Plane Channel Flow Research Articles

Prediction of Cleaning by Means of Computational Fluid Dynamics: Implication of the Pre‐wetting of a Swellable Soil

The cleaning behavior of a pre-wetted soil is studied experimentally and modeled numerically for the prototypical case of plane channel flow. One of the channel walls is soiled with a food-based model soil containing luminescent tracer particles to perform space- and time-resolved investigations of the cleaning process. Pre-wetting is applied for a few minutes before the soil removal is started with flow of Reynolds number up to 20,000. Physical model and simulation are based on a transient boundary condition to represent the behavior of the soil. Pre-wetting is taken into account by an initial removal of a certain amount of soil due to cohesive separation and the subsequent cleaning modeled as being limited by a diffusive process. Compared to a conventional multiphase simulation, the computation time is lowered by about three orders of magnitude. The results obtained with this elementary approach match the experiments astonishingly well.

Exponential growth of lifetime of localized turbulence with its extent in channel flow

The transition to turbulence in wall-bounded flows, for example pipe flow, plane Couette flow and channel flow, initially occurs as localized clusters of turbulence. It is well known that these localized structures are transient and decay in a memoryless process. In this paper we investigate the relationship between the extent of a localized structure and its lifetime. By performing numerical simulations of channel flow with small spanwise periodicity we produce streamwise-localized but spanwise-extended turbulence, turbulent ‘puffs’ instead of spots. We vary the spanwise periodzation length as the parameter of its extent to show that the lifetime of turbulent ‘puffs’ increases double-exponentially as a function of the Reynolds number for any spanwise extent. It is also shown that the lifetime increases exponentially as a function of the spanwise size when it is not large.

Exact coherent states of attached eddies in channel flow

A new set of exact coherent states in the form of a travelling wave is reported in plane channel flow. They are continued over a range in $Re$ from approximately $2600$ up to $30\,000$, an order of magnitude higher than those discovered in the transitional regime. This particular type of exact coherent states is found to be gradually more localised in the near-wall region on increasing the Reynolds number. As larger spanwise sizes $L_{z}^{+}$ are considered, these exact coherent states appear via a saddle-node bifurcation with a spanwise size of $L_{z}^{+}\simeq 50$ and their phase speed is found to be $c^{+}\simeq 11$ at all the Reynolds numbers considered. Computation of the eigenspectra shows that the time scale of the exact coherent states is given by $h/U_{cl}$ in channel flow at all Reynolds numbers, and it becomes equivalent to the viscous inner time scale for the exact coherent states in the limit of $Re\rightarrow \infty$. The exact coherent states at several different spanwise sizes are further continued to a higher Reynolds number, $Re=55\,000$, using the eddy-viscosity approach (Hwang & Cossu, Phys. Rev. Lett., vol. 105, 2010, 044505). It is found that the continued exact coherent states at different sizes are self-similar at the given Reynolds number. These observations suggest that, on increasing Reynolds number, new sets of self-sustaining coherent structures are born in the near-wall region. Near this onset, these structures scale in inner units, forming the near-wall self-sustaining structures. With further increase of Reynolds number, the structures that emerged at lower Reynolds numbers subsequently evolve into the self-sustaining structures in the logarithmic region at different length scales, forming a hierarchy of self-similar coherent structures as hypothesised by Townsend (i.e. attached eddy hypothesis). Finally, the energetics of turbulent flow is discussed for a consistent extension of these dynamical systems notions to high Reynolds numbers.

