Turbulent Wall Layer Research Articles

Effect of a localized MHD body force on near-wall turbulence

The present study focuses on the response of a near-wall turbulent flow subjected to a spatially localized body force. Characteristics of a turbulent channel flow under a Lorentz force distribution are analysed through Direct Numerical Simulation at a low turbulent Reynolds number (4200) and a low magnetic Reynolds number. It is found that streamwise or spanwise magnetic fields alter only slightly the wall layer when the magnetic fields are constrained within the buffer layer. Yet, the turbulent wall layer is greatly affected once the edge of the localized Lorentz force reaches the logarithmic layer. Furthermore, the spanwise magnetic field results in flow relaminarization at much lower magnetic severity parameters than the streamwise magnetic field. We further show that this is basically coming from the specific MHD terms appearing in the shear-stress transport equations. Tables 1, Figs 6, Refs 5.

Compressible Velocity Transformations for Various Noncanonical Wall-Bounded Turbulent Flows

Although several velocity transformations for compressible zero-pressure-gradient boundary layers have been proposed in the past decades, their performance for noncanonical compressible wall-bounded turbulent flows has not been systematically investigated. This work assesses several popular transformations for the velocity profile through their application to several types of noncanonical compressible wall-bounded turbulent flows. Specifically, this work explores direct numerical simulation databases of high-enthalpy boundary layers with dissociation and vibrational excitation, supercritical channel and boundary-layer flows, and adiabatic boundary layers with pressure gradients. The transformations considered include the van Driest (“Turbulent Boundary Layer in Compressible Fluids,” Journal of the Aeronautical Sciences, Vol. 18, No. 3, 1951, pp. 145–216), Zhang et al. (“Mach-Number-Invariant Mean-Velocity Profile of Compressible Turbulent Boundary Layers,” Physical Review Letters, Vol. 109, No. 5, 2012, Paper 054502), Trettel and Larsson (“Mean Velocity Scaling for Compressible Wall Turbulence with Heat Transfer,” Physics of Fluids, Vol. 28, No. 2, 2016, Paper 026102), data-driven (Volpiani et al., “Data-Driven Compressibility Transformation for Turbulent Wall Layers,” Physical Review Fluids, Vol. 5, No. 5, 2020, Paper 052602), and total-stress-based (Griffin et al., “Velocity Transformation for Compressible Wall-Bounded Turbulent Flows with and Without Heat Transfer,” Proceedings of the National Academy of Sciences of the United States of America, Vol. 118, No. 34, 2021, Paper e2111144118) transformations. The Trettel–Larsson transformation collapses velocity profiles of high-enthalpy temporal boundary layers but not the spatial boundary layers considered. For supercritical channel flows, the Trettel–Larsson transformation also performs well over the entire inner layer. None of the aforementioned transformations work for supercritical boundary layers. For all the considered methods, the transformed velocity profiles of boundary layers with weak pressure gradients coincide well with the universal incompressible law of the wall. In summary, all these popular methods fail to deliver uniform performance for noncanonical compressible wall-bounded flows in the logarithmic region, and a more sophisticated version, which accounts for these different physics, is needed. The data-driven and total-stress-based transformations perform well in the viscous sublayer for all the considered flows. Nevertheless, the present assessment provides a useful guideline on the deployment of these transformations on various noncanonical flows.

The late start of the mean velocity overlap log law at – a generic feature of turbulent wall layers in ducts

