Turbulence In Wall-bounded Shear Flows Research Articles

Turbulent–laminar patterns in shear flows without walls

Turbulent–laminar intermittency, typically in the form of bands and spots, is a ubiquitous feature of the route to turbulence in wall-bounded shear flows. Here we study the idealised shear between stress-free boundaries driven by a sinusoidal body force and demonstrate quantitative agreement between turbulence in this flow and that found in the interior of plane Couette flow – the region excluding the boundary layers. Exploiting the absence of boundary layers, we construct a model flow that uses only four Fourier modes in the shear direction and yet robustly captures the range of spatiotemporal phenomena observed in transition, from spot growth to turbulent bands and uniform turbulence. The model substantially reduces the cost of simulating intermittent turbulent structures while maintaining the essential physics and a direct connection to the Navier–Stokes equations. We demonstrate the generic nature of this process by introducing stress-free equivalent flows for plane Poiseuille and pipe flows that again capture the turbulent–laminar structures seen in transition.

レイリー・ベナール・ポアズイユ乱流における熱・運動量輸送

The onset of thermal convection and its effects on heat and momentum transfer have been investigated numerically in pressure-driven turbulent flow in a horizontal plane channel heated from below. At fixed values of the Reynolds number, Re = 2000, 3000, 4000, 6000, the bulk Richardson number Ri is gradually increased to inspect the effects of buoyancy on the wall-bounded turbulent shear flow. Buoyancy-induced streamwise rolls are observed to critically appear at distinct values of the Richardson number depending on the Reynolds number. It is found that the onset of thermal convection can be identified at the critical turbulence Grashof number GrT = 4Ri/Cf2 ≅ 200 being independent of Re, where Cf is the skin friction coefficient. At Re = 2000, moreover, after the onset of the convection GrT > 200 the Stanton number, i.e. dimensionless wall heat flux, is observed to increase due to heat transfer enhancement of the convection with increasing Ri, while the skin friction coefficient decreases in a specific range of Ri.

On the identification of intense Reynolds stress structures in wall-bounded flows using information-limited two-dimensional planar data

Historically the majority of investigations into the transfer of momentum have been focused on the classification and statistical analysis of single points of above average Reynolds stress. These regions of intense Reynolds stress are responsible for the majority of the wall-normal transfer of momentum and most of the production of turbulent kinetic energy. Single point hot-wire measurements were critical in understanding the roll of the regions of intense Reynolds stress, but provide no information about their formation, spatial extent, orientation, nor their position relative to other types of coherent structures. DNS of turbulent wall-bounded shear flows overcomes this information-limit by providing the full three-dimensional (3D) volumetric flow field information, albeit limited to low Reynolds numbers. Advancements in particle image velocimetry (PIV) have enabled the measurement of High Reynolds number turbulent wall-bounded shear flows, with measurements of two instantaneous velocity components in a single two-dimensional (2D) plane (e.g. streamwise wall-normal plane). However, the deductions and conclusions based on these information limited 2D measurements may be detrimentally affected by the lack of information from the third dimension. This study aims to address this issue by using DNS from a turbulent channel flow at Reτ=945 and comparing the geometric characterisation of intense Reynolds stress objects computed from the full 3D flow fields with those computed from the information-limited 2D flow fields that span streamwise wall-normal planes.

The rise of fully turbulent flow.

Over a century of research into the origin of turbulence in wall-bounded shear flows has resulted in a puzzling picture in which turbulence appears in a variety of different states competing with laminar background flow. At moderate flow speeds, turbulence is confined to localized patches; it is only at higher speeds that the entire flow becomes turbulent. The origin of the different states encountered during this transition, the front dynamics of the turbulent regions and the transformation to full turbulence have yet to be explained. By combining experiments, theory and computer simulations, here we uncover a bifurcation scenario that explains the transformation to fully turbulent pipe flow and describe the front dynamics of the different states encountered in the process. Key to resolving this problem is the interpretation of the flow as a bistable system with nonlinear propagation (advection) of turbulent fronts. These findings bridge the gap between our understanding of the onset of turbulence and fully turbulent flows.

