Energy Of Mean Flow Research Articles

Acoustic waves in unbounded shear flows

The linear evolution of acoustic waves in a fluid flow with uniform mean density and uniform shear of velocity is investigated. The process of the mean flow energy extraction by the three-dimensional acoustic waves, stimulated by the non-normal character of the linear dynamics in the shear flow, is analyzed. The thorough examination of the dynamics of different physical variables characterizing the wave evolution is presented. The physics of gaining of the shear energy by acoustic waves is described.

Near-wall Reynolds-stress modelling in noninertial frames of reference

Second-moment closure predictions of fully developed turbulent Poiseuille and Couette flow subjected to spanwise rotation are verified against direct numerical and large eddy simulations in addition to recent experimental results. Near-wall effects are modelled by elliptic relaxation which is used in conjunction with a nonlinear, variable-coefficient pressure strain model consistent with the principle of material frame indifference (MFI) in the limit of two-dimensional turbulence. The dissipation rate model equation is modified in the near-wall region to become explicitly dependent on the imposed system rotation, however, without violating the MFI principle. The model predictions exhibit many of the features due to the Coriolis force on the turbulence and mean flow field, respectively. These include an almost irrotational region in the channel core, augmentation of the turbulence on the unstable side and a corresponding reduction on the stable side, relaminarisation caused not only by stabilising rotation but also by sufficiently high destabilising rotation in Couette flow and, finally, localised regions in the core of the Poiseuille flow where mean flow energy is extracted from the turbulence. The effects of rotational-induced secondary motions on the flow field are also addressed.

Experiments on the Turbulent Boundary Layer on a Thin Cylinder Rotating in an Axial Flow. (2nd Report. Energy Budgets and Power Spectrum).

The budgets of mean flow and turbulent energy as well as the power spectra of u-and v-fluctuating velocities in a three-dimensional turbulent boundary layer on a thin cylinder rotating in a uniform stream are examined experimentally. Advection and diffusion of mean flow energy balance each other in the outer region of the boundary layer, and their absolute values normalized by the characteristic velocity decrease with increasing rotation speed of the cylinder. Over the entire region of the layer, the production and dissipation of turbulent energy are dominant and balance each other, indicating an equilibrium boundary layer. Although advection of turbulent energy is small across the layer, it slightly increases in the outer layer. The power spectra of u-and v-fluctuating velocities measured near the wall exhibit a small peak when the cylinder is rotating, which indicates the existence of an organized eddy structure. There is an increase in the relative population of small-size eddies.

A simulated spectrum of convectively generated gravity waves: Propagation from the tropopause to the mesopause and effects on the middle atmosphere

This work evaluates the interaction of a simulated spectrum of convectively generated gravity waves with realistic middle atmosphere mean winds. The wave spectrum is derived from the nonlinear convection model described by Alexander et al. [1995] that simulated a two‐dimensional midlatitude squall line. This spectrum becomes input to a linear ray tracing model for evaluation of wave propagation as a function of height through climatological background wind and buoyancy frequency profiles. The energy defined by the spectrum as a function of wavenumber and frequency is distributed spatially and temporally into wave packets for the purpose of estimating wave amplitudes at the lower boundary of the ray tracing model. A wavelet analysis provides an estimate of these wave packet widths in space and time. Without this redistribution of energies into wave packets the Fourier analysis alone inaccurately assumes the energy is evenly distributed throughout the storm model domain. The growth with height of wave amplitudes is derived from wave action flux conservation coupled to a convective instability saturation condition. Mean flow accelerations and wave energy dissipation profiles are derived from this analysis and compared to parameterized estimates of gravity wave forcing, providing a measure of the importance of the storm source to global gravity wave forcing. The results suggest that a single large convective storm system like the simulated squall line could provide a significant fraction of the zonal mean gravity wave forcing at some levels, particularly in the mesosphere. The vertical distributions of mean flow acceleration and energy dissipation do not much resemble the parameterized profiles in form because of the peculiarities of the spectral properties of the waves from the storm source. The ray tracing model developed herein provides a tool for examining the role of convectively generated waves in middle atmosphere physics.

