The flow-induced oscillation of an S-shaped buckled flexible filament was explored using the penalty immersed boundary method. As the length and bending rigidity of the filament were varied, three distinct modes emerged: the equilibrium mode, streamwise oscillation (SO) mode and transverse oscillation (TO) mode. A transition region between the SO and TO modes was identified. Notably, the filament exhibited a 3P wake pattern under SO and a 2S wake pattern under TO. The former was induced by fluid–elastic instability, while the latter was attributed to vortex-induced oscillation. The interaction between the filament's motion and vortex shedding was examined for both modes. To elucidate the disparity between the TO of the S-shaped buckled filament and snap-through oscillation (STO), a ball-on-a-hill analogy was introduced. The performance of energy harvesting was evaluated using metrics including the elastic energy and power coefficient. The TO mode was found to show significantly higher energy harvesting performance than the SO and STO modes. The majority of the strain energy was concentrated at the upper and lower midpoints of the filament.

We present new constrained and free-swimming experiments and simulations in the inertial regime, with Reynolds number $\mbox{Re} = O(10^4)$, of a pair of two-dimensional and three-dimensional pitching hydrofoils interacting in a minimal school. The hydrofoils have an out-of-phase synchronisation, and they are varied through in-line, staggered and side-by-side formations within the two-dimensional interaction plane. It is discovered that there is a two-dimensionally stable equilibrium point for a side-by-side formation. This formation is super-stable, meaning that hydrodynamic forces will passively maintain this formation even under external perturbations, and the school as a whole has no net forces acting on it that cause it to drift to one side or the other. Previously discovered one-dimensionally stable equilibria driven by wake vortex interactions are shown to be, in fact, two-dimensionally unstable, at least for an out-of-phase synchronisation. Additionally, it is discovered that a trailing-edge vortex mechanism provides the restorative force to stabilise a side-by-side formation. The stable equilibrium is further verified by experiments and simulations for freely swimming foils where dynamic recoil motions are present. When constrained, swimmers in compact side-by-side formations experience collective efficiency and thrust increases up to 40 % and 100 %, respectively, whereas slightly staggered formations output an even higher efficiency improvement of 84 %, with an 87 % increase in thrust. Freely swimming foils in a stable side-by-side formation show efficiency and speed enhancements of up to 9 % and 15 %, respectively. These newfound schooling performance and stability characteristics suggest that fluid-mediated equilibria may play a role in the control strategies of schooling fish and fish-inspired robots.

This study introduces vector autoregression (VAR) as a linear procedure that can be used for synthesizing turbulence time series over an entire plane, allowing them to be imposed as an efficient turbulent inflow condition in simulations requiring stationary and cross-correlated turbulence time series. VAR is a statistical tool for modelling and prediction of multivariate time series through capturing linear correlations between multiple time series. A Fourier-based proper orthogonal decomposition (POD) is performed on the two-dimensional (2-D) velocity slices from a precursor simulation of a turbulent boundary layer at a momentum thickness-based Reynolds number, $Re_{\theta }=790$. A subset of the most energetic structures in space are then extracted, followed by applying a VAR model to their complex time coefficients. It is observed that VAR models constructed using time coefficients of 5 and 30 most energetic POD modes per wavenumber (corresponding to $66\,\%$ and $97\,\%$ of turbulent kinetic energy, respectively) are able to make accurate predictions of the evolution of the velocity field at $Re_{\theta }=790$ for infinite time. Moreover, the 2-D velocity fields from the POD–VAR when used as a turbulent inflow condition, gave a short development distance when compared with other common inflow methods. Since the VAR model can produce an infinite number of velocity planes in time, this enables reaching statistical stationarity without having to run an extremely long precursor simulation or applying ad hoc methods such as periodic time series.

We introduce a novel recursive procedure to a neural-network-based subgrid-scale (NN-based SGS) model for large eddy simulation (LES) of high-Reynolds-number turbulent flow. This process is designed to allow an SGS model to be applicable to a hierarchy of different grid sizes without requiring expensive filtered direct numerical simulation (DNS) data: (1) train an NN-based SGS model with filtered DNS data at a low Reynolds number; (2) apply the trained SGS model to LES at a higher Reynolds number; (3) update this SGS model with training data augmented with filtered LES (fLES) data, accommodating coarser filter size; (4) apply the updated NN to LES at a further higher Reynolds number; (5) go back to Step (3) until a target (very coarse) filter size divided by the Kolmogorov length scale is reached. We also construct an NN-based SGS model using a dual NN architecture whose outputs are the SGS normal stresses for one NN and the SGS shear stresses for the other NN. The input is composed of the velocity gradient tensor and grid size. Furthermore, for the application of an NN-based SGS model trained with one flow to another flow, we modify the NN by eliminating bias and introducing a leaky rectified linear unit function as an activation function. The present recursive SGS model is applied to forced homogeneous isotropic turbulence (FHIT) and successfully predicts FHIT at high Reynolds numbers. The present model trained from FHIT is also applied to decaying homogeneous isotropic turbulence and shows an excellent prediction performance.

