Turbulent–laminar patterns in shear flows without walls

This paper deals with intrinsic effects of compressibility, i.e. with dilatation fluctuations in response to pressure fluctuations. Three different types of turbulent flows are considered in more detail: homogeneous turbulent shear flow, wall-bounded turbulent shear flow and shock/turbulence interaction. A survey of the present knowledge in this field, mainly based on DNS data, is given. Using the linear inviscid perturbation equations a direct link between fluctuations of dilatation and of velocity in the direction of mean shear is presented for homogeneous shear flow. This relation might form the basis for a more universal pressure-dilatation model. It is conjectured that the insignificance of intrinsic compressibility effects in wall-bounded supersonic shear flow is mainly due to the impermeability constraint of the wall. To this end, a linear stability analysis of supersonic channel flow along cooled, but permeable walls has been performed based on Coleman et al.'s [5] mean flow data. It shows an increase in the moduli of eigenfunctions related to compressibility, like pressure, and in moduli of quantities derived from eigenfunctions such as ‘pressure dilatation’ and squared dilatation. Although these results do not prove our hypothesis they provide hints in this direction. Shock/turbulence interaction is viewed as a source of compressibility. Former DNS data of Hannappel and Friedrich [10] for shock/isotropic turbulence interaction showing the effect of compressibility on the amplification of fluctuations are interpreted based on linear perturbation equations.

Experiments and numerical simulations have shown that turbulence in transitional wall-bounded shear flows frequently takes the form of long oblique bands if the domains are sufficiently large to accommodate them. These turbulent bands have been observed in plane Couette flow, plane Poiseuille flow, counter-rotating Taylor–Couette flow, torsional Couette flow, and annular pipe flow. At their upper Reynolds number threshold, laminar regions carve out gaps in otherwise uniform turbulence, ultimately forming regular turbulent–laminar patterns with a large spatial wavelength. At the lower threshold, isolated turbulent bands sparsely populate otherwise laminar domains, and complete laminarization takes place via their disappearance. We review results for plane Couette flow, plane Poiseuille flow, and free-slip Waleffe flow, focusing on thresholds, wavelengths, and mean flows, with many of the results coming from numerical simulations in tilted rectangular domains that form the minimal flow unit for the turbulent–laminar bands.

Streamwise rolls and accompanying streamwise streaks are ubiquitous in wall-bounded shear flows, both in natural settings, such as the atmospheric boundary layer, as well as in controlled settings, such as laboratory experiments and numerical simulations. The streamwise roll and streak structure has been associated with both transition from the laminar to the turbulent state and with maintenance of the turbulent state. This close association of the streamwise roll and streak structure with the transition to and maintenance of turbulence in wall-bounded shear flow has engendered intense theoretical interest in the dynamics of this structure. In this work, stochastic structural stability theory (SSST) is applied to the problem of understanding the dynamics of the streamwise roll and streak structure. The method of analysis used in SSST comprises a stochastic turbulence model (STM) for the dynamics of perturbations from the streamwise-averaged flow coupled to the associated streamwise-averaged flow dynamics. The result is an autonomous, deterministic, nonlinear dynamical system for evolving a second-order statistical mean approximation of the turbulent state. SSST analysis reveals a robust interaction between streamwise roll and streak structures and turbulent perturbations in which the perturbations are systematically organized through their interaction with the streak to produce Reynolds stresses that coherently force the associated streamwise roll structure. If a critical value of perturbation turbulence intensity is exceeded, this feedback results in modal instability of the combined streamwise roll/streak and associated turbulence complex in the SSST system. In this instability, the perturbations producing the destabilizing Reynolds stresses are predicted by the STM to take the form of oblique structures, which is consistent with observations. In the SSST system this instability exists together with the transient growth process. These processes cooperate in determining the structure of growing streamwise roll and streak. For this reason, comparison of SSST predictions with experiments requires accounting for both the amplitude and structure of initial perturbations as well as the influence of the SSST instability. Over a range of supercritical turbulence intensities in Couette flow, this instability equilibrates to form finite amplitude time-independent streamwise roll and streak structures. At sufficiently high levels of forcing of the perturbation field, equilibration of the streamwise roll and streak structure does not occur and the flow transitions to a time-dependent state. This time-dependent state is self-sustaining in the sense that it persists when the forcing is removed. Moreover, this self-sustaining state rapidly evolves toward a minimal representation of wall-bounded shear flow turbulence in which the dynamics is limited to interaction of the streamwise-averaged flow with a perturbation structure at one streamwise wavenumber. In this minimal realization of the self-sustaining process, the time-dependent streamwise roll and streak structure is maintained by perturbation Reynolds stresses, just as is the case of the time-independent streamwise roll and streak equilibria. However, the perturbation field is maintained not by exogenously forced turbulence, but rather by an endogenous and essentially non-modal parametric growth process that is inherent to time-dependent dynamical systems.