Nonlinear sound–vortex interactions in an inviscid isentropic fluid: A two-fluid model

A new two-fluid model is developed to describe the nonlinear interaction of acoustic waves and vortices. Analytical and computational results are presented for a sound pulse interacting with and being modified by a vortex. A novel numerical method based on a particle-in-cell discretization of the acoustic field is developed and used to study the nonlinear scattering of sound by a cylindrical vortex. Equations for the sound wave packet propagating in an axially symmetric mean flow are integrated analytically. Nonlinear modification of the vortex flow by the high-frequency sound is found to be mediated by growing pressure disturbances generated by the radiative forcing on the high gradient regions of the acoustic pulse. The total energy of the vortex mean flow grows monotonically, as the acoustic component loses its energy. The changes in the kinetic and internal energies of the vortex are greater than the changes in its total energy, although these changes are reversible in lowest order of the nonlinear vortex–acoustic interaction.

Vorticity Transport Analysis of Turbulent Flows

Turbulence closure for the Reynolds averaged Navier-Stokes equations based on vorticity transport theory is investigated. General expressions for the vorticity transport correlation terms in arbitrary two-dimensional mean flows are derived. Direct numerical simulation data for flow in a channel is used to evaluate the modeled terms and set unknown scales. Results are presented for a channel, flat plate boundary layer, and flow over a hill. The computed mean flow and kinetic energy compares well with numerical and physical experiments. The vorticity transport model appears to perform better than conventional Boussinesq eddy viscosity Reynolds stress models near the separated flow region on the leeward side of the hill.

Variability of the flow field in the inner shelf along the central east coast of India during April, 1989

Current measurements collected along the inner shelf off the central east coast of India (water depth ⋍ 50 m) at three stations, A (17°59′N, 83°53.9′E), B (17°00′N, 82°32.1′E) and C (16°31.3′N, 82°21.8′E) during the cruise 211 of R.V. Gaveshani (April 1989) are discussed. Data analysis show there to be two layer circulation at Sta. A: a northerly flowing current, which is restricted to the surface and a southerly flowing current at mid water depth and near the bottom. Southerly flowing current occupies the whole of the water column at Sta. B and northerly flowing current at Sta. C. The striking divergence of the mean velocity field between Stas B and C is due to disparate advection states of the local eddy field. The mean southerly velocity of 6–12 cm s −1 from the surface to 45 m depth at Sta. B, and northerly velocity of 84–121 cm s −1 from the surface to 45 m at Sta. C suggesting that the meridional flow is substantially baroclinic. The onshore-offshore component of the currents is much smaller than the alongshore current at all the stations. The upwelling circulation off Kakinada was generally three layered: the surface offshore flow was confined to approximately the upper 10 m, the compensating onshore flow was maximum at mid depth (about 30 m) and there was weak offshore flow along the bottom. The large values of both K e and K m encountered at Sta. C are consistent with the idea that, in this region, baroclinic instability is converting potential energy of the strong mean flow into fluctuation energy. The overall K e decreases northward from Sta. C to Sta. A represent the interior of the northern anticlockwise gyre.

Nonlinear equilibration of two-dimensional optimal perturbations in viscous shear flow

Two-dimensional perturbations configured for maximum energy growth in laminar viscous shear flow are shown to develop into quasisteady finite amplitude structures, provided that the initial perturbation has sufficient energy and a nearby nonlinear mode exists. For Poiseuille flow, which supports finite amplitude equilibria for Reynolds numbers above ∼2900, an optimal perturbation with initial energy density equal to or greater than 0.1% of the mean flow energy density closely approaches the quasiequilibrium state within 10 advective time units. For Couette flow, which has no finite amplitude solution, the optimal perturbations decay rapidly after reaching maximum amplitude unless the configuration is sufficiently close to a linear mode with slow exponential decay rate. While the quasiequilibrium structure for Poiseuille flow is locally inflectional, it supports only weak instabilities with scales larger than the local region.

Budgets of the Reynolds Stresses in an Axisymmetric Free Jet.

