We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Email your librarian or administrator to recommend adding this journal to your organisation's collection.
You are leaving Cambridge Core and will be taken to this journal's article submission site.
To save this undefined to your undefined account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your undefined account. Find out more about saving content to .
To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.
Three-dimensional bluff body aerodynamics are pertinent across a broad range of engineering disciplines. In three-dimensional bluff body flows, shear layer behaviour has a primary influence on the surface pressure distributions and, therefore, the integrated forces and moments. There currently exists a significant gap in understanding of the flow around canonical three-dimensional bluff bodies such as rectangular prisms and short circular cylinders. High-fidelity numerical experiments using a hybrid turbulence closure that resolves large eddies in separated wakes close this gap and provide new insights into the unsteady behaviour of these bodies. A time-averaging technique that captures the mean shear layer behaviours in these unsteady turbulent flows is developed, and empirical characterizations are developed for important quantities, including the shear layer reattachment distance, the separation bubble pressure, the maximum reattachment pressure, and the stagnation point location. Many of these quantities are found to exhibit a universal behaviour that varies only with the incidence angle and face shape (flat or curved) when an appropriate normalization is applied.
Convective plumes emanating from fixed buoyant sources such as volcanoes, hot springs and oil spills are common in the atmosphere and the ocean. Most of what we know about their dynamics comes from scaling laws, laboratory experiments and numerical simulations. A plume grows laterally during its ascent mainly due to the process of turbulent entrainment of fluid from the environment into the plume. In an unstratified system, nothing hampers the vertical motion of the plume. By contrast, in a stratified system, as the plume rises, it reaches and overshoots the neutral buoyancy height – due to the non-zero momentum at that height. This rising fluid is then dense relative to the environment and slows down, ceases to rise and falls back to the height of the intrusion. For buoyant plumes occurring in the ocean or atmosphere, the rotation of the Earth adds an additional constraint via the conservation of angular momentum. In fact, the effect of rotation is still not well understood, and we addressed this issue in the study reported here. We looked for the steady states of an axisymmetric model in both the rotating and non-rotating cases. At the non-rotating limit, we isolated two regimes of convection depending on the buoyancy flux/momentum flux ratio at the base of the plume, in agreement with scaling laws. However, the inclusion of rotation in the model strongly affects these classical convection patterns: the lateral extension of the plume is confined at the intrusion level by the establishment of a geostrophic balance, and non-trivial swirl speed develops in and around the plume.
The study of flapping-wing aerodynamics faces a large control space with different wing kinematics and deformation. The adjoint-based approach, by solving an inverse problem to obtain simultaneously the sensitivity with respect to all control parameters, has a computational cost independent of the number of control parameters and becomes an efficient tool for the study of problems with a large control space. However, the adjoint equation is typically formulated in a fixed fluid domain. In a continuous formulation, a moving boundary or morphing domain results in inconsistency in the definition of an arbitrary perturbation at the boundary, which leads to ambiguousness and difficulty in the adjoint formulation if control parameters are related to boundary changes (e.g. the control of wing kinematics and dynamic deformation). The unsteady mapping function, as a traditional way to deal with moving boundaries, can in principle be a remedy for this situation. However, the derivation is often too complex to be feasible, even for simple problems. Part of the complexity comes from the unnecessary mapping of the interior mesh, while only mapping of the boundary is needed here. Non-cylindrical calculus, on the other hand, provides a boundary mapping and considers the rest of domain as an arbitrary extension from the boundary. Using non-cylindrical calculus to handle moving boundaries makes the derivation of the adjoint formulation much easier and also provides a simpler final formulation. The new adjoint-based optimization approach is validated for accuracy and efficiency by a well-defined case where a rigid plate plunges normally to an incoming flow. Then, the approach is applied for the optimization of drag reduction and propulsive efficiency of first a rigid plate and then a flexible plate which both flap with plunging and pitching motions against an incoming flow. For the rigid plate, the phase delay between pitching and plunging is the control and considered as both a constant (i.e. a single parameter) and a time-varying function (i.e. multiple parameters). The comparison between its arbitrary initial status and the two optimal solutions (with a single parameter or multiple parameters) reveals the mechanism and control strategy to reach the maximum thrust performance or propulsive efficiency. Essentially, the control is trying to benefit from both lift-induced thrust and viscous drag (by reducing it), and the viscous drag plays a dominant role in the optimization of efficiency. For the flexible plate, the control includes the amplitude and phase delay of the pitching motion and the leading eigenmodes to characterize the deformation. It is clear that flexibility brings about substantial improvement in both thrust performance and propulsive efficiency. Finally, the adjoint-based approach is extended to a three-dimensional study of a rectangular plate in hovering motion for lift performance. Both rigid and flexible cases are considered. The adjoint-based algorithm finds an optimal hovering motion with advanced rotation which has a large leading-edge vortex and strong downwash for lift benefit, and the introduction of flexibility enhances the wake capturing mechanism and generates a stronger downwash to push the lift coefficient higher.
