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1
Wai, OWH, and KW Bedford. "Empirical orthogonal functional analysis of sediment concentration profiles subjected to waves and currents." Marine and Freshwater Research 46, no. 1 (1995): 373. http://dx.doi.org/10.1071/mf9950373.
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Spatial and temporal eigenfunctions for profiles of suspended-sediment concentrations collected during three distinct flow conditions (current-dominated, wave-dominated, and wave- current-dominated) were used to study the non-linear sediment dynamics in the water column. The eigenfunctions were obtained by the method of Empirical Orthogonal Function (EOF) analysis. The variance distribution of the first spatial eigenfunction associated with the largest eigenvalue reflects the characteristic structure of the original profiles, and the second largest spatial eigenfunction indicates the location of possible structural or boundary layer changes in the profiles. The first temporal eigenfunctions for the current- and wave-driven profiles correlate with the turbulence-wave kinetic energy. Because of the complexity of the wave-current flow field, the first two temporal eigenfunctions for the wave-current-driven profiles have a weak relation with the major driving forces. Orthogonal functions can be used to reconstruct sediment concentration profiles efficiently and accurately. To reconstruct 97% of the variation of 10-min averaged profiles in a 2-h data record, only two eigenvalues, and their corresponding orthogonal functions are required, even in the complex wave-current flow field.
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MANDAL, A. C., L. VENKATAKRISHNAN, and J. DEY. "A study on boundary-layer transition induced by free-stream turbulence." Journal of Fluid Mechanics 660 (July 15, 2010): 114–46. http://dx.doi.org/10.1017/s0022112010002600.
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Boundary-layer transition at different free-stream turbulence levels has been investigated using the particle-image velocimetry technique. The measurements show organized positive and negative fluctuations of the streamwise fluctuating velocity component, which resemble the forward and backward jet-like structures reported in the direct numerical simulation of bypass transition. These fluctuations are associated with unsteady streaky structures. Large inclined high shear-layer regions are also observed and the organized negative fluctuations are found to appear consistently with these inclined shear layers, along with highly inflectional instantaneous streamwise velocity profiles. These inflectional velocity profiles are similar to those in the ribbon-induced boundary-layer transition. An oscillating-inclined shear layer appears to be the turbulent spot-precursor. The measurements also enabled to compare the actual turbulent spot in bypass transition with the simulated one. A proper orthogonal decomposition analysis of the fluctuating velocity field is carried out. The dominant flow structures of the organized positive and negative fluctuations are captured by the first few eigenfunction modes carrying most of the fluctuating energy. The similarity in the dominant eigenfunctions at different Reynolds numbers suggests that the flow prevails its structural identity even in intermittent flows. This analysis also indicates the possibility of the existence of a spatio-temporal symmetry associated with a travelling wave in the flow.
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Böberg, L., and U. Brosa. "Onset of Turbulence in a Pipe." Zeitschrift für Naturforschung A 43, no. 8-9 (September 1, 1988): 697–726. http://dx.doi.org/10.1515/zna-1988-8-901.
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AbstractTurbulence in a pipe is derived directly from the Navier-Stokes equation. Analysis of numerical simulations revealed that small disturbances called 'mothers' induce other much stronger disturbances called 'daughters'. Daughters determine the look of turbulence, while mothers control the transfer of energy from the basic flow to the turbulent motion. From a practical point of view, ruling mothers means ruling turbulence. For theory, the mother-daughter process represents a mechanism permitting chaotic motion in a linearly stable system. The mechanism relies on a property of the linearized problem according to which the eigenfunctions become more and more collinear as the Reynolds number increases. The mathematical methods are described, comparisons with experiments are made, mothers and daughters are analyzed, also graphically, with full particulars, and the systematic construction of small systems of differential equations to mimic the non-linear process by means as simple as possible is explained. We suggest that more then 20 but less than 180 essential degrees of freedom take part in the onset of turbulence.
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Lienhard, J. H., and I. Catton. "Heat Transfer Across a Two-Fluid-Layer Region." Journal of Heat Transfer 108, no. 1 (February 1, 1986): 198–205. http://dx.doi.org/10.1115/1.3246887.
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Heat transfer across a two-fluid-layer region is calculated for Rayleigh numbers in excess of the critical value. The method of solution is based on Landau’s comments regarding the onset of turbulence, following Malkus and others. The linear stability problem is solved for its eigenvalues and eigenfunctions, and the eigenfunctions are used to calculate the contribution of secondary motion to heat transfer. Results are obtained in terms of an overall Nusselt number as a function of Rayleigh number, midlayer thickness, and midlayer thermal conductivity.
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Esler, J. G., and T. L. Ashbee. "Universal statistics of point vortex turbulence." Journal of Fluid Mechanics 779 (August 14, 2015): 275–308. http://dx.doi.org/10.1017/jfm.2015.410.
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A new methodology, based on the central limit theorem, is applied to describe the statistical mechanics of two-dimensional point vortex motion in a bounded container $\mathscr{D}$, as the number of vortices $N$ tends to infinity. The key to the approach is the identification of the normal modes of the system with the eigenfunction solutions of the so-called hydrodynamic eigenvalue problem of the Laplacian in $\mathscr{D}$. The statistics of the projection of the vorticity distribution onto these eigenfunctions (‘vorticity projections’) are then investigated. The statistics are used first to obtain the density-of-states function and caloric curve for the system, generalising previous results to arbitrary (neutral) distributions of vortex circulations. Explicit expressions are then obtained for the microcanonical (i.e. fixed energy) probability density functions of the vorticity projections in a form that can be compared directly with direct numerical simulations of the dynamics. The energy spectra of the resulting flows are predicted analytically. Ensembles of simulations with $N=100$, in several conformal domains, are used to make a comprehensive validation of the theory, with good agreement found across a broad range of energies. The probability density function of the leading vorticity projection is of particular interest because it has a unimodal distribution at low energy and a bimodal distribution at high energy. This behaviour is indicative of a phase transition, known as Onsager–Kraichnan condensation in the literature, between low-energy states with no mean flow in the domain and high-energy states with a coherent mean flow. The critical temperature for the phase transition, which depends on the shape but not the size of $\mathscr{D}$, and the associated critical energy are found. Finally the accuracy and the extent of the validity of the theory, at finite $N$, are explored using a Markov chain phase-space sampling method.
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Nizamova, A. D., V. N. Kireev, and S. F. Urmancheev. "Research of eigenfuctions perturbation of the transverse component velocity thermoviscous liquids flow." Multiphase Systems 14, no. 2 (2019): 132–37. http://dx.doi.org/10.21662/mfs2019.2.018.
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The viscous model fluid flow in a plane channel with a linear temperature profile is considered. The problem of the thermoviscous fluid flow stability is solved on the basis of the previously obtained generalized Orr–Sommerfeld equation by the spectral method of decomposition into Chebyshev polynomials. We study the effect of taking into account the linear and exponential dependences of the viscosity of a liquid on temperature on the eigenfunctions of the hydrodynamic stability equation and on perturbations of the transverse velocity of an incompressible fluid in a plane channel when various wall temperatures are specified. Eigenfunctions are found numerically for two eigenvalues of the linear and exponential dependence of viscosity on temperature. Presented pictures of their own functions. The eigenfunctions demonstrate the behavior of the transverse velocity perturbations, their possible growth or attenuation over time. For the given eigenfunctions, perturbations of the transverse flow velocity of a thermoviscous fluid are obtained. It is shown that taking the temperature dependence of viscosity into account affects the eigenfunctions of the equations of hydrodynamic stability and perturbations of the transverse flow velocity. Perturbations of the transverse velocity significantly affect the hydrodynamic instability of the fluid flow. The results show that when considering the unstable eigenvalue over time, the velocity perturbations begin to grow, which leads to turbulence of the flow. The maximum values of the eigenfunctions and perturbations of the transverse velocities are shifted to the hot wall. It is seen that for an unstable eigenvalue, the perturbations of the transverse flow velocity increase over time, and for a stable one, they decay.
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GIBSON, J. F., J. HALCROW, and P. CVITANOVIĆ. "Visualizing the geometry of state space in plane Couette flow." Journal of Fluid Mechanics 611 (September 25, 2008): 107–30. http://dx.doi.org/10.1017/s002211200800267x.
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Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier–Stokes equations. We construct a dynamical 105-dimensional state-space representation of plane Couette flow at Reynolds number Re = 400 in a small periodic cell and offer a new method of visualizing invariant manifolds embedded in such high dimensions. We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Re and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate-Re turbulence. The invariant manifolds partially tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of symmetry-induced heteroclinic connections.
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Pirozzoli, Sergio, Davide Modesti, Paolo Orlandi, and Francesco Grasso. "Turbulence and secondary motions in square duct flow." Journal of Fluid Mechanics 840 (February 14, 2018): 631–55. http://dx.doi.org/10.1017/jfm.2018.66.