J0550202 壁面剪断乱流における熱対流の発生

J0550202 壁面剪断乱流における熱対流の発生

Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow

This paper demonstrates the maintenance of self-sustaining turbulence in a restricted nonlinear (RNL) model of plane Couette flow. The RNL system is derived directly from the Navier-Stokes equations and permits higher resolution studies of the dynamical system associated with the stochastic structural stability theory (S3T) model, which is a second order approximation of the statistical state dynamics of the flow. The RNL model shares the dynamical restrictions of the S3T model but can be easily implemented by reducing a DNS code so that it retains only the RNL dynamics. Comparisons of turbulence arising from DNS and RNL simulations demonstrate that the RNL system supports self-sustaining turbulence with a mean flow as well as structural and dynamical features that are consistent with DNS. These results demonstrate that the simplified RNL system captures fundamental aspects of fully developed turbulence in wall-bounded shear flows and motivate use of the RNL/S3T framework for further study of wall-turbulence.

Turbulence in the highly restricted dynamics of a closure at second order: comparison with DNS

S3T (Stochastic Structural Stability Theory) employs a closure at second order to obtain the dynamics of the statistical mean turbulent state. When S3T is implemented as a coupled set of equations for the streamwise mean and perturbation states, nonlinearity in the dynamics is restricted to interaction between the mean and perturbations. The S3T statistical mean state dynamics can be approximately implemented by similarly restricting the dynamics used in a direct numerical simulation (DNS) of the full Navier-Stokes equations (referred to as the NS system). Although this restricted nonlinear system (referred to as the RNL system) is greatly simplified in its dynamics in comparison to the associated NS, it nevertheless self-sustains a turbulent state in wall-bounded shear flow with structures and dynamics comparable to those observed in turbulence. Moreover, RNL turbulence can be analysed effectively using theoretical methods developed to study the closely related S3T system. In order to better understand RNL turbulence and its relation to NS turbulence, an extensive comparison is made of diagnostics of structure and dynamics in these systems. Although quantitative differences are found, the results show that turbulence in the RNL system closely parallels that in NS and suggest that the S3T/RNL system provides a promising reduced complexity model for studying turbulence in wall-bounded shear flows.

Velocity–vorticity correlation structure in turbulent channel flow

AbstractA new statistical coherent structure (CS), the velocity–vorticity correlation structure (VVCS), using the two-point cross-correlation coefficient $R_{ij}$ of velocity and vorticity components, $u_i$ and $\omega _j~ (i, j = 1, 2, 3)$, is proposed as a useful descriptor of CS. For turbulent channel flow with the wall-normal direction $y$, a VVCS study consists of using $u_i$ at a fixed reference location $y_r$, and using $|R_{ij} (y_r; x, y, z)|\geqslant R_0$ to define a topologically invariant high-correlation region, called $\mathit{VVCS}_{ij}$. The method is applied to direct numerical simulation (DNS) data, and it is shown that the $\mathit{VVCS}_{ij}$ qualitatively and quantitatively captures all known geometrical features of near-wall CS, including spanwise spacing, streamwise length and inclination angle of the quasi-streamwise vortices and streaks. A distinct feature of the VVCS is that its geometry continuously varies with $y_r$. A topological change of $\mathit{VVCS}_{11}$ from quadrupole (for smaller $y_r$) to dipole (for larger $y_r$) occurs at $y^{+}_r=110$, giving a geometrical interpretation of the multilayer nature of wall-bounded turbulent shear flows. In conclusion, the VVCS provides a new robust method to quantify CS in wall-bounded flows, and is particularly suitable for extracting statistical geometrical measures using two-point simultaneous data from hotwire, particle image velocimetry/laser Doppler anemometry measurements or DNS/large eddy simulation data.

Revealing the state space of turbulent pipe flow by symmetry reduction

AbstractSymmetry reduction by the method of slices is applied to pipe flow in order to obtain a quotient of the streamwise translation and azimuthal rotation symmetries of turbulent flow states. Within the symmetry-reduced state space, all travelling wave solutions reduce to equilibria, and all relative periodic orbits reduce to periodic orbits. Projections of these solutions and their unstable manifolds from their infinite-dimensional symmetry-reduced state space onto suitably chosen two- or three-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows. Visualizations of the flow within the slice and its linearization at equilibria enable us to trace out the unstable manifolds, determine close recurrences, identify connections between different travelling wave solutions and find, for the first time for pipe flows, relative periodic orbits that are embedded within the chaotic saddle, which capture turbulent dynamics at transitional Reynolds numbers.

