Thwaites (Aeronaut. Q., vol. 1, 1949, pp. 245–280) developed an approximate method for determining the evolution of laminar boundary layers. The approximation follows from an assumption that the growth of a laminar boundary layer in the presence of pressure gradients could be parameterized solely as a function of the Holstein–Bohlen flow parameter, thus reducing the von Kármán momentum integral to a first-order ordinary differential equation. This method is useful for the analysis of laminar flows, and in computational potential flow solvers to account for the viscous effects. In this work, an approximate method for determining the momentum thickness of a two-dimensional, turbulent boundary layer is proposed following Thwaites’ work. It is shown that the method provides good estimates of the momentum thickness for multiple boundary layers, including both favourable and adverse pressure gradient effects, up to the point of separation. In the limit of high Reynolds numbers, it is possible to derive a criterion for the onset of separation from the proposed model, which is shown to be in agreement with prior empirical observations (Alber, 9th Aerospace Sciences Meeting, 1971). The sensitivity of the separation location with respect to upstream perturbations is also analysed through this model for the NASA/Boeing speed bump and the transonic Bachalo–Johnson bump.

Direct numerical simulations of oscillatory flow over a bed made of ripples have been performed. Two oscillatory flow forcing mechanisms have been compared: (i) a sinusoidal external pressure gradient (pressure-driven flow); and (ii) a sinusoidal velocity boundary conditions on the rippled bed (shear-driven flow). In the second case, the oscillations of the bed are such that when observed from a reference frame fixed with the bed, the free stream follows the same harmonic oscillation as in the pressure-driven case. While the outer layers have the same dynamics in the two cases, close to the bed differences are observed during the cycle, mostly because the large form drag across the ripples cannot be reproduced in the shear-driven case. A comparison against experimental data from an oscillating tray apparatus provides a relatively good agreement for the phase-averaged flow when the same forcing is considered (i.e. a shear-driven flow). The pressure-driven case has a comparable error to the shear-driven numerical results over the crest of the ripples, whereas the discrepancy is larger at the troughs. The discrepancies between the two cases are more limited for time-averaged flow quantities, such as the mean flow pattern and the time-averaged Reynolds stress distribution. This suggests that numerical or experimental shear-driven configurations may capture well the net effects of coastal transport processes (which occur in pressure-driven oscillatory flow), but care should be exercised in interpreting phase-dependent dynamics near the troughs. More work is needed to fully assess the sensitivity to the forcing mechanisms in different flow regimes.

This study aims to quantify how turbulence in a channel flow mixes momentum in the mean sense. We applied the macroscopic forcing method (Mani & Park, Phys. Rev. Fluids, 2021, 054607) to direct numerical simulation (DNS) of a turbulent channel flow at $Re_\tau =180$ using two different forcing strategies that are designed to separately assess the anisotropy and non-locality of momentum mixing. In the first strategy, the leading term of the Kramers–Moyal expansion of the eddy viscosity is quantified, revealing all 81 tensorial coefficients that essentially characterise the local-limit eddy viscosity. The results indicate the following: (1) the eddy viscosity has significant anisotropy, (2) Reynolds stresses are generated by both the mean strain rate and mean rotation rate tensors associated with the momentum field and (3) the local-limit eddy viscosity generates asymmetric Reynolds stress tensors. In the second strategy, the eddy viscosity is quantified as an integration kernel revealing the non-local influence of the mean momentum gradient at each wall-normal coordinate on all nine components of the Reynolds stresses over the channel width. Our results indicate that while the shear component of the Reynolds stress is reasonably reproduced by the local mean gradients, other components of the Reynolds stress are highly non-local. These results provide an understanding of anisotropy and non-locality requirements for closure modelling of momentum transport in attached wall-bounded turbulent flows.