Parallel performance assessment of a pseudo-spectral solver for transition and turbulent boundary layer flows

The computational performance and accuracy of a parallel pseudo-spectral solver is assessed, and the solver is validated for direct numerical simulations (DNS) of bypass transition and turbulent boundary layer flows. The solver shows spatial order of accuracy of 6.8 and 3.7 for laminar and turbulent flow simulations, respectively, and scalability up to 16 thousand processors. DNS predictions for plane channel and flat plate bypass transition flows compare very well with the benchmark results available in the literature. Analysis of the turbulent structures shows counter-rotating longitudinal structures in the sub/buffer layers, which inducing lifting on each other to generate series of sweep and ejection events. These events generate shear stress which transfers energy from mean flow to streamwise turbulent fluctuations. The energy is then redistributed to other turbulent velocity fluctuations via pressure-strain. Temporal DNS of bypass pass transition flow compare very well with spatial DNS using an order of magnitude lower computational cost, thus is identified to be a viable alternative for transition flow physics simulations.

Dominant modes in hydromagnetic stability of channel flow with porous walls

A linear stability analysis of a plane channel flow with porous walls under a uniform cross-flow and an external transverse magnetic field is explored. The physical problem is governed by a system of combined equations of the hydrodynamic and those of Maxwell. The perturbed problem of base state leads to a modified classical Orr-Sommerfeld equation which is solved numerically using the Chebyshev spectral collocation method. The combined effects of the cross-flow Reynolds number and the Hartmann number on the dangerous mode of hydromagnetic stability are investigated.The study shows that, the magnetic field tends to suppress the instability occurred by cross-flow. This stabilizing effect becomes perceptible when the magnetic field produces a mode transition from walls mode to that of the center.

Lattice Boltzmann method simulation of thermal flow dynamics in a channel

Abstract. The objective of this paper is the simulation of thermal flow dynamics in a channel. The mathematical model in a two-dimensional formulation is described by Navier-Stokes equations, continuity and temperature equations. For the numerical simulation of the problem the Lattice Boltzmann method applying the D2Q9 model is used. The validity of this method is tested by comparing the numerical solution to the analytical solution of the planar channel flow and error rates are calculated for various sizes of the computational grid. The test problem of thermal Poiseuille flow in the channel was solved to deactivate the correctness of the developed algorithm. Very good agreement between the exact and numerical solution of this problem is shown.

An assessment of atypical mesh topologies for low-Mach large-eddy simulation

An implicit, low-dissipation, low-Mach, variable density control volume finite element formulation is used to explore foundational understanding of numerical accuracy for large-eddy simulation applications on hybrid meshes. Detailed simulation comparisons are made between low-order hexahedral, tetrahedral, pyramid, and wedge/prism topologies against a third-order, unstructured hexahedral topology. Using smooth analytical and manufactured low-Mach solutions, design-order convergence is established for the hexahedral, tetrahedral, pyramid, and wedge element topologies using a new open boundary condition based on energy-stable methodologies previously deployed within a finite-difference context. A wide range of simulations demonstrate that low-order hexahedral- and wedge-based element topologies behave nearly identically in both computed numerical errors and overall simulation timings. Moreover, low-order tetrahedral and pyramid element topologies also display nearly the same numerical characteristics. Although the superiority of the hexahedral-based topology is clearly demonstrated for trivial laminar, principally-aligned flows, e.g., a 1x2x10 channel flow with specified pressure drop, this advantage is reduced for non-aligned, turbulent flows including the Taylor–Green Vortex, turbulent plane channel flow (Reτ395), and buoyant flow past a heated cylinder. With the order of accuracy demonstrated for both homogenous and hybrid meshes, it is shown that solution verification for the selected complex flows can be established for all topology types. Although the number of elements in a mesh of like spacing comprised of tetrahedral, wedge, or pyramid elements increases as compared to the hexahedral counterpart, for wall-resolved large-eddy simulation, the increased assembly and residual evaluation computational time for non-hexahedral is offset by more efficient linear solver times. Finally, most simulation results indicate that modest polynomial promotion provides a significant increase in solution accuracy.

Spectral/hp element MHD solver

A new MHD solver, based on the Nektar++ spectral/hp element framework, is presented in this paper. The velocity and electric potential quasi-static MHD model is used. The Hartmann flow in plane channel and its stability, the Hartmann flow in rectangular duct, and the stability of Hunt's flow are explored as examples. Exponential convergence is achieved and the resulting numerical values were found to have an accuracy up to $10^{-12}$ for the state flows compared to an exact solution, and $10^{-5}$ for the stability eigenvalues compared to independent numerical results.

