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We provide scaling relations for the Nusselt number $Nu$ and the friction coefficient $C_{S}$ in sheared Rayleigh-B\'enard convection, i.e., in Rayleigh-B\'enard flow with Couette or Poiseuille type shear forcing, by extending the Grossmann & Lohse (2000,2001,2002,2004) 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-B\'enard 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 (1932) 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 $0 \leq Re_S \leq 10^4$., Comment: 22 pages, 9 figures, 2 tables. Submitted to J. Fluid Mech

Noise generated by wind turbines is significantly impacted by its propagation in the atmosphere. Hence, for annoyance issues an accurate prediction of sound propagation is critical to determine noise levels around wind turbines. This study presents a method to predict wind turbine sound propagation based on linearized Euler equations. We compare this approach to the parabolic equation method, which is widely used since it captures the influence of atmospheric refraction, ground reflection, and sound scattering at a low computational cost. Using the linearized Euler equations is more computationally demanding but can reproduce more physical effects as fewer assumptions are made. An additional benefit of the linearized Euler equations is that they provide a time-domain solution. To compare both approaches, we simulate sound propagation in two distinct scenarios. In the first scenario, a wind turbine is situated on flat terrain; in the second, a turbine is situated on a hilltop. The results show that both methods provide similar noise predictions in the two scenarios. We find that while some differences in the propagation results are observed in the second case, the final predictions for a broadband extended source are similar between the two methods.

In this manuscript, it is analysed the onset and evolution of natural convection of an incompressible fluid between spherical shells. The shells are kept at a fixed temperature difference and aspect ratio, and the Rayleigh-Benard convection is driven by different radial gravity profiles. The analysis has been carried out by using a finite difference scheme to solve the three-dimensional Navier-Stokes equations in spherical coordinates. Numerical results are compared with theoretical predictions from linear and non-linear stability analysis, and differ from the expected critical Rayleigh number Ra_c = 1708 by less than 1 percent. In the range of Prandlt numbers Pr studied, and for all the different gravity profiles analysed, the system presents a dependence on its starting condition and flow history. Even in the region just above the onset of convection, two stable states are observed, with qualitative and quantitative differences, and exploring higher values of Ra introduces new modes and time dependency phenomena in the flow. These results are corroborated by spectral analysis.

We develop innovative analytical expressions for the mean wind and potential temperature flux profiles in convective boundary layers (CBLs). CBLs are frequently observed during daytime as the Earth's surface is warmed by solar radiation. Therefore, their modeling is relevant for weather forecasting, climate modeling, and wind energy applications. For CBLs in the convective-roll dominated regime, the mean velocity and potential temperature in the bulk region of the mixed layer are approximately uniform. We propose an analytical expression for the normalized potential temperature flux profile as a function of height, using a perturbation method approach in which we employ the horizontally homogeneous and quasi-stationary characteristics of the surface and inversion layers. The velocity profile in the mixed layer and the entrainment zone is constructed based on insights obtained from the proposed potential temperature flux profile and the convective logarithmic friction law. Combining this with the well-known Monin-Obukhov similarity theory allows us to capture the velocity profile over the entire boundary layer height. The proposed profiles agree excellently with large-eddy simulation results over the range of $-L/z_0 \in [3.6\times10^2, 0.7 \times 10^5]$, where $L$ is the Obukhov length and $z_0$ is the roughness length., Comment: 12 pages, 6 figures

The heat transport in rotating Rayleigh-B\'enard convection (RBC) can be significantly enhanced for moderate rotation, i.e., for an intermediate range of Rossby numbers $Ro$, compared to the non-rotating case. At Rayleigh numbers $Ra\lesssim5\cdot10^8$, the largest heat transport enhancement (HTE) is achieved when the thicknesses of kinetic and thermal boundary layer are equal. However, experimental and numerical observations show that, at larger $Ra$ ($\gtrsim5\cdot10^8$), the maximal HTE starts to deviate from the expected optimal boundary layer ratio and its amplitude decreases drastically. We present data from direct numerical simulations of rotating RBC in a periodic domain in the range of $10^7\leq Ra\leq10^{10}$ and $0\leq Ro^{-1}\leq40$ for Prandtl number $Pr=4.38$ and $6.4$ to identify the reason for the transition to this large $Ra$ regime of HTE. Our analysis reveals that the transition occurs once the bulk flow at the optimal boundary layer ratio changes to geostrophic turbulence for large $Ra$. In that flow state, the vertically coherent vortices, which support HTE by Ekman pumping at smaller $Ra$, dissolve into vertically decorrelated structures in the bulk, such that the enhancing effect of Ekman pumping and the influence of the boundary layer ratio become small. Additionally, more heat leaks out of the Ekman vortices as the fraction of thermal dissipation in the bulk increases. We find that the rotation induced shearing at the plates helps to increase the thermal dissipation in the bulk, and thus acts as a limiting factor for HTE at large $Ra$: beyond a certain ratio of wall shear stress to vortex strength, the heat transport decreases irrespectively of the boundary layer ratio. This $Pr$ dependent threshold, which roughly corresponds to a bulk accounting for $\approx1/3$ of the total thermal dissipation, sets the maximal HTE and the optimal rotation rate at large $Ra$., Comment: Main article: 21 pages, 10 figures; Supplementary material: 6 pages, 4 figures; Published in Phys. Rev. Fluids

