución de onda (amplitud y fase) es afectada perceptiblemente por la disipación usada en el modelo. Las características del primer modo baroclínico de la onda estable son cualitativamente y cuantitativamente similares a las obtenidas por Kang et al. (1982) usando datos hidrográficos de la Corriente de California. La inclusión de topografía de fondo lleva a una moderada redistribución de frecuencias en el espacio de número de onda y a más altas velocidades de grupo.

Nonlinear evolution of a pair of oblique instability waves in a supersonic boundary layer

We study the nonlinear evolution of a pair of oblique instability waves in a supersonic boundary layer over a flat plate in the nonlinear non-equilibrium viscous critical layer regime. The instability wave amplitude is governed by the same integro-differential equation as that derived by Goldstein & Choi (1989) in the inviscid limit and by Wu, Lee & Cowley (1993) with viscous effects included, but the coefficient appearing in this equation depends on the mean flow and linear neutral stability solution of the supersonic boundary layer. This coefficient is evaluated numerically for the Mach number range over which the (inviscid) first mode is the dominant instability. Numerical solutions to the amplitude equation using these values of the coefficient are obtained. It is found that, for insulated and cooled wall conditions and angles corresponding to the most rapidly growing waves, the amplitude ends in a singularity at a finite downstream position over the entire Mach number range regardless of the size of the viscous parameter. The explosive growth of the instability waves provides a mechanism by which the boundary layer can break down. A new feature of the compressible problem is the nonlinear generation of a spanwise-dependent mean distortion of the temperature along with that of the velocity found in the incompressible case.

On Symmetric Stability and Instability of Zonal Mean Flows Near the Equator

Abstract Obserations of longitudinally-averaged zonal flows in the atmosphere and ocean tend to display north–south symmetry about the equator, with a characteristic wind maximum or minimum, and therefore little horizontal wind shear locally near the equator. It is shown that this configuration is required for balanced flow on a sphere to be inertially stable. If dissipation can be neglected, any horizontal wind shear at the equator will cause inertial instability to develop. effectively eliminating the horizontal shear. It follows that the potential vorticity (q) must vanish at the equator for the symmetric circulation to be stable; and that it must increase to the north and decrease to the south. Balanced cross-equatorial flow can occur only if there is a north–south gradient in the torque or the diabatic heating at the equator. These conclusions are obtained under the assumption of a balanced zonal flow; i.e., acceleration and dissipation are explicitly neglected in the meridional momentum equation. Th...