Rotating convection is considered on the tilted $f$-plane where gravity and rotation are not aligned. For sufficiently large rotation rates, $\Omega$, the Taylor–Proudman effect results in the gyroscopic alignment of anisotropic columnar structures with the rotation axis giving rise to rapidly varying radial length scales that vanishes as $\Omega ^{-1/3}$ for $\Omega \rightarrow \infty$. Compounding this phenomenon is the existence of viscous (Ekman) layers adjacent to the impenetrable bounding surfaces that scale as $\Omega ^{-1/2}$. In this investigation, these constraints are relaxed upon utilising a non-orthogonal coordinate representation of the fluid equations where the upright coordinate aligns with rotation axis. This exposes the problem to asymptotic perturbation methods that permit: (i) relaxation of the constraints of gyroscopic alignment; (ii) the filtering of Ekman layers through the uncovering of parameterised velocity pumping boundary conditions; and (iii) the development of reduced quasi-geostrophic systems valid in the limit $\Omega \rightarrow \infty$. Linear stability investigations reveal excellent quantitative agreement between results from parameterised or unapproximated mechanical boundary conditions. For no-slip boundaries, it is demonstrated that the associated Ekman pumping alters convective onset through an enhanced destabilisation of large spatial scales. The range of unstable modes at a fixed thermal forcing is thus significantly extended with a direct dependence on $\Omega$. This holds true even for geophysical and astrophysical regimes characterised by extreme values of the non-dimensional Ekman number $E$. The nonlinear regime is explored via the global heat and momentum transport of single-mode solutions to the quasi-geostrophic systems which indicate $O(1)$ changes which do not scale with the size of $E$.

Recognising that surfactants can impede the drag reduction resulting from superhydrophobic surfaces (SHS), we investigate the impact of spatio–temporal fluctuations in surfactant concentration on the drag-reduction properties of SHS. We model the unsteady transport of soluble surfactant in a channel flow bounded by two SHS. The flow is laminar, pressure driven and the SHS are periodic in the streamwise and spanwise directions. We assume that the channel length is much longer than the streamwise period, the streamwise period is much longer than the channel height and spanwise period, and bulk diffusion is sufficiently strong for cross-channel concentration gradients to be small. By combining long-wave and homogenisation theories, we derive an unsteady advection–diffusion equation for surfactant-flux transport over the length of the channel, which is coupled to a quasi-steady advection–diffusion equation for surfactant transport over individual plastrons. As diffusion over the length of the channel is typically small, the surfactant flux is governed by a nonlinear wave equation. In the fundamental case of the transport of a bolus of surfactant, we predict its propagation speed and describe its nonlinear evolution via interaction with the SHS. The propagation speed can fall below the average streamwise velocity as the surfactant adsorbs and rigidifies the plastrons. Smaller concentrations of surfactant are advected faster than larger ones, so that wave-steepening effects can lead to shock formation in the surfactant-flux distribution. Our asymptotic results reveal how unsteady surfactant transport can affect the spatio–temporal evolution of the slip velocity, drag reduction and effective slip length in SHS channels.

We present the results of an experimental study of buoyancy-driven exchange flows in a vertical pipe, where the lower fluid is Newtonian of low viscosity and the upper fluid has a yield stress. The fluids are initially separated by a gate valve, opened at time $\hat {t}=0$. The fluids are miscible, but away from the diffusive limit. For a sufficiently large ratio $Y$, of the yield stress to the buoyancy stress, no sustained fluid motions arise: the flow is stable. For smaller $Y$ numbers an exchange flow results. Commonly, the less dense fluid penetrates upwards in a central finger, displacing the upper fluid downwards around the walls of the pipe. Three regimes are classified: helical finger, disconnected finger and slug flow. The transition between regimes is governed by increasing relevance of inertial to viscous stresses, in balancing buoyancy. The disconnected finger and slug flow regimes are associated with yielded fluid at the interface and early growth of instabilities. Helical fingers are viscous dominated and evolve slowly until late in the experiments. The scenarios studied represent an idealised set-up for the industrial process of plug cementing. The regimes identified are helpful for industrial process design.

