Turbulent Plane Couette Flow Research Articles

Superposition of system response in modulated turbulent plane Couette flow

Superposition of system response in modulated turbulent plane Couette flow

Turbulent spot formation in three-dimensional yukawa liquids using large-scale molecular dynamics simulation—effect of system size

We have performed classical ‘first principles’ 3D Molecular Dynamics (MD) simulation of plane-Couette flow (PCF) in a 3D Yukawa liquid. The main aim of the present investigation is to study the effect of stream-wise (flow-direction) and span-wise (transverse direction to the flow) width on the subcritical transition to turbulence in plane Couette flow (PCF), separately. In the past, taking very large-aspect ratio systems, subcritical transition to turbulence in PCF has been studied in great detail. However, the effect of stream-wise and span-wise width on the turbulent dynamics separately, have not been studied in detail. In the present work, we have investigated the effect of stream-wise and span-wise width or length and found that the stream-wise length enhances the large-scale energy, which gives rise to strong large-scale flow, and span-wise width enhances the small-scale energy, which gives rise to strong small-scale structures. In other words, topology of the turbulent spot is observed to change with the change in system size. The spectral separation between the modes governing the large and small-scale dynamics improves with the increase in system sizes. The connection between the stream-wise vortices and the stream-wise velocity streaks is investigated in details. We have found that the number stream-wise velocity streaks is one unit larger than the number of stream-wise vortices. A qualitative comparison between the results obtained from the MD simulation and the hydrodynamics experiment is presented in this article. The behavior of the system was observed to vary with the range of interactions among the dust grains.

Modal-based generalised quasilinear approximations for turbulent plane Couette flow

Modal-based generalised quasilinear approximations for turbulent plane Couette flow

Theoretical and numerical analyses of turbulent plane Couette flow controlled using uniform blowing and suction

We theoretically and numerically investigate the control effect of the uniform blowing and suction (UB/US) in a turbulent plane Couette flow under a constant velocity difference condition in terms of the net energy saving. First, the effective range of UB/US amplitude is derived theoretically. According to the theoretical analysis, the effective range of UB/US amplitude decreases as the Reynolds number increases. The upper bound of the ratio between UB/US amplitude and the wall velocity, Vwc/Uw, is about 0.45% even at very low Reynolds numbers, i.e., Rew∼1000, while it is less than 0.1% at practically high Reynolds numbers, i.e., Rew>107. Next, using a direct numerical simulation (DNS), we investigate the control effect of UB/US in a turbulent plane Couette flow at Rew=3200. In the cases with smaller UB/US amplitude, i.e., Vw/Uw≤0.1%, little control effect can be observed. In the cases with larger UB/US amplitude, i.e., Vw/Uw≥0.3%, although not only a significant decrease of the contribution from the Reynolds shear stress (RSS) but also relaminarization of flow field can be observed, the total drag is increased due to the increase of the contribution from the UB/US beyond the decrease of the contribution from the RSS. Based on the present study, we can draw a conclusion that overall drag reduction by UB/US is hardly possible for turbulent Couette flows.

Dynamics of a data-driven low-dimensional model of turbulent minimal Couette flow

Because the Navier–Stokes equations are dissipative, the long-time dynamics of a flow in state space are expected to collapse onto a manifold whose dimension may be much lower than the dimension required for a resolved simulation. On this manifold, the state of the system can be exactly described in a coordinate system parameterising the manifold. Describing the system in this low-dimensional coordinate system allows for much faster simulations and analysis. We show, for turbulent Couette flow, that this description of the dynamics is possible using a data-driven manifold dynamics modelling method. This approach consists of an autoencoder to find a low-dimensional manifold coordinate system and a set of ordinary differential equations defined by a neural network. Specifically, we apply this method to minimal flow unit turbulent plane Couette flow at $Re=400$ , where a fully resolved solutions requires ${O}(10^5)$ degrees of freedom. Using only data from this simulation we build models with fewer than $20$ degrees of freedom that quantitatively capture key characteristics of the flow, including the streak breakdown and regeneration cycle. At short times, the models track the true trajectory for multiple Lyapunov times and, at long times, the models capture the Reynolds stress and the energy balance. For comparison, we show that the models outperform POD-Galerkin models with $\sim$ 2000 degrees of freedom. Finally, we compute unstable periodic orbits from the models. Many of these closely resemble previously computed orbits for the full system; in addition, we find nine orbits that correspond to previously unknown solutions in the full system.