This paper presents three-dimensional numerical simulations of non-colloidal dense suspensions in a wall-bounded shear flow at zero Reynolds number. Simulations rely on a fictitious domain method with a detailed modelling of particle–particle and wall–particle lubrication forces, as well as contact forces including particle roughness and friction. This study emphasizes the effect of walls on the structure, velocity and rheology of a moderately confined suspension (channel gap to particle radius ratio of 20) for a volume fraction range $0.1\leqslant {\it\phi}\leqslant 0.5$. The wall region shows particle layers with a hexagonal structure. The size of this layered zone depends on volume fraction and is only weakly affected by friction. This structure implies a wall slip which is in good accordance with empirical models. Simulations show that this wall slip can be mitigated by reducing particle roughness. For ${\it\phi}\lessapprox 0.4$, wall-induced layering has a moderate impact on the viscosity and second normal stress difference $N_{2}$. Conversely, it significantly alters the first normal stress difference $N_{1}$ and can result in positive $N_{1}$, in better agreement with some experiments. Friction enhances this effect, which is shown to be due to a substantial decrease in the contact normal stress $|{\it\Sigma}_{xx}^{c}|$ (where $x$ is the velocity direction) because of particle layering in the wall region.
This article explores the three-dimensional flow structure of a streamwise-oriented vortex incident on a finite aspect-ratio wing. The vertical positioning of the incident vortex relative to the wing is shown to have a significant impact on the unsteady flow structure. A direct impingement of the streamwise vortex produces a spiralling instability in the vortex just upstream of the leading edge, reminiscent of the helical instability modes of a Batchelor vortex. A small negative vertical offset develops a more pronounced instability while a positive vertical offset removes the instability altogether. These differences in vertical position are a consequence of the upstream influence of pressure gradients provided by the wing. Direct impingement or a negative vertical offset subject the vortex to an adverse pressure gradient that leads to a reduced axial velocity and diminished swirl conducive to hydrodynamic instability. Conversely, a positive vertical offset removes instability by placing the streamwise vortex in line with a favourable pressure gradient, thereby enhancing swirl and inhibiting the growth of unstable modes. In every case, the helical instability only occurs when the properties of the incident vortex fall within the instability threshold predicted by linear stability theory. The influence of pressure gradients associated with separation and stall downstream also have the potential to introduce suction-side instabilities for a positive vertical offset. The influence of the wing is more severe for larger vortices and diminishes with vortex size due to weaker interaction and increased viscous stability. Helical instability is not the only possible outcome in a direct impingement. Jet-like vortices and a higher swirl ratio in wake-like vortices can retain stability upon impact, resulting in the laminar vortex splitting over either side of the wing.
