# Showing papers in "Journal of Fluid Mechanics in 1991"

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TL;DR: In this paper, the authors examined the modeling of the pressure-strain correlation of turbulent flows from a basic theoretical standpoint with a view toward developing improved second-order closure models and proved that for plane homogeneous turbulent flows the equilibrium structure of this hierarchy of models is encapsulated by a relatively simple model which is only quadratically nonlinear in the anisotropy tensor.

Abstract: The modeling of the pressure-strain correlation of turbulence is examined from a basic theoretical standpoint with a view toward developing improved second-order closure models. Invariance considerations along with elementary dynamical systems theory are used in the analysis of the standard hierarchy of closure models. In these commonly used models, the pressure-strain correlation is assumed to be a linear function of the mean velocity gradients with coefficients that depend algebraically on the anisotropy tensor. It is proven that for plane homogeneous turbulent flows the equilibrium structure of this hierarchy of models is encapsulated by a relatively simple model which is only quadratically nonlinear in the anisotropy tensor. This new quadratic model - the SSG model - is shown to outperform the Launder, Reece, and Rodi model (as well as more recent models that have a considerably more complex nonlinear structure) in a variety of homogeneous turbulent flows. Some deficiencies still remain for the description of rotating turbulent shear flows that are intrinsic to this general hierarchy of models and, hence, cannot be overcome by the mere introduction of more complex nonlinearities. It is thus argued that the recent trend of adding substantially more complex nonlinear terms containing the anisotropy tensor may be of questionable value in the modeling of the pressure-strain correlation. Possible alternative approaches are discussed briefly.

1,556 citations

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TL;DR: In this article, the authors performed direct numerical simulations of unsteady channel flow at low to moderate Reynolds numbers on computational boxes chosen small enough so that the flow consists of a doubly periodic array of identical structures.

Abstract: Direct numerical simulations of unsteady channel flow were performed at low to moderate Reynolds numbers on computational boxes chosen small enough so that the flow consists of a doubly periodic (in x and z) array of identical structures. The goal is to isolate the basic flow unit, to study its morphology and dynamics, and to evaluate its contribution to turbulence in fully developed channels. For boxes wider than approximately 100 wall units in the spanwise direction, the flow is turbulent and the low-order turbulence statistics are in good agreement with experiments in the near-wall region. For a narrow range of widths below that threshold, the flow near only one wall remains turbulent, but its statistics are still in fairly good agreement with experimental data when scaled with the local wall stress. For narrower boxes only laminar solutions are found. In all cases, the elementary box contains a single low-velocity streak, consisting of a longitudinal strip on which a thin layer of spanwise vorticity is lifted away from the wall. A fundamental period of intermittency for the regeneration of turbulence is identified, and that process is observed to consist of the wrapping of the wall-layer vorticity around a single inclined longitudinal vortex.

1,097 citations

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TL;DR: In this article, the authors used a direct numerical simulation at resolution 2403 to obtain a statistically stationary three-dimensional homogeneous and isotropic turbulent field at a Reynolds number around 1000 (Rλ ≈ 150).

Abstract: A direct numerical simulation at resolution 2403 is used to obtain a statistically stationary three-dimensional homogeneous and isotropic turbulent field at a Reynolds number around 1000 (Rλ ≈ 150). The energy spectrum displays an inertial subrange. The velocity derivative distribution, known to be strongly non-Gaussian, is found to be close to, but not, exponential. The nth-order moments of this distribution, as well as the velocity structure functions, do not scale with n as predicted by intermittency models. Visualization of the flow confirms the previous finding that the strongest vorticity is organized in very elongated thin tubes. The width of these tubes is of the order of a few dissipation scales, while their length can reach the integral scale of the flow.

912 citations

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Yale University

^{1}TL;DR: In this article, the intermittency of the rate of turbulent energy dissipation e is investigated experimentally, with special emphasis on its scale-similar facets, using a general formulation in terms of multifractals, and interpreting measurements in that light.

Abstract: The intermittency of the rate of turbulent energy dissipation e is investigated experimentally, with special emphasis on its scale-similar facets. This is done using a general formulation in terms of multifractals, and by interpreting measurements in that light. The concept of multiplicative processes in turbulence is (heuristically) shown to lead to multifractal distributions, whose formalism is described in some detail. To prepare proper ground for the interpretation of experimental results, a variety of cascade models is reviewed and their physical contents are analysed qualitatively. Point-probe measurements of e are made in several laboratory flows and in the atmospheric surface layer, using Taylor's frozen-flow hypothesis. The multifractal spectrum f(α) of e is measured using different averaging techniques, and the results are shown to be in essential agreement among themselves and with our earlier ones. Also, long data sets obtained in two laboratory flows are used to obtain the latent part of the f(α) curve, confirming Mandelbrot's idea that it can in principle be obtained from linear cuts through a three-dimensional distribution. The tails of distributions of box-averaged dissipation are found to be of the square-root exponential type, and the implications of this finding for the f(α) distribution are discussed. A comparison of the results to a variety of cascade models shows that binomial models give the simplest possible mechanism that reproduces most of the observations. Generalizations to multinomial models are discussed.