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We study turbulent flows in pressure-driven ducts with square cross-section through direct numerical simulation in a wide enough range of Reynolds number to reach flow conditions which are representative of fully developed turbulence ($Re_{\unicode[STIX]{x1D70F}}\approx 1000$). Numerical simulations are carried out over very long integration times to get adequate convergence of the flow statistics, and specifically to achieve high-fidelity representation of the secondary motions which arise. The intensity of the latter is found to be on the order of 1 %–2 % of the bulk velocity, and approximately unaffected by Reynolds number variation, at least in the range under scrutiny. The smallness of the mean convection terms in the streamwise vorticity equation points to a simple characterization of the secondary flows, which in the asymptotic high-$Re$ regime are approximated with good accuracy by eigenfunctions of the Laplace operator, in the core part of the duct. Despite their effect of redistributing the wall shear stress along the duct perimeter, we find that secondary motions do not have a large influence on the bulk flow properties, and the streamwise velocity field can be characterized with good accuracy as resulting from the superposition of four flat walls in isolation. As a consequence, we find that parametrizations based on the hydraulic diameter concept, and modifications thereof, are successful in predicting the duct friction coefficient.
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Knorr, George. "Symmetries in hydrodynamic turbulence and MHD dynamo theory." Journal of Plasma Physics 56, no. 3 (December 1996): 391–406. http://dx.doi.org/10.1017/s002237780001936x.
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The three-dimensional equations of ideal hydrodynamics and ideal MHD are expanded in eigenfunctions of the curl, and the resulting basic interactions of these nonlinear systems are analysed. As the equations are invariant under time and amplitude reversal, a criterion defining the arrow of time is introduced. A new parameter, the center of energy, serves to characterize a basic interaction. In the 3D Euler equations we find four different interactions and their mirror images, two of which can transport energy to smaller wavenumbers. This can lead to the appearance of structures in turbulent flow, and throws doubt on a derivation of Kolmogorov's law based on a cascading of energy to higher wavenumbers.In energy the corresponding two-dimensional equations, which are isomorphic to the guiding centre model in plasma physics, only one interaction exists, with a strong inverse cascade, which can lead to accumulation of energy in the spatially largest accessible modes. In MHD theory it is possible to separate magnetic from kinetic interactions. The former give again four basic interactions, two being regular and two being inverse cascades. One of these is quite strong, and can lead to the MHD dynamo effect. Kinetic energy can be transferred into magnetic energy. The dynamo effect is is accompanied by alignment of velocity and magnetic fields. We show that stationary velocity fields may lead to exponentially growing magnetic fields and we give an explicit criterion for this instability.
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Gohardehi, Siavash, Saeed Arablu, Hossein Afshin, and Bijan Farhanieh. "Investigation of the effect of turbulence intensity and nozzle exit boundary layer thickness on stability pattern of subsonic jet." Mechanics & Industry 20, no. 1 (2019): 103. http://dx.doi.org/10.1051/meca/2018041.
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In this study, factors affecting the noise generation by instability waves in a subsonic jet with acoustic Mach number of 0.5 are investigated using linear stability analysis. The base flow required for instability analysis is obtained by modeling the jet stream based on the k-ε turbulence model and using the empirical coefficients suggested by Thies and Tam [1]. The resulting base flow profiles are used to solve the linear instability equation, which governs the pressure perturbation for obtaining the eigenvalues and eigenfunctions. The results of linear instability analysis for phase and amplitude of pressure fluctuations are compared against the existing experimental data, which demonstrated the validity of the conducted instability analysis. The effects of turbulence intensity and thickness of the boundary layer at the jet nozzle exit on the results of the linear instability analysis are investigated. The results show that as the turbulence intensity at nozzle exit increases, the frequency range for which the spatial growth rates are positive grows smaller, and except for very low frequencies, this leads to decreased growth rates in both axisymmetric and first azimuthal modes. Also, in both of these modes, an increase in the thickness of the boundary layer at nozzle exit leads to a decrease in perturbation's growth rates in the surveyed frequency ranges.
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Sanghi, Sanjeev, and Nadine Aubry. "Mode interaction models for near-wall turbulence." Journal of Fluid Mechanics 247 (February 1993): 455–88. http://dx.doi.org/10.1017/s0022112093000527.
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Intermittent bursting events, similar to those characterizing the dynamics of near-wall turbulence, have been observed in a low-dimensional dynamical model (Aubry et al. 1988) built from eigenfunctions of the proper orthogonal decomposition (Lumley 1967). In the present work, we investigate the persistency of the intermittent behaviour in higher - but still of relatively low-dimensional dynamical systems. In particular, streamwise variations which were not accounted for in an explicit way in Aubry et al.'s model are now considered. Intermittent behaviour persists but can be of a different nature. Specifically, the non-zero streamwise modes become excited during the eruptive events so that rolls burst downstream into smaller scales. When structures have a finite length, they travel at a convection speed approximately equal to the mean velocity at the top of the layer (y+ ≈ 40). In all cases, intermittency seems to be due to homoclinic cycles connecting hyperbolic fixed points or more complex (apparently chaotic) limit sets. While these sets lie in the zero streamwise modes invariant subspace, the connecting orbits consist of nonzero streamwise modes travelling downstream. Chaotic limit sets connected by quasi-travelling waves have also been observed in a spatio-temporal chaotic regime of the Kuramoto–Sivashinsky equation (Aubry & Lian 1992a). When the limit sets lose their steadiness, the elongated rolls become randomly active, as they probably are in the real flow. A coherent structure study in our resulting flow fields is performed in order to relate our findings to experimental observations. It is shown that streaks, streamwise rolls, horseshoe vortical structures and shear layers, present in our models, are all connected to each other. Finally, criteria to determine a realistic value of the eddy viscosity parameter are developed.
12
Miller, Evan. "Global regularity for solutions of the Navier–Stokes equation sufficiently close to being eigenfunctions of the Laplacian." Proceedings of the American Mathematical Society, Series B 8, no. 12 (May 19, 2021): 129–44. http://dx.doi.org/10.1090/bproc/62.
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In this paper, we will prove a new, scale critical regularity criterion for solutions of the Navier–Stokes equation that are sufficiently close to being eigenfunctions of the Laplacian. This estimate improves previous regularity criteria requiring control on the H ˙ α \dot {H}^\alpha norm of u , u, with 2 ≤ α > 5 2 , 2\leq \alpha >\frac {5}{2}, to a regularity criterion requiring control on the H ˙ α \dot {H}^\alpha norm multiplied by the deficit in the interpolation inequality for the embedding of H ˙ α − 2 ∩ H ˙ α ↪ H ˙ α − 1 . \dot {H}^{\alpha -2}\cap \dot {H}^{\alpha } \hookrightarrow \dot {H}^{\alpha -1}. This regularity criterion suggests, at least heuristically, the possibility of some relationship between potential blowup solutions of the Navier–Stokes equation and the Kolmogorov-Obhukov spectrum in the theory of turbulence.
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Mula, Swathi M., and Charles E. Tinney. "A study of the turbulence within a spiralling vortex filament using proper orthogonal decomposition." Journal of Fluid Mechanics 769 (March 25, 2015): 570–89. http://dx.doi.org/10.1017/jfm.2015.104.
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The stability and turbulence characteristics of a vortex filament emanating from a single-bladed rotor in hover are investigated using proper orthogonal decomposition (POD). The rotor is operated at a tip chord Reynolds number and tip Mach number of 218 000 and 0.23, respectively, and with a blade loading of $C_{T}/{\it\sigma}=0.066$. In-plane components of the velocity field (normal to the axis of the vortex filament) are captured by way of two-dimensional particle image velocimetry with corrections for vortex wander being performed using the ${\it\Gamma}_{1}$ method. The first POD mode alone is found to encompass nearly 75 % of the energy for all vortex ages studied and is determined using a grid of sufficient resolution to avoid numerical integration errors in the decomposition. The findings reveal an equal balance between the axisymmetric and helical modes during vortex roll-up, which immediately transitions to helical mode dominance at all other vortex ages. This helical mode is one of the modes of the elliptic instability. The spatial eigenfunctions of the first few Fourier-azimuthal modes associated with the most energetic POD mode is shown to be sensitive to the choice of the wander correction technique used. Higher Fourier-azimuthal modes are observed in the outer portions of the vortex and appeared not to be affected by the choice of the wander correction technique used.
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Giannakis, Dimitrios, Anastasiya Kolchinskaya, Dmitry Krasnov, and Jörg Schumacher. "Koopman analysis of the long-term evolution in a turbulent convection cell." Journal of Fluid Mechanics 847 (May 29, 2018): 735–67. http://dx.doi.org/10.1017/jfm.2018.297.
