Stability of turbulent channel flow, with application to Malkus's theory

1967 ◽  
Vol 27 (2) ◽  
pp. 253-272 ◽  
Author(s):  
W. C. Reynolds ◽  
W. G. Tiederman
Keyword(s):  
Velocity Profile ◽  
Channel Flow ◽  
Stability Problem ◽  
Turbulent Channel Flow ◽  
Velocity Profiles ◽  
Neutral Stability ◽  
Mean Velocity ◽  
The Mean ◽  
Two Parameter ◽  
Good Agreement

The Orr-Sommerfeld stability problem has been studied for velocity profiles appropriate to turbulent channel flow. The intent was to provide an evaluation of Malkus's theory that the flow assumes a state of maximum dissipation, subject to certain constraints, one of which is that the mean velocity profile is marginally stable. Dissipation rates and neutral stability curves were obtained for a representative two-parameter family of velocity profiles. Those in agreement with experimental profiles were found to be stable; the marginally stable profile of greatest dissipation was not in good agreement with experiments. An explanation for the apparent success of Malkus's theory is offered.

Relationship between the heat transfer law and the scalar dissipation function in a turbulent channel flow

2017 ◽  
Vol 830 ◽  
pp. 300-325 ◽  
Author(s):  
Hiroyuki Abe ◽  
Robert Anthony Antonia
Keyword(s):  
Heat Transfer ◽  
Prandtl Number ◽  
Channel Flow ◽  
Dissipation Rate ◽  
Turbulent Channel Flow ◽  
Scalar Dissipation ◽  
Scalar Dissipation Rate ◽  
Mean Velocity ◽  
The Mean ◽  
Logarithmic Dependence

Integration across a fully developed turbulent channel flow of the transport equations for the mean and turbulent parts of the scalar dissipation rate yields relatively simple relations for the bulk mean scalar and wall heat transfer coefficient. These relations are tested using direct numerical simulation datasets obtained with two isothermal boundary conditions (constant heat flux and constant heating source) and a molecular Prandtl number Pr of 0.71. A logarithmic dependence on the Kármán number $h^{+}$ is established for the integrated mean scalar in the range $h^{+}\geqslant 400$ where the mean part of the total scalar dissipation exhibits near constancy, whilst the integral of the turbulent scalar dissipation rate $\overline{\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}}$ increases logarithmically with $h^{+}$. This logarithmic dependence is similar to that established in a previous paper (Abe & Antonia, J. Fluid Mech., vol. 798, 2016, pp. 140–164) for the bulk mean velocity. However, the slope (2.18) for the integrated mean scalar is smaller than that (2.54) for the bulk mean velocity. The ratio of these two slopes is 0.85, which can be identified with the value of the turbulent Prandtl number in the overlap region. It is shown that the logarithmic $h^{+}$ increase of the integrated mean scalar is intrinsically associated with the overlap region of $\overline{\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}}$, established for $h^{+}$ (${\geqslant}400$). The resulting heat transfer law also holds at a smaller $h^{+}$ (${\geqslant}200$) than that derived by assuming a log law for the mean temperature.

Connections between the Ozmidov scale and mean velocity profile in stably stratified atmospheric surface layers

2016 ◽  
Vol 797 ◽  
Author(s):  
Dan Li ◽  
Scott T. Salesky ◽  
Tirtha Banerjee
Keyword(s):  
Velocity Profile ◽  
Atmospheric Stability ◽  
Length Scale ◽  
Surface Layers ◽  
Mean Velocity ◽  
Stable Conditions ◽  
Stringent Constraint ◽  
Ozmidov Scale ◽  
The Mean ◽  
Good Agreement

The mean velocity profile (MVP) in thermally stratified atmospheric surface layers (ASLs) deviates from the classic logarithmic form. A theoretical framework was recently proposed (Katulet al.Phys. Rev. Lett., vol. 107, 2011, 268502) to link the MVP to the spectrum of turbulence and was found to successfully predict the MVP for unstable stratification. However, the theory failed to reproduce the MVP in stable conditions (Saleskyet al.Phys. Fluids, vol. 25, 2013, 105101), especially when${\it\zeta}>0.2$(where${\it\zeta}$is the atmospheric stability parameter). In the present study, it is demonstrated that this shortcoming is due to the failure to identify the appropriate length scale that characterizes the size of momentum transporting eddies in the stable ASL. Beyond${\it\zeta}\approx 0.2$(near where the original theory fails), the Ozmidov length scale becomes smaller than the distance from the wall$z$and hence is a more stringent constraint for characterizing the size of turbulent eddies. An expression is derived to connect the Ozmidov length scale to the normalized MVP (${\it\phi}_{m}$), allowing${\it\phi}_{m}$to be solved numerically. It is found that the revised theory produces a prediction of${\it\phi}_{m}$in good agreement with the widely used empirical Businger–Dyer relation and two experimental datasets in the stable ASL. The results here demonstrate that the behaviour of${\it\phi}_{m}$in the stable ASL is closely linked to the size of momentum transporting eddies, which can be characterized by the Ozmidov scale under mildly to moderately stable conditions ($0.2<{\it\zeta}<1-2$).

