Viscous gravity currents play a fundamental role in many natural and industrial applications, where practical scenarios often involve the current propagating over rigid curvilinear surfaces. In this study, we employ lubrication theory to develop low-dimensional models for such two-dimensional and axisymmetric propagation, resulting from the release of a finite volume of viscous fluid. A key dimensionless parameter is identified, representing the volume ratio between the released fluid and the curvilinear surface, which governs the current evolution. By simplifying the curvilinear surface with linear–exponential and sinusoidal shapes, we observe distinct flow behaviours. Over linear–exponential surfaces, the current may become trapped, bypass the peak or flow downward, while over sinusoidal surfaces, the propagation is hindered compared with the behaviour over horizontal straight surfaces. The low-dimensional models are validated using the volume of fluid method in computational fluid dynamics, showing consistent predictions of the current evolution over rigid curvilinear surfaces.

The eddy-viscosity model, as initially used to model the mean Reynolds stress, has been widely used in the linear analysis of turbulence recently by direct extension. In this study, the mechanism of the eddy viscosity in improving the prediction of fluctuation structures with linear analysis is clarified by investigating the statistical properties of forcing, eddy-viscosity term and their correlations. From the direct numerical simulation (DNS) results of turbulent channel flows with $Re_{\tau }=186$–$2003$, the spatial correlation of forcing is partially cancelled due to its interaction with eddy-viscosity terms. The stochastic forcing after excluding the eddy-viscosity term is nearly uncorrelated spatially, which better matches the condition where the resolvent modes are consistent with the spectral proper orthogonal decomposition (SPOD) modes from DNS. With the above findings, an optimisation framework is developed by minimising the spatial correlations of the stochastic forcing. The optimised eddy-viscosity profiles nearly overlap with the mean-quantity-based model in the near-wall region, but have different maximum values. Compared with the mean-quantity-based model, the optimised results enhance the consistency between the resolvent and DNS results significantly. Based on the optimised results, a new modelling framework is further abstracted, leaving only one to-be-modelled parameter of the maximum value of the eddy-viscosity profile. This parameter follows distinctive rules with spanwise flow scales, based on which a simplified predictive model is constructed. The resolvent modes predicted by the new model exhibit high consistency with SPOD modes, which are essentially comparable to the optimised results for wide ranges of streamwise and spanwise scales.

Floating fluid-filled membrane breakwater (FFMB) is a temporary structure that can attenuate waves in the deep sea. In this paper the hydrodynamic performance of the FFMB is analysed by using the eigenfunction expansion boundary element method (EEBEM) and physical model experiments. A general motion equation is derived that considers both the dynamic tension and curvature of the membrane. Moreover, an integral expression for the dynamic tension is provided. On this basis, a linear model for solving wave–membrane interaction is established through the EEBEM. Newly designed physical experiments are performed to verify the model and elucidate the nonlinear characteristics of the FFMB. Following verification of the model, this paper investigates the effects of various structural parameters of the FFMB on the wave transmission coefficient, reflection coefficient, horizontal wave force, vertical wave force and dynamic tension. Furthermore, the interrelationship between the structural resonant response and the hydrodynamic performance is elucidated, and the optimal density and filling ratio of the FFMB for engineering applications are proposed. The results demonstrate that the numerical and experimental results are in good agreement, indicating that the model and the motion equation are both practical and highly accurate. By optimizing the structural parameters, the FFMB is capable of effectively attenuating waves within a specific frequency band, while minimizing the wave force.

Understanding the linear growth of disturbances due to external forcing is crucial for flow stability analysis, flow control, and uncertainty quantification. These applications typically require a large number of forward simulations of the forced linearized dynamics, often in a brute-force fashion. When dealing with simple steady-state or periodic base flows, there exist powerful and cost-effective solution operator techniques. Once these solution operators are constructed, they can be used to determine the response to various forcings with negligible computational cost. However, these methods do not apply to problems with arbitrarily time-dependent base flows. This paper develops and investigates reduced-order modelling with time-dependent bases to build low-rank solution operators for forced linearized dynamics with arbitrarily time-dependent base flows. In particular, we use forced optimally time-dependent decomposition (f-OTD), which extracts the time-dependent correlated structures of the flow response to various excitations. Several demonstrations are included to illustrate the utility of the f-OTD low-rank approximation for performing global transient stability analysis. Additionally, we demonstrate the application of f-OTD in computing the post-transient response of linearized Navier–Stokes equations to a large number of impulses, which has applications in flow control.

