Turbulent Plane Couette Flow Research Articles

機械学習を用いた平面クエット乱流における低次元力学系の構築

In a dissipative system, there exists the (global) attractor which has finite fractal dimensions. The flow on the attractor can be parametrized by a finite number of parameters (Temmam 1987). For low-Reynolds-number turbulence in plane Couette flow, we construct precise low-dimensional governing equations on the attractors using machine learning.

Penta-decomposition of instantaneous field in spanwise-rotating turbulent plane Couette flow

Spanwise-rotating turbulent plane Couette flow (RPCF) is one of the fundamental prototypes for wall-bounded turbulent flows in rotating reference frames. In this turbulent problem, there are large-scale roll cells which are widely studied. In this paper, a penta-decomposition method is proposed to separate the instantaneous velocity and the total kinetic energy into five parts, i.e., a mean part, a streamwise part and a cross-flow part of the secondary flow, and a streamwise part and a cross-flow part of the residual field. The transport equations for the last four shares, which contribute the total turbulent kinetic energy, are derived. According to these transport equations, the mechanisms of energy transfer among different fractions of turbulent kinetic energy can be revealed clearly. Our objective is to explore the energy balance and transfer among different fractions of the turbulent kinetic energy in RPCF based on a series of direct numerical simulation databases at a Reynolds number Rew=Uwh/=1300 (here, Uw is half of the wall velocity difference, and h is the channel half-width) and rotation number Ro=2zh/Uw (z is the constant angular velocity in the spanwise direction) in a range of 0Ro0.9. The results show that the energy is transferred between the streamwise part/cross-flow part of secondary flows and the residual field through the correlation between the vorticity of secondary flows and the shear stress of residual field. The rotation term acts as a bridge to transfer the energy between the streamwise part and the cross-flow part of either the secondary flows or the residual field. Moreover, pressure-strain redistribution term also plays an important role in the energy transfer between streamwise part and cross-flow part in residual field. For the streamwise part of residual field, in certain rotate rates, the energy obtained from the streamwise part of secondary flows by the correlation between the vorticity of secondary flows and the shear stress of residual field is larger than that obtained from mean flow through mean shear, implying that the streamwise motions of secondary flows have a significant influence on the streamwise motions of residual field.

The intermittency boundary in stratified plane Couette flow

We study stratified turbulence in plane Couette flow using direct numerical simulations. Two external dimensionless parameters control the dynamics, the Reynolds number $\mathit{Re}=Uh/{\it\nu}$ and the bulk Richardson number $\mathit{Ri}=g{\it\alpha}_{V}Th/U^{2}$, where $U$ and $T$ are half the velocity and temperature difference between the two walls respectively, $h$ is the half channel depth, ${\it\nu}$ is the kinematic viscosity and $g{\it\alpha}_{V}$ is the buoyancy parameter. We focus on spatio-temporal intermittency due to stratification and we explore the boundary between fully developed turbulence and intermittent flow in the $\mathit{Re}{-}\mathit{Ri}$ plane. The structures populating the intermittent flow regime show coexistence between laminar and turbulent patches, and we demonstrate that there are qualitative differences between the previously studied low-$\mathit{Re}$ low-$\mathit{Ri}$ intermittent regime and the high-$\mathit{Re}$ high-$\mathit{Ri}$ intermittent regime. At low-$\mathit{Re}$ low-$\mathit{Ri}$, turbulent regions span the entire gap, whereas at high-$\mathit{Re}$ high-$\mathit{Ri}$, turbulence is confined vertically with complex dynamics arising from interacting turbulent layers. Consistent with a previous investigation of Flores & Riley (Boundary-Layer Meteorol., vol. 129 (2), 2010, pp. 241–259), we present evidence suggesting that intermittency in the asymptotic regime of high-$\mathit{Re}$ Couette flows appears for $L^{+}<200$, where $L^{+}=Lu_{{\it\tau}}/{\it\nu}$, with $L$ being the Monin–Obukhov length scale, $L=u_{{\it\tau}}^{3}/C_{{\it\kappa}}q_{w}$, $q_{w}$ the wall heat flux, $C_{{\it\kappa}}$ the von Kármán constant and $u_{{\it\tau}}=\sqrt{{\it\tau}_{w}/{\it\rho}_{0}}$ the friction velocity determined from the wall shear stress ${\it\tau}_{w}$, where ${\it\rho}_{0}$ is the constant background density. We also consider the mixing as quantified by various versions of the flux Richardson number $\mathit{Ri}_{f}$, defined as the ratio of the conversion rate from kinetic to potential energy to the turbulent kinetic energy injection rate due to shear. We investigate how laminar and turbulent regions separately contribute to the overall mixing. Remarkably, we find that although fluctuations are greatly suppressed in the laminar regions, $\mathit{Ri}_{f}$ does not change significantly compared with its value in turbulent regions. As we observe a tight coupling between the mean temperature and velocity fields, we demonstrate that both Monin–Obukhov self-similarity theory (Monin & Obukhov, Contrib. Geophys. Inst. Acad. Sci. USSR, vol. 151, 1954, pp. 163–187) and the explicit algebraic model of Lazeroms et al. (J. Fluid Mech., vol. 723, 2013, pp. 91–125) predict the mean profiles well. We thus use these models to trace out the boundary between fully developed turbulence and intermittency in the $\mathit{Re}{-}\mathit{Ri}$ plane.