In the previous study an axisymmetric free jet was measured with the aid of a three-dimensional particle tracking velocimeter, and a database of the turbulent statistics was constructed. In this study, by using the database, the budgets of the turbulent kinetic energy and the Reynolds normal and shear stresses are calculated. The dissipation term of the turbulent kinetic energy is estimated as a residual of its budget neglecting the pressure diffusion term. The pressure terms in the budgets of the Reynolds stresses are also obtained as closing terms of each budget equation in which the dissipation term is calculated with an assumption of local isotropy. The mechanism of energy exchange among the stresses is revealed, i.e., the mean flow energy is first fed to the streamwise normal stress through the Reynolds shear stress and then redistributed to the radial and azimuthal normal stresses via the pressure term. The production terms in the budgets of the radial and azimuthal stresses are found to be negative around the flow axis.

Quasi-40-day oscillation and its teleconnection structure together with the possible dependence on conversion of barotropic unstable energy of temporal mean flow

A study is made of the distribution of the diagnostic quantity vector\(\overrightarrow E \) and the teleconnection structure of 30–50 (quasi-40) day oscillation, together with the dependence on the conversion of barotropic unstable energy of mean flow in terms of ECWMF daily 500 hPa grid data in winter, indicating that the energy transportation is closely associated with the westerly jet position, with zonal (meridional) propagation in the strong (weak) wind region, that considerable conversion of barotropic energy occurs at the jet exit region where low-frequency oscillation gains energy from the mean flow, leading to maximum kinetic energy for the oscillation observed there, which is marked by evident barotropy in striking contrast to the baroclinicity at low latitudes and that the teleconnection core is related to the center of action in the atmosphere and bound up with the pattern of the west wind.

Three-dimensional optimal perturbations in viscous shear flow

Transition to turbulence in plane channel flow occurs even for conditions under which modes of the linearized dynamical system associated with the flow are stable. In this paper an attempt is made to understand this phenomena by finding the linear three-dimensional perturbations that gain the most energy in a given time period. A complete set of perturbations, ordered by energy growth, is found using variational methods. The optimal perturbations are not of modal form, and those which grow the most resemble streamwise vortices, which divert the mean flow energy into streaks of streamwise velocity and enable the energy of the perturbation to grow by as much as three orders of magnitude. It is suggested that excitation of these perturbations facilitates transition from laminar to turbulent flow. The variational method used to find the optimal perturbations in a shear flow also allows construction of tight bounds on growth rate and determination of regions of absolute stability in which no perturbation growth is possible.

A theory for the generation of multiscale structure in Langmuir circulations

Extant theories and numerical simulations for Langmuir circulations [Leibovich, Ann. Rev. Fluid Dyn. (1983); Leibovich et al., J. Fluid Mech. (1989)] all show predicted circulation patterns characterized by a single horizontal, cross‐wind scale. Recent research [Smith et al., J. Phys. Oceano. (1987); Weller and Price (1988); Pinkel and Smith (1989), unpublished acoustic data] show multiscale circulation patterns, including the organization of bubbles and phytoplankton, in the oceanic surface layer. An analytic theory for the evolution of two‐dimensional disturbances in neutrally stratified, super‐critical shear flows will be presented, done in collaboration with Chris Bretherton, UW‐Applied Mathematics. This theory shows that initially favored, quickly growing disturbances evolve into catalysts for the transfer of mean flow energy into other scales of motion, producing multiscale circulation fields. This general result directly predicts the production of multiscale Langmuir circulations, which is supporte...