Subcritical transition to turbulence in wall-bounded shear flow is known to be characterized in terms of turbulent patches bounded by laminar regions. In this study spatially localized turbulence structures are investigated in rectangular-duct flow of cross-sectional aspect ratios A_S = 1 - 9 at marginal Reynolds number Ret. The lowest Reynolds number Ret for transient turbulence is determined to demonstrate its dependence on the aspect ratio As. It is found that Ret monotonically decreases from Ret = 730 for As = 1 with increasing As, and for As = 5 it nearly attains to a minimal value Re_T ≒ 670 which is consistent with the Reynolds number for the appearance of a turbulent puff in a plane channel (As = ∞). Streamwise-localized (turbulent puff), or streamwise- and spanwiselocalized (turbulent spot) turbulence are observed at the Reynolds numbers Re = Re_T for distinct As. The localized turbulence exhibits structural 'transition' between a turbulent puff and a turbulent spot around As = 4.

Towards identification of a reliable framework to predict the thermal field in turbulent wall-bounded shear flows

This paper revisits the classical logarithmic velocity profile of wall-bounded turbulent shear flows, with particular emphasis on the persistent scatter in its defining parameters—the von Kármán constant (κ) and the additive constant (B). A combined experimental–theoretical framework is developed to extract these parameters across the three canonical flows: plane channel, circular pipe, and zero-pressure-gradient (ZPG) boundary layer. The authors' derivations show that three different partial differential equations governing these flows in the fully developed state behave similarly as Reτ→∞, particularly within the inertial sublayer, thereby supporting the universality of the log-law, where Reτ denotes the friction Reynolds number. The theoretical approach, based on the slope of the mean velocity profile in logarithmic coordinates, is validated using new hot-wire measurements together with literature data, accounting for wall-curvature effects in pipe flow and flow development in ZPG boundary layers. The analysis indicates that, at sufficiently high Reynolds numbers, the inertial sublayer is governed by the log-law, with an estimated von Kármán constant of κ≈1/e≈0.367879⋯. This value is supported by renormalization group theory and exponential self-similarity arguments, reinforcing that κ is not merely a fitting parameter but a consequence of turbulence dynamics. Furthermore, the present study demonstrates that κ and B cannot be specified independently, as multiple κ–B pairs yield equally valid log-law predictions. A linear κB–B relation, κB = 0.4582 B − 0.3134, is established, reducing data scatter and clarifying the cross-dependence of the two parameters. The present findings highlight the robustness of the logarithmic law and the ongoing challenge of defining its universal constants in wall-bounded turbulent flows, while providing a physical framework for developing and calibrating turbulence models.

Turbulence in wall-bounded shear flow results from a synergistic interaction between linear non-normality and nonlinearity in which non-normal growth of a subset of perturbations configured to transfer energy from the externally forced component of the turbulent state to the perturbation component maintains the perturbation energy, while the subset of energy-transferring perturbations is replenished by nonlinearity. Although it is accepted that both linear non-normality mediated energy transfer from the forced component of the mean flow and nonlinear interactions among perturbations are required to maintain the turbulent state, the detailed physical mechanism by which these processes interact in maintaining turbulence has not been determined. In this work a statistical state dynamics based analysis is performed on turbulent Couette flow at R = 600 and a comparison to DNS is used to demonstrate that the perturbation component in Couette flow turbulence is replenished by a non-normality mediated parametric growth process in which the fluctuating streamwise mean flow has been adjusted to marginal Lyapunov stability. It is further shown that the alternative mechanism in which the subspace of non-normally growing perturbations is maintained directly by perturbation-perturbation nonlinearity does not contribute to maintaining the turbulent state. This work identifies parametric interaction between the fluctuating streamwise mean flow and the streamwise varying perturbations to be the mechanism of the nonlinear interaction maintaining the perturbation component of the turbulent state, and identifies the associated Lyapunov vectors with positive energetics as the structures of the perturbation subspace supporting the turbulence.