Effects of the computational domain on the secondary flow in turbulent plane Couette flow**Project supported by the National Natural Science Foundation of China (Grant Nos. 11221061, 11272013, and 11302006).

A series of direct numerical simulations of the fully developed plane Couette flow at a Reynolds number of 6000 (based on the relative wall speed and half the channel height h) with different streamwise and spanwise lengths are conducted to investigate the effects of the computational box sizes on the secondary flow (SF). Our focuses are the number of counter-rotating vortex pairs and its relationship to the statistics of the mean flow and the SF in the small and moderate computational box sizes. Our results show that the number of vortex pairs is sensitive to the computational box size, and so are the slope parameter, the rate of the turbulent kinetic energy contributed by the SF, and the ratio of the kinetic energy of the SF to the total kinetic energy. However, the averaged spanwise width of each counter-rotating vortex pair in the plane Couette flow is found, for the first time, within 4(1 ± 0.25)h despite the domain sizes.

A comprehensible low-order model for wall turbulence dynamics

Streamwise vortices play an important role in the sustainment of wall turbulence. They are associated with regions with strong Reynolds shear stress production. In turbulent plane Couette flow at low Reynolds numbers, these streamwise vortices fill the whole gap between the plates. Using a low-order model obtained from the Navier-Stokes equations through a two-step Galerkin projection, the dynamics of these streamwise vortices is shown to be similar to the dynamics observed in the near-wall region of turbulent boundary layers [M. Lagha, J. Kim, J. D. Eldredge, and X. Zhong, “Near-wall dynamics of compressible boundary layer,” Phys. Fluids 23, 065109 (2011)]. A spanwise vortex filling the whole gap between the plates and with vorticity opposite in sign to that of the base flow, is tilted in the streamwise direction by the spanwise shear of the streaks. The resultant vortex has a crescent shape and its legs are two streamwise vortices. They regenerate the streaks by the lift-up effect. Through its ability to generate periodic spatio-temporal flow patterns, this model is shown to provide an ideal tool for studying the different mechanisms at work.

Turbulent plane Couette flow at moderately high Reynolds number

AbstractA new set of numerical simulations of turbulent plane Couette flow in a large box of dimension ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}20\pi h,\, 2h,\, 6\pi h$) at Reynolds number $(\mathit{Re}_{\tau }) =125$, 180, 250 and 550 is described and compared with simulations at lower Reynolds numbers, Poiseuille flows and experiments. The simulations present a logarithmic near-wall layer and are used to verify and revise previously known results. It is confirmed that the fluctuation intensities in the streamwise and spanwise directions do not scale well in wall units. The scaling failure occurs both near to and away from the wall. On the contrary, the wall-normal intensity scales in inner units in the near-wall region and in outer units in the core region. The spectral ridge found by Hoyas & Jiménez (Phys. Fluids, vol. 18, 2003, 011702) for the turbulent Poiseuille flow can also be seen in the present flow. Away from the wall, very large-scale motions are found spanning through all the length of the channel. The statistics of these simulations can be downloaded from the webpage of the Chair of Fluid Dynamics.

α effect in a turbulent liquid-metal plane Couette flow.

