Journal of Fluid Mechanics
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Volume 753 - 25 August 2014
The planar X-junction flow: stability analysis and control
Iman Lashgari, Outi Tammisola, Vincenzo Citro, Matthew P. Juniper, Luca Brandt
Published online by Cambridge University Press: 16 July 2014, pp. 1-28
The bifurcations and control of the flow in a planar X-junction are studied via linear stability analysis and direct numerical simulations. This study reveals the instability mechanisms in a symmetric channel junction and shows how these can be stabilized or destabilized by boundary modification. We observe two bifurcations as the Reynolds number increases. They both scale with the inlet speed of the two side channels and are almost independent of the inlet speed of the main channel. Equivalently, both bifurcations appear when the recirculation zones reach a critical length. A two-dimensional stationary global mode becomes unstable first, changing the flow from a steady symmetric state to a steady asymmetric state via a pitchfork bifurcation. The core of this instability, whether defined by the structural sensitivity or by the disturbance energy production, is at the edges of the recirculation bubbles, which are located symmetrically along the walls of the downstream channel. The energy analysis shows that the first bifurcation is due to a lift-up mechanism. We develop an adjustable control strategy for the first bifurcation with distributed suction or blowing at the walls. The linearly optimal wall-normal velocity distribution is computed through a sensitivity analysis and is shown to delay the first bifurcation from $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}=82.5$ to $\mathit{Re}=150$ . This stabilizing effect arises because blowing at the walls weakens the wall-normal gradient of the streamwise velocity around the recirculation zone and hinders the lift-up. At the second bifurcation, a three-dimensional stationary global mode with a spanwise wavenumber of order unity becomes unstable around the asymmetric steady state. Nonlinear three-dimensional simulations at the second bifurcation display transition to a nonlinear cycle involving growth of a three-dimensional steady structure, time-periodic secondary instability and nonlinear breakdown restoring a two-dimensional flow. Finally, we show that the sensitivity to wall suction at the second bifurcation is as large as it is at the first bifurcation, providing a possible mechanism for destabilization.
Role of slipstream instability in formation of counter-rotating vortex rings ahead of a compressible vortex ring
C. L. Dora, T. Murugan, S. De, Debopam Das
Counter-rotating vortex rings (CRVRs) are observed to form ahead of a primary compressible vortex ring that is generated at the open end of a shock tube at sufficiently high Mach numbers. In most of the earlier studies, the embedded shock strength has been asserted as the cause for the formation of CRVRs. In the present study, particle image velocimetry (PIV) measurements and high-order numerical simulations show that CRVRs do not form in the absence of a Mach disk in the sonic under-expanded jet behind the primary vortex ring. Kelvin–Helmholtz-type shear flow instability of the slipstream originating from the triple point of the Mach disk and subsequent eddy pairing, as observed by Rikanati et al. (Phys. Rev. Lett., vol. 96, 2006, art. 174503) in shock-wave Mach reflection, is found to be responsible for CRVR formation. The growth rate of the slipstream in the present problem follows the model proposed by them. The parameters influencing the formation of CRVRs as well as their dynamics is investigated. It is found that the strength of the Mach disk and its duration of persistence results in an exit impulse that determines the number of CRVRs formed.
Electrophoresis of bubbles
Ory Schnitzer, Itzchak Frankel, Ehud Yariv
Smoluchowski's celebrated electrophoresis formula is inapplicable to field-driven motion of drops and bubbles with mobile interfaces. We here analyse bubble electrophoresis in the thin-double-layer limit. To this end, we employ a systematic asymptotic procedure starting from the standard electrokinetic equations and a simple physicochemical interface model. This furnishes a coarse-grained macroscale description where the Debye-layer physics is embodied in effective boundary conditions. These conditions, in turn, represent a non-conventional driving mechanism for electrokinetic flows, where bulk concentration polarization, engendered by the interaction of the electric field and the Debye layer, results in a Marangoni-like shear stress. Remarkably, the electro-osmotic velocity jump at the macroscale level does not affect the electrophoretic velocity. Regular approximations are obtained in the respective cases of small zeta potentials, small ions, and weak applied fields. The nonlinear small-zeta-potential approximation rationalizes the paradoxical zero mobility predicted by the linearized scheme of Booth (J. Chem. Phys., vol. 19, 1951, pp. 1331–1336). For large (millimetre-size) bubbles the pertinent limit is actually that of strong fields. We have carried out a matched-asymptotic-expansion analysis of this singular limit, where salt polarization is confined to a narrow diffusive layer. This analysis establishes that the bubble velocity scales as the $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2/3$ -power of the applied-field magnitude and yields its explicit functional dependence upon a specific combination of the zeta potential and the ionic drag coefficient. The latter is provided to within an $O(1)$ numerical pre-factor which, in turn, is calculated via the solution of a universal (parameter-free) nonlinear flow problem. It is demonstrated that, with increasing field magnitude, all numerical solutions of the macroscale model indeed collapse on the analytic approximation thus obtained. Existing measurements of clean-bubble electrophoresis agree neither with present theory nor with previous models; we discuss this ongoing discrepancy.