0513 矩形ダクトにおける過渡的乱れ(OS5-3 壁乱流:統計,構造,動力学の理解,OS5 壁乱流:統計,構造,動力学の理解)

Subcritical transition to turbulence in wall-bounded shear flow is known to be characterized in terms of turbulent patches bounded by laminar regions. In circular pipe flow, it is observed that turbulence is localized only in the streamwise direction at marginal Reynolds numbers. Such a structure is a so-called turbulent puff. A turbulent puff is known to decay according to a memoryless process, so that its lifetime has an exponential probability distribution. In rectangular-duct flow, when the aspect ratio is small, i.e., As < 4, a turbulent puff is observed; however, it would exhibit difference from a usual puff as in a pipe. In this study, the decay of a turbulent puff in rectangular-duct flow is measured experimentally by using LDV(Laser Doppler Velocimeter) and PIV(Particle Image Velocimetry). It is found that the survival probability of a turbulent puff is exponential as in pipe flow, but its lifetime is very different from that at the comparable Reynolds number in pipe flow.

Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow

AbstractStreamwise rolls and accompanying streamwise streaks are ubiquitous in wall-bounded shear flows, both in natural settings, such as the atmospheric boundary layer, as well as in controlled settings, such as laboratory experiments and numerical simulations. The streamwise roll and streak structure has been associated with both transition from the laminar to the turbulent state and with maintenance of the turbulent state. This close association of the streamwise roll and streak structure with the transition to and maintenance of turbulence in wall-bounded shear flow has engendered intense theoretical interest in the dynamics of this structure. In this work, stochastic structural stability theory (SSST) is applied to the problem of understanding the dynamics of the streamwise roll and streak structure. The method of analysis used in SSST comprises a stochastic turbulence model (STM) for the dynamics of perturbations from the streamwise-averaged flow coupled to the associated streamwise-averaged flow dynamics. The result is an autonomous, deterministic, nonlinear dynamical system for evolving a second-order statistical mean approximation of the turbulent state. SSST analysis reveals a robust interaction between streamwise roll and streak structures and turbulent perturbations in which the perturbations are systematically organized through their interaction with the streak to produce Reynolds stresses that coherently force the associated streamwise roll structure. If a critical value of perturbation turbulence intensity is exceeded, this feedback results in modal instability of the combined streamwise roll/streak and associated turbulence complex in the SSST system. In this instability, the perturbations producing the destabilizing Reynolds stresses are predicted by the STM to take the form of oblique structures, which is consistent with observations. In the SSST system this instability exists together with the transient growth process. These processes cooperate in determining the structure of growing streamwise roll and streak. For this reason, comparison of SSST predictions with experiments requires accounting for both the amplitude and structure of initial perturbations as well as the influence of the SSST instability. Over a range of supercritical turbulence intensities in Couette flow, this instability equilibrates to form finite amplitude time-independent streamwise roll and streak structures. At sufficiently high levels of forcing of the perturbation field, equilibration of the streamwise roll and streak structure does not occur and the flow transitions to a time-dependent state. This time-dependent state is self-sustaining in the sense that it persists when the forcing is removed. Moreover, this self-sustaining state rapidly evolves toward a minimal representation of wall-bounded shear flow turbulence in which the dynamics is limited to interaction of the streamwise-averaged flow with a perturbation structure at one streamwise wavenumber. In this minimal realization of the self-sustaining process, the time-dependent streamwise roll and streak structure is maintained by perturbation Reynolds stresses, just as is the case of the time-independent streamwise roll and streak equilibria. However, the perturbation field is maintained not by exogenously forced turbulence, but rather by an endogenous and essentially non-modal parametric growth process that is inherent to time-dependent dynamical systems.