We calculate the mean electromotive force in plane Couette flows of a nonrotating conducting fluid under the influence of a large-scale magnetic field for driven turbulence. A vertical stratification of the turbulence intensity results in an α effect owing to the presence of horizontal shear. Here we discuss the possibility of an experimental determination of the components of the α tensor using both quasilinear theory and nonlinear numerical simulations. For magnetic Prandtl numbers of the order of unity, we find that in the high-conductivity limit the α effect in the direction of the flow clearly exceeds the component in spanwise direction. In this limit, α runs linearly with the magnetic Reynolds number Rm, while in the low-conductivity limit it runs with the product Rm·Re, where Re is the kinetic Reynolds number, so that for a given Rm the α effect grows with decreasing magnetic Prandtl number. For the small magnetic Prandtl numbers of liquid metals, a common value for the horizontal elements of the α tensor appears, which makes it unimportant whether the α effect is measured in the spanwise or the streamwise directions. The resulting effect should lead to an observable voltage of about 0.5 mV in both directions for magnetic fields of 1 kG and velocity fluctuations of about 1m/s in a channel of 50-cm height (independent of its width).

Novel features of a fully developed mixing-layer between co-flowing laminar and turbulent Couette flows

A new flow configuration has been proposed in which a bilateral mixing-layer exists in the junction between co-flowing laminar and turbulent plane Couette flows. Contrary to a classical plane mixing-layer, the present mixing-layer did neither grow in time nor in streamwise direction. However, the mixing zone varied with the distance from the stationary wall. A direct numerical simulation showed that very-large-scale flow structures were found in the turbulent part of the flow with Reynolds number 1300 based on half the velocity U1 of the fastest-moving wall and half of the distance 2h between the walls. The laminar-turbulent interface exhibited a large-scale meandering motion with frequency 0.014U1/h and wavelength about 25h. Large-scale Taylor-Görtler-like roll cells were observed in the nominally laminar flow region with Reynolds number 260. This tailor-made flow is particularly well suited for explorations of momentum transfer and intermittency in the vicinity of the laminar-turbulent interface.

Dissimilarity of Turbulent Fluxes of Momentum and Heat in Perturbed Turbulent Flows

The dissimilarity between the Reynolds stresses and the heat fluxes in perturbed turbulent channel and plane Couette flows was studied using direct numerical simulation. The results demonstrate that the majority of the dissimilarity was due to the difference between the wall-normal fluxes, while the difference between the streamwise fluxes was lower. The main causes for the dissimilarity were the production terms, followed by the velocity-pressure interaction terms. Further insights into the importance of the velocity-pressure interaction in the origin of the dissimilarity are provided using two-point correlation. Furthermore, an octant conditional averaged dataset reveals that not only the wall-normal heat flux but also the streamwise heat flux is strongly related to the wall-normal gradient of the mean temperature. A simple Reynolds-averaged Navier–Stokes (RANS) heat flux model is proposed as a function of the Reynolds stresses. A comparison of the direct numerical simulation data with an “a priori” prediction suggests that this simple model performs reasonably well.

0501 平板間クエット流における過渡的乱れの発生(OS5-1 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)

The onset of transient turbulence in plane Couette flow is identified as a homoclinic tangency with respect to a simple edge state for the Navier-Stokes equation, i.e., the gentle periodic orbit found by Kawahara & Kida (J. Fluid Mech., vol. 449, 2001, pp. 291-300). The tangency of a pair of distinct homoclinic orbits is numerically discovered at the critical Reynolds number Re ≡ Uh/ν = 241.3 (U, h, and ν being half the difference of the two wall velocities, half the wall separation, and the kinematic viscosity of fluid, respectively) above which transversal homoclinic points appear on the Poincar´e section to generate a transient chaos through a Smale horseshoe.

Dispersion and deposition mechanisms of particles suspended in a turbulent plane Couette flow

Inertial particle transfer in a turbulent plane Couette flow (C flow) was studied using Direct Numerical Simulation (DNS) of the flow combined with a Lagrangian particle tracking approach for particles with Stokes numbers (St) 5, 25 and 125. The particle concentration was assumed low enough, so that the simulations were done under one-way coupling condition.Our numerical simulations show that the particles suspended in a C flow (C particles) have velocities respectively smaller and larger than the carrier fluid velocities in the near-wall and core region. The C particles in comparison to the same sized particles suspended in a Poiseuille flow (P particles) with a same Reynolds number (Re), have larger fluctuating activities in the core, while the opposite was observed in the near-wall region. This fact is due to the Large Scale Structures (LSSs) existing in the core region of the C flows. The LSSs also strengthen the near-wall accumulation of C particles, so that these particles respect to the same sized P particles have more accumulation in the near-wall region. Moreover, the LSSs affect the deposition velocity of particles whereby among the P and C particles with smallest Stokes number (St=5), the C particles have larger deposition velocity while among the those ones with larger Stokes numbers (St=25,125) the opposite was the case.Finally, it was found that although the diffusional deposition mechanism for the C particles is less efficient than for the P ones, the number of deposited C particles during the time is greater than the number of deposited P particles.