The present work aims at developing a spectral model for a passive scalar field and its associated scalar flux in homogeneous anisotropic turbulence. This is achieved using the paradigm of eddy-damped quasi-normal Markovian (EDQNM) closure extended to anisotropic flows. In order to assess the validity of this approach, the model is compared to several detailed direct numerical simulations (DNS) and experiments of shear-driven flows and isotropic turbulence with a mean scalar gradient at moderate Reynolds numbers. This anisotropic modelling is then used to investigate the passive scalar dynamics at very high Reynolds numbers. In the framework of homogeneous isotropic turbulence submitted to a mean scalar gradient, decay and growth exponents for the cospectrum and scalar energies are obtained analytically and assessed numerically thanks to EDQNM closure. With the additional presence of a mean shear, the scaling of the scalar flux and passive scalar spectra in the inertial range are investigated and confirm recent theoretical predictions. Finally, it is found that, in shear-driven flows, the small scales of the scalar second-order moments progressively return to isotropy when the Reynolds number increases.
An experimental investigation of primary and secondary crossflow instability developing in the boundary layer of a $45^{\circ }$ swept wing at a chord Reynolds number of $2.17\times 10^{6}$ is presented. Linear stability theory is applied for preliminary estimation of the flow stability while surface flow visualisation using fluorescent oil is employed to inspect the topological features of the transition region. Hot-wire anemometry is extensively used for the investigation of the developing boundary layer and identification of the statistical and spectral characteristics of the instability modes. Primary stationary, as well as unsteady type-I (z-mode), type-II (y-mode) and type-III modes are detected and quantified. Finally, three-component, three-dimensional measurements of the transitional boundary layer are performed using tomographic particle image velocimetry. This research presents the first application of an optical experimental technique for this type of flow. Among the optical techniques, tomographic velocimetry represents, to date, the most advanced approach allowing the investigation of spatially correlated flow structures in three-dimensional fields. Proper orthogonal decomposition (POD) analysis of the captured flow fields is applied to this goal. The first POD mode features a newly reported structure related to low-frequency oscillatory motion of the stationary vortices along the spanwise direction. The cause of this phenomenon is only conjectured. Its effect on transition is considered negligible but, given the related high energy level, it needs to be accounted for in experimental investigations. Secondary instability mechanisms are captured as well. The type-III mode corresponds to low-frequency primary travelling crossflow waves interacting with the stationary ones. It appears in the inner upwelling region of the stationary crossflow vortices and is characterised by elongated structures approximately aligned with the axis of the stationary waves. The type-I secondary instability consists instead of significantly inclined structures located at the outer upwelling region of the stationary vortices. The much narrower wavelength and higher advection velocity of these structures correlate with the higher-frequency content of this mode. The results of the investigation of both primary and secondary instability from the exploited techniques agree with and complement each other and are in line with existing literature. Finally, they present the first experimental observation of the secondary instability structures under natural flow conditions.
We study the linear stage of the dynamo instability of a turbulent two-dimensional flow with three components $(u(x,y,t),v(x,y,t),w(x,y,t))$ that is sometimes referred to as a 2.5-dimensional (2.5-D) flow. The flow evolves based on the two-dimensional Navier–Stokes equations in the presence of a large-scale drag force that leads to the steady state of a turbulent inverse cascade. These flows provide an approximation to very fast rotating flows often observed in nature. The low dimensionality of the system allows for the realization of a large number of numerical simulations and thus the investigation of a wide range of fluid Reynolds numbers $Re$, magnetic Reynolds numbers $Rm$ and forcing length scales. This allows for the examination of dynamo properties at different limits that cannot be achieved with three-dimensional simulations. We examine dynamos for both large and small magnetic Prandtl-number turbulent flows $Pm=Rm/Re$, close to and away from the dynamo onset, as well as dynamos in the presence of scale separation. In particular, we determine the properties of the dynamo onset as a function of $Re$ and the asymptotic behaviour in the large $Rm$ limit. We are thus able to give a complete description of the dynamo properties of these turbulent 2.5-D flows.