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We analyse the long-time evolution of the three-dimensional flow in a closed cubic turbulent Rayleigh–Bénard convection cell via a Koopman eigenfunction analysis. A data-driven basis derived from diffusion kernels known in machine learning is employed here to represent a regularized generator of the unitary Koopman group in the sense of a Galerkin approximation. The resulting Koopman eigenfunctions can be grouped into subsets in accordance with the discrete symmetries in a cubic box. In particular, a projection of the velocity field onto the first group of eigenfunctions reveals the four stable large-scale circulation (LSC) states in the convection cell. We recapture the preferential circulation rolls in diagonal corners and the short-term switching through roll states parallel to the side faces which have also been seen in other simulations and experiments. The diagonal macroscopic flow states can last as long as 1000 convective free-fall time units. In addition, we find that specific pairs of Koopman eigenfunctions in the secondary subset obey enhanced oscillatory fluctuations for particular stable diagonal states of the LSC. The corresponding velocity-field structures, such as corner vortices and swirls in the midplane, are also discussed via spatiotemporal reconstructions.
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Hack, M. J. Philipp, and Tamer A. Zaki. "Modal and non-modal stability of boundary layers forced by spanwise wall oscillations." Journal of Fluid Mechanics 778 (August 3, 2015): 389–427. http://dx.doi.org/10.1017/jfm.2015.387.
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Modal and non-modal perturbation growth in boundary layers subjected to time-harmonic spanwise wall motion are examined. The superposition of the streamwise Blasius flow and the spanwise Stokes layer can lead to strong modal amplification during intervals of the base-flow period. Linear stability analysis of frozen phases of the base state demonstrates that this growth is due to an inviscid instability, which is related to the inflection points of the spanwise Stokes layer. The generation of new inflection points at the wall and their propagation towards the free stream leads to mode crossing when tracing the most unstable mode as a function of phase. The fundamental mode computed in Floquet analysis has a considerably lower growth rate than the instantaneous eigenfunctions. Furthermore, the algebraic lift-up mechanism that causes the formation of Klebanoff streaks is examined in transient growth analyses. The wall forcing significantly weakens the wall-normal velocity perturbations associated with lift-up. This effect is attributed to the formation of a pressure field which redistributes energy from the wall-normal to the spanwise velocity perturbations. The results from linear theory explain observations from direct numerical simulations of breakdown to turbulence in the same flow configuration by Hack & Zaki (J. Fluid Mech., vol. 760, 2014a, pp. 63–94). When bypass mechanisms are dominant, the flow is stabilized due to the weaker non-modal growth. However, at high amplitudes of wall oscillation, transition is promoted due to fast growth of the modal instability.
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KOCH, WERNER, FABIO P. BERTOLOTTI, ANDREAS STOLTE, and STEFAN HEIN. "Nonlinear equilibrium solutions in a three-dimensional boundary layer and their secondary instability." Journal of Fluid Mechanics 406 (March 10, 2000): 131–74. http://dx.doi.org/10.1017/s0022112099007387.
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The observed nonlinear saturation of crossflow vortices in the DLR swept-plate transition experiment, followed by the onset of high-frequency signals, motivated us to compute nonlinear equilibrium solutions for this flow and investigate their instability to high-frequency disturbances. The equilibrium solutions are independent of receptivity, i.e. the way crossflow vortices are generated, and thus provide a unique characterization of the nonlinear flow prior to turbulence. Comparisons of these equilibrium solutions with experimental measurements exhibit strong similarities. Additional comparisons with results from the nonlinear parabolized stability equations (PSE) and spatial direct numerical simulations (DNS) reveal that the equilibrium solutions become unstable to steady, spatial oscillations with very long wavelengths following a bifurcation close to the leading edge. Such spatially oscillating solutions have been observed also in critical layer theory computations. The nature of the spatial behaviour is herein clarified and shown to be analogous to that encountered in temporal direct numerical simulations. We then employ Floquet theory to systematically study the dependence of the secondary, high-frequency instabilities on the saturation amplitude of the equilibrium solutions. With increasing amplitude, the most amplified instability mode can be clearly traced to spanwise inflectional shear layers that occur in the wake-like portions of the equilibrium solutions (Malik et al. 1994 call it ‘mode I’ instability). Both the frequency range and the eigenfunctions resemble recent experimental measurements of Kawakami et al. (1999). However, the lack of an explosive growth leads us to believe that additional self-sustaining processes are active at transition, including the possibility of an absolute instability of the high-frequency disturbances.
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Derebail Muralidhar, Srikanth, Bérengère Podvin, Lionel Mathelin, and Yann Fraigneau. "Spatio-temporal proper orthogonal decomposition of turbulent channel flow." Journal of Fluid Mechanics 864 (February 11, 2019): 614–39. http://dx.doi.org/10.1017/jfm.2019.48.
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An extension of proper orthogonal decomposition is applied to the wall layer of a turbulent channel flow ($Re_{\unicode[STIX]{x1D70F}}=590$), so that empirical eigenfunctions are defined in both space and time. Due to the statistical symmetries of the flow, the eigenfunctions are associated with individual wavenumbers and frequencies. Self-similarity of the dominant eigenfunctions, consistent with wall-attached structures transferring energy into the core region, is established. The most energetic modes are characterized by a fundamental time scale in the range 200–300 viscous wall units. The full spatio-temporal decomposition provides a natural measure of the convection velocity of structures, with a characteristic value of 12$u_{\unicode[STIX]{x1D70F}}$ in the wall layer. Finally, we show that the energy budget can be split into specific contributions for each mode, which provides a closed-form expression for nonlinear effects.
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Winter, M., T. J. Barber, R. M. Everson, and L. Sirovich. "Eigenfunction analysis of turbulent mixing phenomena." AIAA Journal 30, no. 7 (July 1992): 1681–88. http://dx.doi.org/10.2514/3.11123.
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THEOFILIS, VASSILIOS. "On linear and nonlinear instability of the incompressible swept attachment-line boundary layer." Journal of Fluid Mechanics 355 (January 25, 1998): 193–227. http://dx.doi.org/10.1017/s0022112097007660.
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The stability of an incompressible swept attachment-line boundary layer flow is studied numerically, within the Görtler–Hämmerlin framework, in both the linear and nonlinear two-dimensional regimes in a self-consistent manner. The initial-boundary-value problem resulting from substitution of small-amplitude excitation into the incompressible Navier–Stokes equations and linearization about the generalized Hiemenz profile is solved. A comprehensive comparison of all linear approaches utilized to date is presented and it is demonstrated that the linear initial-boundary-value problem formulation delivers results in excellent agreement with those obtained by solution of either the temporal or the spatial linear stability theory eigenvalue problem for both zero suction and a layer in which blowing is applied. In the latter boundary layer recent experiments have documented the growth of instability waves with frequencies in a range encompassed by that of the unstable Görtler–Hämmerlin linear modes found in our simulations. In order to enable further comparisons with experiment and, thus, assess the validity of the Görtler–Hämmerlin theoretical model, we make available the spatial structure of the eigenfunctions at maximum growth conditions.The condition on smallness of the imposed excitation is subsequently relaxed and the resulting nonlinear initial-boundary-value problem is solved. Extensive numerical experimentation has been performed which has verified theoretical predictions on the way in which the solution is expected to bifurcate from the linear neutral loop. However, it is demonstrated that the two-dimensional model equations considered do not deliver subcritical instability of this flow; this strengthens the conjecture that three-dimensionality is, at least partly, responsible for the observed discrepancy between the linear theory critical Reynolds number and the subcritical turbulence observed either experimentally or in three-dimensional numerical simulations. Further, the present nonlinear computations demonstrate that the unstable flow has its line of maximum amplification in the neighbourhood of the experimentally observed instability waves, in a manner analogous to the Blasius boundary layer. In line with previous eigenvalue problem and direct simulation work, suction is observed to be a powerful stabilization mechanism for naturally occurring instabilities of small amplitude.
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Duggleby, Andrew, Kenneth S. Ball, Mark R. Paul, and Paul F. Fischer. "Dynamical eigenfunction decomposition of turbulent pipe flow." Journal of Turbulence 8 (January 2007): N43. http://dx.doi.org/10.1080/14685240701376316.
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Ball, K. S., L. Sirovich, and L. R. Keefe. "Dynamical eigenfunction decomposition of turbulent channel flow." International Journal for Numerical Methods in Fluids 12, no. 6 (April 5, 1991): 585–604. http://dx.doi.org/10.1002/fld.1650120606.
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Sirovich, Lawrence, and Richard Everson. "Management and Analysis of Large Scientific Datasets." International Journal of Supercomputing Applications 6, no. 1 (April 1992): 50–68. http://dx.doi.org/10.1177/109434209200600104.