On the decay of turbulence in plane Couette flow

The decay of turbulent and laminar oblique bands in the lower transitional range of plane Couette flow is studied by means of direct numerical simulations of the Navier-Stokes equations. We consider systems that are extended enough for several bands to exist, thanks to mild wall-normal under-resolution considered as a consistent and well-validated modelling strategy. We point out a two-stage process involving the rupture of a band followed by a slow regression of the fragments left. Previous approaches to turbulence decay in wall-bounded flows making use of the chaotic transient paradigm are reinterpreted within a spatiotemporal perspective in terms of large deviations of an underlying stochastic process.

Aspects of a turbulent-nonturbulent interface

Transport mechanisms at the interface between a laminar and a turbulent plane Couette flow are examined by means of DNS data of a statistically steady flow field. A mixing-layer is established which neither evolves in the streamwise direction nor in time. This novel flow configuration is perfectly suited for explorations of momentum transfer mechanisms, e.g. turbulent diffusion. Unexpected undulations characterize the large-scale interactions between the turbulent and the nominally non-turbulent part flow. However, surprisingly large velocity fluctuations are observed even in the low-Re half of the flow which otherwise would be laminar.

On the decay of turbulence in plane Couette flow

Upon decreasing the Reynolds number, plane Couette flow first forms alternately turbulent and laminar oblique bands out of featureless turbulence below some upper threshold Ret. These bands exist down to a global stability threshold Reg below which laminar flow ultimately prevails. We study the fragmentation and decay of these bands in systems that are extended enough for several bands to exist. We use direct numerical simulations appropriately tailored to deal with such large systems during long enough durations. We point out a two-stage process involving the rupture of a band and then its slow shrinking. Previous interpretations of turbulence decay in wall-bounded flows within the chaotic transient or spatiotemporal intermittency paradigms are discussed.

A Continuum Mechanical Gradient Theory with an Application to Fully Developed Turbulent Flows

In this study, a gradient theory of grade two, including dissipative boundary conditions based on an axiomatic conception of a nonlocal continuum theory for materials of grade n is presented. The total stress tensor of rank two in the equation of linear momentum contains two higher stress tensors of rank two and three. In the case of isotropic materials, both the tensors of rank two and three are tensor-valued functions of the second order strain rate tensor and its first gradient. So the vector valued differential equation of motion is of order four, where the necessary dissipative boundary conditions are generated by using so-called porosity tensors. This theory is applied to a velocity profile of turbulent plane Couette flow of water. The velocity distribution parameters are identified by a numerical optimization algorithm, using experimental data of velocity profile of plane Couette flow with tow moving vertical walls from the literature. In an experiment, Hagiwara et al. refined flow visualization technique and carried out a two-dimensional particle-tracking-velocimetry (PTV) measurement for the Couette flow. On the basis of this experimental data the material and porosity coefficients are identified by employing the Levenberg–Marquardt numerical algorithm for nonlinear optimization.

On Matrix-Free Computation of 2D Unstable Manifolds

Recently, a flexible and stable algorithm was introduced for the computation of two-dimensional (2D) unstable manifolds of periodic solutions to systems of ordinary differential equations. The main idea of this approach is to represent orbits in this manifold as the solutions of an appropriate boundary value problem (BVP). The BVP is underdetermined, and a one-parameter family of solutions can be found by means of arclength continuation. This family of orbits covers a piece of the manifold. The quality of this covering depends on the way the BVP is discretized, as do the tractability and accuracy of the computation. In this paper, we describe an implementation of the orbit continuation algorithm which relies on multiple shooting and Newton–Krylov continuation. We show that the number of time integrations necessary for each continuation step scales with the number of shooting intervals but not with the number of degrees of freedom of the dynamical system. The number of shooting intervals is chosen based on linear stability analysis to keep the conditioning of the BVP in check. We demonstrate our algorithm with two test systems: a low-order model of shear flow and a well-resolved simulation of turbulent plane Couette flow.