A good understanding of wind turbine noise propagation is relevant to better measure the impact of turbines on the environment. In this study, we developed numerical simulations to study the impact of a 2D hill on the sound propagation for a turbine in non flat terrain. The simulations employed a propagation model derived from the linearized Euler equations, solved in a moving frame following the wavefront. The wind turbine noise was modeled using a point source approach. The flow structure in the atmospheric boundary layer and the wind turbine wake were obtained from large-eddy simulations. We find that the wind turbine wake acts as a waveguide that focuses sound propagation towards the ground. This effect is most pronounced for the flat terrain case and when the turbine is placed on the hill. The sound attenuation is more localized and closer to the turbine when the wind turbine is placed on the hilltop than in the flat case. As the wind speed increases, the sound focusing is observed closer to the turbine. For a turbine placed in front of the hill, the sound propagation is mainly determined by the flow over the hill. This demonstrates that terrain topography can have surprising effects on wind turbine noise propagation., Comment: 14 pages, 17 figures

How do we know how much we know? Quantifying uncertainty associated with our modelling work is the only way we can answer how much we know about any phenomenon. With quantitative science now highly influential in the public sphere and the results from models translating into action, we must support our conclusions with sufficient rigour to produce useful, reproducible results. Incomplete consideration of model-based uncertainties can lead to false conclusions with real world impacts. Despite these potentially damaging consequences, uncertainty consideration is incomplete both within and across scientific fields. We take a unique interdisciplinary approach and conduct a systematic audit of model-related uncertainty quantification from seven scientific fields, spanning the biological, physical, and social sciences. Our results show no single field is achieving complete consideration of model uncertainties, but together we can fill the gaps. We propose opportunities to improve the quantification of uncertainty through use of a source framework for uncertainty consideration, model type specific guidelines, improved presentation, and shared best practice. We also identify shared outstanding challenges (uncertainty in input data, balancing trade-offs, error propagation, and defining how much uncertainty is required). Finally, we make nine concrete recommendations for current practice (following good practice guidelines and an uncertainty checklist, presenting uncertainty numerically, and propagating model-related uncertainty into conclusions), future research priorities (uncertainty in input data, quantifying uncertainty in complex models, and the importance of missing uncertainty in different contexts), and general research standards across the sciences (transparency about study limitations and dedicated uncertainty sections of manuscripts)., Comment: 40 Pages (including supporting information), 3 Figures, 2 Boxes, 1 Table

High-fidelity large-eddy simulations are suitable to obtain insight into the complex flow dynamics in extended wind farms. In order to better understand these flow dynamics, we use dynamic mode decomposition (DMD) to analyze and reconstruct the flow field in large-scale numerically simulated wind farms by large-eddy simulations (LES). Different wind farm layouts are considered, and we find that a combination of horizontal and vertical staggering leads to improved wind farm performance compared to traditional horizontal staggering. We analyze the wind farm flows using the amplitude selection (AP) and sparsity-promoting (SP method) DMD approach. We find that the AP method tends to select modes with a small length scale and a high frequency, while the SP method selects large coherent structures with low frequency. The latter are somewhat reminiscent of modes obtained using proper orthogonal decomposition (POD). We find that a relatively limited number of SP-DMD modes is sufficient to accurately reconstruct the flow field in the entire wind farm, whereas the AP-DMD method requires more modes to achieve an accurate reconstruction. Thus, the SP-DMD method has a smaller performance loss compared to the AP-DMD method in terms of the reconstruction of the flow field.

We evaluate the accuracy of the actuator line model (ALM) approach by performing simulations for the NREL~5~MW wind turbine in uniform inflow using three large eddy simulation codes. The power and thrust coefficients obtained using the three codes agrees within $1\%$ when the grid spacing $\Delta_{\rm grid} \le 5.25$~m, and are cross-validated against blade element momentum (BEM) theory. We find that the results of ALM converge towards BEM theory without the need for tip correction when the numerical resolution is increased. For $\Delta_{\rm grid}=0.98$~m the difference between the power and thrust coefficient obtained using ALM and BEM is $4.5\%$ and $2.1\%$, respectively, although we note that no absolute convergence between ALM and BEM can be obtained as both models use different assumptions, such as the use of a force projection method in the ALM. The difference in the local axial and tangential forces along the blades obtained from ALM simulations using $\Delta_{\rm grid} = 1.97$~m and $\Delta_{\rm grid} = 0.98$~m can be as large as $10\%$. The effect of the number of actuator points on the obtained turbine power and thrust coefficients is limited as the results converge when the spacing between the actuator points is about three times the grid spacing. This insight on the required number of blade points can be used to improve the efficiency of actuator line simulations., Comment: 13 pages, 8 figures

The flow speed-up generated by windbreaks can be used to increase the power production of wind turbines. However, due to the increased drag imposed by the windbreaks, their use in large wind turbine arrays has been questioned. We use large eddy simulations to show that windbreaks can increase the power production of large wind farms. A crucial finding is that windbreaks in a wind farm should be much lower than for a single turbine case. In fact, the optimal windbreak for an isolated turbine can reduce wind farm performance. The optimal windbreak height in a wind farm namely depends on the right balance between flow speed-up over the windbreak and the drag imposed by all windbreaks in the farm. The increased performance is a result of the favorable total pressure flux created by the windbreaks.