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The method of empirical eigenfunctions (Karhunen-Loève procedure) is developed within a framework suitable for dealing with large scientific datasets. It is shown that this furnishes an intrinsic representation of any given database which is always, in a well-defined mathematical sense, the optimal description. The methodology is illustrated by a variety of examples, arising out of current research and taken from pattern recognition, turbulent flow, physiology, and oceanographic flow. In each instance examples of the empirical eigenfunctions are presented.
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Foias, C., O. Manley, and L. Sirovich. "Empirical and Stokes eigenfunctions and the far‐dissipative turbulent spectrum." Physics of Fluids A: Fluid Dynamics 2, no. 3 (March 1990): 464–67. http://dx.doi.org/10.1063/1.857744.
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Sirovich, Lawrence. "Analysis of turbulent flows by means of the empirical eigenfunctions." Fluid Dynamics Research 8, no. 1-4 (October 1991): 85–100. http://dx.doi.org/10.1016/0169-5983(91)90033-f.
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Aubry, Nadine, Philip Holmes, John L. Lumley, and Emily Stone. "The dynamics of coherent structures in the wall region of a turbulent boundary layer." Journal of Fluid Mechanics 192 (July 1988): 115–73. http://dx.doi.org/10.1017/s0022112088001818.
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We have modelled the wall region of a turbulent boundary layer by expanding the instantaneous field in so-called empirical eigenfunctions, as permitted by the proper orthogonal decomposition theorem (Lumley 1967, 1981). We truncate the representation to obtain low-dimensional sets of ordinary differential equations, from the Navier–Stokes equations, via Galerkin projection. The experimentally determined eigenfunctions of Herzog (1986) are used; these are in the form of streamwise rolls. Our model equations represent the dynamical behaviour of these rolls. We show that these equations exhibit intermittency, which we analyse using the methods of dynamical systems theory, as well as a chaotic regime. We argue that this behaviour captures major aspects of the ejection and bursting events associated with streamwise vortex pairs which have been observed in experimental work (Kline et al. 1967). We show that although this bursting behaviour is produced autonomously in the wall region, and the structure and duration of the bursts is determined there, the pressure signal from the outer part of the boundary layer triggers the bursts, and determines their average frequency. The analysis and conclusions drawn in this paper appear to be among the first to provide a reasonably coherent link between low-dimensional chaotic dynamics and a realistic turbulent open flow system.
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Berkooz, Gal, Philip Holmes, and J. L. Lumley. "Intermittent dynamics in simple models of the turbulent wall layer." Journal of Fluid Mechanics 230 (September 1991): 75–95. http://dx.doi.org/10.1017/s002211209100071x.
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We generalize the class of models of the wall layer of Aubry et al. (1988), based on the proper orthogonal decomposition, to permit uncoupled evolution of streamwise and cross-stream disturbances. Since the Reynolds stress is no longer constrained, in the absence of streamwise spatial variations all perturbation velocity components eventually decay to zero. However, their transient behaviour is dominated by ’ghosts’ of the non-trivial fixed points and attracting heteroclinic cycles which are characteristic features of those models based on empirical eigenfunctions whose individual velocity components are fixed. This suggests that the intermittent events observed in Aubry et al. do not arise solely because of the effective closure assumption incorporated in those models, but are rooted deeper in the dynamical phenomenon of the wall region.
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Rempfer, Dietmar, and Hermann F. Fasel. "Evolution of three-dimensional coherent structures in a flat-plate boundary layer." Journal of Fluid Mechanics 260 (February 10, 1994): 351–75. http://dx.doi.org/10.1017/s0022112094003551.
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Using a data base generated by a numerical simulation, the three-dimensional coherent structures of a transitional, spatially evolving boundary layer are determined and their spatio-temporal behaviour is investigated in detail. The coherent structures are calculated by the proper orthogonal decomposition method (POD), which leads to an expansion of the flow field variables into Karhunen-Loéve eigenfunctions. It is shown that the dynamical coherent structures of the flat-plate boundary layer can be described by pairs of eigenfunctions that contain complete information on the spatial evolution of the structures. It is further demonstrated that first-order coherent structures determined by POD correspond to structures that are observed in experiments. In the region of the boundary layer where the spike signals of transition occur, higher-order coherent structures also play an essential role. By considering these higher-order structures as well as their dynamical behaviour in time, a compact description of the flow phenomena in the boundary layer can be obtained. The description of the events occurring at the spike stages of the transitional boundary layer shows, from a coherent structures point of view, striking similarities to the bursting event of fully turbulent boundary layers.
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GORDEYEV, S. V., and F. O. THOMAS. "Coherent structure in the turbulent planar jet. Part 1. Extraction of proper orthogonal decomposition eigenmodes and their self-similarity." Journal of Fluid Mechanics 414 (July 10, 2000): 145–94. http://dx.doi.org/10.1017/s002211200000848x.
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In this paper the coherent structure in the similarity region of the turbulent planar jet is examined experimentally by application of the proper orthogonal decomposition (POD). In particular, twin cross-stream rakes of X-wire probes are used to take cross-spectral measurements with different spanwise separations between the rakes and at several locations throughout the similarity region. The resulting POD spatial eigenfunctions for each of the three velocity components depend on cross-stream spatial coordinate, Strouhal number, and spanwise wavenumber. Corresponding eigenvalue distributions are obtained in Strouhal number–spanwise wavenumber space. Eigenvalue convergence is demonstrated to be rapid. When properly scaled the eigenfunctions and eigenvalues are shown to exhibit self-similarity though the streamwise location at which this commences depends on the particular velocity component. The results suggest that the flow supports a planar structure aligned in the spanwise direction as well as an essentially three-dimensional structure with asymmetrical shape in the cross-stream direction and pseudo-periodically distributed in the spanwise direction. Comparison of the single- and dual-rake implementations of the POD presented in this paper demonstrate that measurements confined to a single plane are incapable of properly extracting the planar modes. Rather, the single-rake implementation results in modes that appear to be a weighted sum of modes corresponding to different spanwise wavenumbers.
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JIMÉNEZ, JAVIER, MARKUS UHLMANN, ALFREDO PINELLI, and GENTA KAWAHARA. "Turbulent shear flow over active and passive porous surfaces." Journal of Fluid Mechanics 442 (August 24, 2001): 89–117. http://dx.doi.org/10.1017/s0022112001004888.
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The behaviour of turbulent shear flow over a mass-neutral permeable wall is studied numerically. The transpiration is assumed to be proportional to the local pressure fluctuations. It is first shown that the friction coefficient increases by up to 40% over passively porous walls, even for relatively small porosities. This is associated with the presence of large spanwise rollers, originating from a linear instability which is related both to the Kelvin–Helmholtz instability of shear layers, and to the neutral inviscid shear waves of the mean turbulent profile. It is shown that the rollers can be forced by patterned active transpiration through the wall, also leading to a large increase in friction when the phase velocity of the forcing resonates with the linear eigenfunctions mentioned above. Phase-lock averaging of the forced solutions is used to further clarify the flow mechanism. This study is motivated by the control of separation in boundary layers.
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Virk, D., M. V. Melander, and F. Hussain. "Dynamics of a polarized vortex ring." Journal of Fluid Mechanics 260 (February 10, 1994): 23–55. http://dx.doi.org/10.1017/s0022112094003423.
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This paper builds on our claim that most vortical structures in transitional and turbulent flows are partially polarized. Polarization is inferred by the application of helical wave decomposition. We analyse initially polarized isolated viscous vortex rings through direct numerical simulation of the Navier-Stokes equations using divergence-free axisymmetric eigenfunctions of the curl operator. Integral measures of the degree of polarization, such as the fractions of energy, enstrophy, and helicity associated with right-handed (or left-handed) eigenfunctions, remain nearly constant during evolution, thereby suggesting that polarization is a persistent feature. However, for polarized rings an axial vortex (tail) develops near the axis, where the local ratio of right- to left-handed vorticities develops significant non-uniformities due to spatial separation of peaks of polarized components. Reconnection can occur in rings when polarized and is clearly discerned from the evolution of axisymmetric vortex surfaces; but interestingly, the location of reconnection cannot be inferred from the vorticity magnitude. The ring propagation velocity Up decreases monotonically as the degree of initial polarization increases. Unlike force-balance arguments, two explanations based on vortex dynamics provided here are not restricted to thin rings and predict reduction in Up correctly. These results reveal surprising differences among the evolutionary dynamics of polarized, partially polarized, and unpolarized rings.
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Imayama, Shintaro, P. Henrik Alfredsson, and R. J. Lingwood. "Experimental study of rotating-disk boundary-layer flow with surface roughness." Journal of Fluid Mechanics 786 (November 24, 2015): 5–28. http://dx.doi.org/10.1017/jfm.2015.634.
