On the interactions between large-scale structure and fine-grained turbulence in a free shear flow I. The development of temporal interactions in the mean

1976 ◽  
Vol 352 (1669) ◽  
pp. 213-247 ◽  
Keyword(s):  
Energy Transfer ◽  
Kinetic Energy ◽  
Shear Layer ◽  
Turbulent Kinetic Energy ◽  
Viscous Dissipation ◽  
Mean Flow ◽  
Fine Grained ◽  
Large Eddy ◽  
The Mean ◽  
Three Components

In this problem a mean turbulent shear layer originally exists, homogeneous in the streamwise direction, formed perhaps by previous instabilities, but in equilibrium with the fine-grained turbulence. At a given time, a large eddy of a fixed horizontal wavenumber is initiated. We study the subsequent time development of the non-equilibrium interactions between the three components of flow as they adjust towards ultimate simultaneous equilibrium, using the integrated energy-balance conservation equations to derive the amplitude equations. This necessarily involves the usual averaging procedure and a conditional or phase-averaging procedure by which the large structure motion is educed from the total fluctuations. In general, the mean flow growth is due to the energy transfer to both fluctuating components, the large eddy gains energy from the mean motion and exchanges energy with the fine-grained turbulence, while the fine-grained turbulence gains energy from the mean flow and exchanges with the large eddy and converts its energy to heat through viscous dissipation of the smallest scales. The closure problem is obtained via the shape assumptions which enter into the interaction integrals. The situation in which the fine-grained turbulent kinetic energy production and viscous dissipation are in local balance is considered, the displacement from equilibrium being due only to the energy transfer from the large eddy. The large eddy shape is taken to be two-dimensional, instability-wavelike, with its vorticity axis perpendicular to the direction of the mean outer stream. Prior to averaging, detailed but approximate calculations of the wave-induced turbulent Reynolds stresses are obtained; the product of these stresses with the appropriate large-eddy rates of strain give the energy transfer mechanism between the two disparate scales of fluctuations. Coupled, nonlinear amplitude or energy density equations for the three components of motion are obtained, the coefficients of which are the interaction integrals guided by the shape assumptions. It is found that for the special case of parallel flow, the energy of the large eddy first undergoes a hydrodynamic-instability type of amplification but eventually decays due to the energy transfer to the fine-grained turbulence, while the turbulent kinetic energy is displaced from an original level of equilibrium to a new one because of the ability of the large eddy to negotiate an indirect energy transfer from the mean flow. For the growing shear layer, approximate considerations show that if the mechanism of energy transfer from the large to the small scale is eventually weakened by the shear layer growth compared to the large-eddy production mechanism so that the amplification and decay process repeats, ‘bursts’ of the remnant of the same large eddy will occur repeatedly until an ultimate equilibrium is reached among the three interacting components of motion. However, for the large eddy whose wavenumber corresponds to that of the initially most amplified case, the ‘bursting’ phenomenon is much less pronounced and equilibrium is very nearly reached at the end of the very first ‘burst’.

Effect of Impeller Speed Perturbation in a Rushton Impeller Stirred Tank

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
Somnath Roy ◽  
Sumanta Acharya
Keyword(s):  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Rotational Speed ◽  
Stirred Tank ◽  
Azimuthal Direction ◽  
Mean Flow ◽  
Eddy Simulation ◽  
Large Eddy ◽  
The Mean ◽  
Rushton Impeller

Flow inside an unbaffled Rushton-impeller stirred tank reactor (STR) is perturbed using a time dependent impeller rotational speed. Large eddy simulation (LES) revealed that the perturbation increased the width of impeller jet compared to the constant rotational speed cases. The turbulent fluctuations were also observed to be enhanced in the perturbed flow and showed higher values of production and convection of turbulent kinetic energy. Changes in the mean flow-field during the perturbation cycle are investigated. The trailing edge vortices were observed to propagate farther both in the radial and azimuthal direction in the perturbed case. Production of turbulent kinetic energy is observed to be related to the breakup of the impeller jet in the perturbed case. Dissipation of turbulent kinetic energy is augmented due to the perturbation ensuring a better mixing at the molecular scale.