scholarly journals Resolvent analysis of turbulent channel flow with manipulated mean velocity profile

2020 ◽  
Vol 15 (3) ◽  
pp. JFST0014-JFST0014
Author(s):  
Riko UEKUSA ◽  
Aika KAWAGOE ◽  
Yusuke NABAE ◽  
Koji FUKAGATA
Keyword(s):  
Velocity Profile ◽  
Channel Flow ◽  
Turbulent Channel Flow ◽  
Mean Velocity

scholarly journals Effects of inlet velocity profiles for thermal stripping phenomena in T-junctions

2021 ◽  
Vol 2116 (1) ◽  
pp. 012026
Author(s):  
Lisa Lampunio ◽  
Yu Duan ◽  
Raad Issa ◽  
Matthew D. Eaton
Keyword(s):  
Velocity Profiles ◽  
Lower Sensitivity ◽  
Detached Eddy Simulation ◽  
Mean Velocity ◽  
Inlet Velocity ◽  
Eddy Simulation ◽  
Delayed Detached Eddy Simulation ◽  
Flow Domain ◽  
The Mean ◽  
Good Agreement

Abstract This paper investigates the effects of different inlet velocities on thermal stripping phenomena within a T-junction. The computational flow domain is modelled using the Improved Delayed Detached Eddy Simulation (IDDES) turbulence model implemented within the commercial CFD code STAR-CCM+ 12.04. The computational model is validated against the OECD-NEA-Vattenfall T-junction Benchmark data. The influence of flat and fully developed inlet velocity profiles is then assessed. The results are in good agreement with the experimental data. The different inlet velocity profiles have a non-negligible effect on the mean wall temperature. The mean velocity shows lower sensitivity to changes in inlet velocity profiles, whose influence is confined mainly to the recirculation zone near the T-junction.

scholarly journals Acceleration in turbulent channel flow: universalities in statistics, subgrid stochastic models and an application

2013 ◽  
Vol 721 ◽  
pp. 627-668 ◽  
Author(s):  
Rémi Zamansky ◽  
Ivana Vinkovic ◽  
Mikhael Gorokhovski
Keyword(s):  
Stochastic Model ◽  
Channel Flow ◽  
Scaling Law ◽  
Stochastic Modelling ◽  
Turbulent Channel Flow ◽  
Mean Value ◽  
Mean Velocity ◽  
Wall Distance ◽  
Eddy Simulation ◽  
The Mean

AbstractThis paper focuses on the characterization and the stochastic modelling of the fluid acceleration in turbulent channel flow. In the first part, the acceleration is studied by direct numerical simulation (DNS) at three Reynolds numbers (${\mathit{Re}}_{\ast } = {u}_{\ast } h/ \nu = 180$, 590 and 1000). It is observed that whatever the wall distance is, the norm of acceleration is log-normally distributed and that the variance of the norm is very close to its mean value. It is also observed that from the wall to the centreline of the channel, the orientation of acceleration relaxes statistically towards isotropy. On the basis of dimensional analysis, a universal scaling law for the acceleration norm is proposed. In the second part, in the framework of the norm/orientation decomposition, a stochastic model of the acceleration is introduced. The stochastic model for the norm is based on fragmentation process which evolves across the channel with the wall distance. Simultaneously the orientation is simulated by a random walk on the surface of a unit sphere. The process is generated in such a way that the mean components of the orientation vector are equal to zero, whereas with increasing wall distance, all directions become equally probable. In the third part, the models are assessed in the framework of large-eddy simulation with stochastic subgrid acceleration model (LES–SSAM), introduced recently by Sabel’nikov, Chtab-Desportes & Gorokhovski (Euro. Phys. J. B, vol. 80, 2011, p. 177–187), and designed to account for the intermittency at subgrid scales. Computations by LES–SSAM and its assessment using DNS data show that the prediction of important statistics to characterize the flow, such as the mean velocity, the energy spectra at small scales, the viscous and turbulent stresses, the distribution of the acceleration can be considerably improved in comparison with standard LES. In the last part of this paper, the advantage of LES–SSAM in accounting for the subgrid flow structure is demonstrated in simulation of particle-laden turbulent channel flows. Compared to standard LES, it is shown that for different Stokes numbers, the particle dynamics and the turbophoresis effect can be predicted significantly better when LES–SSAM is applied.