We report on the mobility and orientation of finite-size, neutrally buoyant prolate ellipsoids (of aspect ratio $\Lambda=4$) in Taylor-Couette flow, using interface resolved numerical simulations. The setup consists of a particle-laden flow in between a rotating inner and a stationary outer cylinder. We simulate two particle sizes $\ell/d=0.1$ and $\ell/d=0.2$, $\ell$ denoting the particle major axis and $d$ the gap-width between the cylinders. The volume fractions are $0.01\%$ and $0.07\%$, respectively. The particles, which are initially randomly positioned, ultimately display characteristic spatial distributions which can be categorised into four modes. Modes $(i)$ to $(iii)$ are observed in the Taylor vortex flow regime, while mode ($iv$) encompasses both the wavy vortex, and turbulent Taylor vortex flow regimes. Mode $(i)$ corresponds to stable orbits away from the vortex cores. Remarkably, in a narrow $\textit{Ta}$ range, particles get trapped in the Taylor vortex cores (mode ($ii$)). Mode $(iii)$ is the transition when both modes $(i)$ and $(ii)$ are observed. For mode $(iv)$, particles distribute throughout the domain due to flow instabilities. All four modes show characteristic orientational statistics. We find the particle clustering for mode ($ii$) to be size-dependent, with two main observations. Firstly, particle agglomeration at the core is much higher for $\ell/d=0.2$ compared to $\ell/d=0.1$. Secondly, the $\textit{Ta}$ range for which clustering is observed depends on the particle size. For this mode $(ii)$ we observe particles to align strongly with the local cylinder tangent. The most pronounced particle alignment is observed for $\ell/d=0.2$ around $\textit{Ta}=4.2\times10^5$. This observation is found to closely correspond to a minimum of axial vorticity at the Taylor vortex core ($\textit{Ta}=6\times10^5$) and we explain why., Comment: 22 pages, 12 figures

Inspired by the hemispherical asymmetry observed in the Earth's inner core, we perform direct numerical simulations to study the effect of the gravity center offset in spherical Rayleigh-B\'enard convection. We find that even a minimal shift of the gravity center has a pronounced influence on the flow structures. When the gravity center is shifted towards the South, the co-latitudinal buoyancy component creates an energetic jet on the Northern side of the inner sphere that is directed towards the outer sphere. As a result, a large-scale meridional circulation is formed. However, surprisingly, the global heat flux is not affected by the changes in the large-scale flow organization induced by the gravity center offset. Our results suggest that the hemispherical core asymmetry is key to model the flow phenomena in the Earth's outer core and mantle.

Direct numerical simulations are employed to reveal three distinctly different flow regions in rotating spherical Rayleigh-B\'enard convection. In the low-latitude region $\mathrm{I}$ vertical (parallel to the axis of rotation) convective columns are generated between the hot inner and the cold outer sphere. The mid-latitude region $\mathrm{II}$ is dominated by vertically aligned convective columns formed between the Northern and Southern hemispheres of the outer sphere. The diffusion-free scaling, which indicates bulk-dominated convection, originates from this mid-latitude region. In the equator region $\mathrm{III}$ the vortices are affected by the outer spherical boundary and are much shorter than in region $\mathrm{II}$. Thermally driven turbulence with background rotation in spherical Rayleigh-B\'enard convection is found to be characterized by three distinctly different flow regions. The diffusion-free scaling, which indicates the heat transfer is bulk-dominated, originates from the mid-latitude region in which vertically aligned vortices are stretched between the Northern and Southern hemispheres of the outer sphere.\ These results show that the flow physics in rotating convection are qualitatively different in planar and spherical geometries. This finding underlines that it is crucial to study convection in spherical geometries to better understand geophysical and astrophysical flow phenomena., Comment: 11pages, 6figures

Conventionally neutral atmospheric boundary layers (CNBLs), which are characterized with zero surface potential temperature flux and capped by an inversion of potential temperature, are frequently encountered in nature. Therefore, predicting the wind speed profiles of CNBLs is relevant for weather forecasting, climate modeling, and wind energy applications. However, previous attempts to predict the velocity profiles in CNBLs have had limited success due to the complicated interplay between buoyancy, shear, and Coriolis effects. Here, we utilize ideas from the classical Monin-Obukhov similarity theory in combination with a local scaling hypothesis to derive an analytic expression for the stability correction function $\psi = -c_\psi (z/L)^{1/2}$, where $c_\psi = 4.2$ is an empirical constant, $z$ is the height above ground, and $L$ is the local Obukhov length based on potential temperature flux at that height, for CNBLs. An analytic expression for this flux is also derived using dimensional analysis and a perturbation method approach. We find that the derived profile agrees excellently with the velocity profile in the entire boundary layer obtained from high-fidelity large eddy simulations of typical CNBLs., Comment: 11 pages, 6 figures, the article has been accepted by Physical Review Letters, see https://journals.aps.org/prl/accepted/2807bYecI411db78409a3879761405c3a75de2a07

Low-level jets (LLJs) are the wind maxima in the lowest 50 to 1000 m of atmospheric boundary layers. Due to their significant influence on the power production of wind farms it is crucial to understand the interaction between LLJs and wind farms. In the presence of an LLJ, there are positive and negative shear regions in the velocity profile. The positive shear regions of LLJs are continuously turbulent, while the negative shear regions have limited turbulence. We present large-eddy simulations of wind farms in which the LLJ is above, below, or in the middle of the turbine rotor swept area. We find that the wakes recover relatively fast when the LLJ is above the turbines. This is due to the high turbulence below the LLJ and the downward vertical entrainment created by the momentum deficit due to the wind farm power production. This harvests the jet's energy and aids wake recovery. However, when the LLJ is below the turbine rotor swept area, the wake recovery is very slow due to the low atmospheric turbulence above the LLJ. The energy budget analysis reveals that the entrainment fluxes are maximum and minimum when the LLJ is above and in the middle of the turbine rotor swept area, respectively. Surprisingly, we find that the negative shear creates a significant entrainment flux upward when the LLJ is below the turbine rotor swept area. This facilitates energy extraction from the jet, which is beneficial for the performance of downwind turbines.