Normal lubrication force between spherical particles immersed in a shear-thickening fluid

In this work, the inverse bi-viscous model [Vazquez-Quesada et al., “Planar channel flow of a discontinuous shear-thickening model fluid: Theory and simulation,” Phys. Fluids 29, 103104 (2017)] is used to describe a shear-thickening fluid. An analytical velocity profile in a planar Poiseuille flow is utilized to calculate an approximate solution to the generalized lubrication force between two close spheres interacting hydrodynamically in such a medium. This approximate analytical expression is compared to the exact numerical solution. The flow topology of the shear-thickening transition within the interparticle gap is also shown and discussed in relation to the behaviour of the lubrication force. The present result can allow in the future to perform numerical simulations of dense particle suspensions immersed in a shear-thickening matrix based on an effective lubrication force acting between pairwise interacting particles. This model may find additional values in representing experimental systems consisting of suspensions in shear thickening media, polymer coated suspensions, and industrial systems such as concrete.

Asymptotic expansion and Padé approximants for gravity-driven flow of a heated granular gas: Competition between inelasticity and forcing, up to Burnett order

The gravity-driven planar channel flow of a heated granular gas is analyzed using a kinetic model with stochastic forcing to understand the roles of inelasticity and forcing on its hydrodynamics and rheology. The closed-form analytical solutions up to the fourth order in gravitational acceleration have been determined to analyze the hydrodynamic and rheological fields as functions of the restitution coefficient (${e}_{n}$) and the Froude number ${\mathrm{Fr}}_{0}$. It is found that the excess temperature [$\mathrm{\ensuremath{\Delta}}T={T}_{max}/T(0)\ensuremath{-}1$, i.e., the deviation of the maximum temperature ${T}_{max}$ of the gas from its centerline value $T(0)$] increases monotonically with decreasing ${e}_{n}$ above a critical value of the Froude number (${\mathrm{Fr}}_{0}g{\mathrm{Fr}}_{0}^{c}$), but has a ``nonmonotonic'' dependence with ${e}_{n}$ [i.e., $\mathrm{\ensuremath{\Delta}}T$ decreases with decreasing ${e}_{n}$ for ${e}_{n}\ensuremath{\in}(1,0.5)$, but increases for ${e}_{n}l0.5$] at ${\mathrm{Fr}}_{0}l{\mathrm{Fr}}_{0}^{c}$. This changeover from nonmonotonic to monotonic dependence of $\mathrm{\ensuremath{\Delta}}T$ with ${e}_{n}$ at ${\mathrm{Fr}}_{0}={\mathrm{Fr}}_{0}^{c}$ also holds for both first (${\mathcal{N}}_{1}$) and second (${\mathcal{N}}_{2}$) normal-stress differences as well as for tangential heat flux (${q}_{x}$). Phase diagrams are constructed in the (${\mathrm{Fr}}_{0},1\ensuremath{-}{e}_{n}$) plane, demarcating two regions in which the dependencies of rarefaction effects ($\mathrm{\ensuremath{\Delta}}T, {\mathcal{N}}_{1}, {\mathcal{N}}_{2}$, and ${q}_{x}$) on ${e}_{n}$ are monotonic and nonmonotonic. The inelasticity plays a ``dual'' role of decreasing (at ${\mathrm{Fr}}_{0}l{\mathrm{Fr}}_{0}^{c}$) and increasing (at ${\mathrm{Fr}}_{0}g{\mathrm{Fr}}_{0}^{c}$) the rarefaction effects with decreasing restitution coefficient ${e}_{n}$ from the elastic limit. This finding, based on the fourth-order solution, is in variance with the leading-order solution that predicts only a nonmonotonic dependence of the above quantities on ${e}_{n}$ for all ${\mathrm{Fr}}_{0}$ [Tij and Santos, J. Stat. Phys. 117, 901 (2004)]. The role of inelasticity on the region of convergence of the asymptotic series solutions are subsequently analyzed by determining the Pad\'e approximants for rheological fields [Rongali and Alam, Phys. Rev. E 98, 012115 (2018)]; the present solution is shown to have a larger range of validity in terms of both inelasticity and Froude number than its leading-order counterpart. Lastly, it is shown that the fourth-order solutions contain all Burnett-order terms (i.e., second order in the gradients of hydrodynamic fields) that can be obtained from the standard Chapman-Enskog expansion.

