Harmonics generation and the mechanics of saturation in flow over an open cavity: a second-order self-consistent description

2017 ◽  
Vol 826 ◽  
pp. 503-521 ◽  
Author(s):  
P. Meliga
Keyword(s):  
Growth Rate ◽  
Second Harmonic ◽  
Open Cavity ◽  
Linear Growth Rate ◽  
Linear Component ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Higher Harmonics ◽  
Self Consistent ◽  
The Mean

The flow over an open cavity is an example of supercritical Hopf bifurcation leading to periodic limit-cycle oscillations. One of its distinctive features is the existence of strong higher harmonics, which results in the time-averaged mean flow being strongly linearly unstable. For this class of flows, a simplified formalism capable of unravelling how exactly the instability grows and saturates is lacking. This study builds on previous work by Mantič-Lugo et al. (Phys. Rev. Lett., vol. 113, 2014, 084501) to fill in the gap using a parametrized approximation of an instantaneous, phase-averaged mean flow, coupled in a quasi-static manner to multiple linear harmonic disturbances interacting nonlinearly with one another and feeding back on the mean flow via their Reynolds stresses. This provides a self-consistent modelling of the mean flow–fluctuation interaction, in the sense that all perturbation structures are those whose Reynolds stresses force the mean flow in such a way that the mean flow generates exactly the aforementioned perturbations. The first harmonic is sought as the superposition of two components, a linear component generated by the instability and aligned along the leading eigenmode of the mean flow, and a nonlinear orthogonal component generated by the higher harmonics, which progressively distorts the linear growth rate and eigenfrequency of the eigenmode. Saturation occurs when the growth rate of the first harmonic is zero, at which point the stabilizing effect of the second harmonic balances exactly the linear instability of the eigenmode. The model does not require any input from numerical or experimental data, and accurately predicts the transient development and the saturation of the instability, as established from comparison to time and phase averages of direct numerical simulation data.

scholarly journals Self-consistent triple decomposition of the turbulent flow over a backward-facing step under finite amplitude harmonic forcing

2019 ◽  
Vol 475 (2225) ◽  
pp. 20190018
Author(s):  
E. Yim ◽  
P. Meliga ◽  
F. Gallaire
Keyword(s):  
Turbulent Flow ◽  
Finite Amplitude ◽  
Navier Stokes ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Backward Facing Step ◽  
Eddy Viscosity Model ◽  
Coherent Interaction ◽  
Self Consistent ◽  
The Mean

We investigate the saturation of harmonically forced disturbances in the turbulent flow over a backward-facing step subjected to a finite amplitude forcing. The analysis relies on a triple decomposition of the unsteady flow into mean, coherent and incoherent components. The coherent–incoherent interaction is lumped into a Reynolds averaged Navier–Stokes (RANS) eddy viscosity model, and the mean–coherent interaction is analysed via a semi-linear resolvent analysis building on the laminar approach by Mantič-Lugo & Gallaire (2016 J. Fluid Mech. 793 , 777–797. ( doi:10.1017/jfm.2016.109 )). This provides a self-consistent modelling of the interaction between all three components, in the sense that the coherent perturbation structures selected by the resolvent analysis are those whose Reynolds stresses force the mean flow in such a way that the mean flow generates exactly the aforementioned perturbations, while also accounting for the effect of the incoherent scale. The model does not require any input from numerical or experimental data, and accurately predicts the saturation of the forced coherent disturbances, as established from comparison to time-averages of unsteady RANS simulation data.

Turbulent Flow Around Rectangular Cylinders with Different Streamwise Aspect Ratios

2021 ◽  
Author(s):  
Sedem Kumahor ◽  
Mark F. Tachie
Keyword(s):  
Aspect Ratio ◽  
High Speed ◽  
Second Harmonic ◽  
Cylinder Surface ◽  
Large Aspect Ratio ◽  
Reynolds Stresses ◽  
Square Cylinder ◽  
Mean Flow ◽  
Aspect Ratios ◽  
The Mean