Polymer diffusive instability leading to elastic turbulence in plane Couette flow

Polymer diffusive instability leading to elastic turbulence in plane Couette flow

Numerical study on turbulence modulation of finite-size particles in plane-Couette flow

Turbulent plane-Couette flow suspended with finite-size spheroidal particles is studied using fully particle-resolved direct numerical simulations. The effects of particle aspect ratio on turbulent arguments and particle statistics are explored, leading to the same conclusions as the previous experimental findings (Wang et al., J. Fluid Mech., vol. 937, 2022, A15). By performing stress analysis, we find that the presence of particles introduces extra stresses to the system and accounts for the global drag increases. The particle-laden flow cases exhibit spectra that are consistent with the scalings $k^{-5/3}$ and $k^{-3}$ in the large and small scales, respectively. While the $k^{-3}$ scaling observed in the particle-laden flow is reminiscent of bubbly flow, an examination of the particle Reynolds number suggests that the mechanism responsible may not be attributable to the pseudo-turbulence induced by particles as in the case of bubbles. In the view of particle statistics, we observe that spherical and non-spherical particles cluster preferentially in the near-wall and the bulk region, respectively, and that the orientations of non-spherical particles are affected by their aspect ratios, especially in the near-wall region. The present numerical results, combined with previous experimental findings in Wang et al. (J. Fluid Mech., vol. 937, 2022, A15), provide in-depth information on both the fluid and the particle phase, contributing to a better understanding of particle suspension in shear flows.

Study of compressible turbulent plane Couette flows via direct numerical simulation

Compressible turbulent plane Couette flows are studied via direct numerical simulation for wall Reynolds numbers up to $Re_w=10\ 000$ and wall Mach numbers up to $M_w=5$ . Various turbulence statistics are compared with their incompressible counterparts at comparable semilocal Reynolds numbers $Re^*_{\tau,c}$ . The skin friction coefficient $C_f$ , which decreases with $Re_w$ , only weakly depends on $M_w$ . On the other hand, the thermodynamic properties (mean temperature, density and others) strongly vary with $M_w$ . Under proper scaling transformations, the mean velocity profiles for the compressible and incompressible cases collapse well and show a logarithmic region with the Kárman constant $\kappa =0.41$ . Compared with wall units, the semilocal units yield a better collapse for the profiles of the Reynolds stresses. While the wall-normal and spanwise Reynolds stress components slightly decrease in the near-wall region, the inner peak of the streamwise component notably increases with increasing $M_w$ – indicating that flow becomes more anisotropic when compressible. In addition, the near-wall turbulence production decreases as $M_w$ increases – due to rapid wall-normal changes of viscosity caused by viscous heating. The streamwise and spanwise energy spectra show that the length scale of near-wall coherent structures does not vary with $M_w$ in semilocal units. Consistent with those in incompressible flows, the superstructures (the large-scale streamwise rollers) with a typical spanwise scale of $\lambda _z/h\approx 1.5{\rm \pi}$ become stronger with increasing $Re_w$ . For the highest $Re_w$ studied, they contribute about $40\,\%$ of the Reynolds shear stress at the channel centre. Interestingly, flow visualization and correlation analysis show that the streamwise coherence of these structures degrades with increasing $M_w$ . In addition, at comparable $Re^*_{\tau,c}$ , the amplitude modulation of these structures on the near-wall small scales is quite similar between incompressible and compressible cases – but much stronger than that in plane Poiseuille flows.

Maximum drag enhancement asymptote in spanwise-rotating viscoelastic plane Couette flow of dilute polymeric solutions

The existence of a maximum drag enhancement (MDE) asymptote at high rotation ( $Ro$ ) and Weissenberg ( $Wi$ ) numbers in turbulent viscoelastic spanwise-rotating plane Couette flow has been demonstrated. Specifically, it is shown that above a critical $Wi$ , drag enhancement plateaus and the MDE asymptote is realized in a broad range of $Ro$ . The mean velocity profiles at MDE appear to closely follow a log-law profile that has a nearly identical slope but different intercepts as a function of $Ro$ . Much like the maximum drag reduction (MDR) asymptote, the logarithmic function in MDE is closely followed if the mean velocity is plotted using the traditional inner variable scaling; however, the logarithmic function is not well defined when examined by the indicator function. Hence, in this study, we have used the logarithmic fit as a visual guide for the mean velocity profile. Last and perhaps the most intriguing finding of this study is that MDE occurs in the elasto-inertial turbulence (EIT) flow state; hence, it is mainly sustained by elastic forces much like the MDR flow state. To that end, a universal picture of elastically induced drag modification asymptotes is emerging, namely these asymptotic states are an inherent property of the elastically sustained EIT flow state.