Simulation of jet cleaning: Diffusion model for swellable soils

A physical-numerical simulation model for the cleaning of swellable soils by diffusive dissolution or cohesive separation of small soil particles, which was developed targeting a low computational effort, is presented and validated. The flow calculation, based on the Reynolds averaged Navier Stokes equations (RANS), and the calculation of the soil transport, modelled following the unsteady RANS concept, are performed successively. The behaviour of the soil is modelled as a transient Dirichlet boundary condition, its parameters being determined in laboratory experiments using plane channel flow. A validation is performed for a coherent water jet impinging vertically on a substrate, which is covered by a starch-based model soil, at a distance of 100mm to the nozzle. Using a nozzle diameter of 1.69mm and pressures in the nozzle ranging from 1.5 to 5bar, corresponding to Reynolds numbers between 17700 and 55600, the wall shear stress in this configuration exceeds the values given in the laboratory tests by two orders of magnitude. Despite this difference, the simplicity of the model and the complexity of the test case, the simulated cleaning times correspond very well to the measured values. Thus, the targeted scalability and transferability of the model to other flow configurations are ensured.

Heat transfer regimes in fully developed plane-channel flows

This paper investigates heat transfer regimes in fully developed plane channel flows without considering the buoyancy effect. Analyzing the governing thermal energy equation, aided by direct numerical simulation (DNS) data, six heat transfer regimes are identified including (i) laminar flow and laminar heat transfer (LamF-LamH), (ii) transitional flow and laminar-like heat transfer (TraF-LamH), (iii) transitional flow and transitional heat transfer (TraF-TraH), (iv) turbulent flow but laminar-like heat transfer (TurF-LamH), (v) turbulent flow and transitional heat transfer (TurF-TraH), and (vi) turbulent flow and turbulent heat transfer (TurF-TurH). One key result is the clarification of a TurF-LamH regime which exists only for low Prandtl number fluid (Pr≪1). A critical non-dimensional number is determined as ReτPr1/2≲50 where Reτ is the Reynolds number defined by the channel half-height δ and the frictional velocity uτ. In the TurF-LamH regime, the simplified thermal energy equation yields a prediction of Nusselt number as Nu≈6.0. The clarification of the TurF-LamH regime provides valuable insight into the understanding of the Nusselt number data for liquid metals, which have very low Prandtl number. Another major finding is that, in the TurF-TurH regime, the Kader-Yaglom-style equation is shown to be better than the traditional power law correlations at predicting Nusselt number, for either low or high Prandtl numbers. No separate correlations are needed for the prediction of Nusselt number at low Prandtl numbers and high Prandtl numbers. Another advantage of the Kader-Yaglom-style equation is that the equation can be theoretically connected to the log-law for the mean velocity and the mean temperature. More importantly, in the TurF-TurH regime the prediction of Kader-Yaglom-style equation can be reliably extended to ultra high Reynolds numbers, for either low or high Prandtl numbers.