In turbulent wall sheared thermal convection, there are three different flow regimes, depending on the relative relevance of thermal forcing and wall shear. In this paper we report the results of direct numerical simulations of such sheared Rayleigh-B\'enard convection, at fixed Rayleigh number $Ra=10^6$, varying the wall Reynolds number in the range $0 \leq Re_w \leq 4000$ and Prandtl number $0.22 \leq Pr \leq 4.6$, extending our prior work by Blass et al. (2020), where $Pr$ was kept constant at unity and the thermal forcing ($Ra$) varied. We cover a wide span of bulk Richardson numbers $0.014 \leq Ri \leq 100$ and show that the Prandtl number strongly influences the morphology and dynamics of the flow structures. In particular, at fixed $Ra$ and $Re_w$, a high Prandtl number causes stronger momentum transport from the walls and therefore yields a greater impact of the wall shear on the flow structures, resulting in an increased effect of $Re_w$ on the Nusselt number. Furthermore, we analyse the thermal and kinetic boundary layer thicknesses and relate their behaviour to the resulting flow regimes. For the largest shear rates and $Pr$ numbers, we observe the emergence of a Prandtl-von Karman log-layer, signalling the onset of turbulent dynamics in the boundary layer. Finally, our results allow to extend the Grossmann-Lohse theory for heat transport in Rayleigh-B\'enard convection to the sheared case, universally describing $Nu(Ra,Pr,Re_w)$.

Modeling the effect of complex terrain on high Reynolds number flows is important to improve our understanding of flow dynamics in wind farms and the dispersion of pollen and pollutants in hilly or mountainous terrain as well as the flow in urban areas. Unfortunately, simulating high Reynolds number flows over complex terrain is still a big challenge. Therefore, we present a simplified version of the wall modeled immersed boundary method by Chester et al. (J. Comput. Phys. 2007; 225: 427-448). By preventing the extrapolation and iteration steps in the original method, the proposed approach is much easier to implement and more computationally efficient. Furthermore, the proposed method only requires information that is available to each processor and thus is much more efficient for simulations performed on a large number of cores. These are crucial considerations for algorithms that are deployed on modern supercomputers and will allow much higher grid resolutions to be considered. We validate our method against wind tunnel measurements for turbulent flows over wall-mounted cubes, a two dimensional ridge, and a three-dimensional hill. We find very good agreement between the simulation results and the measurement data, which shows this method is suitable to model high Reynolds number flows over complex terrain., Comment: 20 pages, 11 figures, accepted for publication in Computer and Fluids

We investigate the large-scale circulation (LSC) of turbulent Rayleigh-B\'enard convection in a large box of aspect ratio $\Gamma =32$ for Rayleigh numbers up to $Ra=10^9$ and at a fixed Prandtl number $Pr=1$. A conditional averaging technique allows us to extract statistics of the LSC even though the number and the orientation of the structures vary throughout the domain. We find that various properties of the LSC obtained here, such as the wall-shear stress distribution, the boundary layer thicknesses and the wind Reynolds number, do not differ significantly from results in confined domains ($\Gamma \approx 1$). This is remarkable given that the size of the structures (as measured by the width of a single convection roll) more than doubles at the highest $Ra$ as the confinement is removed. An extrapolation towards the critical shear Reynolds number of $Re_s^{\textrm{crit}} \approx 420$, at which the boundary layer (BL) typically becomes turbulent, predicts that the transition to the ultimate regime is expected at $Ra_{\textrm{crit}} \approx \mathcal{O}(10^{15})$ in unconfined geometries. This result is in line with the G\"ottingen experimental observations. Furthermore, we confirm that the local heat transport close to the wall is highest in the plume impacting region, where the thermal BL is thinnest, and lowest in the plume emitting region, where the thermal BL is thickest. This trend, however, weakens with increasing $Ra$.

We perform a numerical study of the heat transfer and flow structure of Rayleigh-B\'enard (RB) convection in (in most cases regular) porous media, which are comprised of circular, solid obstacles located on a square lattice. This study is focused on the role of porosity $\phi$ in the flow properties during the transition process from the traditional RB convection with $\phi=1$ (so no obstacles included) to Darcy-type porous-media convection with $\phi$ approaching 0. Simulations are carried out in a cell with unity aspect ratio, for the Rayleigh number $Ra$ from $10^5$ to $10^{10}$ and varying porosities $\phi$, at a fixed Prandtl number $Pr=4.3$, and we restrict ourselves to the two dimensional case. For fixed $Ra$, the Nusselt number $Nu$ is found to vary non-monotonously as a function of $\phi$; namely, with decreasing $\phi$, it first increases, before it decreases for $\phi$ approaching 0. The non-monotonous behaviour of $Nu(\phi)$ originates from two competing effects of the porous structure on the heat transfer. On the one hand, the flow coherence is enhanced in the porous media, which is beneficial for the heat transfer. On the other hand, the convection is slowed down by the enhanced resistance due to the porous structure, leading to heat transfer reduction. For fixed $\phi$, depending on $Ra$, two different heat transfer regimes are identified, with different effective power-law behaviours of $Nu$ vs $Ra$, namely, a steep one for low $Ra$ when viscosity dominates, and the standard classical one for large $Ra$. The scaling crossover occurs when the thermal boundary layer thickness and the pore scale are comparable. The influences of the porous structure on the temperature and velocity fluctuations, convective heat flux, and energy dissipation rates are analysed, further demonstrating the competing effects of the porous structure to enhance or reduce the heat transfer.