Spectral Analysis on Transport Budgets of Turbulent Heat Fluxes in Plane Couette Turbulence

In recent years, scale-by-scale energy transport in wall turbulence has been intensively studied, and the complex spatial and interscale transfer of turbulent energy has been investigated. As the enhancement of heat transfer is one of the most important aspects of turbulence from an engineering perspective, it is also important to study how turbulent heat fluxes are transported in space and in scale by nonlinear multi-scale interactions in wall turbulence as well as turbulent energy. In the present study, the spectral transport budgets of turbulent heat fluxes are investigated based on direct numerical simulation data of a turbulent plane Couette flow with a passive scalar heat transfer. The transport budgets of spanwise spectra of temperature fluctuation and velocity-temperature correlations are investigated in detail in comparison to those of the corresponding Reynolds stress spectra. The similarity and difference between those scale-by-scale transports are discussed, with a particular focus on the roles of interscale transport and spatial turbulent diffusion. As a result, it is found that the spectral transport of the temperature-related statistics is quite similar to those of the Reynolds stresses, and in particular, the inverse interscale transfer is commonly observed throughout the channel in both transport of the Reynolds shear stress and wall-normal turbulent heat flux.

Spectral analysis on dissimilarity between turbulent momentum and heat transfers in plane Couette turbulence

Nonlinear interactions between different scales in turbulence result in both interscale and spatial transport of turbulent energy, and their role in the turbulent heat transfer mechanism is also of practical importance from an engineering viewpoint. In this study, we investigate a turbulent plane Couette flow with passive-scalar heat transfer at the Prandtl number of 0.71 to discuss the similarity/difference between scale interactions in velocity and temperature fields. The constant-temperature-difference boundary condition is used so that the mean velocity and temperature profiles are similar, and then, the roles of interscale and spatial transports are compared for the spectral transport budgets of turbulent energies and temperature-related statistics. We show that turbulent heat transfer occurs at relatively small streamwise length scales compared to momentum transfer, although molecular diffusion is more significant in the temperature field as the Prandtl number is less than 1. Detailed analysis on the transport budgets of temperature-related spectra shows that scale interactions in the temperature field supply more energy to small scales than those in the velocity field. This significant temperature cascade causes more energetic temperature fluctuation at small scales, resulting in the spectral dissimilarity between turbulent heat and momentum transfers.

The effect of modulated driving on non-rotating and rotating turbulent plane Couette flow

Direct numerical simulations of turbulent non-rotating and rotating plane Couette flow with a periodically modulated plate velocity are conducted to study the effect of modulated forcing on turbulent shear flows. The time-averaged shear Reynolds number is fixed at $Re_S = 3 \times 10^4$ , which results in a frictional Reynolds number of approximately $Re_\tau \approx 400$ . The modulating frequency is varied in the range $Wo\in (20,200)$ , while the modulating amplitude is kept fixed at $10\,\%$ of the shear velocity except to demonstrate that varying this parameter has minimal effect. The resulting shear at the plates is found to be independent of the forcing frequency, and equal to the non-modulated baseline. For the non-rotating simulations, two clear flow regions can be seen: a near-wall region that follows Stokes’ theoretical solution, and a bulk region that behaves similar to Stokes’ solutions but with an increased effective viscosity. For high driving frequencies, the amplitude response follows the scaling laws for modulated turbulence of von der Heydt et al. (Phys. Rev. E, vol. 67, 2003, 046308). Cyclonic rotation is not found to modify the system's behaviour in a substantial way, but anti-cyclonic rotation changes significantly the system's response to periodic forcing. We find that the persistent axial inhomogeneities introduced by mild anti-cyclonic rotation make it impossible to measure the propagation of the modulation adequately, while stronger anti-cyclonic rotation creates regions where the modulation travels instantaneously.

Structure of turbulence in planar rough Couette flows

On roughening one of the walls in a planar Couette flow, it was reported that turbulence augments near the opposite wall [Javanappa and Narasimhamurthy, “Turbulent plane Couette flow with a roughened wall,” Phys. Rev. Fluids 6, 104609 (2021)]. The current direct numerical simulation work further explores this interesting phenomenon by investigating the flow dynamics and anisotropic nature of turbulence. For roughening, transverse square ribs are placed only on the bottom wall with streamwise pitch separations s=5r and 10r, where r=0.2h is the rib height and h is the channel half height. The time series of spanwise vorticity fluctuation in the case of s=10r shows the presence of coherent Kelvin–Helmholtz-like structures behind the ribs. Phase analysis using Hilbert transform reveals that the flow within the cavity for the s=5r case is in-phase, while a phase shift is observed for the s=10r case. The visualization of enstrophy production rate (ωiSijωj) reveals that regions of intense positive ones are observed to be topologically “sheet-like,” while the regions of negative ones are found to be “spotty.” Anisotropy tensors and anisotropic invariant maps are used to explore turbulence anisotropy at both large and small scales of motion. It is observed that anisotropy is reduced in both the cases near the vicinity of roughness.