Abstract

Large-eddy simulation and modelling of Taylor–Couette flow

Wall-resolved large-eddy simulations (LES) of the incompressible Navier–Stokes equations together with empirical modelling for turbulent Taylor–Couette (TC) flow are presented. LES were performed with the inner cylinder rotating at angular velocity Ω_i and the outer cylinder stationary. With R_i, R_₀ the inner and outer radii respectively, the radius ratio is η = 0.909 . The subgrid-scale stresses are represented using the stretched-vortex subgrid-scale model while the flow is resolved close to the wall. LES is implemented in the range Re_i = 10⁵-10⁶ where Re_i = Ω_iR_id/v and d = R₀-R_i is the cylinder gap. It is shown that the LES can capture the salient features of the flow, including the quantitative behaviour of spanwise Taylor rolls, the log variation in the inner-cylinder mean-velocity profile and the angular momentum redistribution due to the presence of Taylor rolls. A simple empirical model is developed for the turbulent, TC flow for both a stationary outer cylinder and also for co-rotating cylinders. This consists of near-wall, log-like turbulent wall layers separated by an annulus of constant angular momentum. Model results include the Nusselt number Nu (torque required to maintain the flow) and measures of the wall-layer thickness as functions of both the Taylor number Ta and η. These are compared with results from measurement, direct numerical simulation and the LES. A model extension to rough-wall turbulent flow is described. This shows an asymptotic, fully rough-wall state where the torque is independent of Re_i/Ta, and where Nu ~ Ta^(1/2).

Correlation of pressure fluctuations in turbulent wall layers

Profiles of $\ensuremath{\langle}p\phantom{\rule{0}{0ex}}p{\ensuremath{\rangle}}^{+}$ from direct numerical simulation calculations are correlated for various Reynolds numbers. The correlating functions are unusual; a single function in the outer region and a defect function in the inner region. This is exactly opposite of the correlating functions for the mean velocity.

Implicit Model Equation for Hydraulic Resistance and Heat Transfer including Wall Roughness

Heat transfer to water at supercritical pressure within the core of a supercritical water reactor must be predicted accurately to ensure safe design of the reactor and prevent overheating of the fuel cladding. In the previous work (Laurien, 2012, “Semi-Analytic Prediction of Hydraulic Resistance and Heat Transfer for Pipe Flows of Water at Supercritical Pressure,” Proceedings of the International Conference on Advances in Nuclear Power Plants, ICAPP’12, Chicago, June 24–28), we have demonstrated that the wall shear stress and the wall temperature can be computed in a coupled way by a finite-difference method, taking the wall roughness into account. In the present paper, the classical two-layer model, consisting only of a laminar sublayer and a turbulent wall layer, is extended toward the same task. A set of implicit algebraic equations for the wall shear stress and the wall temperature is derived. It is consistent with the well-established Colebrook equation for rough pipes, which is included as a limiting case for constant properties. The accuracy of the prediction for strongly heated pipe flow is tested by comparison to experiments (Yamagata et al., 1972, “Forced Convective Heat Transfer to Supercritical Water Flowing in Tubes,” Int. J. Heat Mass Transfer, 15(12), 2575–2593) with supercritical water. The high accuracy and the generality of Laurien (2012) “Semi-Analytic Prediction of Hydraulic Resistance and Heat Transfer for Pipe Flows of Water at Supercritical Pressure,” Proceedings of the International Conference on Advances in Nuclear Power Plants, ICAPP’12, Chicago, June 24–28 are not achieved, but with the help of correction factors, the two-layer model has a potential for improved predictions of the hydraulic resistance and the heat transfer of pipe and channel flows at supercritical pressure.

Revisiting the mixing-length hypothesis in the outer part of turbulent wall layers: mean flow and wall friction

AbstractWe reconsider foundations and implications of the mixing length theory as applied to wall-bounded turbulent flows in uniform pressure gradient. Based on recent channel-flow direct numerical simulation (DNS) data at sufficiently high Reynolds number, we find that Prandtl’s hypothesis of linear variation of the mixing length with the wall distance is rather inaccurate, hence overlap arguments are stronger in justifying the formation of a logarithmic layer in the mean velocity profile. Regarding the core region of the wall layer, we find that Clauser’s hypothesis of uniform eddy viscosity is strictly connected with the observed size of the eddy structures, and it delivers surprisingly good agreement with DNS and experiments for channels, pipes, and boundary layers. We show that the analytically derived composite mean velocity profiles can be used to accurately predict skin friction in canonical wall-bounded flows with a minimal number of adjustable parameters directly related to the mean velocity profile, and to obtain some insight into transient growth phenomena.