In this contribution we have briefly introduced the problem of turbulent thermal convection with a particular look at its transition to the ultimate regime and the resolution requirements needed for the direct numerical simulation of this flow., Comment: 10 pages, 6 figures

Rayleigh-B\'enard (RB) convection with free-slip plates and horizontally periodic boundary conditions is investigated using direct numerical simulations. Two configurations are considered, one is two-dimension (2D) RB convection and the other one three-dimension (3D) RB convection with a rotating axis parallel to the plate. We explore the parameter range of Rayleigh numbers Ra from $10^7 to $10^9$ and Prandtl numbers $Pr$ from $1$ to $100$. We show that zonal flow, which was observed, for example, by Goluskin \emph{et al}. \emph{J. Fluid. Mech.} 759, 360-385 (2014) for $\Gamma=2$, is only stable when $\Gamma$ is smaller than a critical value, which depends on $Ra$ and $Pr$. With increasing $\Gamma$, we find a second regime in which both zonal flow and different convection roll states can be statistically stable. For even larger $\Gamma$, in a third regime, only convection roll states are statistically stable and zonal flow is not sustained. For the 3D simulations, we fix $Ra=10^7$ and $Pr=0.71$, and compare the flow for $\Gamma=8$ and $\Gamma = 16$. We demonstrate that with increasing aspect ratio $\Gamma$, zonal flow, which was observed for small $\Gamma=2\pi$ by von Hardenberg \emph{et al}. \emph{Phys. Rev. Lett.} 15, 134501 (2015), completely disappears for $\Gamma=16$. For such large $\Gamma$ only convection roll states are statistically stable. In between, here for medium aspect ratio $\Gamma = 8$, the convection roll state and the zonal flow state are both statistically stable. What state is taken depends on the initial conditions, similarly as we found for the 2D case., Comment: 26 pages, 12 figures

The heat transfer and flow structure in rotating Rayleigh-B\'enard convection are strongly influenced by the Rayleigh ($Ra$), Prandtl ($Pr$), and Rossby ($Ro$) number. For $Pr\gtrsim 1$ and intermediate rotation rates, the heat transfer is increased compared to the non-rotating case. We find that the regime of increased heat transfer is subdivided into a low and a high $Ra$ number regime. For $Ra\lesssim 5\times10^8$ the heat transfer at a given $Ra$ and $Pr$ is highest at an optimal rotation rate, at which the thickness of the viscous and thermal boundary layer is about equal. From the scaling relations of the thermal and viscous boundary layer thicknesses, we derive that the optimal rotation rate scales as $1/Ro_\mathrm{opt} \approx 0.12 Pr^{1/2}Ra^{1/6}$. In the low $Ra$ regime the heat transfer is similar in a periodic domain and cylindrical cells with different aspect ratios, i.e.\ the ratio of diameter to height. This is consistent with the view that the vertically aligned vortices are the dominant flow structure. For $Ra\gtrsim 5\times10^8$ the above scaling for the optimal rotation rate does not hold anymore. It turns out that in the high $Ra$ regime, the flow structures at the optimal rotation rate are very different than for lower $Ra$. Surprisingly, the heat transfer in the high $Ra$ regime differs significantly for a periodic domain and cylindrical cells with different aspect ratios, which originates from the sidewall boundary layer dynamics and the corresponding secondary circulation., Comment: 16 pages, 11 figures, 15 tables

Taylor--Couette (TC) flow is the shear-driven flow between two coaxial independently rotating cylinders. In recent years, high-fidelity simulations and experiments revealed the shape of the streamwise and angular velocity profiles up to very high Reynolds numbers. However, due to curvature effects, so far no theory has been able to correctly describe the turbulent streamwise velocity profile for given radius ratio, as the classical Prandtl--von K\'arm\'an logarithmic law for turbulent boundary layers over a flat surface at most fits in a limited spatial region. Here we address this deficiency by applying the idea of a Monin--Obukhov curvature length to turbulent TC flow. This length separates the flow regions where the production of turbulent kinetic energy is governed by pure shear from that where it acts in combination with the curvature of the streamlines. We demonstrate that for all Reynolds numbers and radius ratios, the mean streamwise and angular velocity profiles collapse according to this separation. We then derive the functional form of the velocity profile. Finally, we match the newly derived angular velocity profile with the constant angular momentum profile at the height of the boundary layer, to obtain the dependence of the torque on the Reynolds number, or, in other words, of the generalized Nusselt number (i.e., the dimensionless angular velocity transport) on the Taylor number.