Towards the accurate prediction of the turbulent flow and heat transfer in low-Prandtl fluids

In the numerical simulation of turbulent heat transfer, a reliable representation of the turbulent flow field is a prerequisite in order to obtain an accurate prediction of the thermal field. In the framework of RANS modelling approach, classical two-equation eddy-viscosity models present intrinsic limitations for applications characterised by complex flow fields where significant turbulence anisotropy can be expected. In this respect, differential Reynolds stress models represent the natural choice for dealing with such highly anisotropic flows. On the other hand, the modelling of the turbulent heat flux term almost universally relies on the so-called Reynolds analogy, due to its simplicity and robustness. Nevertheless, this approach presents several well-known limitations, especially when dealing with low-Prandtl fluids. In this context, algebraic heat flux models are regarded as a promising approach to overcome the shortcomings of the Reynolds analogy. In particular, an implicit algebraic heat flux closure, called as AHFM-NRG, has been proposed for applications involving low-Prandtl fluids. In its original formulation, the algebraic turbulent heat flux closure was coupled with a low-Reynolds linear k-ε model for the closure of the Reynolds stress tensor. In the present work, the applicability of the AHFM-NRG model is extended to an advanced low-Reynolds differential Reynolds stress model. This new version of the model is validated against different planar channel flows at unity and low-Prandtl number; successively, the model is applied to two relatively complex flows, i.e. a planar impinging jet and a bare rod bundle configuration. It is shown that the coupling of the AHFM-NRG with an advanced closure for the Reynolds stresses gives encouraging results for these cases, where an accurate prediction of the anisotropy in the flow field results in a noticeable improvement in the prediction of the turbulent heat transfer.

Elliptic relaxation model for stably stratified turbulence

The article presents a new elliptic relaxation procedure for Second Moment Closure (SMC) turbulence modeling of wall-bounded, stably stratified flows. Specifically, it directly introduces buoyancy effects in the elliptic relaxation equation. The new formulation results in an additional length scale that produces a coupled set of Helmholtz equations. The study then extends the new technique to Reynolds scalar fluxes. Furthermore, it examines the budgets of the different terms of their transport equations near the wall and systematically derives the relevant boundary conditions. Comparison of model predictions with Direct Numerical Simulation (DNS) data for stably stratified plane channel flows available in the literature rounds off this paper.

Effects of Near Wall Modeling in the Improved-Delayed-Detached-Eddy-Simulation (IDDES) Methodology.

The present study aims to assess the effects of two different underlying RANS models on overall behavior of the IDDES methodology when applied to different flow configurations ranging from fully attached (plane channel flow) to separated flows (periodic hill flow). This includes investigating prediction accuracy of first and second order statistics, response to grid refinement, grey area dynamics and triggering mechanism. Further, several criteria have been investigated to assess reliability and quality of the methodology when operating in scale resolving mode. It turns out that irrespective of the near wall modeling strategy, the IDDES methodology does not satisfy all criteria required to make this methodology reliable when applied to various flow configurations at different Reynolds numbers with different grid resolutions. Further, it is found that using more advanced underlying RANS model to improve prediction accuracy of the near wall dynamics results in extension of the grey area, which may delay the transition to scale resolving mode. This systematic study for attached and separated flows suggests that the shortcomings of IDDES methodology mostly lie in inaccurate prediction of the dynamics inside the grey area and demands further investigation in this direction to make this methodology capable of dealing with different flow situations reliably.

Letter: The effects of streamwise system rotation on pressure fluctuations in a turbulent channel flow

In this letter, we report the modulating effects of streamwise system rotation on both the amplitude and the wavenumber of pressure fluctuations in a plane channel flow. The analysis of the pressure field is conducted based on a set of comprehensive direct numerical simulation data of six rotation numbers. It is observed that high pressure fluctuation regions collocate with the Taylor–Görtler-like (TGL) vortex cores. By decomposing the pressure field into rotation-induced and convection-induced parts, it is observed that the rotation-induced part dominates the total pressure fluctuations and facilitates the growth of TGL vortices. Furthermore, through a spectral analysis, it is discovered that the system rotation acts as a “linear amplifier,” which converts high-wavenumber low-amplitude streamwise vorticity fluctuations into low-wavenumber high-amplitude pressure fluctuations.