775 citations

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TL;DR: It is proposed that, in moderate Mach number homogeneous turbulence, the compressible component of the turbulence is in quasi-equilibrium with respect to the incompressible turbulence.

Abstract: It is shown that the dilatational terms that need to be modeled in compressible turbulence include not only the pressure-dilatation term but also another term - the compressible dissipation. The nature of these dilatational terms in homogeneous turbulence is explored by asymptotic analysis of the compressible Navier-Stokes equations. A non-dimensional parameter which characterizes some compressible effects in moderate Mach number, homogeneous turbulence is identified. Direct numerical simulations (DNS) of isotropic, compressible turbulence are performed, and their results are found to be in agreement with the theoretical analysis. A model for the compressible dissipation is proposed; the model is based on the asymptotic analysis and the direct numerical simulations. This model is calibrated with reference to the DNS results regarding the influence of compressibility on the decay rate of isotropic turbulence. An application of the proposed model to the compressible mixing layer has shown that the model is able to predict the dramatically reduced growth rate of the compressible mixing layer.

735 citations

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TL;DR: In this paper, the lateral distributions of depth-mean velocity and boundary shear stress for straight open channels with prismatic complex cross-sections are derived theoretically for channels of any shape, provided that the boundary geometry can be discretized into linear elements.

Abstract: The flow of water in straight open channels with prismatic complex cross-sections is considered. Lateral distributions of depth-mean velocity and boundary shear stress are derived theoretically for channels of any shape, provided that the boundary geometry can be discretized into linear elements. The analytical model includes the effects of bed-generated turbulence, lateral shear turbulence and secondary flows. Experimental data from the Science and Engineering Research Council (SERC) Flood Channel Facility are used to illustrate the relative importance of these three effects on internal shear stresses. New experimental evidence concerning the spatial distribution of Reynolds stresses τyx and τzx is presented for the particular case of compound or two-stage channels. In such channels the vertical distributions of τzx are shown to be highly nonlinear in the regions of strongest lateral shear and the depth-averaged values of τyx are shown to be significantly different from the depth mean apparent shear stresses. The importance of secondary flows in the lateral shear layer region is therefore established. The influence of both Reynolds stresses and secondary flows on eddy viscosity values is quantified. A numerical study is undertaken of the lateral distributions of local friction factor and dimensionless eddy viscosity. The results of this study are then used in the analytical model to reproduce lateral distributions of depth-mean velocity and boundary shear stress in a two stage channel. The work will be of interest to engineers engaged in flood channel hydraulics and overbank flow in particular.

591 citations

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TL;DR: In this paper, the authors describe studies of the turbulence of the liquid in a bubbly, grid-generated turbulent flow field using laser-Doppler and hot-film anemometry.

Abstract: The paper describes studies of the turbulence of the liquid in a bubbly, grid-generated turbulent flow field. Laser-Doppler and hot-film anemometry are used for the experimental investigation. It is found that the turbulent kinetic energy increases strongly with the void fraction α. Roughly speaking, there exist two distinct regimes: the first one corresponds to low value of α, where hydrodynamic interactions between bubbles are negligible, and the second one to higher values, for which, owing to their mutual interactions, the bubbles transfer a greater amount of kinetic energy to the liquid. The Reynolds stress tensor shows that the quasi-isotropy is not altered. At low enough values of α, the difference between the turbulent kinetic energy in the liquid phase and the energy associated with the grid-generated turbulence proves to be approximately equal to the intensity of the pseudo-turbulence, defined as the fluctuating energy that would be induced by the motion of the bubbles under non-turbulent conditions. The one-dimensional spectra exhibit a large range of high frequencies associated with the wakes of the bubbles and the classical dependence.

534 citations

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TL;DR: In this article, a theory of equilibrium statistical mechanics is proposed to explain the emergence of organized structures in two-dimensional turbulent flows by taking into account all the known constants of the motion for the Euler equations.