Turbulent shear flows driven by a combination of a pressure gradient and buoyancy forcing are investigated using direct numerical simulations. Specifically, we consider the set-up of a differentially heated vertical channel subject to a Poiseuille-like horizontal pressure gradient. We explore the response of the system to its three control parameters: the Grashof number $Gr$, the Prandtl number $Pr$, and the Reynolds number $Re$ of the pressure-driven flow. From these input parameters, the relative strength of buoyancy driving to the pressure gradient can be quantified by the Richardson number $Ri=Gr/Re^2$. We compare the response of the mixed vertical convection configuration to that of mixed Rayleigh–Bénard convection, and find a nearly identical behaviour, including an increase in wall friction at higher $Gr$, and a drop in the heat flux relative to natural convection for $Ri=O(1)$. This closely matched response is despite vastly different flow structures in the systems. No large-scale organisation is visible in visualisations of mixed vertical convection – an observation that is confirmed quantitatively by spectral analysis. This analysis, combined with a statistical description of the wall heat flux, highlights how moderate shear suppresses the growth of small-scale plumes and reduces the likelihood of extreme events in the local wall heat flux. Vice versa, starting from a pure shear flow, the addition of thermal driving enhances the drag due to the emission of thermal plumes.

We investigate a turbulent boundary layer (TBL) under uniform pressure force variations, focusing on understanding its response to local pressure force, local pressure force variation (local disequilibrating effect) and upstream history. The flow starts as a zero-pressure-gradient (ZPG) TBL, followed by a uniform increase in the ratio of pressure force to turbulent force in the outer region and concludes with a uniform decrease of the same magnitude. This last zone includes a subzone with a diminishing adverse pressure gradient (APG), followed by an increasing favourable pressure gradient (FPG). In both subzones, the impact remains the same: mean momentum gain and turbulence reduction. In the outer region, the mean flow responds to force balance changes with a considerable delay. The accumulated flow history leads to a FPG TBL at the domain's end with a momentum defect comparable to APG TBLs. Below $y^+=10$, the mean flow responds almost instantaneously to pressure force changes. In the overlap layer, velocity profiles deviate from the conventional logarithmic law of ZPG TBL. Outer-layer turbulence decays more slowly than it increases initially, the latter turbulence increase persisting even after the pressure force begins to decrease. As a result of the slow turbulence decay, the FPG TBL at the domain's end exhibits unusually high outer turbulence levels. Near the wall, turbulence responds with a delay to pressure force changes, partly due to the influence of large-scale turbulence. All these significant cumulative effects of continuous pressure force variation indicate that parameters based solely on local variables cannot fully describe the physics of non-equilibrium TBLs.

High-fidelity simulations are conducted to investigate the turbulent boundary layers around a finite-span NACA0012 wing with a rounded wing-tip geometry at a chord-based Reynolds number of $Re_c=200\,000$ and at various angles of attack up to $10^\circ$. The study aims to discern the differences between the boundary layers on the finite-span wing and those on infinite-span wings at equivalent angles of attack. The finite-span boundary layers exhibit: (i) an altered streamwise and a non-zero spanwise pressure gradient as a result of the variable downwash induced by the wing-tip vortices (an inviscid effect typical of finite-span wings); (ii) differences in the flow history at different wall-normal distances, caused by the variable flow angle in the wall-normal direction (due to constant pressure gradients and variable momentum normal to the wall); (iii) laminar flow entrainment into the turbulent boundary layers near the wing tip (due to a laminar–turbulent interface); and (iv) variations in boundary layer thickness across the span, attributed to the variable wall-normal velocity in that direction (a primarily inviscid effect). These physical effects are then used to explain the differences in the Reynolds stress profiles and other boundary layer quantities, including the reduced near-wall peak of the streamwise Reynolds stress and the elevated Reynolds stress levels near the boundary layer edge, both observed in the finite-span wings. Other aspects of the flow, such as the downstream development of wing-tip vortices and their interactions with the surrounding flow, are reserved for future investigations.

The stirring and mixing of heat and momentum in the ocean surface boundary layer (OSBL) are dominated by 1 to 10 km fluid flows – too small to be resolved in global and regional ocean models. Instead, these processes are parametrized. Two main parametrizations include vertical mixing by surface-forced metre-scale turbulence and overturning by kilometre-scale submesoscale frontal flows and instabilities. In present models, these distinct parametrizations are implemented in tandem, yet ignore meaningful interactions between these two scales that may influence net turbulent fluxes. Using a large-eddy simulation of frontal spin down resolving processes at both scales, this work diagnoses submesoscale and surface-forced turbulence impacts that are the foundation of OSBL parametrizations, following a traditional understanding of these flows. It is shown that frontal circulations act to suppress the vertical buoyancy flux by surface forced turbulence, and that this suppression is not represented by traditional boundary layer turbulence theory. A main result of this work is that current OSBL parametrizations excessively mix buoyancy and overestimate turbulence dissipation rates in the presence of lateral flows. These interactions have a direct influence on the upper ocean potential vorticity and energy budgets with implications for global upper ocean budgets and circulation.