The motion of small non-spherical particles is often studied using the unsteady Stokes equations. Zhang & Stone (J. Fluid Mech., vol. 367, 1998, pp. 329–358) reported an asymptotic treatment for nearly spherical particles, to first order in particle non-sphericity, i.e. $O(\epsilon )$, where $\epsilon$ quantifies the shape deviation from a sphere. Importantly, key physical phenomena are absent at $O(\epsilon )$, including (1) coupling between the torque experienced by the particle and its linear translation, (2) coupling between the force the particle experiences and its rotation and (3) the effect of non-sphericity on the orientation averages of these forces and torques. We present an explicit asymptotic theory to second order in particle non-sphericity, i.e. $O(\epsilon ^2)$, for the force and torque acting on a particle in a general unsteady Stokes flow. The derived analytical formulae apply to particles of arbitrary shape, providing the leading-order asymptotic theory for the three above-mentioned phenomena. The theory is demonstrated for several example nearly spherical particles including a spheroid, a ‘pear-shaped’ particle and a simple model for a SARS-CoV-2 virion. This includes formulae for force and torque as a function of particle orientation and their corresponding orientation averages. Our study reveals that the orientation-averaged forces and torques experienced by a nearly spherical particle cannot be generally represented by a perfect sphere. The reported formulae are validated using finite-amplitude three-dimensional direct numerical simulations of the Navier–Stokes equations. A Mathematica notebook is also provided, facilitating implementation of the theory for particle shapes of the user's choosing.

We investigate the effect of three-dimensionality on the synchronisation characteristics of the wake behind an oscillating circular cylinder at ${\textit {Re}} = 300$. Cylinder oscillations in rotation, transverse translation and streamwise translation are considered. We utilise phase-reduction analysis, which quantifies the phase-sensitivity function of periodic flows, to examine the synchronisation properties. Here, we present an ensemble-based framework for phase-reduction analysis to handle three-dimensional wakes that are not perfectly time-periodic. Based on the phase-sensitivity functions, synchronisability to three types of cylinder oscillations is evaluated. In spite of similar trends, we find that phase-sensitivity functions involving three-dimensional wakes are lower in magnitude compared with those of two-dimensional wakes, which leads to narrower conditions for synchronisation to weak cylinder oscillations. We unveil that the difference between the phase-sensitivity functions of two- and three-dimensional flows is strongly correlated to the amplitude variation of the three-dimensional flow by the cylinder motions. This finding reveals that the cylinder motion modifies the three-dimensionality of the wake as well as the phase of vortex shedding, which leads to reduced phase modulation. The synchronisation conditions of three-dimensional wakes, predicted by phase-reduction analysis, agree with the identification by parametric studies using direct numerical simulations for forced oscillations with small amplitudes. This study presents the potential capability of phase-reduction to study synchronisation characteristics of complex flows.

When a less-viscous solution of a reactant $A$ displaces a more-viscous solution of another reactant $B$, a fast bimolecular $A + B \rightarrow C$ reaction decreasing locally the viscosity can influence the viscous fingering (VF) instability taking place between the two miscible solutions. We show both experimentally and numerically that, for monotonic viscosity profiles, this decrease in viscosity has opposite effects on the fingering pattern depending on the injection flow rate. For high flow rates, the reaction enhances the shielding effect, creating VF patterns with a lower surface density, i.e. thinner fingers covering a smaller area. In contrast, for lower flow rates, the reaction stabilises the VF dynamics, i.e. delays the instability and gives a less-deformed displacement, reaching in some cases an almost-stable displacement. Nonlinear simulations of reactive VF show that these opposite effects at low or high flow rates can only be reproduced if the diffusivity of $A$ is larger than that of $B$, which favours a larger production of the product $C$ and, hence, a larger viscosity decrease. The analysis of one-dimensional viscosity profiles reconstructed on the basis of a one-dimensional reaction–diffusion–advection model confirms that the VF stabilisation at low Péclet number and in the presence of differential diffusion of reactants originates from an optimum reaction-driven decrease in the gradient of the monotonic viscosity profile.