Abstract: We explain the emergence of organised structures in two-dimensional turbulent flows by a theory of equilibrium statistical mechanics. This theory takes into account all the known constants of the motion for the Euler equations. The microscopic states are all the possible vorticity fields, while a macroscopic state is defined as a probability disruption of vorticity at each point of the domain, which describes in a statistical sense the fine-scale vorticity fluctuations. The organised structure appears as a state of maximal entropy, with the constraints of all the constants of the motion. The vorticity field obtained as the local average of this optimal macrostate is a steady solution of the Euler equation. The variational problem provides an explicit relationship between stream function and vorticity, which characterises this steady state. Inertial structures in geophysical fluid dynamics can be predicted, using a generalisation of the theory to potential vorticity.

474 citations

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TL;DR: In this article, the authors explored the usefulness of the wavelet transform for the analysis of turbulent flow fields by examining the wavelets transform properties of a decomposition of turbulent velocity fields into modes that exhibit the localization in a wavenumber and physical space.

Abstract: The usefulness of the wavelet transform for the analysis of turbulent flow fields is explored by examining the wavelet transform properties of a decomposition of turbulent velocity fields into modes that exhibit the localization in a wavenumber and physical space. The calculations are performed on 3D fields from direct numerical simulations of isotropic flow and homogeneous shear flow, and from measurements in two laboratory wind-tunnel experimental velocity signals (boundary layer and wake behind a circular cylinder). The analysis confirmed that there is strong spatial intermittency in nonlinear quantities; their mean spectral behavior results from a delicate balance between large positive and negative excursions. The wavelet analysis is a way to quantify these observations in a standardized fashion by using 'flow-independent eddies' to decompose the velocity field.

422 citations

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TL;DR: In this paper, an extended definition of flamelet regimes based on the existence of a continuous active (not quenched) flame front separating fresh gases and burnt products is introduced.

Abstract: The structure of premixed turbulent flames is a problem of fundamental interest in combustion theory. Possible flame geometries have been imagined and diagrams indicating the corresponding regimes of combustion have been constructed on the basis of essentially intuitive and dimensional considerations. A new approach to this problem is described in the present paper. An extended definition of flamelet regimes based on the existence of a continuous active (not quenched) flame front separating fresh gases and burnt products is first introduced. Direct numerical simulations of flame/vortex interactions using the full Navier–Stokes equations and a simplified chemistry model are then performed to predict flame quenching by isolated vortices. The formulation includes non-unity Lewis number, non-constant viscosity and heat losses so that the effect of stretch, curvature, transient dynamics and viscous dissipation can be accounted for. As a result, flame quenching by vortices (which is one of the key processes in premixed turbulent combustion) may be computed accurately. The effects of curvature and viscous dissipation on flame/vortex interactions may also be characterized by the same simulations. The influence of non-unity Lewis number and of thermo-diffusive processes in turbulent premixed combustion is discussed by comparing flame responses for two values of the Lewis number (Le = 0.8 and 1.2). An elementary (‘spectral’) diagram giving the response of one flame to a vortex pair is constructed. This spectral diagram is then used, along with certain assumptions, to establish a turbulent combustion diagram similar to those proposed by Borghi (1985) or Williams (1985). Results show that flame fronts are much more resistant to quenching by vortices than expected from the classical theories. A cut-off scale and a quenching scale are also obtained and compared with the characteristic scales proposed by Peters (1986). Results show that strain is not the only important parameters determining flame/vortex interaction. Heat losses, curvature, viscous dissipation and transient dynamics have significant effects, especially for small scales and they strongly influence the boundaries of the combustion regimes. It is found, for example, that the Klimov–Williams criterion which is generally advocated to limit the flamelet region, underestimates the size of this region by more than an order of magnitude.

416 citations

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TL;DR: In this article, the motion of a small rigid sphere in a linear shear flow is considered and Saffman's analysis is extended to other asymptotic cases in which the particle Reynolds number based on its slip velocity is comparable with or larger than the square root of the PSR based on the velocity gradient.

Abstract: The motion of a small, rigid sphere in a linear shear flow is considered. Saffman's analysis is extended to other asymptotic cases in which the particle Reynolds number based on its slip velocity is comparable with or larger than the square root of the particle Reynolds number based on the velocity gradient. In all cases, both particle Reynolds numbers are assumed to be small compared to unity. It is shown that, as the Reynolds number based on particle slip velocity becomes larger than the square root of the Reynolds number based on particle shear rate, the magnitude of the inertial migration velocity rapidly decreases to very small values. The latter behaviour suggests that contributions that are higher order in the particle radius may become important in some situations of interest.

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TL;DR: In this article, a large increase or decrease in the resulting displacement thickness, estimated cylinder drag, and associated mixing with the free stream can be achieved, depending on the frequency and amplitude of oscillation.

