Peter A. Monkewitz

Wall-bounded turbulent flows at high Reynolds numbers: Recent advances and key issues

Wall-bounded turbulent flows at high Reynolds numbers have become an increasingly active area of research in recent years. Many challenges remain in theory, scaling, physical understanding, experimental techniques, and numerical simulations. In this paper we distill the salient advances of recent origin, particularly those that challenge textbook orthodoxy. Some of the outstanding questions, such as the extent of the logarithmic overlap layer, the universality or otherwise of the principal model parameters such as the von Karman “constant,” the parametrization of roughness effects, and the scaling of mean flow and Reynolds stresses, are highlighted. Research avenues that may provide answers to these questions, notably the improvement of measuring techniques and the construction of new facilities, are identified. We also highlight aspects where differences of opinion persist, with the expectation that this discussion might mark the beginning of their resolution.

The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows

Low-dimensional empirical Galerkin models are developed for spatially evolving laminar and transitional shear layers, based on a Karhunen–Loeve decomposition of incompressible two- and three-dimensional Navier–Stokes simulations. It is shown that the key to an accurate Galerkin model is a novel analytical pressure-term representation. The effect of the pressure term is elucidated by a modal energy-flow analysis in a mixing layer, which generalizes the framework developed by Rempfer (1991). In convectively unstable shear layers, it is shown in particular that neglecting small energy terms leads to large amplitude errors in the Galerkin model. The effect of the pressure term and small neglected energy flows is very important for a two-dimensional mixing layer, is less pronounced for the three-dimensional analogue, and can be considered as small in an absolutely unstable wake flow.

Self-excited oscillations in the wake of two-dimensional bluff bodies and their control

The onset of Karman-vortex shedding is studied experimentally in the wake of different two-dimensional bluff bodies, namely an oblong cylinder, circular cylinders and plates of rectangular cross-section. Different control measures, such as wake heating, transverse body oscillations and base bleed are investigated. As the steady-periodic Karman shedding has previously been identified as a limit-cycle, i.e. as self-excited oscillations, the experiments are interpreted in the framework of the Stuart–Landau model. The coefficients of the Stuart–Landau equation for the characteristic vortex shedding amplitude, i.e. the linear temporal growth rate, linear frequency and the Landau constant, are fully determined for the two cylinders and in part for the plate. For this purpose transients are generated by suddenly switching transverse body oscillations or base bleed on or off. The analysis of these transients by a refined method based on complex demodulation provides reliable estimates of the model coefficients and yields an experimental validation of the concept that a global instability mode grows or decays as a whole. Also, it is demonstrated that the coefficients of the Stuart–Landau equation are independent of the experimental technique used to produce the transients.

Approach to an asymptotic state for zero pressure gradient turbulent boundary layers

Flat plate turbulent boundary layers under zero pressure gradient at high Reynolds numbers are studied to reveal appropriate scale relations and asymptotic behaviour. Careful examination of the skin-friction coefficient results confirms the necessity for direct and independent measurement of wall shear stress. We find that many of the previously proposed empirical relations accurately describe the local Cf behaviour when modified and underpinned by the same experimental data. The variation of the integral parameter, H, shows consistent agreement between the experimental data and the relation from classical theory. In accordance with the classical theory, the ratio of Δ and δ asymptotes to a constant. Then, the usefulness of the ratio of appropriately defined mean and turbulent time-scales to define and diagnose equilibrium flow is established. Next, the description of mean velocity profiles is revisited, and the validity of the logarithmic law is re-established using both the mean velocity profile and its diagnostic function. The wake parameter, Π, is shown to reach an asymptotic value at the highest available experimental Reynolds numbers if correct values of logarithmic-law constants and an appropriate skin-friction estimate are used. The paper closes with a discussion of the Reynolds number trends of the outer velocity defect which are important to establish a consistent similarity theory and appropriate scaling.

Self-consistent high-Reynolds-number asymptotics for zero-pressure-gradient turbulent boundary layers

The asymptotic behavior of mean velocity and integral parameters in flat plate turbulent boundary layers under zero pressure gradient are studied for Reynolds numbers approaching infinity. Using the classical two-layer approach of Millikan, Rotta, and Clauser with a logarithmic velocity profile in the overlap region between “inner” and “outer” layers, a fully self-consistent leading-order description of the mean velocity profile and all integral parameters is developed. It is shown that this description fits most high Reynolds number data, and in particular their Reynolds number dependence, exceedingly well; i.e., within experimental errors.