Abstract Turbulent flows around a square cylinder and a rectangular cylinder with a streamwise aspect ratio of 5 in a uniform flow were investigated using time-resolved particle image velocimetry. The Reynolds number based on the cylinder height and oncoming flow velocity was 16200. Similarities and differences in the flow dynamics over the cylinders and in the near wake region were examined in terms of the mean flow, Reynolds stresses and triple velocity correlations. The budget of turbulent kinetic energy as well as temporal and spectral analyses were also performed. The results show that the primary, secondary and wake vortexes are smaller for the square cylinder compared to the large aspect ratio cylinder. There are regions of elevated Reynolds stresses and triple velocity correlations along the mean separating streamlines, and the magnitudes of these statistics are an order of magnitude higher over the square cylinder compared to the large aspect ratio cylinder. The topology of the triple velocity correlations shows low-speed ejection and high-speed sweep events, respectively, transporting instantaneous Reynolds normal stresses away from the mean separating streamline into the free-stream and toward the cylinder surface, regardless of aspect ratio. Near the leading and trailing edges of both cylinders, regions of negative turbulence production are observed and the dominant components contributing to this occurrence are discussed. Temporal autocorrelation coefficients of the streamwise and vertical velocity fluctuations show a periodic trend, with a periodicity that is directly linked to the Kármán shedding frequency and its second harmonic.

Note on weakly nonlinear stability theory of a free mixing layer

1987 ◽  
Vol 409 (1837) ◽  
pp. 351-367 ◽  
Keyword(s):  
Growth Rate ◽  
Nonlinear Evolution Equation ◽  
Linear Growth ◽  
Nonlinear Evolution ◽  
Second Harmonic ◽  
Mixing Layer ◽  
Linear Growth Rate ◽  
Mean Flow ◽  
Initial Flow ◽  
Weakly Nonlinear

This paper is concerned with an analysis of the derivation of a nonlinear evolution equation governing the dynamics of a weakly unstable pertur­bation in a mixing layer. It is demonstrated that the Landau constant has no universal character and is determined by the degree of supercriticality of perturbation, whose measure is represented by the linear growth rate of instability γ (which is proportional to the departure from neutral wavenumber). It is shown that the interaction of the fundamental harmonic of the perturbation with the associated distortion of a mean flow is the dominant nonlinear effect. It is this that makes the main contribution to the Landau constant rather than the interaction with the second harmonic as was thought previously. We examine the régimes of both a viscous and a non-stationary critical layer and calculate the Landau constant. It is shown that, except for the case when the supercriticality is very small, γ « v ( v is the inverse of the Reynolds number), the nonlinearity cannot substantially influence the exponential growth of the perturbation predicted by linear theory. When γ « v , however, the nonlinear time exceeds the time of viscous spreading of the initial flow and for a correct formulation of the problem, one has, as was first pointed out by Huerre, to introduce here an artificial force field.

On the stability of an inviscid shear layer which is periodic in space and time

1967 ◽  
Vol 27 (4) ◽  
pp. 657-689 ◽  
Author(s):  
R. E. Kelly
Keyword(s):  
Growth Rate ◽  
Shear Layer ◽  
Finite Amplitude ◽  
Periodic Component ◽  
Periodic Flow ◽  
Flow Profile ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean ◽  
The Stability

In experiments concerning the instability of free shear layers, oscillations have been observed in the downstream flow which have a frequency exactly half that of the dominant oscillation closer to the origin of the layer. The present analysis indicates that the phenomenon is due to a secondary instability associated with the nearly periodic flow which arises from the finite-amplitude growth of the fundamental disturbance.At first, however, the stability of inviscid shear flows, consisting of a non-zero mean component, together with a component periodic in the direction of flow and with time, is investigated fairly generally. It is found that the periodic component can serve as a means by which waves with twice the wavelength of the periodic component can be reinforced. The dependence of the growth rate of the subharmonic wave upon the amplitude of the periodic component is found for the case when the mean flow profile is of the hyperbolic-tangent type. In order that the subharmonic growth rate may exceed that of the most unstable disturbance associated with the mean flow, the amplitude of the streamwise component of the periodic flow is required to be about 12 % of the mean velocity difference across the shear layer. This represents order-of-magnitude agreement with experiment.Other possibilities of interaction between disturbances and the periodic flow are discussed, and the concluding section contains a discussion of the interactions on the basis of the energy equation.

scholarly journals Effects of Rear Angle on the Turbulent Wake Flow between Two in-Line Ahmed Bodies

Atmosphere ◽  
2020 ◽  
Vol 11 (4) ◽  
pp. 328
Author(s):  
Ebenezer Essel ◽  
Subhadip Das ◽  
Ram Balachandar
Keyword(s):  
Wake Flow ◽  
Recirculation Region ◽  
Wake Region ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Slant Angle ◽  
The Road ◽  
Rear Angle ◽  
Turbulent Statistics ◽  
The Mean