Low-level jets (LLJs) are the wind maxima in the lower regions of the atmosphere with a high wind energy potential. Here we use large-eddy simulations to study the effect of LLJ height on the flow dynamics in a wind farm with $10\times4$ turbines. We change the LLJ height and atmospheric thermal stratification by varying the surface cooling rate. We find that the first row power production is higher in the presence of a LLJ compared to a neutral reference case without LLJ. Besides, we show that the first row power production increases with decreasing LLJ height. Due to the higher turbulence intensity, the wind turbine wakes recover faster in a neutral boundary layer than in a stably stratified one. However, for strong thermal stratification with a low-height LLJ, the wake recovery can be faster than for the neutral reference case as energy can be entrained from the LLJ. Flow visualizations reveal that under stable stratification the growth of wind farm's internal boundary layer is restricted and the wind flows around the wind farm. Wind farms extract energy from LLJs through wake meandering and turbulent entrainment depending on the LLJ height. Both effects are advantageous for wake recovery, which is beneficial for the performance of downwind turbines. This finding is confirmed by an energy budget analysis, which reveals a significant increase in the kinetic energy flux in the presence of a LLJ. The jet strength reduces as it passes through consecutive turbine rows. For strong stratification, the combined effect of buoyancy destruction and turbulence dissipation is larger than the turbulent entrainment. Therefore, the power production of turbines in the back of the wind farm is relatively low for strong atmospheric stratifications.

Rely to the comment by Doering et al. in Phys. Rev. Lett. 123, 259401 (2019)

We investigate the interplay between large-scale patterns, so-called superstructures, in the fluctuation fields of temperature $\theta$ and vertical velocity $w$ in turbulent Rayleigh-B\'{e}nard convection at large aspect ratios. Earlier studies suggested that velocity superstructures were smaller than their thermal counterparts in the center of the domain. However, a scale-by-scale analysis of the correlation between the two fields employing the linear coherence spectrum reveals that superstructures of the same size exist in both fields, which are almost perfectly correlated. The issue is further clarified by the observation that in contrast to the temperature, and unlike assumed previously, superstructures in the vertical velocity field do not result in a peak in the power spectrum of $w$. The origin of this difference is traced back to the production terms of the $\theta$- and $w$-variance. These results are confirmed for a range of Rayleigh numbers $Ra = 10^5$--$10^9$, the superstructure size is seen to increase monotonically with $Ra$. Furthermore, the scale distribution of particularly the temperature fluctuations is pronouncedly bimodal. In addition to the large-scale peak caused by the superstructures, there exists a strong small-scale peak. This `inner peak' is most intense at a distance of $\delta_\theta$ from the wall and associated with structures of size $\approx 10 \delta_\theta$, where $\delta_\theta$ is the thermal boundary layer thickness. Finally, based on the vertical coherence relative to a reference height of $\delta_\theta$, a self-similar structure is identified in the velocity field (vertical and horizontal components) but not in the temperature., Comment: 17 pages, 10 figures

In RB convection for fluids with Prandtl number $Pr\gtrsim 1$, rotation beyond a critical (small) rotation rate is known to cause a sudden enhancement of heat transfer which can be explained by a change in the character of the BL dynamics near the top and bottom plates of the convection cell. Namely, with increasing rotation rate, the BL signature suddenly changes from Prandtl--Blasius type to Ekman type. The transition from a constant heat transfer to an almost linearly increasing heat transfer with increasing rotation rate is known to be sharp and the critical Rossby number $Ro_{c}$ occurs typically in the range $2.3\lesssim Ro_{c}\lesssim 2.9$ (for Rayleigh number $Ra=1.3\times 10^9$, $Pr=6.7$, and a convection cell with aspect ratio $\Gamma=\frac{D}{H}=1$, with $D$ the diameter and $H$ the height of the cell). The explanation of the sharp transition in the heat transfer points to the change in the dominant flow structure. At $1/Ro\lesssim 1/Ro_c$ (slow rotation), the well-known LSC is found: a single domain-filling convection roll made up of many individual thermal plumes. At $1/Ro\gtrsim 1/Ro_c$ (rapid rotation), the LSC vanishes and is replaced with a collection of swirling plumes that align with the rotation axis. In this paper, by numerically studying Lagrangian acceleration statistics, related to the small-scale properties of the flow structures, we reveal that this transition between these different dominant flow structures happens at a second critical Rossby number, $Ro_{c_2}\approx 2.25$ (different from $Ro_{c_1}\approx 2.7$ for the sharp transition in the Nusselt number $Nu$; both values for the parameter settings of our present numerical study). When statistical data of Lagrangian tracers near the top plate are collected, it is found that the root-mean-square (rms) values and the kurtosis of the horizontal acceleration of these tracers show a sudden increase at $Ro_{c_2}$., Comment: 16 pages, 9 Figures

We perform direct numerical simulations of wall sheared Rayleigh-B\'enard (RB) convection for Rayleigh numbers up to $Ra=10^8$, Prandtl number unity, and wall shear Reynolds numbers up to $Re_w=10000$. Using the Monin-Obukhov length $L_{MO}$ we identify three different flow states, a buoyancy dominated regime ($L_{MO} \lesssim \lambda_{\theta}$; with $\lambda_{\theta}$ the thermal boundary layer thickness), a transitional regime ($0.5H \gtrsim L_{MO} \gtrsim \lambda_{\theta}$; with $H$ the height of the domain), and a shear dominated regime ($L_{MO} \gtrsim 0.5H$). In the buoyancy dominated regime the flow dynamics are similar to that of turbulent thermal convection. The transitional regime is characterized by rolls that are increasingly elongated with increasing shear. The flow in the shear dominated regime consists of very large-scale meandering rolls, similar to the ones found in conventional Couette flow. As a consequence of these different flow regimes, for fixed $Ra$ and with increasing shear, the heat transfer first decreases, due to the breakup of the thermal rolls, and then increases at the beginning of the shear dominated regime. For $L_{MO} \gtrsim 0.5H$ the Nusselt number $Nu$ effectively scales as $Nu \sim Ra^{\alpha}$, with $\alpha \ll 1/3$ while we find $\alpha \simeq 0.31$ in the buoyancy dominated regime. In the transitional regime the effective scaling exponent is $\alpha > 1/3$, but the temperature and velocity profiles in this regime are not logarithmic yet, thus indicating transient dynamics and not the ultimate regime of thermal convection.