The Malkus theory applied to magnetohydrodynamic turbulent channel flow

1966 ◽  
Vol 25 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Jacques C. J. Nihoul
Keyword(s):  
Magnetic Field ◽  
Velocity Profile ◽  
Stokes Equations ◽  
Turbulent Channel Flow ◽  
Boundary Region ◽  
Transverse Magnetic ◽  
Parallel Plates ◽  
Dimensionless Numbers ◽  
Mean Velocity ◽  
The Mean

The turbulent flow of a weakly conducting liquid between parallel plates in the presence of a transverse magnetic field is investigated. The form of the mean velocity profile is determined by a series of constraints resulting from the boundary conditions and the Navier–Stokes equations and by the Malkus postulates on the spectrum of the mean vorticity gradient. The width of the transition regions near the walls is derived in terms of the governing dimensionless numbers and this expression is checked, in the asymptotic laminar case, against the well-known Hartmann result. A graphical method, exploiting the relation between the boundary region thickness and the smallest scale of motion defined by the Malkus theory is proposed to determine thescaleof the velocity profile, i.e. the flow rate in terms of the pressure gradient and the magnetic field strength.

Resolvent Analysis of Turbulent Friction Drag Reduction by Manipulation of Mean Velocity Profile

2019 ◽  
Author(s):  
Riko Uekusa ◽  
Aika Kawagoe ◽  
Yusuke Nabae ◽  
Koji Fukagata
Keyword(s):  
Velocity Profile ◽  
Drag Reduction ◽  
Turbulent Viscosity ◽  
Turbulent Channel Flow ◽  
The Other ◽  
Friction Drag ◽  
Mean Velocity ◽  
Turbulent Friction ◽  
The Mean ◽  
Friction Drag Reduction

Abstract In the present study, we numerically manipulate the mean velocity profile of a turbulent channel flow and assess the friction drag reduction performance by using resolvent analysis. Building on the implication obtained from Kühnen et al. (Nat. Phys., Vol. 14, 2017, pp. 386–390) that modifying mean velocity profile flat leads to significant drag reduction, we first introduce two functions for turbulent mean velocity, which can express ‘flattened’ profiles: one is derived based on the turbulent viscosity model proposed by Reynolds & Tiederman (J. Fluid Mech., Vol. 658, 2010, pp. 336–382), and the other is based on the mean velocity profile of laminar flow. These functions are used as the mean velocity profile for the resolvent analysis, and the flatness of the resulting profiles is characterized by two different measures. As a result, we confirm that, friction drag reduction is achieved if the turbulent mean velocity profile is ‘flattened’. However, we also find that the flatness of the mean velocity profile in the center of the channel alone is not enough to evaluate the drag reduction performance.

The structure of the viscous sublayer and the adjacent wall region in a turbulent channel flow

1974 ◽  
Vol 65 (3) ◽  
pp. 439-459 ◽  
Author(s):  
Helmut Eckelmann
Keyword(s):  
Channel Flow ◽  
Turbulent Channel Flow ◽  
Streamwise Velocity ◽  
Viscous Sublayer ◽  
Wall Region ◽  
Normal Velocity ◽  
Mean Velocity ◽  
Velocity Fluctuations ◽  
Mean Values ◽  
The Mean

Hot-film anemometer measurements have been carried out in a fully developed turbulent channel flow. An oil channel with a thick viscous sublayer was used, which permitted measurements very close to the wall. In the viscous sublayer between y+ ≃ 0·1 and y+ = 5, the streamwise velocity fluctuations decreased at a higher rate than the mean velocity; in the region y+ [lsim ] 0·1, these fluctuations vanished at the same rate as the mean velocity.The streamwise velocity fluctuations u observed in the viscous sublayer and the fluctuations (∂u/∂y)0 of the gradient at the wall were almost identical in form, but the fluctuations of the gradient at the wall were found to lag behind the velocity fluctuations with a lag time proportional to the distance from the wall. Probability density distributions of the streamwise velocity fluctuations were measured. Furthermore, measurements of the skewness and flatness factors made by Kreplin (1973) in the same flow channel are discussed. Measurements of the normal velocity fluctuations v at the wall and of the instantaneous Reynolds stress −ρuv were also made. Periods of quiescence in the − ρuv signal were observed in the viscous sublayer as well as very active periods where ratios of peak to mean values as high as 30:1 occurred.