Observations and models of deep ocean boundary currents show that they exhibit complex variability, instabilities and eddy shedding, particularly over continental slopes that curve horizontally, for example around coastal peninsulas. In this article the authors investigate the source of this variability by characterizing the properties of baroclinic instability in mean flows over horizontally curved bottom slopes. The classical two-layer quasi-geostrophic solution for linear baroclinic instability over sloping bottom topography is extended to the case of azimuthal mean flow in an annular channel. To facilitate comparison with the classical straight channel instability problem of uniform mean flow, the authors focus on comparatively simple flows in an annulus, namely uniform azimuthal velocity and solid-body rotation. Baroclinic instability in solid-body rotation flow is analytically analogous to the instability in uniform straight channel flow due to several identical properties of the mean flow, including vanishing strain rate and vorticity gradient. The instability of uniform azimuthal flow is numerically similar to straight channel flow instability as long as the mean barotropic azimuthal velocity is zero. Non-zero barotropic flow generally suppresses the instability via horizontal curvature-induced strain and Reynolds stress work. An exception occurs when the ratio of the bathymetric to isopycnal slopes is close to (positive) one, as is often observed in the ocean, in which case the instability is enhanced. A non-vanishing mean barotropic flow component also results in a larger number of growing eigenmodes and in increased non-normal growth. The implications of these findings for variability in deep western boundary currents are discussed.
The changes in discharge in pressure-driven flows through channels with longitudinal grooves have been investigated in the laminar flow regime and in the turbulent flow regime with moderate Reynolds numbers ($Re_{2H}\approx 6000$) using both analytical and numerical methodologies. The results demonstrate that the long-wavelength grooves can increase discharge by 20 %–150 %, depending on the groove amplitude and the type of flow, while the short-wavelength grooves reduce the discharge. It has been shown that the reduced geometry model applies to the analysis of turbulent flows and the performance of grooves of arbitrary form is well approximated by the performance of grooves whose shape is represented by the dominant Fourier mode. The flow patterns, the turbulent kinetic energy as well as the Reynolds stresses were examined to identify the mechanisms leading to an increase in discharge. It is shown that the increase in discharge results from the rearrangement of the bulk fluid movement and not from the suppression of turbulence intensity. The turbulent kinetic energy and the Reynolds stresses are rearranged while their volume-averaged intensities remain the same as in the smooth channel. Analysis of the interaction of the groove patterns on both walls demonstrates that the converging–diverging configuration results in the greatest increase in discharge while the wavy channel configuration results in a reduction in discharge.
Using high-resolution particle image velocimetry, we measure velocity profiles, the wind Reynolds number and characteristics of turbulent plumes in Taylor–Couette flow for a radius ratio of 0.5 and Taylor number of up to $6.2\times 10^{9}$. The extracted angular velocity profiles follow a log law more closely than the azimuthal velocity profiles due to the strong curvature of this ${\it\eta}=0.5$ set-up. The scaling of the wind Reynolds number with the Taylor number agrees with the theoretically predicted $3/7$ scaling for the classical turbulent regime, which is much more pronounced than for the well-explored ${\it\eta}=0.71$ case, for which the ultimate regime sets in at much lower Taylor number. By measuring at varying axial positions, roll structures are found for counter-rotation while no clear coherent structures are seen for pure inner cylinder rotation. In addition, turbulent plumes coming from the inner and outer cylinders are investigated. For pure inner cylinder rotation, the plumes in the radial velocity move away from the inner cylinder, while the plumes in the azimuthal velocity mainly move away from the outer cylinder. For counter-rotation, the mean radial flow in the roll structures strongly affects the direction and intensity of the turbulent plumes. Furthermore, it is experimentally confirmed that, in regions where plumes are emitted, boundary layer profiles with a logarithmic signature are created.