On the size of the energy-containing eddies in the outer turbulent wall layer

AbstractWe investigate the scaling of the energy-containing eddies in the outer part of turbulent wall layers. Their spanwise integral length scales are extracted from a direct numerical simulation (DNS) database, which includes compressible turbulent boundary layers and incompressible turbulent Couette–Poiseuille flows. The results indicate similar behaviour for all classes of flows, with a general increasing trend in the eddy size with the wall distance. A family of scaling relationships are proposed based on simple dimensional arguments, of which the classical mixing length approximation constitutes one example. As in previous studies, we find that the mixing length is in good agreement with the size distribution of the eddies carrying wall-normal velocity, which are active in establishing the mean velocity distribution. However, we find that the eddies associated with wall-parallel motions obey a different scaling, which is controlled by the local mean shear and by an effective eddy diffusivity ${\nu }_{t} = { u}_{\tau }^{\ensuremath{\ast} } \delta $, where ${ u}_{\tau }^{\ensuremath{\ast} } $ is the compressible counterpart of the friction velocity, and $\delta $ is the thickness of the wall layer. The validity of the proposed scalings is checked against DNS data, and the potential implications for the understanding of wall turbulence are discussed.

Wall shear stress modulation in a turbulent flow subjected to imposed unsteadiness with adverse pressure gradient

The modulation characteristics of the turbulent wall shear stress measured in a plane diffuser subjected to imposed velocity oscillations are presented. The measurements reported in this paper pertain to nearly 150 different flows: imposed oscillations with three amplitudes and six frequencies, two different diverging channels and four streamwise positions. Only the most significant data are analyzed and discussed. The imposed unsteadiness affects the time-mean flow under the effect of the adverse pressure gradient (APG), in contrast to the canonical unsteady turbulent wall layers. The laminar viscous solution that adequately describes the amplitude and phase of the wall shear stress in channel flows in the high-imposed frequency regime is no longer valid in unsteady turbulent boundary layers subjected to APG. The latter modifies also the behavior of the phase shifts of the modulations in the turbulent quantities. The time lag of the wall shear stress turbulent intensity decreases as the pressure gradient increases. Some of these structural modifications are explained by the effect of the eddy viscosity that plays a key role in the vorticity diffusion process as the APG increases.

Correlating data: examples from turbulent wall layers

Flow fields in fluid mechanics often contain regions where different physical events occur. The relative size of the regions changes as a parameter is varied. Correlating field data, either physical or DNS calculation experiments, in these situations can be aided by using ideas from matched asymptotic expansions from applied mathematics. A second situation is when two slightly different processes occur in the same spatial region. For this case, a two-term asymptotic expansion is needed. This article discusses how composite expansions and the common part matching behavior are useful in correlating data.

Relation of entropy generation to wall ‘laws’ for turbulent flows

For non-dimensional pressure gradients sufficiently low and Reynolds numbers sufficiently high for the constant shear stress approximation to be valid at the first node of a turbulent wall layer computation, relationships for entropy generation rates are applied to popular wall ‘laws’. The resulting observations lead to a simple approach for evaluating the entropy generation rate per unit area in a computational fluid dynamics calculation for situations which satisfy this approximation.

Particle dispersion and wall-dependent turbulent flow scales: implications for local equilibrium models

Theoretical models for particle dispersion in turbulent wall layers are based on closure assumptions for particle-turbulence correlations which strongly depend on the wall-normal coordinate. This paper presents new data for the near-wall dispersion of inertial particles in fully developed turbulent channel flow, obtained using direct numerical simulation (DNS) with a one-way point-particle approach. The link between wall-dependent flow time scales and particle time scales is discussed addressing further the issue of using integral flow time scales to parametrize particle behavior. This is fundamental to validate closure relations in theoretical local-equilibrium models for particle dispersion in wall-bounded flows, where the reduction of the fluid velocity fluctuation in the wall-normal direction is not accompanied by an equivalent reduction of the particle velocity fluctuation.