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Rotating-disk boundary-layer flow is known to be locally absolutely unstable at $R>507$ as shown by Lingwood (J. Fluid Mech., vol. 299, 1995, pp. 17–33) and, for the clean-disk condition, experimental observations show that the onset of transition is highly reproducible at that Reynolds number. However, experiments also show convectively unstable stationary vortices due to cross-flow instability triggered by unavoidable surface roughness of the disk. We show that if the surface is sufficiently rough, laminar–turbulent transition can occur via a convectively unstable route ahead of the onset of absolute instability. In the present work we compare the laminar–turbulent transition processes with and without artificial surface roughnesses. The differences are clearly captured in the spectra of velocity time series. With the artificial surface roughness elements, the stationary-disturbance component is dominant in the spectra, whereas both stationary and travelling components are represented in spectra for the clean-disk condition. The wall-normal profile of the disturbance velocity for the travelling mode observed for a clean disk is in excellent agreement with the critical absolute instability eigenfunction from local theory; the wall-normal stationary-disturbance profile, by contrast, is distinct and the experimentally measured profile matches the stationary convective instability eigenfunction. The results from the clean-disk condition are compared with theoretical studies of global behaviours in spatially developing flow and found to be in good qualitative agreement. The details of stationary disturbances are also discussed and it is shown that the radial growth rate is in excellent agreement with linear stability theory. Finally, large stationary structures in the breakdown region are described.
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Surkov, S. "EN The wave model of secondary flows and coherent structures in pipes." Refrigeration Engineering and Technology 55, no. 5-6 (March 28, 2020): 273–81. http://dx.doi.org/10.15673/ret.v55i5-6.1655.
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In this article, a theoretical analysis of the flows arising in the cross sections of fluid and gas flows is performed. Such flows are subdivided into secondary flows and coherent structures. From experimental studies it is known that both types of flows are long-lived large-scale movements (LSM) stretched along the flow. The relative stability of the vortices is traditionally explained by the fact that the viscous friction forces that inhibit the rotation are compensated by the intensification of the swirl when moving slowly rotating peripheral layers to the center of the vortex due to longitudinal tension. An analysis of this mechanism made it possible to develop a relatively simple model of vortex structures in which the viscous friction forces and axial expansion are considered to be infinitesimal. Under these assumptions, one can use the equations of motion of an ideal fluid in the variables “stream function - vorticity”. It is shown that under certain assumptions these equations take the form of a wave equation, and the boundary conditions are the condition that the stream function on the solid walls of the flow equals zero. The obtained solutions of the wave equation describe the following special cases: Goertler’s vortices between rotating cylinders, secondary flows in a pipe with a square cross section, swirling flow in a round pipe, paired vortex after bend of the pipe. The physical sense of more complex solutions of the wave equation has become clear relatively recently. Very similar structures were found in experimental studies using orthogonal decomposition (POD) of a turbulent pulsations field. This may mean that the eigenfunctions in the POD correspond to coherent structures that really arise in the flow. The results obtained confirm the hypothesis that secondary flows and coherent structures have a common nature. The solutions obtained in this paper can be used in processing the experiment as eigenfunctions for the orthogonal decomposition method. In addition, they can be used in direct numerical simulation (DNS) of turbulent flows
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LIU, Z., R. J. ADRIAN, and T. J. HANRATTY. "Large-scale modes of turbulent channel flow: transport and structure." Journal of Fluid Mechanics 448 (November 26, 2001): 53–80. http://dx.doi.org/10.1017/s0022112001005808.
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Turbulent flow in a rectangular channel is investigated to determine the scale and pattern of the eddies that contribute most to the total turbulent kinetic energy and the Reynolds shear stress. Instantaneous, two-dimensional particle image velocimeter measurements in the streamwise-wall-normal plane at Reynolds numbers Reh = 5378 and 29 935 are used to form two-point spatial correlation functions, from which the proper orthogonal modes are determined. Large-scale motions – having length scales of the order of the channel width and represented by a small set of low-order eigenmodes – contain a large fraction of the kinetic energy of the streamwise velocity component and a small fraction of the kinetic energy of the wall-normal velocities. Surprisingly, the set of large-scale modes that contains half of the total turbulent kinetic energy in the channel, also contains two-thirds to three-quarters of the total Reynolds shear stress in the outer region. Thus, it is the large-scale motions, rather than the main turbulent motions, that dominate turbulent transport in all parts of the channel except the buffer layer. Samples of the large-scale structures associated with the dominant eigenfunctions are found by projecting individual realizations onto the dominant modes. In the streamwise wall-normal plane their patterns often consist of an inclined region of second quadrant vectors separated from an upstream region of fourth quadrant vectors by a stagnation point/shear layer. The inclined Q4/shear layer/Q2 region of the largest motions extends beyond the centreline of the channel and lies under a region of fluid that rotates about the spanwise direction. This pattern is very similar to the signature of a hairpin vortex. Reynolds number similarity of the large structures is demonstrated, approximately, by comparing the two-dimensional correlation coefficients and the eigenvalues of the different modes at the two Reynolds numbers.
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ARNDT, R. E. A., D. F. LONG, and M. N. GLAUSER. "The proper orthogonal decomposition of pressure fluctuations surrounding a turbulent jet." Journal of Fluid Mechanics 340 (June 10, 1997): 1–33. http://dx.doi.org/10.1017/s0022112097005089.
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It is shown that the pressure signal measured at the outer edge of a jet mixing layer is entirely hydrodynamic in nature and provides a good measure of the large-scale structure of the turbulent flow. Measurement of the pressure signal provides a unique opportunity to utilize proper orthogonal decomposition (POD) to deduce the streamwise structure. Since pressure is a scalar, a significant reduction in the numerical and experimental complexity inherent in the analysis of velocity vector fields results.The POD streamwise eigenfunctions show that the structure associated with any frequency–azimuthal mode number combination displays the general characteristics of amplification–saturation–decay of an instability wave, all within about three wavelengths. High-frequency components saturate early in x and low-frequency components saturate further downstream, indicative of the inhomogeneous character of the flow in the streamwise direction. Application of the POD technique allows the phase velocity to be determined taking into account the inhomogeneity of the flow in the streamwise direction. The phase velocity of each instability wave (POD eigenvector) is constant and equal to 0.58Uj, indicating that the jet structure is non-dispersive.Using the shot-noise decomposition, a characteristic event is constructed. This event is found to contain evidence of both pairings and triplings of vortex structures. The tripling results in a rapid increase in the first asymmetric (m=1) component. On average, pairing occurs once every four Uj/D while tripling occurs once every 13Uj/D.
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Nokes, R. I., and I. R. Wood. "Vertical and lateral turbulent dispersion: some experimental results." Journal of Fluid Mechanics 187 (February 1988): 373–94. http://dx.doi.org/10.1017/s0022112088000473.
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The results of an experimental programme designed to investigate turbulent dispersion of a continuous contaminant source in a wide channel are presented. Both two-dimensional vertical dispersion and the determination of the lateral diffusion coefficient are described. The eigenfunction solution to the turbulent diffusion equation, presented in Nokes et al. (1984) and discussed in greater detail in Nokes (1985), is strongly supported by the results of vertical mixing described here. A variety of source locations are examined in this study and the location of the ideal source, predicted by theory, is verified by the experimental results. For the two smooth-bed flows investigated the depth-averaged values of εz, deduced from the rates of lateral spreading of the plume, lie at the lower end of the range of values obtained by other researchers. Considering only the results obtained in wide channels, the authors demonstrate that previously published values of the lateral diffusion coefficient, non-dimensionalized by the shear velocity u* and the flow depth d are independent of all flow parameters except the friction factor f = 8u*/ū where ū is the mean velocity in the flow. Indeed, above a value of f = 0.055 εz/u*d is also found to be independent of f, and takes a value of 0.134. A brief mathematical analysis of the three-dimensional mixing processes in the near-source region is presented, and utilized to investigate the coupling between the lateral and vertical diffusion processes in this region. Based on these mathematical arguments the experimental results imply that the vertical and lateral diffusion processes are essentially uncoupled in the near-source zone, and thus the lateral diffusivity and longitudinal velocity have similar vertical dependence.
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Gudmundsson, K., and Tim Colonius. "Instability wave models for the near-field fluctuations of turbulent jets." Journal of Fluid Mechanics 689 (November 15, 2011): 97–128. http://dx.doi.org/10.1017/jfm.2011.401.