Abstract: Exploratory experiments have been performed on circular cylinders executing forced rotary oscillations in a steady uniform flow. Flow visualization and wake profile measurements at moderate Reynolds numbers have shown that a considerable amount of control can be exerted over the structure of the wake by such means. In particular, a large increase, or decrease, in the resulting displacement thickness, estimated cylinder drag, and associated mixing with the free stream can be achieved, depending on the frequency and amplitude of oscillation.

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TL;DR: In this paper, the effect of compressibility on a plane mixing layer that is a prototype free shear layer, amenable to study by numerical simulation and experiment, is considered, and the full time-dependent compressible Navier-Stokes equations are solved numerically for a temporally evolving mixing layer employing a mixed spectral and high-order finite difference method.

Abstract: Consideration is given to the effect of compressibility on a plane mixing layer that is a prototype free shear layer, amenable to study by numerical simulation and experiment. The full time-dependent compressible Navier-Stokes equations are solved numerically for a temporally evolving mixing layer employing a mixed spectral and high-order finite difference method. Simulations with random initial conditions confirm the prediction of linear stability theory that at high Mach numbers oblique waves grow faster than two-dimensional waves. Simulations are presented of the nonlinear temporal evolution of the most rapidly amplified linear instability waves.

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TL;DR: In this paper, a free-vortex-type swirling flow is introduced in a long straight circular pipe and the wall shear stresses are measured by a direct method and, except in a short inlet region, are a function only of the swirl intensity and the Reynolds number.

Abstract: Swirling flow through a pipe is a highly complex turbulent flow and is still challenging to predict. An experimental investigation is performed to obtain systematic data about the flow and to understand its physics. A free-vortex-type swirling flow is introduced in a long straight circular pipe. The swirling component decays downstream as a result of wall friction. The velocity distributions are continuously changing as they approach fully developed parallel flow. The swirl intensity Ω, defined as a non-dimensional angular momentum flux, decays exponentially. The decay coefficients, however, are not constant as conventionally assumed, but depend on the swirl intensity. The wall shear stresses are measured by a direct method and, except in a short inlet region, are a function only of the swirl intensity and the Reynolds number. The velocity distributions and all Reynolds stress components are measured at various axial positions in the pipe. The structure of the tangential velocity profile is classified into three regions: core, annular and wall regions. The core region is characterized by a forced vortex motion and the flow is dependent upon the upstream conditions. In the annular region, the skewness of the velocity vector is noticeable and highly anisotropic so that the turbulent viscosity model does not work well here. The tangential velocity is expressed as a sum of free and forced vortex motion. In the wall region the skewness of the flow becomes weak, and the wall law modified by the Monin–Oboukhov formula is applicable. Data on the microscale and the spectrum are also presented and show quite different turbulence structures in the core and the outer regions.

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TL;DR: In this article, a study of both parallel flow and cross-flow in the viscous sublayer generated by a fluid streaming along a grooved surface is presented, with the aim of clarifying the phenomena that underlie the reduction of turbulent drag by such surfaces.

Abstract: A study is undertaken of both parallel flow and cross-flow in the viscous sublayer generated by a fluid streaming along a grooved surface, with the aim of clarifying the phenomena that underlie the reduction of turbulent drag by such surfaces. A quantitative characterization of the effectiveness of different groove profiles in retarding secondary cross-flow is given in terms of the difference of two ‘protrusion heights’. Analytical calculations of limit cases and a boundary-element computer code for the analysis of general profiles are illustrated, and several examples are presented and discussed.

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TL;DR: In this article, a model for a class of water-entry problems characterized by the geometrical property that the impacting body is nearly parallel to the undisturbed water surface and that the impact is so rapid that gravity can be neglected is presented.

Abstract: This paper summarizes and extends some mathematical results for a model for a class of water-entry problems characterized by the geometrical property that the impacting body is nearly parallel to the undisturbed water surface and that the impact is so rapid that gravity can be neglected. Explicit solutions for the pressure distributions are given in the case of two-dimensional flow and a variational formulation is described which provides a simple numerical algorithm for three-dimensional flows. We also pose some open questions concerning the well-posedness and physical relevance of the model for exit problems or when there is an air gap between the impacting body and the water.

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TL;DR: In this article, it is found that cooling (cooling) retards (augments) the spreading process by creating flows that counteract (reinforce) those associated with isothermal spreading for both two-dimensional and axisymmetric configurations.