Topographic control of stratified flows: upstream jets, blocking and isolating layers
Kraig B. Winters, Laurence Armi
Published online by Cambridge University Press: 16 July 2014, pp. 80-103
Optimal solutions to the nonlinear, hydrostatic, Boussinesq equations are developed for steady, density-stratified, topographically controlled flows characterized by blocking and upstream influence. These flows are jet-like upstream of an isolated obstacle and are contained within an asymmetric, thinning stream tube that is accelerated as it passes over the crest. A stagnant, nearly uniform-density isolating layer, surrounded by a bifurcated uppermost streamline, separates the accelerated flow from an uncoupled flow above. The flows are optimal in the sense that, for a given stratification, the solutions maximize the topographic rise above the blocking level required for hydraulic control while minimizing the total energy of the flow. Hydraulic control is defined mathematically by the asymmetry of the accelerated flow as it passes the crest. A subsequent analysis of the Taylor–Goldstein equation shows that these sheared, non-uniformly stratified flows are indeed subcritical upstream, critical at the crest, and supercritical downstream with respect to gravest-mode, long internal waves. The flows obtained are relevant to arrested wedge flows, selective withdrawal, stratified towing experiments, tidal flow over topography and atmospheric flows over mountains.
Test of the anomalous scaling of passive temperature fluctuations in turbulent Rayleigh–Bénard convection with spatial inhomogeneity
Xiaozhou He, Xiao-dong Shang, Penger Tong
The scaling properties of the temperature structure function (SF) and temperature–velocity cross-structure function (CSF) are investigated in turbulent Rayleigh–Bénard convection (RBC). The measured SFs and CSFs exhibit good scaling in space and time and the resulting SF and CSF exponents are obtained both at the centre of the convection cell and near the sidewall. A universal relationship between the CSF exponent and the thermal dissipation exponent is found, confirming that the anomalous scaling of passive temperature fluctuations in turbulent RBC is indeed caused by the spatial intermittency of the thermal dissipation field. It is also found that the difference in the functional form of the measured SF and CSF exponents at the two different locations in the cell is caused by the change of the geometry of the most dissipative structures in the (inhomogeneous) temperature field from being sheetlike at the cell centre to filament-like near the sidewall. The experiment thus provides direct evidence showing that the universality features of turbulent cascade are linked to the degree of anisotropy and inhomogeneity of turbulent statistics.
Shear-imposed falling film
Arghya Samanta
The study of a film falling down an inclined plane is revisited in the presence of imposed shear stress. Earlier studies regarding this topic (Smith, J. Fluid Mech., vol. 217, 1990, pp. 469–485; Wei, Phys. Fluids, vol. 17, 2005a, 012103), developed on the basis of a low Reynolds number, are extended up to moderate values of the Reynolds number. The mechanism of the primary instability is provided under the framework of a two-wave structure, which is normally a combination of kinematic and dynamic waves. In general, the primary instability appears when the kinematic wave speed exceeds the speed of dynamic waves. An equality criterion between their speeds yields the neutral stability condition. Similarly, it is revealed that the nonlinear travelling wave solutions also depend on the kinematic and dynamic wave speeds, and an equality criterion between the speeds leads to an analytical expression for the speed of a family of travelling waves as a function of the Froude number. This new analytical result is compared with numerical prediction, and an excellent agreement is achieved. Direct numerical simulations of the low-dimensional model have been performed in order to analyse the spatiotemporal behaviour of nonlinear waves by applying a constant shear stress in the upstream and downstream directions. It is noticed that the presence of imposed shear stress in the upstream (downstream) direction makes the evolution of spatially growing waves weaker (stronger).