The present study investigates streamwise ($\overline {u^2}$) energy-transfer mechanisms in the inner and outer regions of turbulent boundary layers (TBLs). Particular focus is placed on the $\overline {u^2}$ production, its inter-component and wall-normal transport as well as dissipation, all of which become statistically significant in the outer region with increasing friction Reynolds number ($Re_{\tau }$). These properties are analysed using published data sets of zero, weak and moderately strong adverse-pressure-gradient (APG) TBLs across a decade of $Re_{\tau }$, revealing similarity in energy-transfer pathways for all these TBLs. It is found that both the inner and outer peaks of $\overline {u^2}$ are always associated with local maxima in the $\overline {u^2}$ production and its inter-component transport, and the regions below/above each of these peaks are always dominated by wall-ward/away-from-wall transport of $\overline {u^2}$, thereby classifying the $\overline {u^2}$ profiles into four distinct regimes. This classification reveals existence of phenomenologically similar energy-transfer mechanisms in the ‘inner’ and ‘outer’ regions of moderately strong APG TBLs, which meet at an intermediate location coinciding with the minimum in $\overline {u^2}$ profiles. Conditional averaging suggests existence of similar phenomena even in low $Re_{\tau }$ canonical and/or weak APG TBLs, albeit with the outer-region mechanisms weaker than those in the inner region. This explains the absence of their $\overline {u^2}$ outer peak and the dominance of $\overline {u^2}$ wall-normal transport away from the wall, which potentially originates from the inner region. Given that the wall-ward/away-from-wall transport of $\overline {u^2}$ is governed by the $Q_4$(sweeps)/$Q_2$(ejections) quadrants of the Reynolds shear stress, it is argued that the emergence of the $\overline {u^2}$ outer peak corresponds with the statistical dominance of $Q_4$ events in the outer region. Besides unravelling the dynamical significance of $Q_2$ and $Q_4$ events in the outer region of TBLs, the present analysis also proposes new phenomenological arguments for testing on canonical wall-turbulence data at very high $Re_{\tau }$.

This study uses high-fidelity simulations (direct numerical simulation or large-eddy simulation) and experimental datasets to analyse the effect of non-equilibrium streamwise mean pressure gradients (adverse or favourable), including attached and separated flows, on the statistics of boundary-layer wall-pressure fluctuations. The datasets collected span a wide range of Reynolds numbers ($Re_\theta$ from 300 to 23 400) and pressure gradients (Clauser parameter from $-0.5$ to 200). The datasets are used to identify an optimal set of variables to scale the wall-pressure spectrum: edge velocity, boundary layer thickness and the peak magnitude of Reynolds shear stress. Using the present datasets, existing semi-empirical models of the wall-pressure spectrum are shown unable to capture effects of strong, non-equilibrium adverse pressure gradients, due to inappropriate scaling of the wall pressure using the wall shear stress, calibration with limited types of flows and dependency on model parameters based on the friction velocity, which reduces to zero at the detachment point. To address these shortcomings, a generalized wall-pressure spectral model is developed with parameters that characterize the extent of the logarithmic layer and the strength of the wake. Derived from the local mean streamwise velocity profile, these two parameters inherently carry the effect of the Reynolds number, as well as those of the non-equilibrium pressure gradient and its history. Comparison with existing models shows that the proposed model behaves well and is more accurate in strong-pressure-gradient flows and in separated-flow regions.