We study single bubble deformation statistics in an homogeneous and isotropic turbulent flow by means of direct numerical simulations. We consider bubbles at low Weber number ($We <3$) that have not been broken. We show that we can reproduce bubble deformations with a linear dynamics for each spherical harmonic mode. Inferring the coefficients of the linear model from the DNS data, we find that the natural frequency corresponds to the Rayleigh frequency, derived in a quiescent flow. However, the effective damping increases by a factor 7 compared with the quiescent case, at Taylor Reynolds number $\textit {Re}_\lambda = 55$. Looking at the flow structure around the bubble, we argue that the enhanced damping originates from a thick boundary layer surrounding the bubble. We demonstrate that the effective forcing, originating from the turbulent flow forcing on the bubble surface, is independent of bubble deformability. Therefore, the interface deformations are only one-way coupled to the flow. From this model we conclude that bubbles break rather from turbulent fluctuations than from a resonant mechanism. Eventually, we investigate the pressure modes’ statistics in the absence of bubbles and compare them with the effective forcing statistics. We show that both fields share the same probability distribution function, characterized by exponential tails, and a characteristic time scale corresponding to the eddy turnover time at the mode scale.

We establish a theoretical framework for predicting friction and heat transfer coefficients in variable-property forced air convection. Drawing from concepts in high-speed wall turbulence, which also involves significant temperature, viscosity and density variations, we utilize the mean momentum balance and mean thermal balance equations to develop integral transformations that account for the impact of variable fluid properties. These transformations are then applied inversely to predict the friction and heat transfer coefficients, leveraging the universality of passive scalars transport theory. Our proposed approach is validated using a comprehensive dataset from direct numerical simulations (DNS), covering both heating and cooling conditions up to a friction Reynolds number $\textit {Re}_\tau \approx 3200$. The predicted friction and heat transfer coefficients closely match the DNS data with accuracy margin 1–2 %, representing a significant improvement over the current state of the art.

In this study we consider a freely decaying, stably stratified homogeneous magnetohydrodynamic turbulent plasma with a weak vertical background magnetic field ($\boldsymbol {B}_0=B_0\hat {\boldsymbol {z}}),$ aligned with the density gradient of strength $N$ (i.e. Brunt–Väisälä frequency). Both linear theory and direct numerical simulations (DNS) are used to analyse the flow dynamics for a Boussinesq fluid with unitary magnetic and thermal Prandtl numbers. We implemented a normal mode decomposition emphasizing different types of motions depending on whether both the Froude $F_r$ and Alfvén–Mach $M$ numbers are small or only $F_r$ is small but $M$ is finite. In the former case, there is a non-propagating (NP) mode and fast modes: Alfvén waves with frequency $\omega _a$ and magnetogravity waves with frequency $\omega _{ag}$. In the latter case, there are fast gravity waves with frequency $\omega _g$ and slow modes: NP mode and slow Alfvén waves. The numerical simulations carried out are started from initial isotropic conditions with zero initial magnetic and density fluctuations, so that the initial energy of the NP mode is strictly zero, for $0< B_0/(L_iN)\leqslant 0.12$ and a weak mean magnetic field ($B_0=0.2$ or $B_0=0.4),$ where $L_i$ denotes the isotropic integral length scale. The DNS results indicate a weak turbulence regime for which $F_r$ is small and $M$ is finite. It is found that the vertical magnetic energy as well as the energy of the NP mode are drastically reduced as $N$ increases, while there is instead a forward cascade even for the magnetic field. The contribution coming from the energy of fast (gravity) waves does not exceed $50\,\%,$ while that coming from the energy of the NP mode does not exceed $10\,\%.$ Vertical motions are more affected by the effect of stratification than by the effect of the mean magnetic field, while it is the opposite for horizontal motions. We show that the spectrum of slow (Alfvén) waves and fast (gravity) waves tends to follow the power law $k_\perp ^{-3}$ for a wide range of time, $3< t<20$. At high vertical (or horizontal) wavenumbers, the main contribution to total energy comes from the energy of slow Alfvén waves. At large and intermediate horizontal (or vertical) scales, the spectra of the energy of NP mode exhibit a flat shape.