Numerical simulation of sand waves in a turbulent open channel flow
Ali Khosronejad, Fotis Sotiropoulos
We develop a coupled hydro-morphodynamic numerical model for carrying out large-eddy simulation of stratified, turbulent flow over a mobile sand bed. The method is based on the curvilinear immersed boundary approach of Khosronejad et al. (Adv. Water Resour., vol. 34, 2011, pp. 829–843). We apply this method to simulate sand wave initiation, growth and evolution in a mobile bed laboratory open channel, which was studied experimentally by Venditti & Church (J. Geophys. Res., vol. 110, 2005, F01009). We show that all the major characteristics of the computed sand waves, from the early cross-hatch and chevron patterns to fully grown three-dimensional bedforms, are in good agreement with the experimental data both qualitatively and quantitatively. Our simulations capture the measured temporal evolution of sand wave amplitude, wavelength and celerity with good accuracy and also yield three-dimensional topologies that are strikingly similar to what was observed in the laboratory. We show that near-bed sweeps are responsible for initiating the instability of the initially flat sand bed. Stratification effects, which arise due to increased concentration of suspended sediment in the flow, also become important at later stages of the bed evolution and need to be taken into account for accurate simulations. As bedforms grow in amplitude and wavelength, they give rise to energetic coherent structures in the form of horseshoe vortices, which transport low-momentum near-bed fluid and suspended sediment away from the bed, giving rise to characteristic 'boil' events at the water surface. Flow separation off the bedform crestlines is shown to trap sediment in the lee side of the crestlines, which, coupled with sediment erosion from the accelerating flow over the stoss side, provides the mechanism for continuous bedform migration and crestline rearrangement. The statistical and spectral properties of the computed sand waves are calculated and shown to be similar to what has been observed in nature and previous numerical simulations. Furthermore, and in agreement with recent experimental findings (Singh et al., Water Resour. Res., vol. 46, 2010, pp. 1–10), the spectra of the resolved velocity fluctuations above the bed exhibit a distinct spectral gap whose width increases with distance from the bed. The spectral gap delineates the spectrum of turbulence from the low-frequency range associated with very slowly evolving, albeit energetic, coherent structures induced by the migrating sand waves. Overall the numerical simulations reproduce the laboratory observations with good accuracy and elucidate the physical phenomena governing the interaction between the turbulent flow and the developing mobile bed.
The influence of nonlinear bottom friction on the properties of decaying cyclonic and anticyclonic vortex structures in a shallow rotated fluid
S. V. Kostrykin, A. A. Khapaev, I. G. Yakushkin
The problem of the decay of intense vortices in a shallow rotated neutrally stratified fluid is considered using simulations with a modified model of von Kármán type and laboratory experiments. The numerical model describes a forced axisymmetric vortex, vertically confined, but infinite in the horizontal plane. It may be used for comparisons with laboratory experiments, in which a quasi-turbulent eddy flow is generated, using magnetohydrodynamic forcing. A detailed analysis of simulations of the free decay of the flow from an initial state, given either by an arbitrary Poiseuille or by a forced stationary profile of vorticity, is provided. Based on this analysis, three different regimes of decay of intense anticyclones in the parameter space of the Ekman and initial Rossby numbers are found. It is shown that anticyclones with large enough Rossby and small enough Ekman numbers may decay to a non-trivial stationary state, or at least they decay much slower than cyclones of the same intensity. The laboratory experiments show much slower decay of intense anticyclones than weak anticyclones or cyclones, and also a dominance of anticyclones over cyclones during the initial stage of decay. These observations qualitatively agree with theoretical predictions.