As shown by Wenzel et al. (J. Fluid Mech., vol. 930, 2022, A1), the Eckert number $Ec$ defined using the difference between recovery temperature $\bar{T}_r$ and wall temperature $\bar{T}_w$ can be understood as a meaningful quantity to compare heat-transfer effects inside compressible turbulent boundary layers (for a calorically perfect gas), no matter whether these are caused by different Mach-number or wall-temperature conditions. While the named study deduced this comparative behaviour of $Ec$ from an integral perspective in a strict sense, Cogo et al. (J. Fluid Mech., vol. 974, 2023, A10) performed a systematic parameter study based on the previous findings to look at wall-normal profiles. They have shown that the diabatic parameter $\varTheta$, being equivalent to $Ec$, is capable of categorizing heat-transfer effects for cases at different Mach numbers, even to some extent for some of the wall-normal profiles. Building on this progress, the present paper provides a comprehensive classification of both existing and newly computed super- and hypersonic direct numerical simulation data at various wall temperature conditions into heated cases, adiabatic cases or weakly/moderately/strongly/quasi-incompressibly cooled cases. Hereby, the classification is largely based on the wall-normal position of the temperature peak occurring in cooled boundary-layer cases, which is one of the determining factors for the topological characteristics of diabatic boundary-layer profiles. Integrating high-enthalpy data into the analysis allowed us to confirm the reliability of the proposed classification also in more complex scenarios, where the calorically perfect gas assumption no longer applies and additional heat-transfer mechanisms come into play. While the Eckert number is shown to well characterize heat-transfer effects on most important temperature-related quantities for a wide range of Mach numbers and $\bar {T}_w/\bar {T}_r$ conditions, also the local Reynolds number $Re_{\tau }$ is shown to notably affect the strength of heat-transfer effects. Since both $Ec$ and $Re_{\tau }$ can be determined in advance – or estimated to a reasonable extent – a key advantage of the classification scheme is to allow for an effective a priori estimation of the extent to which heat-transfer effects are to be expected for a given compressible turbulent boundary-layer configuration.

The influence of outer large-scale motions (LSMs) on near-wall structures in compressible turbulent channel flows is investigated. To separate the compressibility effects, velocity fluctuations are decomposed into solenoidal and dilatational components using the Helmholtz decomposition method. Solenoidal velocity fluctuations manifest as near-wall streaks and outer large-scale structures. The spanwise drifting of near-wall solenoidal streaks is found to be driven by the outer LSMs, while LSMs have a trivial influence on the spanwise density of solenoidal streaks, consistent with the outer LSM impacts found in incompressible flows (Zhou et al., J. Fluid Mech., vol. 940, 2022, p. A23). Dilatational motions are characterized by the near-wall small-scale travelling-wave packets and the large-scale parts in the outer region. The streamwise advection velocity of the near-wall structures remains at $16 \sim 18u_{\tau }$, hardly influenced by Mach numbers, Reynolds numbers and wall temperatures. The spanwise drifting of near-wall dilatational structures, quantified by the particle image velocimetry method, follows a mechanism distinct from solenoidal streaks. This drifting velocity is notably larger than those of the solenoidal streaks, and the influence of outer LSMs is not the primary trigger for this drifting.

The scaling relations mapping the turbulence statistics in compressible turbulent boundary layers (TBLs) onto their incompressible counterparts are of fundamental significance for turbulence modelling, such as the Morkovin scaling for velocity fields, while for pressure fluctuation fields, a corresponding scaling relation is currently absent. In this work, the underlying scaling relations of pressure fluctuations about Mach number ($M$) contained in their generation mechanisms are explored by analysing a series of direct numerical simulation data of compressible TBLs over a wide Mach number range $(0.5\leq M \leq 8.0)$. Based on the governing equation of pressure fluctuations, they are decomposed into components according to the properties of source terms. It is notable that the intensity of the compressible component, predominantly originating from the acoustic mode, obeys a monotonic distribution about the Mach number and wall distance; further, the intensity of the rest of the pressure components, which are mainly generated by the vorticity mode, demonstrates a uniform distribution consistent with its incompressible counterpart. Moreover, the coupling between the two components is negligibly weak. Based on the scaling relations, semiempirical models for the fluctuation intensity of both pressure and its components are constructed. Hence, a mapping relation is obtained that the profiles of pressure fluctuation intensities in compressible TBLs can be mapped onto their incompressible counterparts by removing the contribution from the acoustic mode, which can be provided by the model. The intrinsic scaling relation can provide some basic insight for pressure fluctuation modelling.

The influence of symmetry-breaking effects of ridge-type roughness on secondary currents in turbulent channel flow is investigated using direct numerical simulations. The ridges have triangular cross-section, which is systematically varied from isosceles to right-angled triangle, introducing an imbalance to the slopes of the ridges’ lateral surfaces while the streamwise homogeneity of the surfaces is maintained. In all cases, secondary current vortices are produced, but asymmetric ridge cross-sections break the symmetry of these vortices. As a result of the asymmetry-induced misalignment and imbalance in the secondary current vortices, net spanwise flow emerges. The magnitude of the spanwise flow increases with the slope ratio of the ridge lateral surfaces and significantly modifies the mean flow topology, leading to the merging of critical points in the case of the right-angled triangular ridge shape. Within the cavities, the net spanwise flow is accompanied by a non-zero mean spanwise pressure gradient, while from the perspective of the outer flow, the scalene ridge surfaces have a similar effect as a wall that is slowly moving in the spanwise direction. Overall, the present results suggest the existence of a special type of Prandtl's secondary currents of the second kind, namely those that result in net spanwise flow.