Progress in roughness research, mapping any given roughness geometry to its fluid dynamic behaviour, has been hampered by the lack of accurate and direct measurements of skin-friction drag, especially in open systems. The Taylor--Couette (TC) system has the benefit of being a closed system, but its potential for characterizing irregular, realistic, 3-D roughness has not been previously considered in depth. Here, we present direct numerical simulations (DNSs) of TC turbulence with sand grain roughness mounted on the inner cylinder. The model proposed by Scotti (\textit{Phys. Fluids}, vol. 18, 031701, 2006) has been improved to simulate a random rough surface of monodisperse sand grains, which is characterized by the equivalent sand grain height $k_s$. Taylor numbers range from $Ta = 1.0\times 10^7$(corresponding to $Re_\tau = 82$) to $Ta = 1.0\times 10^9$($Re_\tau = 635$). We focus on the influence of the roughness height $k_s^+$ in the transitionally rough regime, through simulations of TC with rough surfaces, ranging from $k_s^+=5$ up to $k_s^+ = 92$, where the superscript `$+$' indicates non-dimensionalization in viscous units. We find that the downwards shift of the logarithmic layer, due to transitionally rough sand grains exhibits remarkably similar behavior to that of the Nikuradse (\textit{VDI-Forschungsheft} 361, 1933) data of sand grain roughness in pipe flow, regardless of the Taylor number dependent constants of the logarithmic layer.

The possible transition to the so-called ultimate regime, wherein both the bulk and the boundary layers are turbulent, has been an outstanding issue in thermal convection, since the seminal work by Kraichnan [Phys. Fluids 5, 1374 (1962)]. Yet, when this transition takes place and how the local flow induces it is not fully understood. Here, by performing two-dimensional simulations of Rayleigh-B\'enard turbulence covering six decades in Rayleigh number Ra up to $10^{14}$ for Prandtl number Pr $=1$, for the first time in numerical simulations we find the transition to the ultimate regime, namely at $\textrm{Ra}^*=10^{13}$. We reveal how the emission of thermal plumes enhances the global heat transport, leading to a steeper increase of the Nusselt number than the classical Malkus scaling $\textrm{Nu} \sim \textrm{Ra}^{1/3}$ [Proc. R. Soc. London A 225, 196 (1954)]. Beyond the transition, the mean velocity profiles are logarithmic throughout, indicating turbulent boundary layers. In contrast, the temperature profiles are only locally logarithmic, namely within the regions where plumes are emitted, and where the local Nusselt number has an effective scaling $\textrm{Nu} \sim \textrm{Ra}^{0.38}$, corresponding to the effective scaling in the ultimate regime., Comment: 6 pages, 4figures

We report the observation of superstructures, i.e.\ very large-scale and long living coherent structures in highly turbulent Rayleigh-B\'enard convection up to Rayleigh $Ra=10^9$. We perform direct numerical simulations in horizontally periodic domains with aspect ratios up to $\Gamma=128$. In the considered $Ra$ number regime the thermal superstructures have a horizontal extend of six to seven times the height of the domain and their size is independent of $Ra$. Many laboratory experiments and numerical simulations have focused on small aspect ratio cells in order to achieve the highest possible $Ra$. However, here we show that for very high $Ra$ integral quantities such as the Nusselt number and volume averaged Reynolds number only converge to the large aspect ratio limit around $\Gamma \approx 4$, while horizontally averaged statistics such as standard deviation and kurtosis converge around $\Gamma \approx 8$, and the integral scale converges around $\Gamma \approx 32$, and the peak position of the temperature variance and turbulent kinetic energy spectra only around $\Gamma \approx 64$., Comment: 7 pages, 4 figures

Computational codes for direct numerical simulations of Rayleigh-B\'enard (RB) convection are compared in terms of computational cost and quality of the solution. As a benchmark case, RB convection at $Ra=10^8$ and $Pr=1$ in a periodic domain, in cubic and cylindrical containers is considered. A dedicated second-order finite-difference code (AFID/RBflow) and a specialized fourth-order finite-volume code (Goldfish) are compared with a general purpose finite-volume approach (OpenFOAM) and a general purpose spectral-element code (Nek5000). Reassuringly, all codes provide predictions of the average heat transfer that converge to the same values. The computational costs, however, are found to differ considerably. The specialized codes AFID/RBflow and Goldfish are found to excel in efficiency, outperforming the general purpose flow solvers Nek5000 and OpenFOAM by an order of magnitude with an error on the Nusselt number $Nu$ below $5\%$. However, we find that $Nu$ alone is not sufficient to assess the quality of the numerical results: in fact, instantaneous snapshots of the temperature field from a near wall region obtained for deliberately under-resolved simulations using Nek5000 clearly indicate inadequate flow resolution even when $Nu$ is converged. Overall, dedicated special purpose codes for RB convection are found to be more efficient than general purpose codes., Comment: 12 pages, 5 figures