Pore-resolved direct numerical simulations have been performed to investigate the turbulent open-channel flow over a rough and permeable sediment bed, represented by a mono-disperse random sphere pack. After a careful validation, eleven cases were simulated to systemically sample a parameter space spanned by a friction Reynolds number $Re_\tau \in [150, 500]$ and a permeability Reynolds number $Re_K \in [0, 2.8]$. By varying the ratio of flow depth to sphere diameter within a range of $h/D \in \{ 3,5,10,\infty \}$, the influence of both Reynolds numbers on the flow field and the turbulence structure could be investigated independently. The simulation results are analysed within a time–space double-averaging framework, whereas flow visualizations provide insight into instantaneous fields. Based on the drag distribution, we propose a consistent interface description, which can be used to define both near-interface and outer-flow coordinates. In these near-interface coordinates, the profiles of the mean velocity and the total shear stress collapse. Furthermore, the proposed interface definition yields outer-layer coordinates, in which the flow and turbulence statistics over a rough and permeable bed reveal similarity to a smooth-wall flow at a similar $Re_\tau$. Within the parameter space, $Re_\tau$ has a strong influence on the wake region of the velocity profile. In contrast, $Re_K$ changes the wall-blocking effect and the shear intensity, which is reflected by the turbulence structure and vortex orientation in the near-interface region. As streamwise velocity streaks disappear and the vortex inclination increases with higher $Re_K$, differences between near-interface and outer-layer turbulence structure are reduced.

Settling velocity statistics for dilute, non-Brownian homogeneous suspensions of polydisperse spheres having a log-normal size distribution are generated from Stokesian dynamics simulations, as a function of the total volume fraction $\phi$ and normalised width $\alpha$ of the particle size distribution. Several hundred instantaneous configurations are averaged to obtain reliable statistics. The paper reports data for the average and fluctuating settling velocity of each particle class in a suspension that is widely polydisperse – previous work was limited to only two or three classes, and the average settling velocity of each particle class was in most cases not reported – and provides an assessment of the accuracy of the analytical models proposed by Batchelor, Richardson & Zaki, Davis & Gecol and Masliyah–Lockett–Bassoon in predicting the simulation data. A limited comparison with dynamic simulations in which the particle microstructure is allowed to evolve in time is also included.

In this work, the shape of a bluff body is optimized to mitigate velocity fluctuations of turbulent wake flows based on large-eddy simulations (LES). The Reynolds-averaged Navier–Stokes method fails to capture velocity fluctuations, while direct numerical simulations are computationally prohibitive. This necessitates using the LES method for shape optimization given its scale-resolving capability and relatively affordable computational cost. However, using LES for optimization faces challenges in sensitivity estimation as the chaotic nature of turbulent flows can lead to the blowup of the conventional adjoint-based gradient. Here, we propose using the regularized ensemble Kalman method for the LES-based optimization. The method is a statistical optimization approach that uses the sample covariance between geometric parameters and LES predictions to estimate the model gradient, circumventing the blowup issue of the adjoint method for chaotic systems. Moreover, the method allows for the imposition of smoothness constraints with one additional regularization step. The ensemble-based gradient is first evaluated for the Lorenz system, demonstrating its accuracy in the gradient calculation of the chaotic problem. Further, with the proposed method, the cylinder is optimized to be an asymmetric oval, which significantly reduces turbulent kinetic energy and meander amplitudes in the wake flows. The spectral analysis methods are used to characterize the flow field around the optimized shape, identifying large-scale flow structures responsible for the reduction in velocity fluctuations. Furthermore, it is found that the velocity difference in the shear layer is decreased with the shape change, which alleviates the Kelvin–Helmholtz instability and the wake meandering.

Modern machine-learning techniques are generally considered data-hungry. However, this may not be the case for turbulence as each of its snapshots can hold more information than a single data file in general machine-learning settings. This study asks the question of whether nonlinear machine-learning techniques can effectively extract physical insights even from as little as a single snapshot of turbulent flow. As an example, we consider machine-learning-based super-resolution analysis that reconstructs a high-resolution field from low-resolution data for two examples of two-dimensional isotropic turbulence and three-dimensional turbulent channel flow. First, we reveal that a carefully designed machine-learning model trained with flow tiles sampled from only a single snapshot can reconstruct vortical structures across a range of Reynolds numbers for two-dimensional decaying turbulence. Successful flow reconstruction indicates that nonlinear machine-learning techniques can leverage scale-invariance properties to learn turbulent flows. We also show that training data of turbulent flows can be cleverly collected from a single snapshot by considering characteristics of rotation and shear tensors. Second, we perform the single-snapshot super-resolution analysis for turbulent channel flow, showing that it is possible to extract physical insights from a single flow snapshot even with inhomogeneity. The present findings suggest that embedding prior knowledge in designing a model and collecting data is important for a range of data-driven analyses for turbulent flows. More broadly, this work hopes to stop machine-learning practitioners from being wasteful with turbulent flow data.