An inconsistence-free integral-based dynamic one- and two-parameter mixed model

ABSTRACTThe development of the dynamic procedure as well as the deeper understanding of the link between filtering, modelling and numerics, allowed Large Eddy Simulation (LES) to make great progresses during the last years. Among several modelling approaches, the scale-similar-based modelling is based on the observation that the smallest resolved scales are the most active in the interaction with the unresolved ones. Owing to the low dissipation introduced by the scale-similar models (SSMs), the coupling with the eddy-viscosity model is often used in the so-called mixed models. Dynamic version of mixed models is historically based on the application of the test-filtering on the differential form of the filtered momentum equation. Such an approach is used for both the one and the two-coefficients mixed models. The use of the differential form of the filtered equations produces the well-known mathematical inconsistence caused by the need to extract arbitrarily the model functions out of filtering. It is known that, along with the eddy viscosity assumption, the magnitude of the Germano identity error (GIE) is strongly influenced. The mathematical inconsistence in the extraction of the dynamic eddy viscosity coefficient was recently superseded by using the new integral-based formulation. However, owing to the intrinsic limits of the Smagorinsky model, also in those results, the GIE is still remarkable therefore, the present paper presents a new formulation to the integral-based dynamic procedure for both one and two-coefficients mixed models (IDMM). The original contributions of the present paper can be summarised: (1) A theoretical analysis comparing the spectral errors for the differential and integral-based SSM, assessing that the errors are less relevant for the integral form; (2) The implementation of one and two parameters IDMM for the simulation of turbulence in a plane channel flow, assessing the reduction of the GIE and the good behaviour of the statistics that are compared with those of the other LES codes used in the LESinItaly project.

Correlation coefficients of thermodynamic fluctuations in compressible aerodynamic turbulence

Thermodynamic fluctuations of pressure, density, temperature or entropy$\{p^{\prime },\unicode[STIX]{x1D70C}^{\prime },T^{\prime },s^{\prime }\}$in compressible aerodynamic turbulence, although generated by the flow, are fundamentally related to one another by the thermodynamic equation of state. Ratios between non-dimensional root-mean-square (r.m.s.) levels ($\text{CV}_{p^{\prime }}:=\bar{p}^{-1}\,p_{rms}^{\prime }$,$\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}:=\bar{\unicode[STIX]{x1D70C}}^{-1}\,\unicode[STIX]{x1D70C}_{rms}^{\prime }$,$\text{CV}_{T^{\prime }}:=\bar{T}^{-1}\,T_{rms}^{\prime }$), along with all possible 2-moment correlation coefficients$\{c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }},c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }},c_{p^{\prime }T^{\prime }},c_{s^{\prime }\unicode[STIX]{x1D70C}^{\prime }},c_{s^{\prime }T^{\prime }},c_{s^{\prime }p^{\prime }}\}$, represent, in the sense of Bradshaw (Annu. Rev. Fluid Mech., vol. 9, 1977, pp. 33–54), the thermodynamic turbulence structure of the flow. We use direct numerical simulation (DNS) data, both for plane channel flow and for sustained homogeneous isotropic turbulence, to determine the range of validity of the leading-order, formally$O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$, approximations of the exact relations between thermodynamic turbulence structure parameters. Available DNS data are mapped on the$(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,\text{CV}_{T^{\prime }},c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }})$-plane and their loci, identified using the leading-order approximations, highlight specific behaviour for different flows or flow regions. For the particular case of sustained compressible homogeneous isotropic turbulence, it is shown that the DNS data collapse onto a single curve corresponding to$c_{s^{\prime }T^{\prime }}\approxeq 0.2$(for air flow), while the approximation$c_{s^{\prime }p^{\prime }}\approxeq 0$fits reasonably well wall turbulence DNS data, providing building blocks towards the construction of simple phenomenological models.