The mean flow in a turbulent boundary layer (TBL) deviates from the canonical law of the wall (LoW) when influenced by a pressure gradient. Consequently, LoW-based near-wall treatments are inadequate for such flows. Chen et al. (J. Fluid Mech., vol. 970, 2023, A3) derived a Navier–Stokes-based velocity transformation that accurately describes the mean flow in TBLs with arbitrary pressure gradients. However, this transformation requires information on total shear stress, which is not always readily available, limiting its predictive power. In this work, we invert the transformation and develop a predictive near-wall model. Our model includes an additional transport equation that tracks the Lagrangian integration of the total shear stress. Particularly noteworthy is that the model introduces no new parameters and requires no calibration. We validate the developed model against experimental and computational data in the literature, and the results are favourable. Furthermore, we compare our model with equilibrium models. These equilibrium models inevitably fail when there are strong pressure gradients, but they prove to be sufficient for boundary layers subjected to weak, moderate and even moderately high pressure gradients. These results compel us to conclude that history effects in mean flow, which negatively impact the validity of equilibrium models, can largely be accounted for by the material time derivative term and the pressure gradient term, both of which require no additional modelling.

Roughness of the surface underlying the atmospheric boundary layer causes departures of the near-surface scalar and momentum transport in comparison with aerodynamically smooth surfaces. Here, we investigate the effect of $56\times 56$ homogeneously distributed roughness elements on bulk properties of a turbulent Ekman flow. Direct numerical simulation in combination with an immersed boundary method is performed for fully resolved, three-dimensional roughness elements. The packing density is approximately $10\,\%$ and the roughness elements have a mean height in wall units of $10 \lesssim H^+ \lesssim 40$. According to their roughness Reynolds numbers, the cases are transitionally rough, although the roughest case is on the verge of being fully rough. We derive the friction of velocity and of the passive scalar through vertical integration of the respective balances. Thereby, we quantify the enhancement of turbulent activity with increasing roughness height and find a scaling for the friction Reynolds number that is verified up to $Re_\tau \approx 2700$. The higher level of turbulent activity results in a deeper logarithmic layer for the rough cases and an increase of the near-surface wind veer in spite of higher $Re_\tau$. We estimate the von Kármán constant for the horizontal velocity $\kappa _{m}=0.42$ (offset $A=5.44$) and for the passive scalar $\kappa _{h}=0.35$ (offset $\mathbb {A}=4.2$). We find an accurate collapse of the data under the rough-wall scaling in the logarithmic layer, which also yields a scaling for the roughness parameters $z$-nought for momentum ($z_{0{m}}$) and the passive scalar ($z_{0{h}}$).

Different types of neural networks have been used to solve the flow sensing problem in turbulent flows, namely to estimate velocity in wall-parallel planes from wall measurements. Generative adversarial networks (GANs) are among the most promising methodologies, due to their more accurate estimations and better perceptual quality. This work tackles this flow sensing problem in the vicinity of the wall, addressing for the first time the reconstruction of the entire three-dimensional (3-D) field with a single network, i.e. a 3-D GAN. With this methodology, a single training and prediction process overcomes the limitation presented by the former approaches based on the independent estimation of wall-parallel planes. The network is capable of estimating the 3-D flow field with a level of error at each wall-normal distance comparable to that reported from wall-parallel plane estimations and at a lower training cost in terms of computational resources. The direct full 3-D reconstruction also unveils a direct interpretation in terms of coherent structures. It is shown that the accuracy of the network depends directly on the wall footprint of each individual turbulent structure. It is observed that wall-attached structures are predicted more accurately than wall-detached ones, especially at larger distances from the wall. Among wall-attached structures, smaller sweeps are reconstructed better than small ejections, while large ejections are reconstructed better than large sweeps as a consequence of their more intense footprint.