The unsteady flow physics of wind-turbine wakes under dynamic forcing conditions are critical to the modelling and control of wind farms for optimal power density. Unsteady forcing in the streamwise direction may be generated by unsteady inflow conditions in the atmospheric boundary layer, dynamic induction control of the turbine or streamwise surge motions of a floating offshore wind turbine due to floating-platform oscillations. This study seeks to identify the dominant flow mechanisms in unsteady wakes forced by a periodic upstream inflow condition. A theoretical framework for the problem is derived, which describes travelling-wave undulations in the wake radius and streamwise velocity. These dynamics encourage the aggregation of tip vortices into large structures that are advected along in the wake. Flow measurements in the wake of a periodically surging turbine were obtained in an optically accessible towing-tank facility, with an average diameter-based Reynolds number of 300 000 and with surge-velocity amplitudes of up to 40 % of the mean inflow velocity. Qualitative agreement between trends in the measurements and model predictions is observed, supporting the validity of the theoretical analyses. The experiments also demonstrate large enhancements in the recovery of the wake relative to the steady-flow case, with wake-length reductions of up to 46.5 % and improvements in the available power at 10 diameters downstream of up to 15.7 %. These results provide fundamental insights into the dynamics of unsteady wakes and serve as additional evidence that unsteady fluid mechanics can be leveraged to increase the power density of wind farms.

We provide scaling relations for the Nusselt number $Nu$ and the friction coefficient $C_{S}$ in sheared Rayleigh–Bénard convection, i.e. in Rayleigh–Bénard flow with Couette- or Poiseuille-type shear forcing, by extending the Grossmann & Lohse (J. Fluid Mech., vol. 407, 2000, pp. 27–56, Phys. Rev. Lett., vol. 86, 2001, pp. 3316–3319, Phys. Rev. E, vol. 66, 2002, 016305, Phys. Fluids, vol. 16, 2004, pp. 4462–4472) theory to sheared thermal convection. The control parameters for these systems are the Rayleigh number $Ra$, the Prandtl number $Pr$ and the Reynolds number $Re_S$ that characterises the strength of the imposed shear. By direct numerical simulations and theoretical considerations, we show that, in turbulent Rayleigh–Bénard convection, the friction coefficients associated with the applied shear and the shear generated by the large-scale convection rolls are both well described by Prandtl's (Ergeb. Aerodyn. Vers. Gött., vol. 4, 1932, pp. 18–29) logarithmic friction law, suggesting some kind of universality between purely shear-driven flows and thermal convection. These scaling relations hold well for $10^6 \leq Ra \leq 10^8$, $0.5 \leq Pr \leq 5.0$, and $0 \leq Re_S \leq 10^4$.

Most of the existing theories on electrophoresis are based on the consideration of a weak applied electric field and ions as point charges, which create a mean electric potential and neglect ion–solvent interactions. These theories cannot demonstrate the dependence of electrophoretic mobility on the applied electric field (nonlinear electrophoresis), reversal in mobility with increasing ion concentration and/or surface charge density or counterion saturation in the electric double layer. In this study we consider a modified electrokinetic model to analyse nonlinear electrophoresis by taking into account the finite ion size effects and ion–ion electrostatic correlations. In this approach, the mean-field-based model is extended to capture the many-body phenomena by considering the non-local electrostatic contribution in the ion free energy functional and the ion–ion hydrodynamic steric interactions are incorporated through the volume exclusion effect in the electrochemical potential. The viscosity of the medium is considered to vary with the local ionic volume fraction. Stronger correlations for multivalent counterions create ion layering, charge density oscillation and mobility reversal. Such phenomena are captured by the present continuum model. The ion crowding attenuates the growth of the electrophoretic mobility with the electric field. At a higher range of the imposed electric field, the ion concentration in the electric double layer enhances, which modifies both the overscreening and ion crowding processes.