This study conducts experimental investigations into wake-induced vibration (WIV) of a circular cylinder placed downstream of an oscillating cylinder. Surprisingly, it is observed that the previously identified WIV phenomenon, characterized by a sustained increase in amplitude at higher reduced velocities, does not occur when the upstream cylinder oscillates at large amplitudes. Instead, a different phenomenon, which we refer to as the ‘wake-captured vibration’, becomes dominant. The experiments reveal a negative correlation between the vortex-induced vibration amplitude response of the upstream cylinder and the WIV amplitude response of the downstream cylinder. Through a quasi-steady and linear instability analysis, the study demonstrates that the previously proposed wake-displacement mechanism may not be applicable for predicting the cylinder WIV response in the wake of an oscillating body. This is because the lift force gradients across the wake, measured through stationary cylinder experiments, decrease significantly when the upstream cylinder vibrates at higher amplitudes. Consequently, actively controlled vibration experiments are conducted to systematically map the hydrodynamic properties of the downstream cylinder vibrating in the wake of an oscillating cylinder. The findings align with observations from free-vibration experiments, and help to explain the amplitude and frequency response of WIV. Additionally, wake visualization through particle image velocimetry is conducted to provide further insights into the complex wake and vortex–body interactions.

We experimentally investigate the effect of Reynolds number ($Re$) on the turbulence induced by the motion of bubbles in a quiescent Newtonian fluid at small $Re$. The energy spectra, $E(k)$, are determined from the decaying turbulence behind the bubble swarm obtained using particle image velocimetry. We show that when $Re \sim O(100)$, the slope of the normalized energy spectra is no longer independent of the gas volume fraction and the $k^{-3}$ subrange is significantly narrower, where $k$ is the wavenumber. This is further corroborated using second-order longitudinal velocity structure function and spatial correlation of the velocity behind the bubble swarm. On further decreasing the bubble Reynolds number ($O(1) < Re < O(10)$), the signature $k^{-3}$ of the energy spectra for the bubble-induced turbulence is replaced by $k^{-5/3}$ scaling. Thus, we provide experimental evidence to the claim by Mazzitelli et al. (Phys. Fluids, vol. 15, 2003, pp. L5–L8) that at low Reynolds numbers the normalized energy spectra of the bubble-induced turbulence will no longer show the $k^{-3}$ scaling because of the absence of bubble wake and that the energy spectra will depend on the number of bubbles, thus being non-universal.

This experimental study employs Bayesian optimisation to maximise the cross-flow (transverse) flow-induced vibration (FIV) of an elastically mounted thin elliptical cylinder by implementing axial (or angular) flapping motions. The flapping amplitude was in proportion to the vibration amplitude, with a relative phase angle imposed between the angular and transverse displacements of the cylinder. The control parameter space spanned over the ranges of proportional gain and phase difference of $0 \leq K_p^* \leq 5$ and $0 \leq \phi _d \leq 360^\circ$, respectively, over a reduced velocity range of $3.0 \leqslant {U^*} = U/({{f_{nw}}} b) \leqslant 8.5$. The corresponding Reynolds number range was $1250 \leqslant {{Re}} =(U b)/\nu \leqslant 3580$. Here, $U$ is the free stream velocity, $b$ is the major cross-sectional diameter of the cylinder, ${{f_{nw}}}$ is the natural frequency of the system in quiescent fluid (water) and $\nu$ is the kinematic viscosity of the fluid. It was found that the controlled body rotation extended the wake-body synchronisation across the entire ${U^*}$ range tested, with a larger amplitude response than the non-rotating case for all flow speeds. Interestingly, two new wake-body synchronisation regimes were identified, which have not been reported in previous studies. As this geometry acts as a ‘hard-oscillator’ for ${U^*} \geqslant 6.3$, an adaptive gain (i.e. one that varies as a function of oscillation amplitude) was also implemented, allowing the body vibration, achieved for a non-rotating cylinder using increasing ${U^*}$ increments, to be excited from rest. The findings of the present study hold potential implications for the use of FIV as a means to efficiently extract energy from free-flowing water sources, a topic of increasing interest over the last decade.