Interactions Between the Shear Layer Emanating From Rectangular Cylinders and the Near Wake Region

2021 ◽  
Author(s):  
Sedem Kumahor ◽  
Samuel Addai ◽  
Mark F. Tachie
Keyword(s):  
Kinetic Energy ◽  
Shear Layer ◽  
Turbulent Kinetic Energy ◽  
Leading Edge ◽  
Wake Region ◽  
Mean Flow ◽  
Near Wake ◽  
Aspect Ratios ◽  
The Mean ◽  
Rectangular Cylinders

Abstract The interactions between the separated shear layer and the near wake region of rectangular cylinders of varying streamwise extents in a uniform flow are investigated using time resolved particle image velocimetry. The streamwise aspect ratios (AR) tested were 1 and 5, and the Reynolds number based on the oncoming flow velocity and cylinder height is 16200. The effects of varying AR on the mean flow, turbulent kinetic energy and Reynolds shear stresses are studied. Furthermore, the unsteady characteristics of the separation bubbles are examined in terms of frequency spectra analysis. The mean flow topology shows flow separation at the leading edge is not affected by the streamwise aspect ratios. However, the primary, secondary and wake vortexes show significant differences. Mean flow reattaches over the cylinder at 4.30 cylinder heights in the AR5 case while there is no mean reattachment in the AR1 case. The magnitudes of turbulent kinetic energy and Reynolds shear stress in the wake region are an order of magnitude higher in AR1 compared to AR5. Depending on the streamwise location, the vortex shedding motions in the near wake region reflect the dominant and second harmonic of the shear layer shedding frequency measured near the leading edge.

On the structure of shear stress and turbulent kinetic energy flux across the roughness layer of a gravel-bed channel flow

2009 ◽  
Vol 638 ◽  
pp. 423-452 ◽  
Author(s):  
EMMANUEL MIGNOT ◽  
D. HURTHER ◽  
E. BARTHELEMY
Keyword(s):  
Shear Stress ◽  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Channel Flow ◽  
Mean Flow ◽  
Local Flow ◽  
Gravel Bed ◽  
Wall Flows ◽  
The Mean ◽  
Conditional Statistics

This study examines the structure of shear stress and turbulent kinetic energy (TKE) flux across the roughness layer of a uniform, fully rough gravel-bed channel flow (ks+ ≫ 100, δ/k = 20) using high-resolution acoustic Doppler velocity profiler measurements. The studied gravel-bed roughness layer exhibits a complex random multi-scale roughness structure in strong contrast with conceptualized k- or d-type roughness in standard rough-wall flows. Within the roughness layer, strong spatial variability of all time-averaged flow quantities are observed affecting up to 40% of the boundary layer height. This variability is attributed to the presence of bed zones with emanating bed protuberances (or gravel clusters) acting as local flow obstacles and bed zones of more homogenous roughness of densely packed gravel elements. Considering the strong spatial mean flow variability across the roughness layer, a spatio-temporal averaging procedure, called double averaging (DA), has been applied to the analysed flow quantities. Three aspects have been addressed: (a) the DA shear stress and DA TKE flux in specific bed zones associated with three classes of velocity profiles as previously proposed in Mignot, Barthélemy & Hurther (J. Fluid Mech., vol. 618, 2009, p. 279), (b) the global and per class DA conditional statistics of shear stress and associated TKE flux and (c) the contribution of large-scale coherent shear stress structures (LC3S) to the TKE flux across the roughness layer. The mean Reynolds and dispersive shear structure show good agreement between the protuberance bed zones associated with the S-shape/accelerated classes and recent results obtained in standard k-type rough-wall flows (Djenidi et al., Exp. Fluids, vol. 44, 2008, p. 37; Pokrajac, McEwan & Nikora, Exp. Fluids, vol. 45, 2008, p. 73). These gravel-bed protuberances act as local flow obstacles inducing a strong turbulent activity in their wake regions. The conditional statistics show that the Reynolds stress contribution is fairly well distributed between sweep and ejection events, with threshold values ranging from H = 0 to H = 8. However, the TKE flux across the roughness layer primarily results from the residual shear stress between ejection and sweep of very high magnitude (H = 10–20) and of small turbulent scale. Although LC3S are seen to penetrated the interfacial roughness layer, their TKE flux contribution is found to be negligible compared to the very energetic small-scale sweep events. These sweeps are dominantly produced in the bed zones of local gravel protuberances where the velocity profiles are inflexional of S-shape type and the mean flow properties are of mixing-layer flow type as previously shown in Mignot et al. (2009).