Multiscale Large Eddy States in Weakly Stratified Planetary Boundary Layers

We first discuss observations of two classes of two-dimensional large eddy states in weakly stratified atmospheric boundary layers. One class is characterized by large eddies with a single horizontal scale. The other contains multiscale large eddies, with horizontal wavelengths on the order of the boundary layer depth (as generally supported by spectra), coexisting with scales that are several times this depth (as generally seen in cloud street patterns). We then present a theory which describes these different large eddy states as the result of wave–wave interactions between growing disturbances in the planetary boundary layer. We develop nonlinear, coupled evolution equations that govern the temporal evolution of a set of three isolated disturbances of initial eigenmode form in a general unstable flow. These equations model both wave/wave and wave/mean flow interactions. They imply that in unstable flow regimes, modes with relatively large growth rates as predicted by linear theory act as catalysts for the transfer of mean flow energy into modes for which linear theory would predict slow growth or even decay. This energy transfer results in the acceleration of the growth of these initially slow modes. In essence, relatively fast growing instabilities to an unstable mean state are themselves unstable. Because of this indiscriminate catalytic property, a wide range of waves can compete for prominence other than the primary instabilities to a given mean state. We therefore explore four broad classes of wave–wave interactions. This produces predicted large eddy velocity fields which we relate to two classes of observed large eddy states. The predicted multiscale states may be described as a long wavelength modulation of short wavelength roll vortices. Finally, we suggest that the determining factor in selecting a large eddy state in the boundary layer for given synoptic conditions is the presence or absence of sources of waves to the boundary layer that complement the dynamically or convectively driven boundary layer roll vortices. These additional waves could be due to other instability mechanisms, or topographic effects. They would project onto the eigenstates of the boundary layer, introducing relatively well defined scales in the flow field along with the dominant boundary layer disturbances, thereby selecting a set of waves for interaction.

TURBULENT ENTRAINMENT IN PLANE BUOYANT SURFACE JET

The relationship between the entrainment coefficient E=Ve/Uc and the Richardson number Ric at a plane buoyant surface jet was investigated experimentally and theoretically. The entrainment process is well correlated with the dynamics of quasiordered structures. The shear layer Richardson number Rih reflecting their evolution was used to sort the experimental data without scattering. The slope of an approximate curve changes drastically at Rih=0.12. Furthermore, its relationship was analyzed by an integral formulation using the mean flow energy equation as an additional egaution. Theoretical results could be seen to give a good approximation of the relationship between E and Ris obtained experimentally and to evaluate the development of a buoyant surface jet.

Cellular Secondary Currents in Straight Conduit

An investigation of cellular secondary currents, i.e., three‐dimensional flow patterns, is very important in hydraulic engineering because these currents might cause three‐dimensional sediment distributions and bed configurations such as sand ribbons in straight rivers. It is, however, fairly difficult to measure secondary currents of water flows, even by making use of hot‐film anemometers, because the velocity of secondary currents is within 5% of the mainstream velocity. Thus, on the basis of the hypothesis that the existence of free surface may not be an essential cause of secondary currents, the present study has investigated experimentally the turbulent structure of their currents in air conduit. Considering an essential interaction between secondary currents and bedform, longitudinal ridge elements were attached onto both lower and upper bottoms of the conduit, which simulated the longitudinal ridges and trough of river bedform, i.e., sand ribbons. All three components of the velocity were measured accurately by hot‐wire anemometers. The structure of secondary currents was examined through the equations of mean flow vorticity and mean flow energy.

Mean flow energy content of boundary layer down-stream of vertical-axis wind-turbine simulators

True scale models of vertical-axis wind turbines do not function at the small scales which are necessary for turbine array simulation. In past model tests vertical-axis wind turbines have been viewed as devices which have a given drag coefficient and a certain intensity of turbulence in their wakes. These features were modelled in the wind tunnel by stationary simulators when the optimum spacing in vertical-axis wind-turbine arrays was studied. In this paper, it is suggested that rotating vertical-axis machines may be spaced more closely than indicated by past model tests, owing to the enhanced mixing of laterally deflected wakes with the surrounding flow. This wake deflection has not been modelled in the past. In the present study the mean flow energy content of the boundary layer was measured in a wind tunnel downstream of groups of stationary simulators when the turbulence intensity and the drag were modelled both with and without reproduction of the deflected wake of rotating machines. Comparison of the results shows the mean flow energy content of the boundary layer to be up to ∼ 10% higher when wake deflection is reproduced. The results of past model tests are expected, therefore, to be conservative.