The reactive Navier–Stokes equations with adaptive mesh refinement and a detailed chemical reactive mechanism (11 species, 27 steps) were adopted to investigate a detonation engine considering the injection and supersonic mixing processes. Flame acceleration and deflagration-to-detonation transition (DDT) in a premixed/inhomogeneous supersonic hydrogen–air mixture with and without transverse jet obstacles were addressed. Results demonstrate the difficulty in undergoing DDT in the premixed/inhomogeneous supersonic mixture within a smooth chamber. By contrast, multiple transverse jets injected into the chamber aid detonation transition by introducing perturbed vortices, shock waves and a suitable blockage ratio. Increasing distance between the leading shock and the flame tip impedes detonation transition due to an insufficient blockage ratio. The extremely perturbed distributions of fuel-lean and fuel-rich mixtures lead to more complicated flame structures. Also, a larger flame thickness appears in the inhomogeneous mixture compared with the premixed mixture, resulting in a lower combustion temperature. The key findings are that the DDT, detonation quenching and reinitiation are generated in the inhomogeneous supersonic mixture, but both DDT mechanisms are ascribed to a strong Mach stem with the Zel'dovich gradient mechanism. Additionally, the obtained results demonstrate that an intensely fuel-lean mixture (equivalence ratio = 0.15) results in a partially decoupled flame front. However, detonation reinitiation and subsequent self-sustained detonation occur when a fierce shock wave propagates through a highly sensitive mixture, even within a smaller and elongated area. Moreover, the inhomogeneous mixture also augments the propagation speed and detonation cell structure instabilities and delays the sonic point resulting from the extending non-equilibrium reaction.

In the astrophysics community it is common practice to model collisionless dust, entrained in a gas flow, as a pressureless fluid. However, a pressureless fluid is fundamentally different from a collisionless fluid – the latter of which generically possess a non-zero anisotropic pressure or stress tensor. In this paper we derive a fluid model for collisionless dust, entrained in a turbulent gas, starting from the equations describing the motion of individual dust grains. We adopt a covariant formulation of our model to allow for the geometry and coordinate systems prevalent in astrophysics, and provide a closure valid for the accretion disc context. We show that the continuum mechanics properties of a dust fluid corresponds to a higher-dimensional anisotropic Maxwell fluid, after the extra dimensions are averaged out, with a dynamically important rheological stress tensor. This higher-dimensional treatment has the advantage of keeping the dust velocity and velocity of the fluid seen, and their respective moments, on the same footing. This results in a simplification of the constitutive relation describing the evolution of the dust rheological stress.

Prandtl's secondary flows of the second kind generated by laterally varying roughness are studied using the linearised Reynolds-averaged Navier–Stokes approach proposed by Zampino et al. (J. Fluid Mech., vol. 944, 2022, p. A4). The momentum equations are coupled to the Spalart–Allmaras model while the roughness is captured by adapting established strategies for homogeneous roughness to heterogeneous surfaces. Linearisation of the governing equations yields a framework that enables a rapid exploration of the parameter space associated with heterogeneous surfaces, in the limiting case of small spanwise variations of the roughness properties. Channel flow is considered, with longitudinal high- and low-roughness strips arranged symmetrically. By varying the strip width, it is found that linear mechanisms play a dominant role in determining the size and intensity of secondary flows. In this setting, secondary flows may be interpreted as the time-averaged output response of the turbulent mean flow subjected to a steady forcing produced by the wall heterogeneity. In fact, the linear model predicts that secondary flows are most intense when the strip width is about 0.7 times the half-channel height, in excellent agreement with available data. Furthermore, a unified framework to analyse combinations of heterogeneous roughness properties and laterally varying topographies, common in applications, is discussed. Noting that the framework assumes small spanwise variations of the surface properties, two separate secondary-flow-inducing source mechanisms are identified, i.e. the lateral variation of the virtual origin from which the turbulent structure develops and the lateral variation of the streamwise velocity slip, capturing the acceleration/deceleration perceived by the bulk flow over troughs and crests of non-planar topographies.