Stratified shear flow: experiments in an inclined duct
Colin R. Meyer, P. F. Linden
We present results of experiments on stratified shear flow in an inclined duct. The duct connects two reservoirs of fluid with different densities, and contains a counterflow with a dense layer flowing beneath a less dense layer moving in the opposite direction. We identify four flow states in this experiment, depending on the fractional density differences, characterised by the dimensionless Atwood number, and the angle of inclination $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\theta $ , which is defined to be positive (negative) when the along-duct component of gravity reinforces (opposes) the buoyancy-induced pressure differences across the ends of the duct. For sufficiently negative angles and small fractional density differences, the flow is observed to be laminar ( $\mathsf{L}$ state), with an undisturbed density interface separating the two layers. For positive angles and/or high fractional density differences, three other states are observed. For small angles of inclination, the flow is wave-dominated and exhibits Holmboe modes ( $\mathsf{H}$ state) on the interface, with characteristic cusp-like wave breaking. At the highest positive angles and density differences, there is a turbulent ( $\mathsf{T}$ state) high-dissipation interfacial region typically containing Kelvin–Helmholtz (KH)-like structures sheared in the direction of the mean shear and connecting both layers. For intermediate angles and density differences, an intermittent state ( $\mathsf{I}$ state) is found, which exhibits a rich range of spatio-temporal behaviour and an interfacial region that contains features of KH-like structures and of the other two lower-dissipation states: thin interfaces and Holmboe-like structures. We map the state diagram of these flows in the Atwood number– $\theta $ plane and examine the force balances that determine each of these states. We find that the $\mathsf{L}$ and $\mathsf{H}$ states are hydraulically controlled at the ends of the duct and the flow is determined by the pressure difference associated with the density difference between the reservoirs. As the inclination increases, the along-slope component of the buoyancy force becomes more significant and the $\mathsf{I}$ and $\mathsf{T}$ states are associated with increasing dissipation within the duct. We replot the state space in the Grashof number– $\theta $ phase plane and find the transition to the $\mathsf{T}$ state is governed by a critical Grashof number. We find that the corresponding buoyancy Reynolds number of the transition to the $\mathsf{T}$ state is of the order of 100, and that this state is also found to be hydraulically controlled at the ends of the duct. In this state the dissipation balances the force associated with the along-slope component of buoyancy and the counterflow has a critical composite Froude number.
Steady streaming in a two-dimensional box model of a passive cochlea
Elisabeth Edom, Dominik Obrist, Leonhard Kleiser
Acoustic stimulation of the cochlea leads to a travelling wave in the cochlear fluids and on the basilar membrane (BM). It has long been suspected that this travelling wave leads to a steady streaming flow in the cochlea. Theoretical investigations suggested that the steady streaming might be of physiological relevance. Here, we present a quantitative study of the steady streaming in a computational model of a passive cochlea. The structure of the streaming flow is illustrated and the sources of streaming are closely investigated. We describe a source of streaming which has not been considered in the cochlea by previous authors. This source is also related to a steady axial displacement of the BM which leads to a local stretching of this compliant structure. We present theoretical predictions for the streaming intensity which account for these new phenomena. It is shown that these predictions compare well with our numerical results and that there may be steady streaming velocities of the order of millimetres per second. Our results indicate that steady streaming should be more relevant to low-frequency hearing because the strength of the streaming flow rapidly decreases for higher frequencies.
A parametric study of the coalescence of liquid drops in a viscous gas
James E. Sprittles, Yulii D. Shikhmurzaev
The coalescence of two liquid drops surrounded by a viscous gas is considered in the framework of the conventional model. The problem is solved numerically with particular attention paid to resolving the very initial stage of the process which only recently has become accessible both experimentally and computationally. A systematic study of the parameter space of practical interest allows the influence of the governing parameters in the system to be identified and the role of viscous gas to be determined. In particular, it is shown that the viscosity of the gas suppresses the formation of toroidal bubbles predicted in some cases by early computations where the gas' dynamics was neglected. Focusing computations on the very initial stages of coalescence and considering the large parameter space allows us to examine the accuracy and limits of applicability of various 'scaling laws' proposed for different 'regimes' and, in doing so, reveal certain inconsistencies in recent works. A comparison with experimental data shows that the conventional model is able to reproduce many qualitative features of the initial stages of coalescence, such as a collapse of calculations onto a 'master curve' but, quantitatively, overpredicts the observed speed of coalescence and there are no free parameters to improve the fit. Finally, a phase diagram of parameter space, differing from previously published ones, is used to illustrate the key findings.