This work presents models for the behaviour of both upstream- and downstream-travelling waves in screeching elliptical jets. Proper orthogonal decomposition is performed on experimental velocity data in both the major and minor axis planes, for an aspect ratio $AR=2$ converging elliptical jet operating at nozzle pressure ratios of $2.6$ and $3.4$. From this decomposition, the radial and axial structure of the guided-jet mode (GJM) and the Kelvin–Helmholtz instability are educed. Linear-stability analysis (LSA) is performed using both the experimentally obtained mean flow, and one obtained using Reynolds-averaged Navier–Stokes (RANS) at matched conditions. It is shown that the wavenumber predicted by LSA for both waves are within the range of experimentally observed wavenumbers. Furthermore, the model accurately predicts the structure of these waves at multiple axial locations, using either the experimental or RANS mean flow. Most critically, it is demonstrated that the GJM is only predicted to be neutrally stable at the screech frequency for a relatively limited streamwise domain, the size and location of which is dependent on the nozzle pressure ratio. A comparison with the amplitude envelope for the GJM extracted from the experimental measurements indicates that the maximum fluctuations associated with the GJM are collocated with this region of the flow that is predicted to support the GJM. While there have been extensive discussions about the frequency dependence of the GJM, this is the first demonstration that its existence is highly dependent on streamwise position within the flow.

Colloidal gels have strong industrial relevance as they can behave as liquids or solids. The latter allows them to support a buoyant weight against gravity. However, the system is intrinsically out of equilibrium, which means that the colloids must eventually settle out of the suspension. The process of settling has been captured theoretically, but the presence of a delay time during which the gel appears relatively unaffected by gravity has not. Here, we modify existing frameworks to capture this delay, by treating the gel as a continuum with viscous response that is based on the local bond density. We can solve our model numerically to obtain the evolution of the colloid density profile and recover qualitatively the accumulation of a dense layer on top of the settling gel, as is observed experimentally in depletion gels. This numerical study is complemented by a theoretical analysis that allows us to identify an emergent time and length scale that set the dynamics of the gel. Our model provides a solid foundation for future studies that incorporate hydrodynamic erosion and tackle industrially relevant geometries.

A cylindrical liquid thread readily destabilizes into a series of drops due to capillary instability, which is also responsible for undesirable bead-on-fibre structures observed when coating a thin fibre. In this experimental study, we show how a falling liquid thread can be stabilized by internally distorting the cross-sectional shape using two vertically hung fibres. Below a critical flow rate $Q_c$, the dual-fibre system deforms the falling thread into a smooth column with a non-circular cross-section, thereby suppressing instability. Above $Q_{{c}}$, the cylindrical thread is left undeformed by the fibres and destabilizes into beads connected by a stable, non-cylindrical film. An empirical stability threshold is identified showing that flow delays the onset of capillary instability when compared with a marginally stable quasi-static coating. When the flow is unstable $Q>Q_c$, the bead velocity $v$ obeys a simple scaling law that is well supported by our experiments over a large parameter range. This suppression technique can be extended to other slender geometries, such as a ribbon, which shows similar qualitative results but exhibits a different stability threshold due to spontaneous dewetting about its short edge.

We employ direct numerical simulations to investigate the heat transfer and flow structures in turbulent Rayleigh–Bénard convection in both cylindrical cells and laterally periodic domains, spanning an unprecedentedly wide range of aspect ratios $0.075 \leqslant \varGamma \leqslant 32$. We focus on Prandtl number ${Pr}=1$ and Rayleigh numbers ${{Ra}}=2\times 10^7$ and ${{Ra}}=10^8$. In both cases, with increasing aspect ratio, the heat transfer first increases, then reaches a maximum (which is more pronounced for the cylindrical case due to confinement effects), and then slightly goes down again before it finally saturates at the large aspect ratio limit, which is achieved already at $\varGamma \approx 4$. Already for $\varGamma \gtrsim 0.75$, the heat transfers in both cylindrical and laterally periodic domains become identical. The large-$\varGamma$ limit for the volume-integrated Reynolds number and the boundary layer thicknesses are also reached at $\varGamma \approx 4$. However, while the integral flow properties converge at $\varGamma \approx 4$, the confinement of a cylindrical domain impacts the temperature and velocity variance distributions up to $\varGamma \approx 16$, as thermal superstructures cannot form close to the sidewall.