Suspensions of finite-size neutrally buoyant spheres in turbulent duct flow

We study the turbulent square duct flow of dense suspensions of neutrally buoyant spherical particles. Direct numerical simulations (DNS) are performed in the range of volume fractions $\unicode[STIX]{x1D719}=0{-}0.2$, using the immersed boundary method (IBM) to account for the dispersed phase. Based on the hydraulic diameter a Reynolds number of 5600 is considered. We observe that for $\unicode[STIX]{x1D719}=0.05$ and 0.1, particles preferentially accumulate on the corner bisectors, close to the corners, as also observed for laminar square duct flows of the same duct-to-particle size ratio. At the highest volume fraction, particles preferentially accumulate in the core region. For plane channel flows, in the absence of lateral confinement, particles are found instead to be uniformly distributed across the channel. The intensity of the cross-stream secondary flows increases (with respect to the unladen case) with the volume fraction up to $\unicode[STIX]{x1D719}=0.1$, as a consequence of the high concentration of particles along the corner bisector. For $\unicode[STIX]{x1D719}=0.2$ the turbulence activity is reduced and the intensity of the secondary flows reduces to below that of the unladen case. The friction Reynolds number increases with $\unicode[STIX]{x1D719}$ in dilute conditions, as observed for channel flows. However, for $\unicode[STIX]{x1D719}=0.2$ the mean friction Reynolds number is similar to that for $\unicode[STIX]{x1D719}=0.1$. By performing the turbulent kinetic energy budget, we see that the turbulence production is enhanced up to $\unicode[STIX]{x1D719}=0.1$, while for $\unicode[STIX]{x1D719}=0.2$ the production decreases below the values for $\unicode[STIX]{x1D719}=0.05$. On the other hand, the dissipation and the transport monotonically increase with $\unicode[STIX]{x1D719}$. The interphase interaction term also contributes positively to the turbulent kinetic energy budget and increases monotonically with $\unicode[STIX]{x1D719}$, in a similar way as the mean transport. Finally, we show that particles move on average faster than the fluid. However, there are regions close to the walls and at the corners where they lag behind it. In particular, for $\unicode[STIX]{x1D719}=0.05,0.1$, the slip velocity distribution at the corner bisectors seems correlated to the locations of maximum concentration: the concentration is higher where the slip velocity vanishes. The wall-normal hydrodynamic and collision forces acting on the particles push them away from the corners. The combination of these forces vanishes around the locations of maximum concentration. The total mean forces are generally low along the corner bisectors and at the core, also explaining the concentration distribution for $\unicode[STIX]{x1D719}=0.2$.

Optimal mixing in three-dimensional plane Poiseuille flow at high Péclet number

We consider a passive zero-mean scalar field organised into two layers of different concentrations in a three-dimensional plane channel flow subjected to a constant along-stream pressure gradient. We employ a nonlinear direct-adjoint-looping method to identify the optimal initial perturbation of the velocity field with given initial energy which yields ‘maximal’ mixing by a target time horizon, where maximal mixing is defined here as the minimisation of the spatially integrated variance of the concentration field. We verify in three-dimensional flows the conjecture by Foures et al. (J. Fluid Mech., vol. 748, 2014, pp. 241–277) that the initial perturbation which maximises the time-averaged energy gain of the flow leads to relatively weak mixing, and is qualitatively different from the optimal initial ‘mixing’ perturbation which exploits classical Taylor dispersion. We carry out the analysis for two different Reynolds numbers ($Re=U_{m}h/\unicode[STIX]{x1D708}=500$ and $Re=3000$, where $U_{m}$ is the maximum flow speed of the unperturbed flow, $h$ is the channel half-depth and $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the fluid) demonstrating that this key finding is robust with respect to the transition to turbulence. We also identify the initial perturbations that minimise, at chosen target times, the ‘mix-norm’ of the concentration field, i.e. a Sobolev norm of negative index in the class introduced by Mathew et al. (Physica D, vol. 211, 2005, pp. 23–46). We show that the ‘true’ variance-based mixing strategy can be successfully and practicably approximated by the mix-norm minimisation since we find that the mix-norm-optimal initial perturbations are far less sensitive to changes in the target time horizon than their optimal variance-minimising counterparts.