Abstract: A viscous-liquid drop spreads on a smooth horizontal surface, which is uniformly heated or cooled. Lubrication theory is used to study thin drops subject to capillary, thermocapillary and gravity forces, and a variety of contact-angle-versus-speed conditions. It is found for isothermal drops that gravity is very important at large times and determines the power law for unlimited spreading. Predictions compare well with the experimental data on isothermal spreading for both two-dimensional and axisymmetric configurations. It is found that heating (cooling) retards (augments) the spreading process by creating flows that counteract (reinforce) those associated with isothermal spreading. For zero advancing contact angle, heating will prevent the drop from spreading to infinity. Thus, the heat transfer serves as a sensitive control on the spreading.

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Open University

^{1}TL;DR: In this paper, the authors investigated how the vibrations of a non-rotating liquid droplet are affected by the asphericity and additional restoring forces that the levitating field introduces and showed that the expected single frequency of the fundamental mode is split into either three, when there is an axis of rotational symmetry, or five unequally spaced bands.

Abstract: In experiments to measure the surface energy of a magnetically levitated molten metal droplet by observation of its oscillation frequencies, Rayleigh's equation is usually used. This assumes that the equilibrium shape is a sphere, and the surface restoring force is due only to surface tension. This work investigates how the vibrations of a non-rotating liquid droplet are affected by the asphericity and additional restoring forces that the levitating field introduces. The calculations show that the expected single frequency of the fundamental mode is split into either three, when there is an axis of rotational symmetry, or five unequally spaced bands. Frequencies, on average, are higher than those of an unconstrained droplet; the surface tension appears to be increased over its normal value. This requires a small correction to be made in all analyses of surface energy. A frequency sum rule is derived from a simplified model of the magnetic field which allows the corresponding Rayleigh frequency to be evaluated from the observed frequencies of the fundamental and translational modes. A more detailed analysis shows a similar correction but one that is also sensitive to the position of the droplet in the field.

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TL;DR: In this article, the effect of particle inertia and drift due to body forces on particle dispersion has been investigated in the presence of particle drift and particle drift in the absence of body forces.

Abstract: Measurements of heavy particle dispersion have been made using direct numerical simulations of isotropic turbulence. The parameters affecting the dispersion of solid particles, namely particle inertia and drift due to body forces were investigated separately. In agreement with the theoretical studies of Reeks, and Pismen & Nir, the effect of particle inertia is to increase the eddy diffusivity over that of the fluid (in the absence of particle drift). The increase in the eddy diffusivity of particles over that of the fluid was between 2 and 16%, in reasonable agreement with the increases reported in Reeks, and Pismen & Nir. The effect of a deterministic particle drift is shown to decrease unequally the dispersion in directions normal and parallel to the particle drift direction. Eddy diffusivities normal and parallel to particle drift are shown to be in good agreement with the predictions of Csanady and the experimental measurements of Wells & Stock.

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TL;DR: In this paper, a simplified model of the oscillations of a gas bubble in a slightly compressible liquid is discussed by means of simplified model based on the assumption of a spatially uniform internal pressure.

Abstract: Several aspects of the oscillations of a gas bubble in a slightly compressible liquid are discussed by means of a simplified model based on the assumption of a spatially uniform internal pressure. The first topic considered is the linear initial-value problem for which memory effects and the approach to steady state are analysed. Large-amplitude oscillations are studied next in the limit of large and small thermal diffusion lengths obtaining, in the first case, an explicit expression for the internal pressure, and, in the second one, an integral equation of the Volterra type. The validity of the assumption of uniform pressure is then studied analytically and numerically. Finally, the single-bubble model is combined with a simple averaged-equation model of a bubbly liquid and the propagation of linear and weakly nonlinear pressure waves in such a medium is considered.

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TL;DR: In this paper, a model ramjet combustor facility is studied, and the authors show that the instability is associated with large-scale flamefront motions which are driven by periodic vortex shedding at the instability frequency.