Energetics of mode-1 internal waves interacting with topographic ridges are investigated using high-resolution two-dimensional simulations at spatial scales of $O(100)$ m that span between classical laboratory-scale ($O(10)$ m) and field-scale simulations ($O(1000\unicode{x2013}10\,000)$ m). This paper focuses on the energetics of wave–topography interaction, with emphasis on systematically examining the partitioning of the incident wave energy as a function of wave forcing and topographic parameters. Partitioning of energy into the transmitted, reflected and dissipated components is quantified as a function of wave Froude number $Fr=U_0/c_{ph}$ ($U_0=$ velocity amplitude of forcing and $c_{ph}=$ internal wave celerity), slope criticality $=\gamma /s$, where $\gamma =$ topographic slope and $s=$ wave characteristic slope, and the ratio of topographic height $h_t$ to water depth $d$. As $Fr$ increases, dense fluid from the base of the stratified water column surges upslope with significant vertical inertia, leading to the formation of internal boluses that plunge over and onto the downstream side of the ridge, resulting in elevated dissipation. Results show that non-hydrostatic contributions to the total energy flux are significant (up to 50 %). Analysis of the energy flux budget shows that transmitted energy flux decreases monotonically as $\gamma /s$ increases for any given $Fr$ and $h_t/d$. At critical slopes ($\gamma /s=1$), the transmitted energy flux scales as a linear function of $h_t/d$, with a mild dependence on $Fr$, a key result that can be useful in energy flux parameterizations. Reflected energy flux exhibits a nonlinear dependence on the ridge height, increasing sharply when $h_t/d > 0.5$. Dissipation is enhanced at critical slopes, with a plateau evident for $\gamma /s \ge 1$ and $h_t/d = 0.5$ for all $Fr$.

Porous membranes are thin solid structures that allow the flow to pass through their tiny openings, called pores. Flow inertia may play a significant role in several filtration flows of natural and engineering interest. Here, we develop a predictive macroscopic model to describe solvent and solute flows past thin membranes for non-negligible inertia. We leverage homogenization theory to link the solvent velocity and solute concentration to the jumps of solvent stress and solute flux across the membrane. Within this framework, the membrane acts as a boundary separating two distinct fluid regions. These jump conditions rely on several coefficients, stemming from closure problems at the microscopic pore scale. Two approximations for the advective terms of Navier–Stokes and advection–diffusion equations are introduced to include inertia in the microscopic problem. The approximate inertial terms couple the micro- and macroscopic fields. Here, this coupling is solved numerically using an iterative fixed-point procedure. We compare the resulting models against full-scale simulations, with a good agreement both in terms of averaged values across the membrane and far-field values. Eventually, we develop a strategy based on unsupervised machine learning to improve the computational efficiency of the iterative procedure. The extension of homogenization towards weak-inertia flow configurations as well as the performed data-driven approximation may find application in preliminary analyses as well as optimization procedures towards the design of filtration systems, where inertia effects can be instrumental in broadening the spectrum of permeability and selectivity properties of these filters.

Experiments and numerical simulations of inertial particles in underexpanded jets are performed. The structure of the jet is controlled by varying the nozzle pressure ratio, while the influence of particles on emerging shocks and rarefaction patterns is controlled by varying the particle size and mass loading. Ultra-high-speed schlieren and Lagrangian particle tracking are used to experimentally determine the two-phase flow quantities. Three-dimensional simulations are performed using a high-order, low-dissipative discretization of the gas phase while particles are tracked individually in a Lagrangian manner. A simple two-way coupling strategy is proposed to handle interphase exchange in the vicinity of shocks. Velocity statistics of each phase are reported for a wide range of pressure ratios, particle sizes and volume fractions. An upstream shift of the Mach disk in the presence of particles reveals significant two-way coupling even at low mass loading. A semi-analytic model that predicts the extent of the Mach disk shift is presented based on a one-dimensional Fanno flow that takes into account volume displacement by particles and interphase exchange due to drag and heat transfer. The per cent shift in Mach disk is found to scale with the mass loading, nozzle pressure ratio and interphase slip velocity and inversely with the particle diameter.