The generalized Onsager model for a binary gas mixture
V. Kumaran, S. Pradhan
The Onsager model for the secondary flow field in a high-speed rotating cylinder is extended to incorporate the difference in mass of the two species in a binary gas mixture. The base flow is an isothermal solid-body rotation in which there is a balance between the radial pressure gradient and the centrifugal force density for each species. Explicit expressions for the radial variation of the pressure, mass/mole fractions, and from these the radial variation of the viscosity, thermal conductivity and diffusion coefficient, are derived, and these are used in the computation of the secondary flow. For the secondary flow, the mass, momentum and energy equations in axisymmetric coordinates are expanded in an asymptotic series in a parameter $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\epsilon = (\Delta m/ m_{av})$ , where $\Delta m$ is the difference in the molecular masses of the two species, and the average molecular mass $m_{av}$ is defined as $m_{av}= (\rho _{w1} m_1 + \rho _{w2} m_2)/\rho _w$ , where $\rho _{w1}$ and $\rho _{w2}$ are the mass densities of the two species at the wall, and $\rho _w = \rho _{w1} + \rho _{w2}$ . The equation for the master potential and the boundary conditions are derived correct to $O(\epsilon ^2)$ . The leading-order equation for the master potential contains a self-adjoint sixth-order operator in the radial direction, which is different from the generalized Onsager model (Pradhan & Kumaran, J. Fluid Mech., vol. 686, 2011, pp. 109–159), since the species mass difference is included in the computation of the density, viscosity and thermal conductivity in the base state. This is solved, subject to boundary conditions, to obtain the leading approximation for the secondary flow, followed by a solution of the diffusion equation for the leading correction to the species mole fractions. The $O(\epsilon )$ and $O(\epsilon ^2)$ equations contain inhomogeneous terms that depend on the lower-order solutions, and these are solved in a hierarchical manner to obtain the $O(\epsilon )$ and $O(\epsilon ^2)$ corrections to the master potential. A similar hierarchical procedure is used for the Carrier–Maslen model for the end-cap secondary flow. The results of the Onsager hierarchy, up to $O(\epsilon ^2)$ , are compared with the results of direct simulation Monte Carlo simulations for a binary hard-sphere gas mixture for secondary flow due to a wall temperature gradient, inflow/outflow of gas along the axis, as well as mass and momentum sources in the flow. There is excellent agreement between the solutions for the secondary flow correct to $O(\epsilon ^2)$ and the simulations, to within 15 %, even at a Reynolds number as low as 100, and length/diameter ratio as low as 2, for a low stratification parameter $\mathcal{A}$ of 0.707, and when the secondary flow velocity is as high as 0.2 times the maximum base flow velocity, and the ratio $2 \Delta m / (m_1 + m_2)$ is as high as 0.5. Here, the Reynolds number $\mathit{Re}= \rho _w \varOmega R^2 / \mu $ , the stratification parameter $\mathcal{A}= \sqrt{m \varOmega ^2 R^2 / (2k_BT)}$ , $R$ and $\varOmega $ are the cylinder radius and angular velocity, $m$ is the molecular mass, $\rho _w$ is the wall density, $\mu $ is the viscosity and $T$ is the temperature. The leading-order solutions do capture the qualitative trends, but are not in quantitative agreement.
Effects of radiation in turbulent channel flow: analysis of coupled direct numerical simulations
R. Vicquelin, Y. F. Zhang, O. Gicquel, J. Taine
The role of radiative energy transfer in turbulent boundary layers is carefully analysed, focusing on the effect on temperature fluctuations and turbulent heat flux. The study is based on direct numerical simulations (DNS) of channel flows with hot and cold walls coupled to a Monte-Carlo method to compute the field of radiative power. In the conditions studied, the structure of the boundary layers is strongly modified by radiation. Temperature fluctuations and turbulent heat flux are reduced, and new radiative terms appear in their respective balance equations. It is shown that they counteract turbulence production terms. These effects are analysed under different conditions of Reynolds number and wall temperature. It is shown that collapsing of wall-scaled profiles is not efficient when radiation is considered. This drawback is corrected by the introduction of a radiation-based scaling. Finally, the significant impact of radiation on turbulent heat transfer is studied in terms of the turbulent Prandtl number. A model for this quantity, based on the new proposed scaling, is developed and validated.