Understanding the wake characteristics between two in-line vehicles is essential for improving and developing new strategies for reducing in-cabin air pollution. In this study, Ahmed bodies are used to investigate the effects of the rear slant angle of a leading vehicle on the mean flow and turbulent statistics between two vehicles. The experiments were conducted with a particle image velocimetry at a fixed Reynolds number, R e H = 1.7 × 10 4 , and inter-vehicle spacing distance of 0.75 L , where H and L are the height and length of the model. The rear slant angles investigated were a reference square back, high-drag angle ( α = 25 ° ) and low-drag angle ( α = 35 ° ). The mean velocities, Reynolds stresses, production of turbulent kinetic energy and instantaneous swirling strength are used to provide physical insight into the wake dynamics between the two bodies. The results indicate that the recirculation region behind the square back Ahmed body increases while those behind the slant rear-end bodies decreases in the presence of a follower. For the square back models, the dominant motion in the wake region is a strong upwash of jet-like flow away from the road but increasing the rear slant angle induces a stronger downwash flow that suppresses the upwash and dominates the wake region.

scholarly journals Moisture Modes and the Eastward Propagation of the MJO

2013 ◽  
Vol 70 (1) ◽  
pp. 187-192 ◽  
Author(s):  
Adam Sobel ◽  
Eric Maloney
Keyword(s):  
Growth Rate ◽  
Moisture Gradient ◽  
Moist Static Energy ◽  
Mean Flow ◽  
Considerable Uncertainty ◽  
Low Level ◽  
Zonal Advection ◽  
The Mean ◽  
Frictional Convergence ◽  
Static Energy

Abstract The authors discuss modifications to a simple linear model of intraseasonal moisture modes. Wind–evaporation feedbacks were shown in an earlier study to induce westward propagation in an eastward mean low-level flow in this model. Here additional processes, which provide effective sources of moist static energy to the disturbances and which also depend on the low-level wind, are considered. Several processes can act as positive sources in perturbation easterlies: zonal advection (if the mean zonal moisture gradient is eastward), modulation of synoptic eddy drying by the MJO-scale wind perturbations, and frictional convergence. If the sum of these is stronger than the wind–evaporation feedback—as observations suggest may be the case, though with considerable uncertainty—the model produces unstable modes that propagate weakly eastward relative to the mean flow. With a small amount of horizontal diffusion or other scale-selective damping, the growth rate is greatest at the largest horizontal scales and decreases monotonically with wavenumber.

scholarly journals A Framework for Parameterizing Eddy Potential Vorticity Fluxes

2012 ◽  
Vol 42 (4) ◽  
pp. 539-557 ◽  
Author(s):  
David P. Marshall ◽  
James R. Maddison ◽  
Pavel S. Berloff
Keyword(s):  
Stress Tensor ◽  
Potential Vorticity ◽  
Large Scale ◽  
Eddy Diffusivity ◽  
Linear Instability ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Conservation Of Energy ◽  
The Mean ◽  
Energy And Momentum

Abstract A framework for parameterizing eddy potential vorticity fluxes is developed that is consistent with conservation of energy and momentum while retaining the symmetries of the original eddy flux. The framework involves rewriting the residual-mean eddy force, or equivalently the eddy potential vorticity flux, as the divergence of an eddy stress tensor. A norm of this tensor is bounded by the eddy energy, allowing the components of the stress tensor to be rewritten in terms of the eddy energy and nondimensional parameters describing the mean shape and orientation of the eddies. If a prognostic equation is solved for the eddy energy, the remaining unknowns are nondimensional and bounded in magnitude by unity. Moreover, these nondimensional geometric parameters have strong connections with classical stability theory. When applied to the Eady problem, it is shown that the new framework preserves the functional form of the Eady growth rate for linear instability. Moreover, in the limit in which Reynolds stresses are neglected, the framework reduces to a Gent and McWilliams type of eddy closure where the eddy diffusivity can be interpreted as the form proposed by Visbeck et al. Simulations of three-layer wind-driven gyres are used to diagnose the eddy shape and orientations in fully developed geostrophic turbulence. These fields are found to have large-scale structure that appears related to the structure of the mean flow. The eddy energy sets the magnitude of the eddy stress tensor and hence the eddy potential vorticity fluxes. Possible extensions of the framework to ensure potential vorticity is mixed on average are discussed.

scholarly journals A Geometric Interpretation of Eddy Reynolds Stresses in Barotropic Ocean Jets

2016 ◽  
Vol 46 (8) ◽  
pp. 2285-2307 ◽  
Author(s):  
Talia Tamarin ◽  
James R. Maddison ◽  
Eyal Heifetz ◽  
David P. Marshall
Keyword(s):  
Stress Tensor ◽  
Ray Tracing ◽  
Analytic Solutions ◽  
Geometric Interpretation ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Nonlinear Simulation ◽  
Fully Nonlinear ◽  
Geometric Decomposition ◽  
The Mean