Abstract: Low-frequency combustion instabilities are studied in a model ramjet combustor facility. The facility is two-dimensional, and is comprised of a long inlet duct, a dump combustor cavity with variable size capability, and an exhaust nozzle. The flame is observed to be unstable over a wide range of operating conditions. Acoustic pressure and velocity measurements are made at various locations in the system. They show that the inlet duct acts as a long-wavelength acoustic resonator. However, the instability frequency does not lock to any particular value. This result suggests that the instability mechanism is not purely acoustic in nature. Schlieren imaging reveals that the instability is associated with large-scale flame-front motions which are driven by periodic vortex shedding at the instability frequency. Vortices are generated at the dump in phase with the acoustic velocity fluctuations in the inlet duct. The unsteady heat addition process closely follows the vortex history: the vortices form, grow in size, convect through the combustor cavity, impinge on the exhaust nozzle, break down to small scales and burn. C2 and CH radical spectroscopy is used to determine the phase relation between heat release and pressure in the reaction zone. Rayleigh's criterion is thereby shown to be satisfied. Next, the crucial question of how the oscillation frequency is determined is addressed. Inlet velocity and combustor length are systematically varied to assess the role of vortices by modification of their characteristic lifetime. The influence of the acoustic feedback time is also studied by shortening the inlet duct. The results show that the instability frequency is controlled by both vortex kinetics in the combustor and acoustic response of the inlet section. Therefore, the instability may be considered as a mixed acoustic-convective mode. Finally, combining Rayleigh's criterion with a global feedback loop equation, it is found that the resonant frequencies are selected according to the restriction
\[
\frac{1}{4N-1} < \frac{\tau_{\rm v}}{\tau_{\rm f}} < \frac{3}{4N-3},
\]
where N is the mode of oscillation and τv is the time for vortices to be convected from inlet to exhaust with τf being the feedback time taken for a pressure disturbance to travel up the inlet system and back.

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TL;DR: In this article, the effects of low-Reynolds-number turbulent boundary layers flowing over a smooth flat surface in nominally zero pressure gradients were investigated, and three different types of tripping devices were chosen: a wire, distributed grit and cylindrical pins.

Abstract: An investigation was undertaken to improve our understanding of low-Reynolds-number turbulent boundary layers flowing over a smooth flat surface in nominally zero pressure gradients. In practice, such flows generally occur in close proximity to a tripping device and, though it was known that the flows are affected by the actual low value of the Reynolds number, it was realized that they may also be affected by the type of tripping device used and variations in free-stream velocity for a given device. Consequently, the experimental programme was devised to investigate systematically the effects of each of these three factors independently. Three different types of device were chosen: a wire, distributed grit and cylindrical pins. Mean-flow, broadband-turbulence and spectral measurements were taken, mostly for values of Rθ varying between about 715 and about 2810. It was found that the mean-flow and broadband-turbulence data showed variations with Rθ, as expected. Spectra were plotted using scaling given by Perry, Henbest & Chong (1986) and were compared with their models which were developed for high-Reynolds-number flows. For the turbulent wall region, spectra showed reasonably good agreement with their model. For the fully turbulent region, spectra did show some appreciable deviations from their model, owing to low-Reynolds-number effects. Mean-flow profiles, broadband-turbulence profiles and spectra were found to be affected very little by the type of device used for Rθ ≈ 1020 and above, indicating an absence of dependence on flow history for this Rθ range. These types of measurements were also compared at both Rθ ≈ 1020 and Rθ ≈ 2175 to see if they were dependent on how Rθ was formed (i.e. the combination of velocity and momentum thickness used to determine Rθ). There were noticeable differences for Rθ ≈ 1020, but these differences were only convincing for the pins, and there was a general overall improvement in agreement for Rθ ≈ 2175.

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TL;DR: The linear-eddy approach for modeling molecular mixing in turbulent flow involves stochastic simulation on a one-dimensional domain with sufficient resolution to include all physically relevant lengthscales as discussed by the authors.

Abstract: The linear-eddy approach for modelling molecular mixing in turbulent flow involves stochastic simulation on a one-dimensional domain with sufficient resolution to include all physically relevant lengthscales. In each realization, molecular diffusion is implemented deterministically, punctuated by a sequence of instantaneous, statistically independent ‘rearrangement events’ (measure-preserving maps) representing turbulent stirring. These events emulate the effect of compressive strain on the scalar field. An inertial-range similarity law is incorporated.The model reproduces key features of scalar power spectra, including dependences of spectra! amplitudes and transition wavenumbers on Reynolds and Schmidt numbers. Computed scaling exponents governing scalar power spectra, higher-order fluctuation statistics such as structure functions, and the spatial distribution of scalar level crossings are close to measured exponents. It is inferred that the characterization of stirring as a sequence of independent events (the model analogue of eddies) leads to a useful representation of mixing-field microstructure.

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TL;DR: In this article, a general set of conservation equations and constitutive integrals for the dynamic properties of the rapid flow of a granular material consisting of slightly inelastic and slightly rough spherical particles is derived by following an approach used in the kinetic theory of dense gases.