Statistics of the trajectories of molecules diffusing via Brownian motion in a turbulent flow are extracted from simulations of stationary isotropic turbulence, using a postprocessing approach applicable in both forward and backward reference frames. Detailed results are obtained for Schmidt numbers ($Sc$) from 0.001 to 1000 at Taylor-scale Reynolds numbers up to 1000. The statistics of displacements of single molecules compare well with the earlier theoretical work of Saffman (J. Fluid Mech. vol. 8, 1960, pp. 273–283) except for the scaling of the integral time scale of the fluid velocity following the molecular trajectories. For molecular pairs we extend Saffman’s theory to include pairs of small but finite initial separation, which is in excellent agreement with numerical results provided that data are collected at sufficiently small times. At intermediate times the separation statistics of molecular pairs exhibit a more robust Richardson scaling behaviour than for the fluid particles. The forward scaling constant is very close to 0.55, whereas the backward constant is approximately 1.53–1.57, with a weak Schmidt number dependence, although no scaling exists if $Sc\ll 1$ at the Reynolds numbers presently accessible. An important innovation in this work is to demonstrate explicitly the practical utility of a Lagrangian description of turbulent mixing, where molecular displacements and separations in the limit of small backward initial separation can be used to calculate the evolution of scalar fluctuations resulting from a known source function in space. Lagrangian calculations of the production and dissipation rates of the scalar fluctuations are shown to agree very well with Eulerian results for the case of passive scalars driven by a uniform mean gradient. Although the Eulerian–Lagrangian comparisons are made only for $Sc\sim O(1)$, the Lagrangian approach is more easily extended to both very low and very high Schmidt numbers. The well-known scalar dissipation anomaly is accordingly also addressed in a Lagrangian context.
The unsteady behaviour of a massively separated, pressure-induced turbulent separation bubble (TSB) is investigated experimentally using high-speed particle image velocimetry (PIV) and piezo-resistive pressure sensors. The TSB is generated on a flat test surface by a combination of adverse and favourable pressure gradients. The Reynolds number based on the momentum thickness of the incoming boundary layer is 5000 and the free stream velocity is $25~\text{m}~\text{s}^{-1}$. The proper orthogonal decomposition (POD) is used to separate the different unsteady modes in the flow. The first POD mode contains approximately 30 % of the total kinetic energy and is shown to describe a low-frequency contraction and expansion, called ‘breathing’, of the TSB. This breathing is responsible for a variation in TSB size of approximately 90 % of its average length. It also generates low-frequency wall-pressure fluctuations that are mainly felt upstream of the mean detachment and downstream of the mean reattachment. A medium-frequency unsteadiness, which is linked to the convection of large-scale vortices in the shear layer bounding the recirculation zone and their shedding downstream of the TSB, is also observed. When scaled with the vorticity thickness of the shear layer and the convection velocity of the structures, this medium frequency is very close to the characteristic frequency of vortices convected in turbulent mixing layers. The streamwise position of maximum vertical turbulence intensity generated by the convected structures is located downstream of the mean reattachment line and corresponds to the position of maximum wall-pressure fluctuations.
Rotating Rayleigh–Bénard convection, the flow in a rotating fluid layer heated from below and cooled from above, is used to analyse the transition to the geostrophic regime of thermal convection. In the geostrophic regime, which is of direct relevance to most geo- and astrophysical flows, the system is strongly rotating while maintaining a sufficiently large thermal driving to generate turbulence. We directly simulate the Navier–Stokes equations for two values of the thermal forcing, i.e. $Ra=10^{10}$ and $Ra=5\times 10^{10}$, at constant Prandtl number $Pr=1$, and vary the Ekman number in the range $Ek=1.3\times 10^{-7}$ to $Ek=2\times 10^{-6}$, which satisfies both requirements of supercriticality and strong rotation. We focus on the differences between the application of no-slip versus stress-free boundary conditions on the horizontal plates. The transition is found at roughly the same parameter values for both boundary conditions, i.e. at $Ek\approx 9\times 10^{-7}$ for $Ra=1\times 10^{10}$ and at $Ek\approx 3\times 10^{-7}$ for $Ra=5\times 10^{10}$. However, the transition is gradual and it does not exactly coincide in $Ek$ for different flow indicators. In particular, we report the characteristics of the transitions in the heat-transfer scaling laws, the boundary-layer thicknesses, the bulk/boundary-layer distribution of dissipations and the mean temperature gradient in the bulk. The flow phenomenology in the geostrophic regime evolves differently for no-slip and stress-free plates. For stress-free conditions, the formation of a large-scale barotropic vortex with associated inverse energy cascade is apparent. For no-slip plates, a turbulent state without large-scale coherent structures is found; the absence of large-scale structure formation is reflected in the energy transfer in the sense that the inverse cascade, present for stress-free boundary conditions, vanishes.