Sediment erosion by Görtler vortices: the scour-hole problem

Experimental results on sediment erosion (scour) by a plane turbulent wall jet, issuing from a sluice gate, are presented which show clearly – it seems for the first time – that the turbulent wall layer is destabilized by the concave curvature of the water/sediment interface. The streamwise Görtler vortices which emerge create sediment streaks or longitudinal sediment ridges. The analysis of the results in terms of Görtler instability of the wall layer indicates that the strength of these curvature-excited streamwise vortices is such that the sediment transport is primarily due to turbulence created by these vortices. Their contribution to the wall shear stress is taken to be of the same form as the normal turbulent wall shear stress. For this reason, the model developed by Hogg et al. (J. Fluid Mech. Vol. 338, 1997, p. 317) remains valid; only the numerical coefficients are affected. The logarithmic dependency of the time evolution of the scour-hole depth predicted by this model is shown to be in good agreement with experiments. New scaling laws for the quasi-steady state depth and the associated time, inspired by the Hogg et al. (1997) model are proposed. Furthermore, it is emphasized that at least two scouring regimes must be distinguished: a short-time regime after which a quasi-steady state is reached, followed by a long-time regime, leading to an asymptotic state of virtually no sediment transport.

Effect of Roughness on Wall-Bounded Turbulence

Direct numerical simulation of turbulent incompressible plane-channel flow between a smooth wall and one covered with regular three-dimensional roughness elements is performed. While the impact of roughness on the mean-velocity profile of turbulent wall layers is well understood, at least qualitatively, the manner in which other features are affected, especially in the outer layer, has been more controversial. We compare results from the smooth- and rough-wall sides of the channel for three different roughness heights of h + = 5.4, 10.8, and 21.6 for Reτ of 400, to isolate the effects of the roughness on turbulent statistics and the instantaneous turbulence structure at large and small scales. We focus on the interaction between the near-wall and outer-layer regions, in particular the extent to which the near-wall behavior influences the flow further away from the surface. Roughness tends to increase the intensity of the velocity and vorticity fluctuations in the inner layer. In the outer layer, although the roughness alters the velocity fluctuations, the vorticity fluctuations are relatively unaffected. The higher-order moments and the energy budgets demonstrate significant differences between the smooth-wall and rough-wall sides in the processes associated with the wall-normal fluxes of the Reynolds shear stresses and turbulence kinetic energy. The length scales and flow dynamics in the roughness sublayer, the spatially inhomogeneous layer within which the flow is directly influenced by the individual roughness elements, are also examined. Alternative mechanisms involved in producing and maintaining near-wall turbulence in rough-wall boundary layers are also considered. We find that the strength of the inner/outer-layer interactions are greatly affected by the size of the roughness elements.

Instability in a viscous flow driven by streamwise vortices

Unsteady separation processes at large finite, Reynolds number, Re, are considered, as well as the possible relation to existing descriptions of boundary-layer separation in the limit Re → ∞. The model problem is a fundamental vortex-driven three-dimensional flow, believed to be relevant to bursting near the wall in a turbulent boundary layer. Bursting is known to be associated with streamwise vortex motion, but the vortex/wall interactions that drive the near-wall flow toward breakdown have not yet been fully identified. Here, a simulation of symmetric counter-rotating vortices is used to assess the influence of sustained pumping action on the development of a viscous wall layer. The calculated solutions describe a three-dimensional flow at finite Re that is independent of the streamwise coordinate and consists of a crossflow plane motion, with a developing streamwise flow. The unsteady problem is constructed to mimic a typical cycle in turbulent wall layers and numerical solutions are obtained over a range of Re. Recirculating eddies develop rapidly in the near-wall flow, but these eddies are eventually bisected by alleyways which open up from the external flow region to the wall. At sufficiently high Re, an oscillation was found to develop in the streamwise vorticity field near the alleyways with a concurrent evolution of a local spiky behaviour in the wall shear. Above a critical value of Re, the oscillation grows rapidly in amplitude and eventually penetrates the external flow field, suggesting the onset of an unstable wall-layer breakdown. Local zones of severely retarded streamwise velocity are computed which are reminiscent of the low-speed streaks commonly observed in turbulent boundary layers. A number of other features also bear a resemblance to observed coherent structure in the turbulent wall layer.