Abstract: A general set of conservation equations and constitutive integrals for the dynamic properties of the rapid flow of a granular material consisting of slightly inelastic and slightly rough spherical particles is derived by following an approach used in the kinetic theory of dense gases. By taking moments of the translational and rotational particle velocities in the general transport moment equation and making the Enskog approximation, the singlet velocity distribution function is determined. As a result, the constitutive relations and coefficients such as stresses, energy fluxes, rates of translational and rotational energy interchanges, shear viscosity, spin viscosity, bulk viscosity and ‘thermal’ conductivities are obtained. The present theory incorporates the kinetic as well as the collisional contributions for stresses and energy fluxes. Thus, it is appropriate for dilute as well as dense concentrations of solids. For the case of simple shear flow, there is favourable agreement between the theoretical predictions of stresses and both the experimental measurements and the results from computer simulations.

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TL;DR: The linear theory for water waves impinging obliquely on a vertically sided porous structure is examined in this article, where the reflection and transmission coefficients are significantly altered and they are calculated using a plane-wave assumption.

Abstract: The linear theory for water waves impinging obliquely on a vertically sided porous structure is examined. For normal wave incidence, the reflection and transmission from a porous breakwater has been studied many times using eigenfunction expansions in the water region in front of the structure, within the porous medium, and behind the structure in the down-wave water region. For oblique wave incidence, the reflection and transmission coefficients are significantly altered and they are calculated here. Using a plane-wave assumption, which involves neglecting the evanescent eigenmodes that exist near the structure boundaries (to satisfy matching conditions), the problem can be reduced from a matrix problem to one which is analytic. The plane-wave approximation provides an adequate solution for the case where the damping within the structure is not too great. An important parameter in this problem is Γ 2 = ω 2 h ( s - i f )/ g , where ω is the wave angular frequency, h the constant water depth, g the acceleration due to gravity, and s and f are parameters describing the porous medium. As the friction in the porous medium, f , becomes non-zero, the eigenfunctions differ from those in the fluid regions, largely owing to the change in the modal wavenumbers, which depend on Γ 2 . For an infinite number of values of ΓF 2 , there are no eigenfunction expansions in the porous medium, owing to the coalescence of two of the wavenumbers. These cases are shown to result in a non-separable mathematical problem and the appropriate wave modes are determined. As the two wavenumbers approach the critical value of Γ 2 , it is shown that the wave modes can swap their identity.

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TL;DR: In this article, the authors derived an evolution equation for two-dimensional disturbances of a uniform viscous liquid in a uniformly heated inclined plate and derived a linear theory to describe the competition among the instabilities, and derived the finite-amplitude behaviour that determines the propensity for dryout.

Abstract: A layer of volatile viscous liquid drains down a uniformly heated inclined plate. Long-wave instabilities of the uniform film are studied by deriving an evolution equation for two-dimensional disturbances. This equation incorporates viscosity, gravity, surface tension, thermocapillarity, and evaporation eifects. The linear theory derived from this describes the competition among the instabilities. Numerical solution of the evolution equation describes the finite-amplitude behaviour that determines the propensity for dryout of the film. Among the phenomena that appear are the tendency to wave breaking, the creation of secondary structures, and the preemption of dryout by mean flow.

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TL;DR: In this paper, a modified version of Rayleigh's instability criterion was proposed to explain the apparent difference between unstable cyclonic and anticyclonic vortices, in which the cyclonic core has a triangular shape, with three smaller cyclonic satellite vortice at its sides.

Abstract: Laboratory experiments on barotropic vortices in a rotating fluid revealed that the instability behaviour of cyclonic and anticyclonic vortices is remarkably different. Depending on its initial vorticity distribution, the cyclonic vortex has in a number of experiments been observed to be unstable to wavenumber-2 perturbations, leading to the gradual formation of a stable tripolar vortex structure. This tripole consists of an elongated cyclonic core vortex adjoined by two anticyclonic satellite vortices.In contrast, the anticyclonic vortex shows a rather explosive instability behaviour, in the sense that it is observed to immediately split up into two dipoles. Under somewhat different circumstances the higher-order mode-3 instability is observed, in which the anticyclonic core has a triangular shape, with three smaller cyclonic satellite vortices at its sides.A modified version of Rayleigh's instability criterion offers a qualitative explanation for this apparent difference between unstable cyclonic and anticyclonic vortices.

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TL;DR: In this paper, a wide range of direct numerical simulations of forced-dissipative, differentially rotating two-dimensional turbulence are presented, in order to delineate the broad dependence of flow type on forcing parameters.