0117 矩形ダクト流における局在乱れ構造(OS1 噴流,後流および剥離流れ現象の探求と技術革新,オーガナイズドセッション)

Subcritical transition to turbulence in wall-bounded shear flow is known to be characterized in terms of turbulent patches bounded by laminar regions. In this study spatially localized turbulence structures are investigated in rectangular-duct flow of cross-sectional aspect ratios A_S = 1 - 9 at marginal Reynolds number Ret. The lowest Reynolds number Ret for transient turbulence is determined to demonstrate its dependence on the aspect ratio As. It is found that Ret monotonically decreases from Ret = 730 for As = 1 with increasing As, and for As = 5 it nearly attains to a minimal value Re_T ≒ 670 which is consistent with the Reynolds number for the appearance of a turbulent puff in a plane channel (As = ∞). Streamwise-localized (turbulent puff), or streamwise- and spanwiselocalized (turbulent spot) turbulence are observed at the Reynolds numbers Re = Re_T for distinct As. The localized turbulence exhibits structural 'transition' between a turbulent puff and a turbulent spot around As = 4.

Geometry of the turbulence in wall-bounded shear flows: periodic orbits

The dynamical theory of moderate Reynolds number turbulence triangulates the infinite-dimensional Navier–Stokes state space by sets of exact solutions (equilibria, relative equilibria, periodic orbits, etc), which form a rigid backbone that enables us to describe and predict the sinuous motions of a turbulent fluid. We report on the determination of a set of unstable periodic orbits from close recurrences of the turbulent flow. A few equilibria that closely resemble frequently observed but unstable coherent structures are used for constructing a low-dimensional state-space projection from the extremely high-dimensional data sets. The turbulent flow can then be visualized as a sequence of close passages to unstable periodic orbits, i.e the time-recurrent dynamical coherent structures typical of a turbulent flow.

STREAMWISE VORTICES AND REYNOLDS STRESSES OF WALL-BOUNDED TURBULENT SHEAR FLOWS

Reynolds Equation is the most commonly used governing equation in turbulence. However, its application in wall-bounded turbulent shear flows may involve a defect. In general, Reynolds averaging should be ensemble averaging and Reynolds stresses are supposed to express all the actions of turbulence on the mean field. In statistically steady three-dimensional flows, Reynolds stresses are usually defined as correlations of temporal velocity fluctuations so that they cannot contain the influences of steady components of streamwise vortices. This is believed to be one of the reasons why many closure models in RANS meet problems in flows where streamwise vortices play significant roles. In this paper, Spatial-Temporal (S-T) averaged Reynolds stresses were defined, which separates the turbulence actions caused by temporal or spatial velocity fluctuations. DNS data for a fully developed channel flow were then used to check balancing of equations. Comparison showed that the balancing errors in the S-T averaged Reynolds equations were obviously smaller than those in the temporal averaged one, in particular, in the near wall region where the streamwise vortices located. Thus, a combination of traditional model with a supplemental model expressing influences of streamwise vortices might be a way out to improve the turbulence modeling.

Statistics of velocity gradients in wall-bounded shear flow turbulence

The statistical properties of velocity gradients in a wall-bounded turbulent channel flow are discussed on the basis of three-dimensional direct numerical simulations. Our analysis is concentrated on the trend of the statistical properties of the local enstrophy ω 2 ( x , t ) and the energy dissipation rate ϵ ( x , t ) with increasing distance from the wall. We detect a sensitive dependence of the largest amplitudes of both fields (which correspond with the tail of the distribution) on the spectral resolution. The probability density functions of each single field as well as their joint distribution vary significantly with increasing distance from the wall. The largest fluctuations of the velocity gradients are found in the logarithmic layer. This is in agreement with recent experiments which observe a bursting of hairpin vortex packets into the logarithmic region.

Equilibrium and travelling-wave solutions of plane Couette flow

We present 10 new equilibrium solutions to plane Couette flow in small periodic cells at low Reynolds numberReand two new travelling-wave solutions. The solutions are continued under changes ofReand spanwise period. We provide a partial classification of the isotropy groups of plane Couette flow and show which kinds of solutions are allowed by each isotropy group. We find two complementary visualizations particularly revealing. Suitably chosen sections of their three-dimensional physical space velocity fields are helpful in developing physical intuition about coherent structures observed in low-Returbulence. Projections of these solutions and their unstable manifolds from their ∞-dimensional state space on to suitably chosen two- or three-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows.