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AbstractPrevious work has shown that aspects of the evolution of large-scale structures, particularly in forced and transitional mixing layers and jets, can be described by linear and nonlinear stability theories. However, questions persist as to the choice of the basic (steady) flow field to perturb, and the extent to which disturbances in natural (unforced), initially turbulent jets may be modelled with the theory. For unforced jets, identification is made difficult by the lack of a phase reference that would permit a portion of the signal associated with the instability wave to be isolated from other, uncorrelated fluctuations. In this paper, we investigate the extent to which pressure and velocity fluctuations in subsonic, turbulent round jets can be described aslinearperturbations to the mean flow field. The disturbances are expanded about the experimentally measured jet mean flow field, and evolved using linear parabolized stability equations (PSE) that account, in an approximate way, for the weakly non-parallel jet mean flow field. We utilize data from an extensive microphone array that measures pressure fluctuations just outside the jet shear layer to show that, up to an unknown initial disturbance spectrum, the phase, wavelength, and amplitude envelope of convecting wavepackets agree well with PSE solutions at frequencies and azimuthal wavenumbers that can be accurately measured with the array. We next apply the proper orthogonal decomposition to near-field velocity fluctuations measured with particle image velocimetry, and show that the structure of the most energetic modes is also similar to eigenfunctions from the linear theory. Importantly, the amplitudes of the modes inferred from the velocity fluctuations are in reasonable agreement with those identified from the microphone array. The results therefore suggest that, to predict, with reasonable accuracy, the evolution of the largest-scale structures that comprise the most energetic portion of the turbulent spectrum of natural jets, nonlinear effects need only be indirectly accounted for by considering perturbations to the mean turbulent flow field, while neglecting any non-zero frequency disturbance interactions.
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Caraballo, E., J. Little, M. Debiasi, and M. Samimy. "Development and Implementation of an Experimental-Based Reduced-Order Model for Feedback Control of Subsonic Cavity Flows." Journal of Fluids Engineering 129, no. 7 (January 22, 2007): 813–24. http://dx.doi.org/10.1115/1.2742724.
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This work is focused on the development of a reduced-order model based on experimental data for the design of feedback control for subsonic cavity flows. The model is derived by applying the proper orthogonal decomposition (POD) in conjunction with the Galerkin projection of the Navier-Stokes equations onto the resulting spatial eigenfunctions. The experimental data consist of sets of 1000 simultaneous particle image velocimetry (PIV) images and surface pressure measurements taken in the Gas Dynamics and Turbulent Laboratory (GDTL) subsonic cavity flow facility at the Ohio State University. Models are derived for various individual flow conditions as well as for their combinations. The POD modes of the combined cases show some of the characteristics of the sets used. Flow reconstructions with 30 modes show good agreement with experimental PIV data. For control design, four modes capture the main features of the flow. The reduced-order model consists of a system of nonlinear ordinary differential equations for the modal amplitudes where the control input appears explicitly. Linear and quadratic stochastic estimation methods are used for real-time estimation of the modal amplitudes from real-time surface pressure measurements.
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IQBAL, M. O., and F. O. THOMAS. "Coherent structure in a turbulent jet via a vector implementation of the proper orthogonal decomposition." Journal of Fluid Mechanics 571 (January 4, 2007): 281–326. http://dx.doi.org/10.1017/s0022112006003351.
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The coherent structure in the near-field of an axisymmetric turbulent jet at a Reynolds number of 3.8 × 105 and Mach number of 0.3 is experimentally characterized by a vector implementation of the proper orthogonal decomposition (POD). The POD eigenfunctions and associated eigenvalues are extracted at several selected streamwise locations in the initial region. The focus on the near-field is motivated by its importance in numerous technical applications. Results show a rapid energy convergence with POD mode number. Examination of the relative energy contained in the combined azimuthal and radial components of the POD modes reveals that it is comparable to that in the streamwise component. The streamwise evolution of the eigenvalue spectra is characterized by a remarkable variation in the azimuthal mode number energy distribution, leading to the dominance of azimuthal mode m = 1 beyond the end of the jet core. In contrast, a scalar implementation using only the streamwise component shows the dominance of mode m = 2 which is consistent with previous scalar implementations of the POD. For a given azimuthal mode number, the eigenvalue spectra exhibit a broad peak which occurs at a constant value of Strouhal number based on local shear layer momentum thickness and local jet maximum velocity. The phase information required for a local reconstruction of the jet structure is obtained by projecting the POD eigenmodes onto instantaneous realizations of the flow at fixed streamwise locations. The instantaneous realizations are obtained by utilizing cross-stream arrays of multi-sensor probes in conjunction with linear stochastic estimation (LSE). Results clearly show the local dynamic behaviour of each component of the jet structure.
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GÜNTHER, AXEL, and PHILIPP RUDOLF VON ROHR. "Large-scale structures in a developed flow over a wavy wall." Journal of Fluid Mechanics 478 (March 10, 2003): 257–85. http://dx.doi.org/10.1017/s0022112002003488.
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We address – motivated in part by the findings of Gong et al. (1996) and Miller (1995) – the role of streamwise-oriented large-scale structures in a developed flow between a sinusoidal bottom wall and a flat top wall. Particle image velocimetry (PIV) is used to examine the spatial variation of the velocity in different planes of the flow through a water channel with an aspect ratio of 12:1. The wave amplitude is equal to one tenth of the wall wavelength, Λ, and Reynolds numbers between 500 and 7300, defined with the bulk velocity and the half-height of the channel, are considered. To examine streamwise-oriented structures, the spanwise variation of the velocity field is studied in a plane parallel to the top wall, and in one that intersects the wavy surface at an uphill location. From a proper orthogonal decomposition (POD) of the streamwise velocity fluctuations, we obtain the dominant eigenfunctions with a characteristic spanwise scale of O(1.5Λ), which agrees with the scale of perturbations for the streamwise velocity at laminar conditions. A decomposition of the turbulent velocity field close to the uphill section of the wavy surface reveals smaller structures at a location that coincides with the Reynolds shear stress maximum.
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CITRINITI, J. H., and W. K. GEORGE. "Reconstruction of the global velocity field in the axisymmetric mixing layer utilizing the proper orthogonal decomposition." Journal of Fluid Mechanics 418 (September 10, 2000): 137–66. http://dx.doi.org/10.1017/s0022112000001087.
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Experimental data are presented from 138 synchronized channels of hot-wire anemometry in an investigation of the large-scale, or coherent, structures in a high Reynolds number fully developed, turbulent axisymmetric shear layer. The dynamics of the structures are obtained from instantaneous realizations of the streamwise velocity field in a single plane, x/D = 3, downstream of a round jet nozzle. The Proper Orthogonal Decomposition (POD) technique is applied to an ensemble of these realizations to determine optimal representations of the velocity field, in a mean-square sense, in terms of an orthogonal basis. The coefficients of the orthogonal functions, which describe the temporal evolution of the POD eigenfunctions, are determined by projecting instantaneous realizations of the velocity field onto the basis.Evidence is presented to show that with a partial reconstruction of the velocity field, using only the first radial POD mode, the large-scale structure is objectively educed from the turbulent field. Further, it is shown that only five azimuthal Fourier modes (0,3,4,5,6) are necessary to represent the evolution of the large-scale structure. The results of the velocity reconstruction using the POD provide evidence for azimuthally coherent structures that exist near the potential core. In addition to the azimuthal structures near the potential core, evidence is also found for the presence of counter-rotating, streamwise vortex pairs (or ribs) in the region between successive azimuthally coherent structures as well as coexisting for short periods with them. The large-scale structure cycle, which includes the appearance of the ring structure, the advection of fluid by the ribs in the braid region and their advection toward the outside of the layer by a following ring structure, repeats approximately every one integral time scale. One surprising result was that the most spatially correlated structure in the flow, the coherent ring near the potential core which ejects fluid in the streamwise direction in a volcano-like eruption, is also the one with the shortest time scale.
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Wheless, G. H., and G. T. Csanady. "Instability waves on the air–sea interface." Journal of Fluid Mechanics 248 (March 1993): 363–81. http://dx.doi.org/10.1017/s0022112093000801.
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We used a compound matrix method to integrate the Orr–Sommerfeld equation in an investigation of short instability waves (λ < 6 cm) on the coupled shear flow at the air–sea interface under suddenly imposed wind (a gust model). The method is robust and fast, so that the effects of external variables on growth rate could easily be explored. As expected from past theoretical studies, the growth rate proved sensitive to air and water viscosity, and to the curvature of the air velocity profile very close to the interface. Surface tension had less influence, growth rate increasing somewhat with decreasing surface tension. Maximum growth rate and minimum wave speed nearly coincided for some combinations of fluid properties, but not for others.The most important new finding is that, contrary to some past order of magnitude estimates made on theoretical grounds, the eigenfunctions at these short wavelengths are confined to a distance of the order of the viscous wave boundary-layer thickness from the interface. Correspondingly, the perturbation vorticity is high, the streamwise surface velocity perturbation in typical cases being five times the orbital velocity of free waves on an undisturbed water surface. The instability waves should therefore be thought of as fundamentally different flow structures from free waves: given their high vorticity, they are akin to incipient turbulent eddies. They may also be expected to break at a much lower steepness than free waves.
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Kerhervé, F., P. Jordan, A. V. G. Cavalieri, J. Delville, C. Bogey, and D. Juvé. "Educing the source mechanism associated with downstream radiation in subsonic jets." Journal of Fluid Mechanics 710 (August 31, 2012): 606–40. http://dx.doi.org/10.1017/jfm.2012.378.