scholarly journals The streamwise turbulence intensity in the intermediate layer of turbulent pipe flow

2015 ◽  
Vol 774 ◽  
pp. 324-341 ◽  
Author(s):  
J. C. Vassilicos ◽  
J.-P. Laval ◽  
J.-M. Foucaut ◽  
M. Stanislas
Keyword(s):  
Reynolds Number ◽  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Intermediate Layer ◽  
High Reynolds Number ◽  
Logarithmic Derivative ◽  
Turbulent Pipe Flow ◽  
Mean Flow ◽  
The Mean ◽  
High Reynolds

The spectral model of Perryet al. (J. Fluid Mech., vol. 165, 1986, pp. 163–199) predicts that the integral length scale varies very slowly with distance to the wall in the intermediate layer. The only way for the integral length scale’s variation to be more realistic while keeping with the Townsend–Perry attached eddy spectrum is to add a new wavenumber range to the model at wavenumbers smaller than that spectrum. This necessary addition can also account for the high-Reynolds-number outer peak of the turbulent kinetic energy in the intermediate layer. An analytic expression is obtained for this outer peak in agreement with extremely high-Reynolds-number data by Hultmarket al. (Phys. Rev. Lett., vol. 108, 2012, 094501;J. Fluid Mech., vol. 728, 2013, pp. 376–395). Townsend’s (The Structure of Turbulent Shear Flows, 1976, Cambridge University Press) production–dissipation balance and the finding of Dallaset al. (Phys. Rev. E, vol. 80, 2009, 046306) that, in the intermediate layer, the eddy turnover time scales with skin friction velocity and distance to the wall implies that the logarithmic derivative of the mean flow has an outer peak at the same location as the turbulent kinetic energy. This is seen in the data of Hultmarket al. (Phys. Rev. Lett., vol. 108, 2012, 094501;J. Fluid Mech., vol. 728, 2013, pp. 376–395). The same approach also predicts that the logarithmic derivative of the mean flow has a logarithmic decay at distances to the wall larger than the position of the outer peak. This qualitative prediction is also supported by the aforementioned data.

Large-eddy simulation of a three-dimensional shear-driven turbulent boundary layer

2000 ◽  
Vol 423 ◽  
pp. 175-203 ◽  
Author(s):  
CHANDRASEKHAR KANNEPALLI ◽  
UGO PIOMELLI
Keyword(s):  
Boundary Layer ◽  
Kinetic Energy ◽  
Turbulent Boundary Layer ◽  
Turbulent Kinetic Energy ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Quasi Equilibrium ◽  
Mean Flow ◽  
Curvature Effects ◽  
Large Eddy

A three-dimensional shear-driven turbulent boundary layer over a flat plate generated by moving a section of the wall in the transverse direction is studied using large-eddy simulations. The configuration is analogous to shear-driven boundary layer experiments on spinning cylinders, except for the absence of curvature effects. The data presented include the time-averaged mean flow, the Reynolds stresses and their budgets, and instantaneous flow visualizations. The near-wall behaviour of the flow, which was not accessible to previous experimental studies, is investigated in detail. The transverse mean velocity profile develops like a Stokes layer, only weakly coupled to the streamwise flow, and is self-similar when scaled with the transverse wall velocity, Ws. The axial skin friction and the turbulent kinetic energy, K, are significantly reduced after the imposition of the transverse shear, due to the disruption of the streaky structures and of the outer-layer vortical structures. The turbulent kinetic energy budget reveals that the decrease in production is responsible for the reduction of K. The flow then adjusts to the perturbation, reaching a quasi-equilibrium three-dimensional collateral state. Following the cessation of the transverse motion, similar phenomena take place again. The flow eventually relaxes back to a two-dimensional equilibrium boundary layer.