Subharmonic resonance, pairing and shredding in the mixing layer

An instability-wave analysis is presented to describe the spatial evolution of a fundamental mode and its subharmonic on an inviscid parallel mixing layer. It incorporates explicitly the weakly nonlinear interaction between the two modes. The computational finding that the development of the subharmonic, leading eventually to pairing or shredding, crucially depends on its phase relation with the fundamental is fully confirmed. Furthermore it is shown that a critical fundamental amplitude has to be reached before the (spatial) subharmonic becomes phase locked with the fundamental and exhibits a modified growth rate. Then the analysis is exploited to explain the occurrence of amplitude modulations in ‘natural’ mixing layers and to estimate the width of the subharmonic spectral peaks. Also, the case of oblique subharmonic waves is briefly touched upon. In the last part, ways are explored to model non-parallel effects, i.e. to handle the saturation of the rapidly growing subharmonic. Using this wave description, the role of mode interaction in the ‘vortex pairing’ and ‘shredding’ process is assessed.

Non-premixed jet flame pulsations near extinction

A systematic experimental study was aimed at elucidating the conditions for which regular axisymmetric pulsations of the anchoring base of diluted propane and methane jet diffusion flames are observed. These self-excited oscillations result in the periodic streamwise contraction and extension of the flame length. The key experimental parameters governing this type of instability, which was only found near the extinction limit, are identified as the reactant stream Lewis numbers, the oxidizer-to-fuel-stream velocity ratio, and the initial mixture strength. For the diluted propane and methane flames studied, the tendency for pulsations increases with decreasing initial mixture strength, increasing reactant Lewis numbers (effective Lewis number typically >1), and decreasing oxidizer stream velocity (for fixed fuel velocity). Finally, possible connections between these experimental findings and combustion instabilities predicted by recent theoretical analyses are briefly discussed.

Convective versus absolute instability in mixed Rayleigh–Bénard–Poiseuille convection

Transition from convective to absolute instability in Rayleigh–Benard convection in the presence of a uni-directional Poiseuille flow is studied. The evaluation of the long-time behaviour of the Green function in the horizontal plane allows the determination of regions of convective and absolute instability in the Rayleigh–Reynolds number plane as a function of Prandtl number. It is found that the mode reaching zero group velocity at the convective–absolute transition always corresponds to transverse rolls, while the system remains convectively unstable with respect to pure streamwise (longitudinal) rolls for all non-zero Reynolds numbers. Finally, the roll pattern within the entire wave packet and in particular near its centre is elucidated and possible connections between experiments and our findings are discussed.

Linear global modes in spatially developing media

Selection criteria for self-excited global modes in doubly infinite one-dimensional domains are examined in the context of the linearized Ginzburg-Landau equation with slowly varying coefficients. Following Lynn & Keller (1970), uniformly valid approximations are sought in the complex plane in a region containing all relevant turning points. A mapping transformation is introduced to reduce the original Ginzburg-Landau equation to an exactly solvable comparison equation which qualitatively preserves the geometry of the Stokes line network. The specific case of two turning points with counted multiplicity is analysed in detail, particular attention being paid to the allowable configurations of the Stokes line network. It is shown that all global modes are either of type-1, with two simple turning points connected by a common Stokes line, or of type-2, with a single double-turning point. Explicit approximations are derived in both instances, for the global frequencies and associated eigenfunctions. It is argued, on geometrical grounds, that type-1 global modes may, in principle, be more unstable than type-2 global modes. This paper is a continuation and extension of the earlier study of Chomaz, Huerre & Redekopp (1991), where only type-2 global modes were investigated via a local WKBJ approximation scheme.

Nonlinear modelling of vortex shedding control in cylinder wakes

Abstract Karman vortex shedding behind a cylinder placed at right angles to a uniform flow is known to be a limit cycle oscillation that results from the saturation of a global instability of the wake flow. In this paper we study the feedback control of Karman vortex shedding for Reynolds numbers (based on cylinder diameter) close to the critical value of Re c ≈ 47 using “single input - single output” (SISO) proportional control. A model is presented that combines the linear streamwise global mode amplitude equation and the nonlinear spanwise Ginzburg-Landau equation and correctly models the three-dimensional effects observed in the controlled wake of finite length cyclinders. In particular it is demonstrated that for long cylinders vortex shedding can only be suppressed at the spanwise location of the sensor even though the actuation occurs uniformly over the entire span. At a fixed streamwise position the spanwise variation of the shedding angle is thereby given by the “hole solution” of Nozaki and Bekki, J. Phys. Soc. Jpn. 53 (1984) 1581.