AbstractBarotropic eddy fluxes are analyzed through a geometric decomposition of the eddy stress tensor. Specifically, the geometry of the eddy variance ellipse, a two-dimensional visualization of the stress tensor describing the mean eddy shape and tilt, is used to elucidate eddy propagation and eddy feedback on the mean flow. Linear shear and jet profiles are analyzed and theoretical results are compared against fully nonlinear simulations. For flows with zero planetary vorticity gradient, analytic solutions for the eddy ellipse tilt and anisotropy are obtained that provide a direct relationship between the eddy tilt and the phase difference of a normal-mode solution. This allows a straightforward interpretation of the eddy–mean flow interaction in terms of classical stability theory: the initially unstable jet gives rise to eddies that are tilted “against the shear” and extract energy from the mean flow; once the jet stabilizes, eddies become tilted “with the shear” and return their energy to the mean flow. For a nonzero planetary vorticity gradient, ray-tracing theory is used to predict ellipse geometry and its impact on eddy propagation within a jet. An analytic solution for the eddy tilt is found for a Rossby wave on a constant background shear. The ray-tracing results broadly agree with the eddy tilt diagnosed from a fully nonlinear simulation.

3D Reynolds Stress Modelling of Turbulent Flow in Linear Turbine Cascade

1997 ◽  
Author(s):  
M. Kanniche ◽  
R. Boudjemadi ◽  
F. Déjean ◽  
F. Archambeau
Keyword(s):  
Reynolds Stress ◽  
Eddy Viscosity ◽  
Strain Tensor ◽  
Secondary Flows ◽  
Turbulence Closure ◽  
Turbine Cascade ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Principal Axes ◽  
The Mean

The flow in a linear turbine cascade (Gregory-Smith et al. (1990)) is numerically investigated using a Reynolds Stress Turbulence closure. A particular attention is given to secondary flows where the normal Reynolds stresses are expected to play an important role. The most classical turbulence closure, the k-epsilon model uses the Boussinesq Eddy Viscosity concept which assumes an isotropic turbulent viscosity. The Reynolds stresses are then related to local velocity gradients by this isotropic eddy viscosity. Corollary, the principal axes of the Reynolds stress tensor are colinear with those of the mean strain tensor. The advantage of Reynolds Stress Turbulence closure is the calculation of Reynolds stresses by their own individual transport equations. This leads to a more realistic description of the turbulence and of its dependance on the mean flow. The most classical Second Order turbulence model (Launder et al. (1975)) is applied to a linear turbine cascade, and the results are compared to secondary velocity and turbulence measurements at cross-passage planes.

On the mean structure of sharp-fin-induced shock wave/turbulent boundary layer interactions over a cylindrical surface

2019 ◽  
Vol 865 ◽  
pp. 212-246 ◽  
Author(s):  
J. D. Pickles ◽  
B. R. Mettu ◽  
P. K. Subbareddy ◽  
V. Narayanaswamy
Keyword(s):  
Boundary Layer ◽  
Shock Wave ◽  
Growth Rate ◽  
Cylindrical Surface ◽  
Separated Flow ◽  
Oblique Shock Wave ◽  
Mean Flow ◽  
Downstream Distance ◽  
Separation Shock ◽  
The Mean

Interactions between an oblique shock wave generated by a sharp fin placed on a cylindrical surface and the incoming boundary layer are investigated to unravel the mean features of the resulting shock/boundary layer interaction (SBLI) unit. This fin-on-cylinder SBLI unit has several unique features caused by the three-dimensional (3-D) relief offered by the cylindrical surface that noticeably alter the shock structure. Complementary experimental and computational studies are made to delineate both the surface and off-body flow features of the fin-on-cylinder SBLI unit and to obtain a detailed understanding of the mechanisms that dictate the mean flow and wall pressure features of the SBLI unit. Results show that the fin-on-cylinder SBLI exhibits substantial deviation from quasi-conical symmetry that is observed in planar fin SBLI. Furthermore, the separated flow growth rate appears to decrease with downstream distance and the separation size is consistently smaller than the planar fin SBLI with the same inflow and fin configurations. The causes for the observed diminution of the separated flow and its downstream growth rate were investigated in the light of changes caused by the cylinder curvature on the inviscid as well as separation shock. It was found that the inviscid shock gets progressively weakened in the region close to the triple point with downstream distance due to the 3-D relief effect from cylinder curvature. This weakening of the inviscid shock feeds into the separation shock, which is also independently impacted by the 3-D relief, to result in the observed modifications in the fin-on-cylinder SBLI unit.

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