On grid-generated turbulence in the near- and far field regions
Juan C. Isaza, Ricardo Salazar, Zellman Warhaft
Using a conventional bi-planar turbulence-generating grid, we confirm the recent findings (Valente & Vassilicos, Phys. Rev. Lett., vol. 108, 2012, art. 214503) that show there is a turbulence decay region close to the generating grid that departs from the 'classical' turbulence decay (Comte-Bellot & Corrsin, J. Fluid Mech., vol. 25, 1966, pp. 657–682). In this 'near-field' region, the turbulence energy decays more rapidly than in the far-field and it exhibits unusual scaling properties. Based on the velocity decay laws, we show that for our conventional grid, the near-field extends from $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}x/M \sim 6$ to $x/M \sim 12$ where $x$ is the downstream distance from the grid and $M$ is the mesh size. This corresponds to $1.1 \le x/x* \le 2.3$ where $x*$ is the wake interaction length scale (Mazellier & Vassilicos, Phys. Fluids, vol. 22, 2010, art. 075101). However, other statistics (velocity derivatives and length-scale ratios) indicate that the extent of the initial period depends on the grid mesh Reynolds number, $R_M$ , extending further for higher values of $R_M$ . In the near-field the turbulence approaches isotropy both at the large and small scales but there still is inhomogeneity in the derivative statistics. The derivative skewness also departs from values observed at comparable Reynolds numbers in the far-field decay region, and in other turbulent flows at comparable Reynolds numbers. Two values of $R_M$ were studied: $42 \times 10^3$ and $76 \times 10^3$ .
Trapped modes and scattering for oblique waves in a two-layer fluid
M. I. Romero Rodríguez, P. Zhevandrov
The system describing time-harmonic motions of a two-layer fluid in the linearised shallow-water approximation is considered. It is assumed that the depth is constant, with a cylindrical protrusion (an underwater ridge) of small height. For obliquely incident waves, the system reduces to a pair of coupled ordinary differential equations. The values of frequency for which wave propagation in the unperturbed system is possible are bounded from below by a cutoff, to the left of which no propagating modes exist. Under the perturbation, a trapped mode appears to the left of the cutoff and, if a certain geometric requirement is imposed upon the shape of the perturbation (for example, if the ridge is a rectangular barrier of certain width), a trapped mode appears whose frequency is embedded in the continuous spectrum. When these geometric conditions are not satisfied, the embedded trapped mode transforms into a complex pole of the reflection and transmission coefficients of the corresponding scattering problem, and the phenomenon of almost total reflection is observed when the frequency coincides with the real part of the pole. Exact formulae for the trapped modes are obtained explicitly in the form of infinite series in powers of the small parameter characterising the perturbation. The results provide a theoretical understanding of analogous phenomena observed numerically in the literature for the full problem for the potentials in a two-layer fluid in the presence of submerged cylinders, and furnish explicit formulae for the frequencies at which total reflection occurs and the trapped modes exist.
Thermoacoustic instability – a dynamical system and time domain analysis
Taraneh Sayadi, Vincent Le Chenadec, Peter J. Schmid, Franck Richecoeur, Marc Massot
This study focuses on the Rijke tube problem, which includes features relevant to the modelling of thermoacoustic coupling in reactive flows: a compact acoustic source, an empirical model for the heat source and nonlinearities. This thermoacoustic system features both linear and nonlinear flow regimes with complex dynamical behaviour. In order to synthesize accurate time series, we tackle this problem from a numerical point of view, and start by proposing a dedicated solver designed for dealing with the underlying stiffness – in particular, the retarded time and the discontinuity at the location of the heat source. Stability analysis is performed on the limit of low-amplitude disturbances by using the projection method proposed by Jarlebring (PhD thesis, Technische Universität Braunschweig, 2008), which alleviates the problems arising from linearization with respect to the retarded time. The results are then compared with the analytical solution of the undamped system and with the results obtained from Galerkin projection methods commonly used in this setting. This analysis provides insight into the consequences of the various assumptions and simplifications that justify the use of Galerkin expansions based on the eigenmodes of the unheated resonator. We demonstrate that due to the presence of a discontinuity in the spatial domain, the eigenmodes in the heated case predicted by using Galerkin expansion show spurious oscillations resulting from the Gibbs phenomenon. Finally, time series in the fully nonlinear regime, where a limit cycle is established, are analysed and dominant modes are extracted. By comparing the modes of the linear regime to those of the nonlinear regime, we are able to illustrate the mean-flow modulation and frequency switching, which appear as the nonlinearities become significant and ultimately affect the form of the limit cycle. Analysis of the saturated limit cycles shows the presence of higher-frequency modes, which are linearly stable but become significant through nonlinear growth of the signal. This bimodal effect is not exhibited when the coupling between different frequencies is not accounted for. In conclusion, a dedicated solver for capturing thermoacoustic instability is proposed and methods for analysing linear and nonlinear regions of the resulting time series are introduced.