The role of displacement thickness fluctuations in hydroacoustics, and the jet-drive mechanism of the flug organ pipe

This paper re-examines a proposal due to Liepmann in which the hydro­acoustic effects of a turbulent boundary layer are represented in terms of the displacement thickness fluctuations. The influence of the curvature of the surface that supports the boundary layer is discussed, and in particular the asymptotic condition is obtained under which Liepmann’s formalism is applicable in the vicinity of leading and trailing edges. This is important for the theoretical treatment of the interaction of nominally steady flows with wall cavities, slots in aerofoils, splitter plates, etc. Displacement thickness fluctuations in the form of Tollmien-Schlichting waves gene­ rated at a leading edge by a disturbance, such as an incident sound wave, are shown to result in a conversion of mean flow energy into sound. At a trailing edge, however, acoustic-mean flow interaction results in the absorption of acoustic energy. A consequence of the leading-edge effect is that it provides an energy transfer mechanism which is capable of main­taining edge tone and cavity oscillations, and this is illustrated by applica­tion of the theory to the flue organ pipe. In this case encouraging support for the asymptotic analysis is provided by a comparison with recently published experimental data.

Major and minor stratospheric warmings and their interactions on the troposphere

Analyses of evolutions of the kinetic and thermal energy associated with the major and minor stratospheric warmings in the winters of 1976–77 and 1975–76 respectively indicate that the predominant ultra-long waves in the stratosphere oscillated at periods of 10–20 days, whereas in the troposphere the predominant long waves oscillated at periods of 8 to 12 days. These tropospheric long waves are almost out-of-phase with the stratospheric ultra-long waves for the minor warming, but in-phase for the major warming. The kinetic energy of the zonal mean flow in the stratosphere for the minor warming is much greater than that for the major warming, indicating that the occurrence of a major warming depends on the magnitude of the kinetic energy of the zonal mean flow relative to that of the meridional convergence of the poleward flux of sensible heat. In both the major and minor warmings, most of the stratospheric eddy kinetic energy is contained in waves of wavenumbers 1 and 2, whereas the stratospheric available potential energy is primarily contained in waves of wavenumber 1. The kinetic energy associated with waves of wavenumber 1 appeared to be 180° out-of-phase with those of wavenumber 2, indicating that nonlinear transfer of kinetic energy occurred between waves of wavenumbers 1 and 2. The occurrences of wind reversals were accompanied by decouplings of the stratospheric and tropospheric motions, and blockings in the troposphere.

The theory ot the index cycle in the general circulation of the atmosphere

Abstract The theory of the index cycle presented here is based on the action of the turbulent Coriolis force. Unlike the ordinary Coriolis force this force does work at a rate equal to the energy exchange between the kinetic energy of the regular motions and that of the turbulent motions. It is shown that in a rotating, differentially-heated fluid the turbulent Coriolis force generates a particular type of instability which we call “gyroturbulent instability”. Gyroturbulent instability is associated with barotropic transfer within the kinetic energy spectrum of a rotating turbulent fluid, and is connected with the conversion of kinetic energy between eddy energy and energy of mean flow as a result of the action of the turbulent Coriolis force. Gyroturbulent instability generates specific oscillations in the general circulation of the atmosphere between the state of high index with weak turbulence and that of low index with strong turbulence (the index cycle). In laboratory rotating annulus experiments, th...

Kinematic wave approach to hydraulic jumps with waves

Rayleigh's classical approach to hydraulic jumps is discussed and extended to allow for the added presence of finite amplitude disturbance waves. In this paper the decrease of mean flow energy through the jump discontinuity is accounted for by the sudden appearance of enhanced downstream radiating waves. Without appealing to turbulent dissipation, the proposed model, which applies only to weak inviscid bores, relates the mean height, the mean speed, the energy density, and the wavenumber fore and aft of the jump through the conservation laws for total mass, total momentum, total energy, and crest number. The model, in shallow water, completely describes the undulatory character of the wavy downstream flow, yielding results in agreement with observed features of bores in nature. Deep water jumps, although somewhat speculative, are also briefly described, with the aim of stimulating further discussion. The relationships between the present hydraulic model and some work of Benjamin and Lighthill and others are also cited for completeness. The basic notions are then extended to wave discontinuities in arbitrary continuous media using ideas derived from kinematic theory, and a further stability analysis points to the conditions under which the postulated jumps exist.