Direct Assessment of the Accuracy of Stereo PIV in Turbulent Channel Flow

Volume 1 ◽  
2004 ◽  
Author(s):  
K. T. Christensen ◽  
Y. Wu
Keyword(s):  
Reynolds Number ◽  
Channel Flow ◽  
Single Point ◽  
Outer Region ◽  
Turbulence Models ◽  
Turbulent Channel Flow ◽  
Wall Region ◽  
Mean Velocity ◽  
Velocity Gradients ◽  
The Mean

Stereo particle-image velocimetry (PIV) has become a widely-used method for studying complex flows because it allows one to acquire instantaneous, three-component velocity data on a planar domain with high spatial resolution. However, the accuracy of such measurements must be carefully evaluated before stereo PIV data can be faithfully used in the development of sophisticated turbulence models, assessment of appropriate computational boundary conditions, and in the validation of advanced computations. To this end, the accuracy of stereo PIV is assessed directly in an actual turbulent environment: two-dimensional turbulent channel flow. This flow is a challenging test of stereo PIV because the turbulent velocity fluctuations are quite small compared to the mean (typically less than ten percent of the mean velocity) and strong velocity gradients exist in the near-wall region. Measurements are made in the streamwise–wall-normal plane along the channel’s spanwise centerline using both stereoscopic and conventional 2-D PIV. A large ensemble of statistically independent velocity realizations are acquired with each method at a friction Reynolds number Reτ = u*h/ν = 934. Single-point statistics are computed from the experimental data and compared to statistics determined from a direct numerical simulation (DNS) of turbulent channel flow at a nearly-identical friction Reynolds number of 940 [5]. Excellent agreement is found in the outer region of the flow (y/h &gt; 0.15, where h is the half-height of the channel). For y/h &lt; 0.15, both the conventional and stereo PIV results differ from the DNS data. These differences are most-likely a manifestation of errors associated with strong velocity gradients and intense turbulent events present in this region of the flow.

Velocity statistics in turbulent channel flow up to

2014 ◽  
Vol 742 ◽  
pp. 171-191 ◽  
Author(s):  
Matteo Bernardini ◽  
Sergio Pirozzoli ◽  
Paolo Orlandi
Keyword(s):  
Reynolds Number ◽  
Kinetic Energy ◽  
Velocity Profile ◽  
Outer Layer ◽  
Energy Production ◽  
Turbulent Channel Flow ◽  
Streamwise Velocity ◽  
Mean Velocity ◽  
The Mean ◽  
Velocity Spectra

AbstractThe high-Reynolds-number behaviour of the canonical incompressible turbulent channel flow is investigated through large-scale direct numerical simulation (DNS). A Reynolds number is achieved ($Re_{\tau } = h/\delta _v \approx 4000$, where $h$ is the channel half-height, and $\delta _v$ is the viscous length scale) at which theory predicts the onset of phenomena typical of the asymptotic Reynolds number regime, namely a sensible layer with logarithmic variation of the mean velocity profile, and Kolmogorov scaling of the velocity spectra. Although higher Reynolds numbers can be achieved in experiments, the main advantage of the present DNS study is access to the full three-dimensional flow field. Consistent with refined overlap arguments (Afzal & Yajnik, J. Fluid Mech. vol. 61, 1973, pp. 23–31; Jiménez & Moser, Phil. Trans. R. Soc. Lond. A, vol. 365, 2007, pp. 715–732), our results suggest that the mean velocity profile never achieves a truly logarithmic profile, and the logarithmic diagnostic function instead exhibits a linear variation in the outer layer whose slope decreases with the Reynolds number. The extrapolated value of the von Kármán constant is $k \approx 0.41$. A near logarithmic layer is observed in the spanwise velocity variance, as predicted by Townsend’s attached eddy hypothesis, whereas the streamwise variance seems to exhibit a shoulder, perhaps being still affected by low-Reynolds-number effects. Comparison with previous DNS data at lower Reynolds number suggests enhancement of the imprinting effect of outer-layer eddies onto the near-wall region. This mechanisms is associated with excess turbulence kinetic energy production in the outer layer, and it reflects in flow visualizations and in the streamwise velocity spectra, which exhibit sharp peaks in the outer layer. Associated with the outer energy production site, we find evidence of a Kolmogorov-like inertial range, limited to the spanwise spectral density of $u$, whereas power laws with different exponents are found for the other spectra. Finally, arguments are given to explain the ‘odd’ scaling of the streamwise velocity variances, based on the analysis of the kinetic energy production term.

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