Lagrangian and Eulerian view of the bursting period

Low-dimensional models for the turbulent wall layer display an intermittent phenomenon with an ejection phase and a sweep phase that strongly resembles the bursting phenomenon observed in experimental flows. The probability distribution of inter-burst times has the observed shape [E. Stone and P. J. Holmes, Physica D 37, 20 (1989); SIAM J. Appl. Math. 50, 726 (1990); Phys. Lett. A 5, 29 (1991); P. J. Holmes and E. Stone, in Studies in Turbulence, edited by T. B. Gatski, S. Sarkar, and C. G. Speziale (Springer, Heidelberg, 1992)]. However, the time scales both for bursts and interburst durations are unrealistically long, a fact that was not appreciated until recently. We believe that the long time scales are due to the model’s inclusion of only a single coherent structure, when in fact a succession of quasi-independent structures are being swept past the sensor in an experiment. A simple statistical model of this situation restores the magnitude of the observed bursting period, although there is a great deal of flexibility in the various parameters involved.

Active control in the turbulent wall layer of a minimal flow unit

Direct simulations of flow in a channel with complex, time-dependent wall geometries facilitate an investigation of smart skin control in a turbulent wall layer (with skin friction drag reduction as the goal). The test bed is a minimal flow unit, containing one pair of coherent structures in the near-wall region: a high- and a low-speed streak. The controlling device consists of an actuator, Gaussian in shape and approximately twelve wall units in height, that emerges from one of the channel walls. Raising the actuator underneath a low-speed streak effects an increase in drag, raising it underneath a high-speed streak effects a reduction – indicating a mechanism for control. In the high-speed region, fast-moving fluid is lifted by the actuator away from the wall, allowing the adjacent low-speed region to expand and thereby lowering the average wall shear stress. Conversely, raising an actuator underneath a low-speed streak allows the adjacent high-speed region to expand, which increases skin drag.

Structure of near wall turbulence downstream of a wall mounted protrusion: an interesting Reynolds stress suppression phenomena

A local suppression in the generation of near wall Reynolds stress is achieved by modifying the buffer region and sublayer (y+ <30) of a turbulent pipe flow with a 16.4 wall unit high wall mounted protrusion. Multi-component, multi-point, time resolved laser Doppler velocimetry measurements are made in the undisturbed and modified ARL/PSU glycerin tunnel pipe flow at a Reynolds number of approximately 10000. A downstream converging flow field is produced by the divergence of the approaching mean flow around the protrusion. A pair of counter-rotating vortices, ≈ 15 wall units in diameter with common flow down, are generated by the protrusion and also contribute to the wall directed flow convergence. The convergence region is 15 wall units high and more than 100 wall units long and appears to decouple the near wall region from the outer turbulent wall layer. Locally, turbulent velocity fluctuations in the form of Reynolds stress producing events, sweeps and ejections, are retarded within this region. This results in a reduction in near wall uv Reynolds stress and local wall shear. Interestingly, the counter-rotating vortices act to increase turbulent diffusion in a manner which is uncorrelated with Reynolds stress generation.

On the relation between low-dimensional models and the dynamics of coherent structures in the turbulent wall layer

In this paper we establish some rigorous connections between the dynamics of coherent structures in the wall region of the turbulent boundary layer and the low-dimensional models of the type studied by Aubry et al. (1988). An important first step is to determine what sort of connection is feasible. We choose to study the energy budget of the models in comparison with the energy budget of the real flow. This is done by comparing the respective kinetic energy equations. In the process we reexamine some of the assumptions and approximations of Aubry et al. (1988) and perform order of magnitude analyses to determine when they hold. We find that, for the models developed in that paper, involving modes which do not vary in the streamwise direction, the energy production lies within positive, experimentally determined, bounds. Moreover, the dissipation due to neglected modes may be reproduced correctly with an order 1 value of the Heisenberg parameter α, as assumed by Aubry et al.

On modeling three-dimensional turbulent wall layers

A Reynolds stress model with a modification to the ε equation is shown to reproduce the essential new features observed by Moin et al. [Phys. Fluids A 2, 1846 (1990)] in three-dimensional turbulent channel flow. Perhaps the most significant of these features is an initial drop of turbulent kinetic energy; previous turbulence models do not predict this. The effect of the three-dimensional modification on the infinite swept-wing boundary layer is assessed. It is not large and is primarily restricted to a region near to the surface. A three-dimensional homogeneous shear flow is formulated and turbulence statistics are computed by solving a simple Reynolds stress model.