Lubrication flows appear in many applications in engineering, biophysics and nature. Separation of surfaces and minimisation of friction and wear is achieved when the lubricating fluid builds up a lift force. In this paper we analyse soft lubricated contacts by treating the solid walls as viscoelastic: soft materials are typically not purely elastic, but dissipate energy under dynamical loading conditions. We present a method for viscoelastic lubrication and focus on three canonical examples, namely Kelvin–Voigt, standard linear and power law rheology. It is shown how the solid viscoelasticity affects the lubrication process when the time scale of loading becomes comparable to the rheological time scale. We derive asymptotic relations between the lift force and the sliding velocity, which give scaling laws that inherit a signature of the rheology. In all cases the lift is found to decrease with respect to purely elastic systems.
In this experimental–theoretical investigation, we consider a turbulent plume generated by an isothermal wall in a closed cavity and the formation of heat stratification in the interior. The buoyancy of the plume near the wall and the temperature stratification are measured across a vertical plane with the temperature laser induced fluorescence method, which is shown to be accurate and efficient (precision of $0.2\,^{\circ }$C) for experimental studies on convection. The simultaneous measurement of the velocity field with particle image velocimetry allows for the calculation of the flow characteristics such as the Richardson number and Reynolds stress. This enables us to give a refined description of the wall plume, as well as the circulation and evolution of the stratification in the interior. The wall plume is found to have an inner layer close to the heated boundary with a laminar transport of hardly mixed fluid which causes a relatively warm top layer and an outer layer with a transition from laminar to turbulent at a considerable height. The measured entrainment coefficient is found to be dramatically influenced by the increase in stratification of the ambient fluid. To model the flow, the entrainment model of Morton, Taylor & Turner (Proc. R. Soc. Lond. A, vol. 234 (1196), 1956, pp. 1–23) has first been adapted to the case of an isothermal wall. Differences due to their boundary condition of a constant buoyancy flux, modelled with salt by Cooper & Hunt (J. Fluid Mech., vol. 646, 2010, pp. 39–58), turn out to be small. Next, to include the laminar–turbulent transition of the boundary layer, a hybrid model is constructed which is based on the similarity solutions reported by Worster & Leitch (J. Fluid Mech., vol. 156, 1985, pp. 301–319) for the laminar part and the entrainment model for the turbulent part. Finally, the observed variation of the global entrainment coefficient, which is due to the increased presence of an upper stratified layer with a relatively low entrainment coefficient, is incorporated into both models. All models show reasonable agreement with experimental measurements for the volume, momentum and buoyancy fluxes as well as for the evolution of the stratification in the interior. In particular, the introduction of the variable entrainment coefficient improves all models significantly.
Channel bifurcations are a fundamental element of a broad variety of flowing freshwater environments worldwide, such as braiding and anabranching rivers, river deltas and alluvial fans. River bifurcations often develop asymmetrical configurations with uneven discharge partition and a bed elevation gap between the downstream anabranches. This has been reproduced by one-dimensional (1-D) analytical theories which, however, rely on the empirical calibration of one or more parameters and cannot provide a clear and detailed physical explanation of the observed dynamics. We propose a novel two-dimensional (2-D) solution for the flow and bed topography in channel bifurcations based on an innovative application to a multi-thread channel configuration of the 2-D steady linear solution developed decades ago to study river bars and meandering in single thread river settings. The resonant value of the upstream channel aspect ratio, corresponding to the theoretical resonance condition of regular river meanders (Blondeaux & Seminara, J. Fluid Mech., vol. 157, 1985, pp. 449–470) is the key parameter discriminating between symmetrical and asymmetrical bifurcations, in quantitative agreement with experimental observations and numerical simulations, and qualitatively matching field observations. Only when the aspect ratio of the upstream channel of the bifurcation exceeds resonance, is the bifurcation node able to trigger the upstream development of a steady alternate bar pattern, thus creating an unbalanced configuration. Ultimately, the work provides an analytical explanation of the intrinsic legacy between bifurcation asymmetry and the phenomenon of 2-D upstream morphodynamic influence discovered by Zolezzi & Seminara (J. Fluid Mech., vol. 438, 2001, pp. 183–211).