Wall temperature has a significant effect on shock wave/turbulent boundary layer interactions (STBLIs) and has become a non-negligible factor in the design process of hypersonic vehicles. In this paper, direct numerical simulations are conducted to investigate the wall temperature effects on STBLIs over a 34° compression ramp at Mach number 6. Three values of the wall-to-recovery-temperature ratio (0.50, 0.75 and 1.0) are considered in the simulations. The results show that the size of the separation bubble declines significantly as the wall temperature decreases. This is because the momentum profile of the boundary layer becomes fuller with wall cooling, which means the near-wall fluid has a greater momentum to suppress flow separation. An equation based on the free-interaction theory is proposed to predict the distributions of the wall pressure upstream of the corner at different wall temperatures. The prediction results are generally consistent with the simulation results (Reynolds number Reτ ranges from 160 to 675). In addition, the low-frequency unsteadiness is studied through the weighted power spectral density of the wall pressure and the correlation between the upstream and downstream. The results indicate that the low-frequency motion of the separation shock is mainly driven by the downstream mechanism and that wall cooling can significantly suppress the low-frequency unsteadiness, including the strength and streamwise range of the low-frequency motions.

This study proposes a novel super-resolution (or SR) framework for generating high-resolution turbulent boundary layer (TBL) flow from low-resolution inputs. The framework combines a super-resolution generative adversarial neural network (SRGAN) with down-sampling modules (DMs), integrating the residual of the continuity equation into the loss function. The DMs selectively filter out components with excessive energy dissipation in low-resolution fields prior to the super-resolution process. The framework iteratively applies the SRGAN and DM procedure to fully capture the energy cascade of multi-scale flow structures, collectively termed the SRGAN-based energy cascade reconstruction framework (EC-SRGAN). Despite being trained solely on turbulent channel flow data (via ‘zero-shot transfer’), EC-SRGAN exhibits remarkable generalization in predicting TBL small-scale velocity fields, accurately reproducing wavenumber spectra compared to direct numerical simulation (DNS) results. Furthermore, a super-resolution core is trained at a specific super-resolution ratio. By leveraging this pretrained super-resolution core, EC-SRGAN efficiently reconstructs TBL fields at multiple super-resolution ratios from various levels of low-resolution inputs, showcasing strong flexibility. By learning turbulent scale invariance, EC-SRGAN demonstrates robustness across different TBL datasets. These results underscore the potential of EC-SRGAN for generating and predicting wall turbulence with high flexibility, offering promising applications in addressing diverse TBL-related challenges.

Serrations are commonly employed to mitigate the turbulent boundary layer trailing-edge noise. However, significant discrepancies persist between model predictions and experimental observations. In this paper, we show that this results from the frozen turbulence assumption. A fully developed turbulent boundary layer over a flat plate is first simulated using the large-eddy simulation method, with the turbulence at the inlet generated using the digital filter method. The space–time correlations and spectral characteristics of wall pressure fluctuations are examined. The simulation results demonstrate that the coherence function decays in the streamwise direction, deviating from the constant value of unity assumed in the frozen turbulence assumption. By considering an exponential decay function, we relax the frozen turbulence assumption and develop a prediction model that accounts for the intrinsic non-frozen nature of turbulent boundary layers. To facilitate a direct comparison with frozen models, a correction coefficient is introduced to account for the influence of non-frozen turbulence. The comparison between the new and original models demonstrates that the new model predicts lower noise reductions, aligning more closely with the experimental observations. The physical mechanism underlying the overprediction of the noise model assuming frozen turbulence is discussed. The overprediction is due to the decoherence of the phase variation along the serrated trailing edge. Consequently, the ratio of the serration amplitude to the streamwise frequency-dependent correlation length is identified as a crucial parameter in determining the correct prediction of far-field noise.

High-dimensional dynamical systems projected onto a lower-dimensional manifold cease to be deterministic and are best described by probability distributions in the projected state space. Their equations of motion map onto an evolution operator with a deterministic component, describing the projected dynamics, and a stochastic one representing the neglected dimensions. This is illustrated with data-driven models for a moderate-Reynolds-number turbulent channel. It is shown that, for projections in which the deterministic component is dominant, relatively ‘physics-free’ stochastic Markovian models can be constructed that mimic many of the statistics of the real flow, even for fairly crude operator approximations, and this is related to general properties of Markov chains. Deterministic models converge to steady states, but the simplified stochastic models can be used to suggest what is essential to the flow and what is not.