Turbulent boundary layers absent mean shear

2017 ◽  
Vol 835 ◽  
pp. 217-251 ◽  
Author(s):  
Blair A. Johnson ◽  
Edwin A. Cowen
Keyword(s):  
Boundary Layer ◽  
Kinetic Energy ◽  
Turbulent Boundary Layer ◽  
Turbulent Kinetic Energy ◽  
Temporal Frequency ◽  
Synthetic Jet ◽  
Mean Flow ◽  
Frequency Spectra ◽  
Empirical Constant ◽  
The Mean

We perform an experimental study to investigate the turbulent boundary layer above a stationary solid glass bed in the absence of mean shear. High Reynolds number $(Re_{\unicode[STIX]{x1D706}}\sim 300)$ horizontally homogeneous isotropic turbulence is generated via randomly actuated synthetic jet arrays (RASJA – Variano & Cowen J. Fluid Mech. vol. 604, 2008, pp. 1–32). Each of the arrays is controlled by a spatio-temporally varying algorithm, which in turn minimizes the formation of secondary mean flows. One array consists of an $8\times 8$ grid of jets, while the other is a $16\times 16$ array. Particle image velocimetry measurements are used to study the isotropic turbulent region and the boundary layer formed beneath as the turbulence encounters a stationary wall. The flow is characterized with statistical metrics including the mean flow and turbulent velocities, turbulent kinetic energy, integral scales and the turbulent kinetic energy transport equation, which includes the energy dissipation rate, production and turbulent transport. The empirical constant in the Tennekes (J. Fluid Mech. vol. 67, 1975, pp. 561–567) model of Eulerian frequency spectra is calculated based on the dissipation results and temporal frequency spectra from acoustic Doppler velocimetry measurements. We compare our results to prior literature that addresses mean shear free turbulent boundary layer characterizations via grid-stirred tank experiments, moving-bed experiments, rapid-distortion theory and direct numerical simulations in a forced turbulent box. By varying the operational parameters of the randomly actuated synthetic jet array, we also find that we are able to control the turbulence levels, including integral length scales and dissipation rates, by changing the mean on-times in the jet algorithm.

Non-equilibrium scaling laws in axisymmetric turbulent wakes

2015 ◽  
Vol 781 ◽  
pp. 166-195 ◽  
Author(s):  
T. Dairay ◽  
M. Obligado ◽  
J. C. Vassilicos
Keyword(s):  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Scaling Laws ◽  
Reynolds Shear Stress ◽  
Mean Flow ◽  
Turbulent Dissipation ◽  
Turbulent Wakes ◽  
The Mean ◽  
Energy Equations ◽  
Non Equilibrium

We present a combined direct numerical simulation and hot-wire anemometry study of an axisymmetric turbulent wake. The data lead to a revised theory of axisymmetric turbulent wakes which relies on the mean streamwise momentum and turbulent kinetic energy equations, self-similarity of the mean flow, turbulent kinetic energy, Reynolds shear stress and turbulent dissipation profiles, non-equilibrium dissipation scalings and an assumption of constant anisotropy. This theory is supported by the present data up to a distance of 100 times the wake generator’s size, which is as far as these data extend.

Note on Turbulent Kinetic Energy Production for Reynolds-Averaged Navier–Stokes Models

2016 ◽  
Vol 138 (11) ◽  
Author(s):  
Solkeun Jee ◽  
Gorazd Medic ◽  
Georgi Kalitzin
Keyword(s):  
Kinetic Energy ◽  
Strain Rate ◽  
Turbulent Kinetic Energy ◽  
Energy Equation ◽  
Boussinesq Approximation ◽  
Navier Stokes ◽  
Mean Flow ◽  
Production Term ◽  
The Mean ◽  
Reynolds Averaged Navier Stokes

Linear eddy-viscosity Reynolds-Averaged Navier–Stokes (RANS) turbulence models are based on the Boussinesq approximation that asserts the Reynolds stresses to be linearly dependent on the mean strain rate. Using the Boussinesq approximation for the Reynolds stress yields a production term in the turbulent kinetic energy equation that is proportional to the square of the magnitude of the strain rate tensor. For some flows, this relation to the strain causes overproduction of turbulence. Widely used ad hoc modifications of the production term using vorticity lead to an inconsistent energy balance in the mean flow kinetic energy equation, violating the energy conservation. In this note, how to obtain a consistent RANS framework for a given production term modification is shown.