Inertial wave dynamics in a rotating liquid metal
Tobias Vogt, Dirk Räbiger, Sven Eckert
The dynamics of free and forced inertial waves inside cylinders of different aspect ratios ( $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}A=H_0/2R_0$ ) were investigated experimentally in this study. The liquid metal GaInSn was chosen as the fluid in order to enable a contactless stimulation of the flow by means of alternating electromagnetic fields. A rotating magnetic field generates the rotating motion of the liquid, whereas periodic modulations of the field strength and short pulses excite specific wave modes. Ultrasound Doppler velocimetry was used to record the flow structure and to identify inertial waves in the set-up. Our experiments demonstrate selective excitation of different inertial wave modes by deliberate variation of the magnetic field parameters. Furthermore, it was found that turbulent perturbations in the boundary layers of the swirling flow are able to induce an inertial wave mode that survives over a long time. Experiments at the fundamental resonance have shown that multiple harmonic wave modes appeared simultaneously. The measured inertial wave frequencies were compared to the predictions of the linear inviscid theory.
Large-eddy simulation of turbulence and particle dispersion inside the canopy roughness sublayer
Ying Pan, Marcelo Chamecki, Scott A. Isard
Modelling the dispersion of small particles such as fungal spores, pollens and small seeds inside and above plant canopies is important for many applications. Transport of these particles is driven by strongly inhomogeneous and non-Gaussian turbulent flows inside the canopy roughness sublayer, the region that extends from the ground to approximately three canopy heights. A large-eddy simulation (LES) approach is refined to study particle dispersion within and above the canopy region. Effects of plant reconfiguration are parameterized through a velocity-dependent drag coefficient, which is shown to be critical for accurate reproduction of velocity statistics and mean spore concentrations. The model yields predictions of turbulence statistics that are in good agreement with measurements. This is particularly true of the stress fractions carried by strong events, as revealed by standard quadrant analysis of the resolved velocity fluctuations, which is a known weakness of earlier LES studies of canopy flow using a constant drag coefficient. Experimental data on spore dispersal inside and above a maize canopy are reproduced successfully as well. Characteristics of the particle plume are analysed using LES results, and a pre-existing theoretical framework is adapted to model particle dispersal above the canopy. The results suggest that the plume above the canopy can be approximated using a simple analytical solution if the fraction of spores that escape the canopy region is known. Source height and gravitational settling have strong effects on the plume inside the canopy region and consequently determine the escape fraction. These effects are parameterized in the theoretical model by using the escape fraction to rescale the source strength.
Viscous Marangoni migration of a drop in a Poiseuille flow at low surface Péclet numbers
On Shun Pak, Jie Feng, Howard A. Stone
The motion of a spherical drop with a bulk-insoluble surfactant immersed in a background flow in the limits of low surface Péclet number and low Reynolds number is investigated. We develop a reciprocal theorem that applies to any prescribed background flow and provide a specific example of an unbounded Poiseuille flow. Analytical formulas for the migration velocity of the drop are obtained perturbatively in powers of the surface Péclet number. We show that the redistribution of surfactant due to the background flow acts to retard the motion of the drop, with the magnitude of this slip velocity being independent of the drop's position in the Poiseuille flow. Moreover, a surfactant-induced cross-streamline migration of the drop occurs towards the centre of the Poiseuille flow, with its magnitude depending linearly on the distance of the drop from the centre of the Poiseuille flow.
Boundary layer flow and bed shear stress under a solitary wave – CORRIGENDUM
P. L.-F. Liu, Y. S. Park, E. A. Cowen
Published online by Cambridge University Press: 29 July 2014, p. 553