An analytical model for the streamwise velocity space-time correlations in turbulent flows is derived and applied to the special case of velocity fluctuations in large wind farms. The model is based on the Kraichnan-Tennekes random sweeping hypothesis, capturing the decorrelation in time while including a mean wind velocity in the streamwise direction. In the resulting model, the streamwise velocity space-time correlation is expressed as a convolution of the pure space correlation with an analytical temporal decorrelation kernel. Hence, the spatio-temporal structure of velocity fluctuations in wind farms can be derived from the spatial correlations only. We then explore the applicability of the model to predict spatio-temporal correlations in turbulent flows in wind farms. Comparisons of the model with data from a large eddy simulation of flow in a large, spatially periodic wind farm are performed, where needed model parameters such as spatial and temporal integral scales and spatial correlations are determined from the large eddy simulation. Good agreement is obtained between the model and large eddy simulation data showing that spatial data may be used to model the full temporal structure of fluctuations in wind farms., Comment: Submitted to Wind Energy

We describe a generalization of the Coupled Wake Boundary Layer (CWBL) model for wind-farms that can be used to evaluate the performance of wind-farms under arbitrary wind inflow directions whereas the original CWBL model (Stevens et al., J. Renewable and Sustainable Energy 7, 023115 (2015)) focused on aligned or staggered wind-farms. The generalized CWBL approach combines an analytical Jensen wake model with a "top-down" boundary layer model coupled through an iterative determination of the wake expansion coefficient and an effective wake coverage area for which the velocity at hub-height obtained using both models converges in the "deep-array" portion (fully developed region) of the wind-farm. The approach accounts for the effect of the wind direction by enforcing the coupling for each wind direction. Here we present detailed comparisons of model predictions with LES results and field measurements for the Horns Rev and Nysted wind-farms operating over a wide range of wind inflow directions. Our results demonstrate that two-way coupling between the Jensen wake model and a "top-down" model enables the generalized CWBL model to predict the "deep-array" performance of a wind-farm better than the Jensen wake model alone. The results also show that the new generalization allows us to study a much larger class of wind-farms than the original CWBL model, which increases the utility of the approach for wind-farm designers., Comment: 17 pages, 11 figures

We present results from large eddy simulations (LES) of extended wind-farms for several turbine configurations with a range of different spanwise and streamwise spacing combinations. The results show that for wind-farms arranged in a staggered configuration with spanwise spacings in the range $\approx[3.5,8]$D, where $D$ is the turbine diameter, the power output in the fully developed regime depends primarily on the geometric mean of the spanwise and streamwise turbine spacings. In contrast, for the aligned configuration the power output in the fully developed regime strongly depends on the streamwise turbine spacing and shows weak dependence on the spanwise spacing. Of interest to the rate of wake recovery, we find that the power output is well correlated with the vertical kinetic energy flux, which is a measure of how much kinetic energy is transferred into the wind-turbine region by the mean flow. A comparison between the aligned and staggered configurations reveals that the vertical kinetic energy flux is more localized along turbine rows for aligned wind-farms than for staggered ones. This additional mixing leads to a relatively fast wake recovery for aligned wind-farms., Comment: 12 pages, 8 figures

Recent large eddy simulations have led to improved parameterizations of the effective roughness height of wind farms. This effective roughness height can be used to predict the wind velocity at hub-height as function of the geometric mean of the spanwise and streamwise turbine spacings and the turbine loading factors. Meyers and Meneveau (Wind Energy 15, 305-317 (2012)) used these parameterizations to make predictions for the optimal wind turbine spacing in infinitely large wind farms. They found that for a realistic cost ratio between the turbines and the used land surface the optimal turbine spacing may be considerably larger than used in conventional wind farms. Here we extend this analysis by taking the length of the wind farm, i.e.\ the number of rows in the downstream direction, into account and show that the optimal turbine spacing strongly depends on the wind farm length. For small to moderately sized wind farms the model predictions are consistent with spacings found in operational wind farms. For much larger wind farms the extended optimal spacing found for infinite wind farms is confirmed., Comment: 13 pages, 9 figures

The AFiD code, an open source solver for the incompressible Navier-Stokes equations ({\color{blue}\burl{http://www.afid.eu}}), has been ported to GPU clusters to tackle large-scale wall-bounded turbulent flow simulations. The GPU porting has been carried out in CUDA Fortran with the extensive use of kernel loop directives (CUF kernels) in order to have a source code as close as possible to the original CPU version; just a few routines have been manually rewritten. A new transpose scheme, which is not limited to the GPU version only and can be generally applied to any CFD code that uses pencil distributed parallelization, has been devised to improve the scaling of the Poisson solver, the main bottleneck of incompressible solvers. The GPU version can reduce the wall clock time by an order of magnitude compared to the CPU version for large meshes. Due to the increased performance and efficient use of memory, the GPU version of AFiD can perform simulations in parameter ranges that are unprecedented in thermally-driven wall-bounded turbulence. To verify the accuracy of the code, turbulent Rayleigh-B\'enard convection and plane Couette flow are simulated and the results are in good agreement with the experimental and computational data that published in previous literatures., Comment: 33 pages, 8 figures, submitted