Reynolds Stress Field and Turbulent Kinetic Energy Budget in a Repeating Compressor Stage

2020 ◽  
Author(s):  
Ahmad Nawab ◽  
Feng Wang ◽  
Luca di Mare ◽  
John J. Adamczyk
Keyword(s):  
Kinetic Energy ◽  
Pressure Gradient ◽  
Turbulent Kinetic Energy ◽  
Turbulence Model ◽  
Turbulence Modelling ◽  
Compressor Stage ◽  
Mean Flow ◽  
Rans Simulation ◽  
Dissipation Rates ◽  
The Mean

Abstract Turbulence modelling in compressor passages continues to be a challenging problem. In order to better understand the shortcomings of turbulence modelling, a LES and a RANS computation were performed of a repeating compressor stage. The computation was carried out near the aerodynamic design point of the compressor stage, in order to minimise the challenge posed to the turbulence model. The use of a repeating stage configuration removes the need to specify the statistics of the incoming turbulent field; the statistics become an output of the simulation and not an input. This is a critical fact that greatly increases the credibility of the current LES compressor simulation over many previous simulations. As the computations are performed at mid-span, radial gradients can safely be assumed to be small, thus removing issues associated with capturing flow features attributed to 3D geometry. The flow field is assumed to be incompressible, which is required in order to achieve a true repeating stage environment. The RANS computation is based on a state-of-the-art turbulence model. At the same flow coefficient, the RANS simulation yielded a total pressure rise very near that of the LES simulation. However, there are nontrivial differences in the flow details. The mean flow and Reynolds shear stress boundary layer profiles are in good agreement in regions of favourable pressure gradient, but significant differences exist in the presence of adverse pressure-gradients. The turbulent kinetic energy profiles however are in poor agreement throughout the flow. The mean flow production rates predicted by the RANS computation are largely similar to those of the LES simulation forward of mid-chord where the pressure gradient is favourable. A notable exception is the leading-edge region where the LES predicts negative production i.e. a net transfer of energy to the time-mean flow, and the region aft of mid-chord where the pressure gradient is adverse. Outside of the viscous sub-layer, the dissipation rates are also predicted correctly by the RANS simulation forward of midchord where the pressure gradient is favourable. Aft of mid-chord however, there are significant differences in the dissipation rates.

scholarly journals Estimating Turbulent Dissipation from Microstructure Shear Measurements Using Maximum Likelihood Spectral Fitting over the Inertial and Viscous Subranges

2016 ◽  
Vol 33 (4) ◽  
pp. 713-722 ◽  
Author(s):  
Cynthia E. Bluteau ◽  
Nicole L. Jones ◽  
Gregory N. Ivey
Keyword(s):  
Maximum Likelihood ◽  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Empirical Model ◽  
Integration Method ◽  
Mean Flow ◽  
Likelihood Estimator ◽  
Turbulent Dissipation ◽  
Spectral Fitting ◽  
The Mean

AbstractA technique is presented to derive the dissipation of turbulent kinetic energy ϵ by using the maximum likelihood estimator (MLE) to fit a theoretical or known empirical model to turbulence shear spectral observations. The commonly used integration method relies on integrating the shear spectra in the viscous range, thus requiring the resolution of the highest wavenumbers of the turbulence shear spectrum. With current technology, the viscous range is not resolved at sufficiently large wavenumbers to estimate high ϵ; however, long inertial subranges can be resolved, making spectral fitting over both this subrange and the resolved portion of the viscous range an attractive method for deriving ϵ. The MLE takes into account the chi-distributed properties of the spectral observations, and so it does not rely on the log-transformed spectral observations. This fitting technique can thus take advantage of both the inertial and viscous subranges, a portion of both, or simply one of the subranges. This flexibility allows a broad range of ϵ to be resolved. The estimated ϵ is insensitive to the range of wavenumbers fitted with the model, provided the noise-dominated portion of the spectra and the low wavenumbers impacted by the mean flow are avoided. For W kg−1, the MLE fitting estimates agree with those obtained by integrating the spectral observations. However, with increasing ϵ the viscous subrange is not fully resolved and the integration method progressively starts to underestimate ϵ compared with the values obtained from fitting the spectral observations.

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