Large-scale coherent rolls are observed frequently in unstably stratified turbulent wall-bounded flows where they influence strongly the turbulent transport and the mean flow properties. Here we address the question of their genesis by means of a linear stability analysis of the turbulent mean flow where a model of turbulent Reynolds stresses is embedded in the linear stability operator. We use the unstably stratified turbulent channel flow as a testbed for the analysis. We show that the onset of large-scale convection is associated to the linear instability of the mean flow to large-scale streamwise-uniform coherent rolls of aspect ratio$A \approx 3\unicode{x2013}3.3$when the friction Richardson number exceeds$|Ri_{\tau,c}|=0.86$. This corresponds to critical Rayleigh numbers that increase with the Reynolds number approximately as$Ra_c \approx 0.04\,Re_b^{1.8}$. These results are consistent with those obtained in recent direct numerical simulations performed in the same setting. It is also found that if turbulent Reynolds stresses are not modelled in the linear operator used in the stability analysis, then predictions are not consistent with direct numerical simulations results.

Turbulent shear flows can be divided into three groups according to the number of fixed boundaries: 1. Free turbulent shear flows are said to be free, i.e. remote from walls, like jets, wakes, mixing layers and plumes. Jets and wakes differ from each other only by the sign of the momentum creating them. Free turbulent shear flows play a significant role in many kinds of engineering equipments, in rivers and in atmosphere: jets in propulsive devices for rockets and aircrafts and pneumatic control systems, wakes behind aeroplanes and submarines, dispersion in rivers, jet streams in the atmosphere are few examples. Most of 2D shear layers are regions in which there is one single predominant direction, the streamwise direction, while the shear stresses and diffusion fluxes are significant in the direction perpendicular to it. Most of shear layers, but not all, can be considered wiihin the framework of the boundary layer approximation, with only one small length scale, 8, in a direction perpendicular to the streamwise direction. Finally free shear layers include significant regions with both strong and weak strain making them rather sensitive to turbulence modeling which is more challenging there than in the vicinity of a wall where the flow is rather shear-dominated. 2. Turbulent shear flows bounded by one free and one fixed boundary, such as boundary layers or wall-jets, where the wall may be straight or curved, permeable or impermeable. The velocity of the ambiant flow may be in the same or in the opposite direction of the shear flow. 3. Turbulent shear flows bounded by two or more fixed boundaries, such as or channel flows, pipe flows (a pipe is a closed duct of circular cross section), duct flows (a duct has a non circular cross section).

Vortex dynamics offers a tractable avenue for explaining the dynamics of coherent structures, which appear to be the dominant features of turbulent shear flows and are the key to predicting and controlling turbulence phenomena such as mixing, entrainment, chemical reaction, drag, and aerodynamic noise. With this in mind, a series of investigations have been undertaken in our group; herein we provide a summary of these studies to date. In particular, we survey coherent structure eduction in free and wall-bounded shear flows, our new generic definition of a vortex, incompressible and compressible vortex reconnection, internal core dynamics within vortices, core dynamics-induced mixing layer transition, helical wave decomposition to extract polarized vorticity wavepackets from turbulent flows, and dynamical coupling of coherent structures with fine-scale turbulence. In addition to reviewing specific contributions in these topics, the results are discussed in context of their generic relevance to turbulence physics.

Separation behaviour in front of a two-dimensional fence

This article presents the finite size analysis of two consecutive crossovers leading laminar-turbulent bands to uniform wall turbulence in transitional plane Couette flow. Direct numerical simulations and low order modeling simulations of the flow are performed. The kinetic energy E of the turbulent flow and the order parameter M, a measure of the spatially organised modulation of turbulence, are sampled. These two quantities are processed in view of analytical results from the phenomenology of phase transitions. The first crossover concerns the loss of spatial organisation of turbulence in the flow. In the band phase, the order parameter M decreases continuously with the Reynolds number R toward a small value, while its response function displays a maximum at the crossover. The increase of the maximum of the response function with domain size is consistent with a polynomial law , . A first critical Reynolds number Rc,1 can be defined as the value at which the maximum of the response function is reached. In the uniform phase, the order parameter M and its variance σ decrease toward zero following mean field scalings as R is increased. The kinetic energy E is an affine function of R except in a small range where a sharp increase is detected, which corresponds to the second crossover. A second critical Reynolds number Rc,2 can be defined as the center of this sharp increase range. In this range, spatial and temporal coexistence of the uniform turbulence phase and laminar-turbulent bands phase is observed. This sharp increase is concomitant with a maximum of the response function of the kinetic energy. The finite size analysis reveals that the jump does not steepen and that the maximum of response function of E saturates as size is increased. The first crossover is formally identical to a critical phenomenon in condensed matter. The second crossover is in agreement with a first order phase transition smeared by finite noise. The analytical analysis of this phenomenon assuming a non interacting gas of fronts between domains of the two phases provides a scaling of the response function consistent with that of E. In our context, this amounts to the statistics of the grain boundaries between domains of banded turbulence and uniform turbulence. We eventually discuss how this formalism could explain a breakdown of the orientation order of the bands, with a grain boundary between domains of different orientations. This breakdown may occur in extremely large size domains and could affect the order parameter and its response function.