Heated or cooled fluids at supercritical pressure show large variations in thermophysical properties, such as the density, dynamic viscosity and molecular Prandtl number, which strongly influence turbulence characteristics. To investigate this, direct numerical simulations were performed of a turbulent flow at supercritical pressure (CO$_{2}$ at 8 MPa) in an annulus with a hot inner wall and a cold outer wall. The pseudo-critical temperature lies close to the inner wall, which results in strong thermophysical property variations in that region. The turbulent shear stress and the turbulent intensities significantly decrease near the hot inner wall, but increase near the cold outer wall, which can be partially attributed to the mean dynamic viscosity and density stratification. This leads to decreased production of turbulent kinetic energy near the inner wall and vice versa near the outer wall. However, by analysing a transport equation for the coherent streak flank strength, it was found that thermophysical property fluctuations significantly affect streak evolution. Near the hot wall, thermal expansion and buoyancy tend to decrease streak coherence, while the viscosity gradient that exists across the streaks interacts with mean shear to act as either a source or a sink in the evolution equation for the coherent streak flank strength. The formation of streamwise vortices on the other hand is hindered by the torque that is the result of the kinetic energy and density gradients. Near the cold wall, the results are reversed, i.e. the coherent streak flank strength and the streamwise vortices are enhanced due to the variable density and dynamic viscosity. The results show that not only the mean stratification but also the large instantaneous thermophysical property variations that occur in heated or cooled fluids at supercritical pressure have a significant effect on turbulent structures that are responsible for the self-regeneration process in near-wall turbulence. Thus, instantaneous density and dynamic viscosity fluctuations are responsible for decreased (or increased) turbulent motions in heated (or cooled) fluids at supercritical pressure.
This paper concerns the generation of large-scale flows in forced two-dimensional systems. A Kolmogorov flow with a sinusoidal profile in one direction (driven by a body force) is known to become unstable to a large-scale flow in the perpendicular direction at a critical Reynolds number. This can occur in the presence of a ${\it\beta}$-effect and has important implications for flows observed in geophysical and astrophysical systems. It has recently been termed ‘zonostrophic instability’ and studied in a variety of settings, both numerically and analytically. The goal of the present paper is to determine the effect of magnetic field on such instabilities using the quasi-linear approximation, in which the full fluid system is decoupled into a mean flow and waves of one scale. The waves are driven externally by a given random body force and move on a fast time scale, while their stress on the mean flow causes this to evolve on a slow time scale. Spatial scale separation between waves and mean flow is also assumed, to allow analytical progress. The paper first discusses purely hydrodynamic transport of vorticity including zonostrophic instability, the effect of uniform background shear and calculation of equilibrium profiles in which the effective viscosity varies spatially, through the mean flow. After brief consideration of passive scalar transport or equivalently kinematic magnetic field evolution, the paper then proceeds to study the full magnetohydrodynamic system and to determine effective diffusivities and other transport coefficients using a mixture of analytical and numerical methods. This leads to results on the effect of magnetic field, background shear and ${\it\beta}$-effect on zonostrophic instability and magnetically driven instabilities.
We consider the linear stability of Hill’s vortex with respect to axisymmetric perturbations. Given that Hill’s vortex is a solution of a free-boundary problem, this stability analysis is performed by applying methods of shape differentiation to the contour dynamics formulation of the problem in a three-dimensional axisymmetric geometry. This approach allows us to systematically account for the effect of boundary deformations on the linearized evolution of the vortex under the constraint of constant circulation. The resulting singular integro-differential operator defined on the vortex boundary is discretized with a highly accurate spectral approach. This operator has two unstable and two stable eigenvalues complemented by a continuous spectrum of neutrally stable eigenvalues. By considering a family of suitably regularized (smoothed) eigenvalue problems solved with a range of numerical resolutions, we demonstrate that the corresponding eigenfunctions are in fact singular objects in the form of infinitely sharp peaks localized at the front and rear stagnation points. These findings thus refine the results of the classical analysis by Moffatt & Moore (J. Fluid Mech., vol. 87, 1978, pp. 749–760).