Abstract: Results from a wide range of direct numerical simulations of forced-dissipative, differentially rotating two-dimensional turbulence are presented, in order to delineate the broad dependence of flow type on forcing parameters. For most parameter values the energy spectra of simulations forced at low wavenumbers are markedly steeper than the classical k−3 enstrophy inertial-range prediction, and although k−3 spectra can be produced under certain circumstances, the regime is not robust, and the Kolmogorov constant is not universal unless a slight generalization is made in the phenomenology. Long-lived, coherent vortices form in many cases, accompanied by steep energy spectra and a higher than Gaussian vorticity kurtosis. With the addition of differential rotation (the β-effect), a small number of fairly distinct flow regimes are observed. Coherent vortices weaken and finally disappear as the strength of the β-effect increases, concurrent with increased anisotropy and decreased kurtosis. Even in the absence of coherent vortices and with a Gaussian value of the kurtosis, the spectra remain relatively steep, although not usually as steep as for the non-rotating cases. If anisotropy is introduced at low wavenumbers, the anisotropy is transferred to all wavenumbers in the inertial range, where the dynamics are isotropic.For those simulations that are forced at relatively high wavenumbers, a well resolved and very robust k−5/3 energy inertial range is observed, and the Kolmogorov constant appears universal. The low-wavenumber extent of the reverse energy cascade is essentially limited by the β-effect, which produces an effective barrier in wavenumber space at which energy accumulates, and by frictional effects which must be introduced to achieve equilibrium. Anisotropy introduced at large scales remains largely confined to the low wavenumbers, rather than being cascaded to small scales. When there is forcing at both large and small scales (which is of relevance to the Earth's atmosphere), energy and enstrophy inertial ranges coexist, with an upscale energy transfer and downscale enstrophy transfer in the same wavenumber interval, without the need for any dissipation mechanism between forcing scales.

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TL;DR: In this article, the authors examined the correction to Darcy's law due to weak convective inertia of the pore fluid and derived general formulae for all constitutive coefficients that can be calculated by numerical solution of certain canonical cell problems.

Abstract: Using the theory of homogenization we examine the correction to Darcy's law due to weak convective inertia of the pore fluid. General formulae are derived for all constitutive coefficients that can be calculated by numerical solution of certain canonical cell problems. For isotropic and homogeneous media the correction term is found to be cubic in the seepage velocity, hence remains small even for Reynolds numbers which are not very small. This implies that inertia, if it is weak, is of greater importance locally than globally. Existing empirical knowledge is qualitatively consistent with our conclusion since the linear law of Darcy is often accurate for moderate flow rates.

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TL;DR: In this paper, the authors used a two-colour laser-Doppler anemometer to measure Axial and radial velocity components of turbulent flow in a circular pipe for the range of Reynolds numbers Reδ = U0δ/ν (U0 = amplitude of cross-sectional mean velocity, δ = (2ν/ω)½) = Stokes layer thickness) from 550 to 2000 and Stokes parameters Λ = R/δ (R = radius of the pipe) from 5 to 10.

Abstract: Experimental results on flow-field statistics are presented for turbulent oscillatory flow in a circular pipe for the range of Reynolds numbers Reδ = U0δ/ν (U0 = amplitude of cross-sectional mean velocity, δ = (2ν/ω)½) = Stokes layer thickness) from 550 to 2000 and Stokes parameters Λ = R/δ (R = radius of the pipe) from 5 to 10. Axial and radial velocity components were measured simultaneously using a two-colour laser-Doppler anemometer, providing information on ensemble-averaged velocity profiles as well as various turbulence statistics for different phases during the cycle. In all flows studied, turbulence appeared explosively towards the end of the acceleration phase of the cycle and was sustained throughout the deceleration phase. During the turbulent portion of the cycle, production of turbulence was restricted to the wall region of the pipe and was the result of turbulent bursts. The statistics of the resulting turbulent flow showed a great deal of similarity to results for steady turbulent pipe flows; in particular the three-layer description of the flow consisting of a viscous sublayer, a logarithmic layer (with von Karman constant = 0.4) and an outer wake could be identified at each phase if the corresponding ensemble-averaged wall-friction velocities were used for normalization. Consideration of similarity laws for these flows reveals that the existence of a logarithmic layer is a dimensional necessity whenever at least two of the scales R, u*/ω and ν/u* are widely separated; with the exact structure of the flow being dependent upon the parameters u*/Rω and u2*/ων. During the initial part of the acceleration phase, production of turbulence as well as turbulent Reynolds stresses were reduced to very low levels and the velocity profiles were in agreement with laminar theory. Nevertheless, the fluctuations retained a small but finite energy. In Part 2 of this paper, the major features observed in these experiments are used as a guideline, in conjunction with direct numerical simulations of the ‘perturbed’ Navier–Stokes equations for oscillatory flow in a channel, to identify the nature of the instability that is most likely to be responsible for transition in this class of flows.