A sparse optimal closure for a reduced-order model of wall-bounded turbulence

In the present study, a set of physics-informed and data-driven approaches are examined towards the development of an accurate reduced-order model for a turbulent plane Couette flow. Based on the utilisation of the proper orthogonal decomposition (POD), a particular focus is given to the development of a reduced-order model where the number of POD modes are not large enough to cover the full dynamics of the given turbulent state, the situation directly relevant to the reduced-order modelling for turbulent flows. Starting from the conventional Galerkin projection approach ignoring the truncation error, three approaches enhanced by both physics and data are examined: (1) sparse regression of the POD-Galerkin dynamics; (2) Galerkin projection with an empirical eddy-viscosity model; (3) Galerkin projection with an optimal eddy viscosity obtained from a newly proposed sparse regression – an idea applying the sparse identification of nonlinear dynamics framework to an eddy-viscosity model. The sparse regression of the POD-Galerkin dynamics does not perform well, as the number of POD modes for the given chaotic dynamics appears to be too small. While the unsatisfactory performance of the Galerkin projection model with an empirical eddy viscosity is observed, the newly proposed approach, which combines the concept of an optimal eddy-viscosity closure with a sparse regression, more accurately approximates the chaotic dynamics than the other reduced-order models considered. This is demonstrated with the mean and time scales of the POD mode amplitudes as well as the first- and second-order turbulence statistics.

Modeling the pressure strain correlation in turbulent flows using deep neural networks

The pressure strain correlation plays a critical role in the Reynolds stress transport modeling. Accurate modeling of the pressure strain correlation leads to the proper prediction of turbulence stresses and subsequently the other terms of engineering interest. However, classical pressure strain correlation models are often unreliable for complex turbulent flows. Machine learning–based models have shown promise in turbulence modeling, but their application has been largely restricted to eddy viscosity–based models. In this article, we outline a rationale for the preferential application of machine learning and turbulence data to develop models at the level of Reynolds stress modeling. As an illustration, we develop data-driven models for the pressure strain correlation for turbulent channel flow using neural networks. The input features of the neural networks are chosen using physics-based rationale. The networks are trained with the high-resolution DNS data of turbulent channel flow at different friction Reynolds numbers (Reλ). The testing of the models is performed for unknown flow statistics at other Reλ and also for turbulent plane Couette flows. Based on the results presented in this article, the proposed machine learning framework exhibits considerable promise and may be utilized for the development of accurate Reynolds stress models for flow prediction.

Direct Numerical Simulation of Stratified Turbulent Flows and Passive Tracer Transport on HPC Systems: Comparison of CPU Architectures

In this paper we assess the influence of CPU architectures commonly used in HPC systems on the efficiency of the implementation of algorithms used for direct numerical simulation (DNS) of turbulent flows. We consider a stably stratified turbulent plane Couette flow as a benchmark problem supplemented with the additional transport of passive substances. The comparison includes the Intel Xeon, AMD Rome x86 CPU architecture processors and the Huawei Kunpeng ARM CPU processor. We discuss the role of memory-oriented optimizations on the efficiency of tracer transport implementation on each platform.

Collapse of transitional wall turbulence captured using a rare events algorithm

This text presents one of the first successful applications of a rare events sampling method for the study of multistability in a turbulent flow without stochastic energy injection. The trajectories of collapse of turbulence in plane Couette flow, and their probability and rate of occurrence are systematically computed using adaptive multilevel splitting (AMS). The AMS computations are performed in a system of size $L_x\times L_z=24\times 18$ at Reynolds number $R=370$ with an acceleration by a factor ${O}(10)$ with respect to direct numerical simulations (DNS) and in a system of size $L_x\times L_z=36\times 27$ at Reynolds number $R=377$ with an acceleration by a factor ${O}(10^3)$ . The AMS results are validated by a comparison with DNS in the smaller system. Visualisations indicate that turbulence collapses because the self-sustaining process of turbulence fails locally. The streamwise vortices decay first in streamwise elongated holes, leaving streamwise invariant streamwise velocity tubes that experience viscous decay. These holes then extend in the spanwise direction. The examination of more than a thousand trajectories in the $(E_{k,x}=\int u_x^2/2\,\textrm {d}^3\boldsymbol {x},E_{k,y-z}=\int (u_y^2/2+u_z^2/2)\,\textrm {d}^3\boldsymbol {x})$ plane in the smaller system confirms the faster decay of streamwise vortices and shows concentration of trajectories. This hints at an instanton phenomenology in the large size limit. The computation of turning point states, beyond which laminarisation is certain, confirms the hole formation scenario and shows that it is more pronounced in larger systems. Finally, the examination of non-reactive trajectories indicates that both the vortices and the streaks reform concomitantly when the laminar holes close.