The failure of normal-mode linear stability analysis to predict a transition Reynolds number (Retr) in pipe flow and subcritical transition in plane Poiseuille flow (PPF) has led to the search of other scenarios to explain transition to turbulence in both flows. In this work, various results associated with linear and nonlinear mechanisms of both flows are presented. The results that combine analytical and experimental approaches indicate the strong link between the mechanisms governing the transition of both flows. It is demonstrated that the linear transient growth mechanism is based on the existence of a pair of least stable nearly parallel modes (having opposite phases and almost identical amplitude distributions). The analysis that has been applied previously to pipe flow is extended here to a fully developed channel flow predicting the shape of the optimized initial disturbance (a pair of counter-rotating vortices, CVP), time for maximum energy amplification and the dependence of the latter on Re. The results agree with previous predictions based on many modes. Furthermore, the shape of the optimized initial disturbance is similar in both flows and has been visualized experimentally. The analysis reveals that in pipe flow, the transient growth is a consequence of two opposite running modes decaying with an equal decay rate whereas in PPF it is due to two stationary modes decaying with different decay rates. In the first nonlinear scenario, the breakdown of the CVPs (produced by the linear transient growth mechanism) into hairpin vortices is followed experimentally. The associated scaling laws, relating the minimal disturbance amplitude required for the initiation of hairpins and the Re, are found experimentally for both PPF and pipe flow. The scaling law associated with PPF agrees well with the previous predictions of Chapman, whereas the scaling of the pipe flow is the same as the one previously obtained by Hof et al. indicating transition to a turbulent state. In the second nonlinear scenario, the base flow of pipe when it is mildly deviated from the Poiseuille profile by an axisymmetric distortion is examined. The nonlinear features reveal a Retr of approximately 2000 associated with the bifurcation between two deviation solutions.

Hairpin vortices have been found to be one of the predominant coherent structures in both transitional and turbulent wall-bounded shear flows. Here, we study, both experimentally and numerically, the evolution of a hairpin vortex from its birth, through its growth as it interacts with the mean shear, to its breakdown. The vortices are introduced in a subcritical laminar boundary layer by fluid injection through a long and thin slot aligned with the free stream direction. With the Reynolds number and slot geometry fixed, the control parameter of this experiment is the ratio of the injection velocity to the free stream velocity. This investigation is relevant to both the study of near-wall turbulent coherent structures and by-pass transition.

In this paper, we revisit the eddy viscosity formulation to highlight a number of important issues that have direct implications for the prediction of near-wall turbulence. For steady wall-bounded turbulent flows, we make the equilibrium assumption between rates of production ($P$) and dissipation ($\epsilon $) of turbulent kinetic energy ($k$) in the near-wall region to propose that the eddy viscosity should be given by${\nu }_{t} \approx \epsilon / {S}^{2} $, where$S$is the mean shear rate. We then argue that the appropriate velocity scale is given by$\mathop{(S{T}_{L} )}\nolimits ^{- 1/ 2} {k}^{1/ 2} $where${T}_{L} = k/ \epsilon $is the turbulence (decay) time scale. The difference between this velocity scale and the commonly assumed velocity scale of${k}^{1/ 2} $is subtle but the consequences are significant for near-wall effects. We then extend our discussion to show that the fundamental length and time scales that capture the near-wall behaviour in wall-bounded shear flows are the shear mixing length scale${L}_{S} = \mathop{(\epsilon / {S}^{3} )}\nolimits ^{1/ 2} $and the mean shear time scale$1/ S$, respectively. With these appropriate length and time scales (or equivalently velocity and time scales), the eddy viscosity can be rewritten in the familiar form of the$k$–$\epsilon $model as${\nu }_{t} = \mathop{(1/ S{T}_{L} )}\nolimits ^{2} {k}^{2} / \epsilon $. We use the direct numerical simulation (DNS) data of turbulent channel flow of Hoyas & Jiménez (Phys. Fluids, vol. 18, 2006, 011702) and the turbulent boundary layer flow of Jiménezet al. (J. Fluid Mech.vol. 657, 2010, pp. 335–360) to perform‘a priori’tests to check the validity of the revised eddy viscosity formulation. The comparisons with the exact computations from the DNS data are remarkable and highlight how well the equilibrium assumption holds in the near-wall region. These findings could prove to be useful in near-wall modelling of turbulent flows.