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AbstractThis work belongs to the ongoing debate surrounding the mechanism responsible for low-angle sound emission from subsonic jets. The flow, simulated by large eddy simulation (Bogey & Bailly, Comput. Fluids, vol. 35 (10), 2006a, pp. 1344–1358), is a Mach 0.9 jet with Reynolds number, based on the exit diameter, of $4\ensuremath{\times} 1{0}^{5} $. A methodology is implemented to educe, explore and model the flow motions associated with low-angle sound radiation. The eduction procedure, which is based on frequency–wavenumber filtering of the sound field and subsequent conditional analysis of the turbulent jet, provides access to space- and time-dependent (hydrodynamic) pressure and velocity fields. Analysis of these shows the low-angle sound emission to be underpinned by dynamics comprising space and time modulation of axially coherent wavepackets: temporally localized energization of wavepackets is observed to be correlated with the generation of high-amplitude acoustic bursts. Quantitative validation is provided by means of a simplified line-source Ansatz (Cavalieri et al. J. Sound Vib., vol. 330, 2011b, pp. 4474–4492). The dynamic nature of the educed field is then assessed using linear stability theory (LST). The educed pressure and velocity fields are found to compare well with LST: the radial structures of these match the corresponding LST eigenfunctions; the axial evolutions of their fluctuation energy are consistent with the LST amplification rates; and the relative amplitudes of the pressure and velocity fluctuations, which are educed independently of one another, are consistent with LST.
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Poulier, P. L., D. Fournier, L. Gizon, and T. L. Duvall. "Acoustic wave propagation through solar granulation: Validity of effective-medium theories, coda waves." Astronomy & Astrophysics 643 (November 2020): A168. http://dx.doi.org/10.1051/0004-6361/202039201.
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Context. The frequencies, lifetimes, and eigenfunctions of solar acoustic waves are affected by turbulent convection, which is random in space and in time. Since the correlation time of solar granulation and the periods of acoustic waves (∼5 min) are similar, the medium in which the waves propagate cannot a priori be assumed to be time independent. Aims. We compare various effective-medium solutions with numerical solutions in order to identify the approximations that can be used in helioseismology. For the sake of simplicity, the medium is one dimensional. Methods. We consider the Keller approximation, the second-order Born approximation, and spatial homogenization to obtain theoretical values for the effective wave speed and attenuation (averaged over the realizations of the medium). Numerically, we computed the first and second statistical moments of the wave field over many thousands of realizations of the medium (finite-amplitude sound-speed perturbations are limited to a 30 Mm band and have a zero mean). Results. The effective wave speed is reduced for both the theories and the simulations. The attenuation of the coherent wave field and the wave speed are best described by the Keller theory. The numerical simulations reveal the presence of coda waves, trailing the ballistic wave packet. These late arrival waves are due to multiple scattering and are easily seen in the second moment of the wave field. Conclusions. We find that the effective wave speed can be calculated, numerically and theoretically, using a single snapshot of the random medium (frozen medium); however, the attenuation is underestimated in the frozen medium compared to the time-dependent medium. Multiple scattering cannot be ignored when modeling acoustic wave propagation through solar granulation.
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Guiho, F., F. Alizard, and J. Ch Robinet. "Instabilities in oblique shock wave/laminar boundary-layer interactions." Journal of Fluid Mechanics 789 (January 15, 2016): 1–35. http://dx.doi.org/10.1017/jfm.2015.729.
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The interaction of an oblique shock wave and a laminar boundary layer developing over a flat plate is investigated by means of numerical simulation and global linear-stability analysis. Under the selected flow conditions (free-stream Mach numbers, Reynolds numbers and shock-wave angles), the incoming boundary layer undergoes separation due to the adverse pressure gradient. For a wide range of flow parameters, the oblique shock wave/boundary-layer interaction (OSWBLI) is seen to be globally stable. We show that the onset of two-dimensional large-scale structures is generated by selective noise amplification that is described for each frequency, in a linear framework, by wave-packet trains composed of several global modes. A detailed analysis of both the eigenspectrum and eigenfunctions gives some insight into the relationship between spatial scales (shape and localization) and frequencies. In particular, OSWBLI exhibits a universal behaviour. The lowest frequencies correspond to structures mainly located near the separated shock that emit radiation in the form of Mach waves and are scaled by the interaction length. The medium frequencies are associated with structures mainly localized in the shear layer and are scaled by the displacement thickness at the impact. The linear process by which OSWBLI selects frequencies is analysed by means of the global resolvent. It shows that unsteadiness are mainly associated with instabilities arising from the shear layer. For the lower frequency range, there is no particular selectivity in a linear framework. Two-dimensional numerical simulations show that the linear behaviour is modified for moderate forcing amplitudes by nonlinear mechanisms leading to a significant amplification of low frequencies. Finally, based on the present results, we draw some hypotheses concerning the onset of unsteadiness observed in shock wave/turbulent boundary-layer interactions.
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MA, B., C. W. H. VAN DOORNE, Z. ZHANG, and F. T. M. NIEUWSTADT. "On the spatial evolution of a wall-imposed periodic disturbance in pipe Poiseuille flow at Re = 3000. Part 1. Subcritical disturbance." Journal of Fluid Mechanics 398 (November 10, 1999): 181–224. http://dx.doi.org/10.1017/s0022112099006199.
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We have performed a numerical study on the transition of a cylindrical pipe flow under the influence of a localized disturbance in the form of periodic suction and blowing (PSB) applied at the pipe wall. We focus here on the so-called receptivity problem where the spatial evolution of this disturbance is studied as it travels downstream through the pipe. The study is carried out by means of two techniques: an eigenmode expansion solution (EES) and a full nonlinear direct numerical simulation (DNS). The EES is based on an analytical expansion in terms of the eigenfunctions of the linear operator which follows from the equations of motion expressed in a cylindrical coordinate system. The DNS is formulated in terms of a spectral element method.We restrict ourselves to a so-called subcritical disturbance, i.e. the flow does not undergo transition. For very small amplitudes of the PSB disturbance the results of the EES and DNS techniques agree excellently. For larger amplitudes nonlinear interactions come into play which are neglected in the EES method. Nevertheless, the results of both methods are consistent with the following transition scenario. The PSB excites a flow perturbation that has the same angular wavenumber and frequency as the imposed disturbance itself. This perturbation is called the fundamental mode. By nonlinear self-interaction of this fundamental mode higher-order harmonics, both in the angular wavenumber and frequency, are generated. It is found that the harmonic with angular wavenumber 2, i.e. twice the wavenumber of the fundamental mode, and with zero frequency grows strongly by a linear process known as transient growth. As a result the (perturbed) pipe flow downstream of the disturbance region develops extended regions of low velocity, known as low-speed streaks. At large disturbance amplitudes these low-speed streaks show the development of high wavenumber oscillations and it is expected that at even higher disturbance amplitudes these oscillations become unstable and turbulent flow will set in.Our result agrees (at least qualitatively) with the transition scenario in a plane Poiseuille flow as discussed by Reddy et al. (1998) and Elofson & Alfredson (1998).
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Lefauve, Adrien, J. L. Partridge, Qi Zhou, S. B. Dalziel, C. P. Caulfield, and P. F. Linden. "The structure and origin of confined Holmboe waves." Journal of Fluid Mechanics 848 (June 5, 2018): 508–44. http://dx.doi.org/10.1017/jfm.2018.324.
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Finite-amplitude manifestations of stratified shear flow instabilities and their spatio-temporal coherent structures are believed to play an important role in turbulent geophysical flows. Such shear flows commonly have layers separated by sharp density interfaces, and are therefore susceptible to the so-called Holmboe instability, and its finite-amplitude manifestation, the Holmboe wave. In this paper, we describe and elucidate the origin of an apparently previously unreported long-lived coherent structure in a sustained stratified shear flow generated in the laboratory by exchange flow through an inclined square duct connecting two reservoirs filled with fluids of different densities. Using a novel measurement technique allowing for time-resolved, near-instantaneous measurements of the three-component velocity and density fields simultaneously over a three-dimensional volume, we describe the three-dimensional geometry and spatio-temporal dynamics of this structure. We identify it as a finite-amplitude, nonlinear, asymmetric confined Holmboe wave (CHW), and highlight the importance of its spanwise (lateral) confinement by the duct boundaries. We pay particular attention to the spanwise vorticity, which exhibits a travelling, near-periodic structure of sheared, distorted, prolate spheroids with a wide ‘body’ and a narrower ‘head’. Using temporal linear stability analysis on the two-dimensional streamwise-averaged experimental flow, we solve for three-dimensional perturbations having two-dimensional, cross-sectionally confined eigenfunctions and a streamwise normal mode. We show that the dispersion relation and the three-dimensional spatial structure of the fastest-growing confined Holmboe instability are in good agreement with those of the observed confined Holmboe wave. We also compare those results with a classical linear analysis of two-dimensional perturbations (i.e. with no spanwise dependence) on a one-dimensional base flow. We conclude that the lateral confinement is an important ingredient of the confined Holmboe instability, which gives rise to the CHW, with implications for many inherently confined geophysical flows such as in valleys, estuaries, straits or deep ocean trenches. Our results suggest that the CHW is an example of an experimentally observed, inherently nonlinear, robust, long-lived coherent structure which has developed from a linear instability. We conjecture that the CHW is a promising candidate for a class of exact coherent states underpinning the dynamics of more disordered, yet continually forced stratified shear flows.