Velocity and temperature dissimilarity in fully developed turbulent channel and plane Couette flows

The natural dissimilarity or decorrelation of stream-wise velocity and temperature fluctuations in fully developed turbulent channel and plane Couette flows was studied using direct numerical simulation (DNS). For both of the flow configurations, a Reynolds number of about 150 was used based on the friction velocity and half the distance between walls. Buoyancy effects were neglected, and only results with a molecular Prandtl number, Pr, equal to 1 are presented. The boundary conditions for the thermal field were a uniform source of energy in the domain and isothermal wall temperature for the channel and Couette flow, respectively. The importance of those events responsible for wall-normal turbulent fluxes in the generation of axial velocity and temperature dissimilarity was examined using conditional probability. It was found that the dissimilarity in the whole domain was higher in Couette than in channel flow. It was also found that for wall-normal turbulent fluxes (momentum and heat), the averaged dissimilarity in the whole domain was slightly more correlated with those events in the second or fourth quadrant, according to the quadrant analysis technique. For channel flow, the importance of both kinds of events was similar, while for Couette flow there was a predominance in the generation of dissimilarity by those events in the fourth quadrant. Also, for both flow configurations and throughout the wall-normal direction, it was found that in the buffer region there was a predominance of events in the fourth quadrant associated with dissimilarity for both wall-normal turbulent fluxes. In the frequency domain, the distribution of energy showed that there was a high-frequency shift experienced from the wall towards the centerline by the temperature spectrum with regards to the axial velocity spectrum, for which the action of the fluctuations of the wall-normal velocity was the main cause. In the central region of the flow, on the other hand, there was a global convergence of all spectra towards the pressure spectrum, with this convergence lower for Couette flow. Finally, it is shown that the dissimilarity in developed conditions is caused by the greater correlation existing for the temperature fluctuation with the instantaneous axial pressure gradient than for the velocity fluctuation with the instantaneous axial pressure gradient.

On modelling transitional turbulent flows using under-resolved direct numerical simulations: the case of plane Couette flow

Direct numerical simulations have proven of inestimable help to our understanding of the transition to turbulence in wall-bounded flows. While the dynamics of the transition from laminar flow to turbulence via localised spots can be investigated with reasonable computing resources in domains of limited extent, the study of the decay of turbulence in conditions approaching those in the laboratory requires consideration of domains so wide as to exclude the recourse to fully resolved simulations. Using Gibson's C++ code ChannelFlow, we scrutinize the effects of a controlled lowering of the numerical resolution on the decay of turbulence in plane Couette flow at a quantitative level. We show that the number of Chebyshev polynomials describing the cross-stream dependence can be drastically decreased while preserving all the qualitative features of the solution. In particular, the oblique turbulent band regime experimentally observed in the upper part of the transitional range is extremely robust. In terms of Reynolds numbers, the resolution lowering is seen to yield a regular downward shift of the upper and lower thresholds Rt and Rg where the bands appear and break down. The study is illustrated with the results of two preliminary experiments.

A streamwise constant model of turbulence in plane Couette flow

Streamwise and quasi-streamwise elongated structures have been shown to play a significant role in turbulent shear flows. We model the mean behaviour of fully turbulent plane Couette flow using a streamwise constant projection of the Navier–Stokes equations. This results in a two-dimensional three-velocity-component (2D/3C) model. We first use a steady-state version of the model to demonstrate that its nonlinear coupling provides the mathematical mechanism that shapes the turbulent velocity profile. Simulations of the 2D/3C model under small-amplitude Gaussian forcing of the cross-stream components are compared to direct numerical simulation (DNS) data. The results indicate that a streamwise constant projection of the Navier–Stokes equations captures salient features of fully turbulent plane Couette flow at low Reynolds numbers. A systems-theoretic approach is used to demonstrate the presence of large input–output amplification through the forced 2D/3C model. It is this amplification coupled with the appropriate nonlinearity that enables the 2D/3C model to generate turbulent behaviour under the small-amplitude forcing employed in this study.

Transient turbulence in plane Couette flow

Plane Couette flow, the flow between two parallel plates moving in opposite directions, belongs to the group of shear flows where turbulence occurs while the laminar profile is stable. Experimental and numerical studies show that at intermediate Reynolds numbers turbulence is transient and that the lifetimes are distributed exponentially. However, these studies have remained inconclusive about a divergence in lifetimes above a critical Reynolds number. The extensive numerical results for flow in a box of width 2pi and length 8pi presented here cover observation times up to 12,000 units and show that while the lifetimes increase rapidly with Reynolds number, they do not indicate a divergence and therefore no transition to persistent turbulence.