Depth-limited overturning wave shape affects water turbulence and sediment suspension. Experiments have shown that wind affects shoaling and overturning wave shape, with uncertain mechanism. Here, we study wind effects (given by the wind Reynolds number) on solitary wave shoaling and overturning with the two-phase direct numerical simulations model Basilisk run in two dimensions on steep bathymetry for fixed wave Reynolds number and Bond number. For all wind, the propagating solitary wave sheds a two-dimensional turbulent air wake and has nearly uniform speed with minimal wave energy changes over the rapidly varying bathymetry. Wave-face slope is influenced by wind, and shoaling wave shape changes are consistent with previous studies. As overturning jet impacts, wind-dependent differences in overturn shape are quantified. The non-dimensional breakpoint location and overturn area have similar wind dependence as previous experience, whereas the overturn aspect ratio has opposite wind dependence. During shoaling, the surface viscous stresses are negligible relative to pressure. Surface tension effects are also small but grow rapidly near overturning. In a wave frame of reference, surface pressure is low in the lee and contributes 2–5 % to the velocity potential rate of change in the surface dynamic boundary condition, which, integrated over time, changes the wave shape. Reasons why the overturn aspect ratio is different than in experiment and why a stronger simulated wind is required are explored. The dramatic wind effects on overturning jet area, and thus to the available overturn potential energy, make concrete the implications of wind-induced changes to wave shape.

We introduce a continuum framework for the energetics of particle-size segregation in bidisperse granular flows. Building on continuum segregation equations and a recent segregation flux model, the proposed framework offers general analytical expressions to study the physics of granular flows from a mechanical energy perspective. We demonstrate the framework's applicability by examining the energetics of shear-driven granular flows. Numerical experiments with varying frictional coefficients and particle-size ratios reveal two distinct phases in the energetics, marked by the separate onset of particle segregation and diffusive remixing. Furthermore, our numerical simulations alongside previous experimental results show that the bulk Richardson number $Ri$, defined as the potential energy to kinetic energy ratio at steady state, follows the scaling relationship $Ri\equiv \hat {E}^{(s)}_{gp} / \hat {E}^{(s)}_{k} \propto Pe^{-1/2}_{sr}$ for $0.1 \leq Ri\leq ~10^{3}$ and $10^{-4} \leq Pe_{sr} \leq ~300$, the segregation–rheology Péclet number. Finally, we present a Péclet-number-dependent theoretical expression for the degree of mixing (or segregation), validated by the compiled numerical and experimental dataset. Our findings hint that the bulk segregation–mixing state can be predicted and controlled using the segregation Péclet number $Pe$ and $Pe_{sr}$, both determined from known system parameters, providing an instrumental tool for engineering and geophysical applications.

Direct numerical simulation (DNS) is performed to explore turbulent Rayleigh–Bénard convection in spherical shells. Our simulations cover six distinct values of radius ratio, $\eta = r_i/r_o = 0.2$, 0.3, 0.4, 0.5, 0.6 and 0.8, under the assumption of a centrally condensed mass with gravity profile $g \sim 1/r^{2}$; where $r_i$, $r_o$ and $r$ denote the inner shell radius, the outer shell radius and the local radial coordinate, respectively. The Prandtl number is kept constant at unity while the Rayleigh number ($Ra$) is varied from $3 \times 10^{3}$ to $5 \times 10^8$. Our primary aim is to analyze how the radius ratio influences the global transport properties and flow physics. To gain insights into the scaling behaviour of the Nusselt number ($Nu$) and the Reynolds number ($Re$) with respect to $Ra$ and $\eta$, we apply the Grossmann–Lohse (GL) theory (Grossmann & Lohse, J. Fluid Mech., vol. 407, 2000, pp. 27–56) to the system. It is observed that the scaling exponents for $Nu$ and $Re$ in relation to $Ra$ are more significant for smaller $\eta$ values, suggesting that the simulations with smaller $\eta$ reach the classical $Nu\sim Ra^{1/3}$ regime at a relatively lower $Ra$. This observation could also imply the systems with smaller $\eta$ might transition to the ultimate regime earlier at a smaller $Ra$. Based on our extensive DNS data, we establish that the thickness of the inner thermal boundary, $\lambda _{\vartheta }^{i}$, follows a scaling relationship of $\lambda _{\vartheta }^{i} \sim \eta ^{1/2}$. This relationship, in turn, leads to a scaling law for $Nu$ in the form of $Nu \sim f(\eta ) Ra^{\gamma }$, where the function $f(\eta )$ is defined as $f(\eta ) = {\eta ^{1/2}}/{(1+\eta ^{4/3})}$, and the exponent $\gamma$ depends on both $Ra$ and $\eta$. Additionally, we characterize and explain the asymmetry in the velocity field by introducing the separate Reynolds numbers for the inner and outer shells. The asymmetry of the kinetic and thermal energy dissipation rates in the inner and outer boundary layers (BLs) is also quantified.