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Brand, E., and J. F. Gibson. "A doubly localized equilibrium solution of plane Couette flow." Journal of Fluid Mechanics 750 (June 5, 2014). http://dx.doi.org/10.1017/jfm.2014.285.
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AbstractWe present an equilibrium solution of plane Couette flow that is exponentially localized in both the spanwise and streamwise directions. The solution is similar in size and structure to previously computed turbulent spots and localized, chaotically wandering edge states of plane Couette flow. A linear analysis of dominant terms in the Navier–Stokes equations shows how the exponential decay rate and the wall-normal overhang profile of the streamwise tails are governed by the Reynolds number and the dominant spanwise wavenumber. Perturbations of the solution along its leading eigenfunctions cause rapid disruption of the interior roll-streak structure and formation of a turbulent spot, whose growth or decay depends on the Reynolds number and the choice of perturbation.
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"The effect of global-scale, steady-state convection and elastic-gravitational asphericities on helioseismic oscillations." Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences 339, no. 1655 (June 15, 1992): 431–96. http://dx.doi.org/10.1098/rsta.1992.0048.
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In this paper we derive a theory, based on quasi-degenerate perturbation theory, that governs the effect of global-scale, steady-state convection and associated static asphericities in the elastic-gravitational variables (adiabatic bulk modulus κ, density ρ, and gravitational potential ∅) on helioseismic eigenfrequencies and eigenfunctions and present a formalism with which this theory can be applied computationally. The theory rests on three formal assumptions: (1) that convection is temporally steady in a frame corotating with the Sun, (2) that accurate eigenfrequencies and eigenfunctions can be determined by retaining terms in the seismically perturbed equations of motion only to first order in p -mode displacement, and (3) that we are justified in retaining terms only to first order in convective velocity (this is tantamount to assuming that the convective flow is anelastic). The most physically unrealistic assumption is (1), and we view the results of this paper as the first step toward a more general theory governing the seismic effects of time-varying fields. Although the theory does not govern the seismic effects of non-stationary flows, it can be used to approximate the effects of unsteady flows on the acoustic wavefield if the flow is varying smoothly in time. The theory does not attempt to model seismic modal amplitudes since these are governed, in part, by the exchange of energy between convection and acoustic motions which is not a part of this theory. However, we show how theoretical wavefields can be computed given a description of the stress field produced by a source process such as turbulent convection. The basic reference model that will be perturbed by rotation, convection, structural asphericities, and acoustic oscillations is a spherically symmetric, nonrotating, non-magnetic, isotropic, static solar model that, when subject to acoustic oscillations, oscillates adiabatically. We call this the SNRNMAIS model. An acoustic mode of the SNRNMAIS model is denoted by k = ( n, l, m ), where n is the radial order, l is the harmonic degree, and m is the azimuthal order of the mode. The main result of the paper is the general matrix element H m'm n'n,l'l for steady-state convection satisfying the anelastic condition with static structural asphericities. It is written in terms of the radial, scalar eigenfunctions of the snrnmais model, resulting in equations (90)—(110). We prove Rayleigh’s principle in our derivation of quasi-degenerate perturbation theory which, as a by-product, yields the general matrix element. Within this perturbative method, modes need not be exactly degenerate in the SNRNMAIS solar model to couple, only nearly so. General matrix elements compose the hermitian supermatrix Z . The eigenvalues of the supermatrix are the eigenfrequency perturbations of the convecting, aspherical model and the eigenvector components of Z are the expansion coefficients in the linear combination forming the eigenfunctions in which the eigenfunctions of the SNRNMAIS solar model act as basis functions. The properties of the Wigner 3 j symbols and the reduced matrix elements composing H m'm n'n,l'l produce selection rules governing the coupling of SNRNMAIS modes that hold even for time-varying flows. We state selection rules for both quasidegenerate and degenerate perturbation theories. For example, within degenerate perturbation theory, only odd-degree s toroidal flows and even degree structural asphericities, both with s ≤ 2 l , will couple and/or split acoustic modes with harmonic degree l . In addition, the frequency perturbations caused by a toroidal flow display odd symmetry with respect to the degenerate frequency when ordered from the minimum to the maximum frequency perturbation. We consider the special case of differential rotation, the odd-degree, axisymmetric, toroidal component of general convection, and present the general matrix element and selection rules under quasi-degenerate perturbation theory. We argue that due to the spacing of modes that satisfy the selection rules, quasi-degenerate coupling can, for all practical purposes, be neglected in modelling the effect of low-degree differential rotation on helioseismic data. In effect, modes that can couple through low-degree differential rotation are too far separated in frequency to couple strongly. This is not the case for non-axisymmetric flows and asphericities where near degeneracies will regularly occur, and couplings can be relatively strong especially among SNRNMAIS modes within the same multiplet. All derivations are performed and all solutions are presented in a frame corotating with the mean solar angular rotation rate. Equation (18) shows how to transform the eigenfrequencies and eigenfunctions in the corotating frame into an inertial frame. The transformation has the effect that each eigenfunction in the inertial frame is itself time varying. That is, a mode of oscillation, which is defined to have a single frequency in the corotating frame, becomes multiply periodic in the inertial frame.
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Tzou, D. Y. "Instability of Nanofluids in Natural Convection." Journal of Heat Transfer 130, no. 7 (May 16, 2008). http://dx.doi.org/10.1115/1.2908427.
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Abstract Instability of natural convection in nanofluids is investigated in this work. As a result of Brownian motion and thermophoresis of nanoparticles, the critical Rayleigh number is shown to be much lower, by one to two orders of magnitude, as compared to that for regular fluids. The highly promoted turbulence, in the presence of nanoparticles for as little as 1% in volume fraction, significantly enhances heat transfer in nanofluids, which may be much more pronounced than the enhancement of the effective thermal conductivity alone. Seven dominating groups are extracted from the nondimensional analysis. By extending the method of eigenfunction expansions in conjunction with the method of weighted residuals, closed-form solutions are derived for the Rayleigh number to justify such remarkable change by the nanoparticles at the onset of instability.
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Slepyshev, A. A. "Vertical Transfer of Momentum by Inertia-Gravity Internal Waves on a Two-Dimensional Shear Flow." Morskoy gidrofizicheskiy zhurnal 37, no. 4 (August 2021). http://dx.doi.org/10.22449/0233-7584-2021-4-391-404.
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Purpose. The paper is aimed at investigating the momentum vertical transfer by inertia-gravity internal waves on a two-dimensional flow with a vertical shear of velocity, and also at studying the Stokes drift of liquid particles and the mean current effect on it. Methods and Results. Free internal waves in an infinite basin of constant depth are considered in the Boussinesq approximation with the regard for the Earth rotation. Two components of the mean current velocity depend on the vertical coordinate. The equation for the vertical velocity amplitude has complex coefficients; therefore the eigenfunction and the wave frequency are complex. The corresponding boundary value problem is solved numerically by the implicit Adams scheme of the third order of accuracy. The wave frequency at a fixed wavenumber was found by the shooting method. It was determined that the frequency imaginary part was small and could be either negative or positive depending on a wave number and a mode number. Thus, both weak attenuation and weak amplification of an internal wave are possible. The vertical wave momentum fluxes are nonzero and can exceed the corresponding turbulent fluxes. The Stokes drift velocity, transverse to the wave direction, is nonzero and less than the longitudinal velocity. The vertical component of the Stokes drift velocity is also nonzero and four orders of magnitude less than the longitudinal component. The signs of the vertical component of the Stokes drift velocity for the waves with the frequencies 10 and 16 cycle/h are opposite, since the signs of their frequency imaginary parts are different; and the vertical component of the Stokes drift velocity is proportional to the wave frequency imaginary part. Conclusions. The vertical momentum wave flux of inertia-gravity internal waves differs from zero in the presence of the current whose velocity component, transverse to the wave propagation direction, depends on the vertical coordinate. The component of the Stokes drift velocity, transverse to the wave propagation direction, is nonzero and less than the longitudinal one. The vertical component of the Stokes drift velocity is also nonzero and can contribute to formation of the vertical fine structure