Ascending–descending and direct–inverse cascades of Reynolds stresses in turbulent Couette flow

The interaction between small- and large-scale structures and the coexisting bottom-up and top-down processes are studied in a turbulent plane Couette flow, where space-filling longitudinal rolls appear at relatively low values of the Reynolds number $Re$ . A direct numerical simulation database at $Re_\tau =101$ is built to replicate the highest $Re$ considered in recent experimental work by Kawata & Alfredsson (Phys. Rev. Lett., vol. 120, 2018, 244501). Our study is based on the exact budget equations for the second-order structure function tensor $\left \langle {\delta u_i \delta u_j} \right \rangle$ , i.e. the anisotropic generalized Kolmogorov equations (AGKE). The AGKE study production, redistribution, transport and dissipation of every Reynolds stress tensor component, considering simultaneously the physical space and the space of scales, and properly define the concept of scale in the inhomogeneous wall-normal direction. We show how the large-scale energy-containing motions are involved in the production and redistribution of the turbulent fluctuations. Both bottom-up and top-down interactions occur, and the same is true for direct and inverse cascading. The wall-parallel components $\left \langle {\delta u \delta u} \right \rangle$ and $\left \langle {\delta w \delta w} \right \rangle$ show that both small and large near-wall scales feed the large scales away from the wall. The wall-normal component $\left \langle {\delta v \delta v} \right \rangle$ is different, and shows a dominant top-down dynamics, being produced via pressure-strain redistribution away from the wall and transferred towards near-wall larger scales via an inverse cascade. The off-diagonal component shows a top-down interaction, with both direct and inverse cascades, albeit the latter takes place within a limited range of scales.

Turbulent plane Couette flow with a roughened wall

Using direct numerical simulations, we studied the effects of roughness in a turbulent plane Couette flow where one wall is rough and the other one is smooth. Two different roughness configurations are simulated, and the results are compared with the case where both walls are smooth. Upon roughening a wall, roughness would generate vortical structures near the vicinity of the roughness elements and enhance turbulence locally. In the current study, however, we report that turbulence is enhanced near both the rough and the smooth walls.

Residual-based closure model for density-stratified incompressible turbulent flows

This paper presents a locally and dynamically adaptive residual-based closure model for density stratified incompressible flows. The method is based on the three-level form of the Variational Multiscale (VMS) modeling paradigm applied to the system of incompressible Navier–Stokes equations and an energy conservation equation for the relative temperature field. The velocity, pressure, and relative temperature fields are additively decomposed into overlapping scales which leads to a set of coupled mixed-field sub-problems for the coarse- and the fine-scales. In the hierarchical application of the VMS method, the fine-scale velocity and relative temperature fields are further decomposed, leading to a nested system of two-way coupled fine-scale level-I and level-II variational subproblems. A direct application of bubble functions approach to the fine-scale variational equations helps derive fine-scale models that are nonlinear and time dependent. Embedding the derived model from the level-II variational equation in the level-I variational equation helps stabilize the convection-dominated mixed-field thermodynamic subproblem. Locally resolving the unconstrained level-I variational equation yields the residual-based turbulence model which is a function of the residual of the Euler–Lagrange equations of the conservation of momentum, mass, and energy. The derived model accommodates forward- and back-scatter of energy and entropy and embeds sub-grid scale physics in the computable scales of the problem. The steps of the derivation show that it is essential to apply the concept of scale separation systematically to the coupled system of equations and it is critical to preserve the coupling between flow and thermal phases in the fine-scale variational equations. The method has been implemented with hexahedral and tetrahedral elements with equal order interpolations for the velocity, pressure, and temperature fields. Several canonical flow cases are presented that include Rayleigh–Bénard instability, Rayleigh–Taylor instability, and turbulent plane Couette flow with stable stratification.