Improvement of the nonlinear eddy diffusivity model for rotational turbulent heat transfer at various rotating axes

This paper presents an improved turbulent heat transfer model for analysis of a turbulent heat transfer in a rotational wall-bounded shear flow. Since a turbulent heat transfer with rotation has often been encountered in physical or engineering problems, a turbulence model should be reconstructed to adequately predict a turbulent heat transfer with rotation. In order to predict turbulent heat transfer with rotation, models of both the velocity and thermal fields are needed. In the velocity field, the modified nonlinear eddy diffusivity for a momentum model (NLEDMM) was proposed to accurately predict rotational wall-bounded turbulent shear flows with arbitrary rotating axes. As for a thermal field, the turbulence model was only improved to predict a heated channel turbulent flow with spanwise rotation. Thus, a turbulence model, which can be applied in heated rotational turbulent shear flows with arbitrary rotating axes, should be modified. In order to improve the turbulent heat transfer model, the adequate data, which are used for evaluation of turbulence model performance are arranged by conducting direct numerical simulations in this study. Since there are few reports on streamwise and wall-normal rotating channel flows with heat transfer, DNSs are carried out under conditions of streamwise and wall-normal rotating axes. In the present study, based on our DNS databases of streamwise and wall-normal rotating channel flows, existing nonlinear two-equation heat-transfer turbulence models were evaluated. Using the results of this evaluation, we then improved the modeled expression of turbulent heat flux for heat transfer in flows with arbitrary rotating axes, i.e. a nonlinear eddy diffusivity for the heat model (NLEDHM) is proposed.

Mechanics and Prediction of Turbulent Drag Reduction with Polymer Additives

This article provides a review of recent progress in understanding and predicting polymer drag reduction (DR) in turbulent wall-bounded shear flows. The reduction in turbulent friction losses by the dilute addition of high–molecular weight polymers to flowing liquids has been extensively studied since the phenomenon was first observed over 60 years ago. Although it has long been reasoned that the dynamical interactions between polymers and turbulence are responsible for DR, it was not until recently that progress had been made to begin to elucidate these interactions in detail. These advancements come largely from numerical simulations of viscoelastic turbulent flows and detailed turbulence measurements in flows of dilute polymer solutions using laser-based optical techniques. This review presents a selective overview of the current state of the numerics and experimental techniques and their impact on understanding the mechanics and prediction of polymer DR. It includes a discussion of areas in which our understanding is incomplete, warranting further study.

Pressure-velocity joint measurements of a wall-bounded turbulent shear flow *

The unsteady behavior of the large-scale vortical structures buried in a wall-bounded turbulent shear layer flow was extensively investigated using pressure-velocity joint measurements. The wall pressure fluctuations and flow field velocity fluctuations were measured simultaneously by using a microphone and an X-type hotwire, respectively. The spatially and temporally strong coupling between the convecting flow structures and the wall pressure fluctuations were meticulously investigated in terms of the continuous wavelet transform, cross-correlation and coherence of the wall pressure and flow field. The characteristics of the large-scale vortical structures, e.g., the shedding frequency, averaged convection velocity, convective motion, and structure pattern were revealed.

Nonlinear eddy diffusivity models reflecting buoyancy effect for wall-shear flows and heat transfer

The main objective of this study is to construct new nonlinear eddy diffusivity models reflecting buoyant effects in wall-bounded turbulent shear flows and heat transfer. It is now well known that the turbulent heat-fluxes, which are the key quantities for the prediction of turbulent flows with buoyancy, are not modeled accurately by employing the conventional turbulence models using eddy diffusivities. In order to appropriately predict wall-bounded turbulent flows with buoyancy, an innovative turbulence heat-transfer model with eddy diffusivities which are composed of the k– ε two-equation model for velocity field and the k θ – ε θ two-equation model for thermal field, must be constructed. Consequently, we should improve the modeled expressions for Reynolds stresses and turbulent heat-fluxes reflecting the buoyant effect in wall-bounded turbulent shear flows. The existing two-equation turbulence models were evaluated on the basis of the DNS data of channel flows with buoyancy. Using the results of evaluation, we constructed new modeled expressions for Reynolds stresses and turbulent heat-fluxes in explicit algebraic models, and reconstructed the nonlinear two-equation turbulence models for the buoyancy-affected wall-shear flows and heat transfer, including the newly proposed nonlinear eddy diffusivity for a momentum model (NLEDM) and the nonlinear eddy diffusivity for heat model (NLEDHM). The proposed nonlinear two-equation turbulence models reflecting buoyancy effect appropriately predict wall-bounded turbulent shear flows with buoyancy given by DNS.