Describing the evolution of a wind turbine's wake from a top-hat profile near the turbine to a Gaussian profile in the far wake is a central feature of many engineering wake models. Existing approaches, such as super-Gaussian wake models, rely on a set of tuning parameters that are typically obtained from fitting high-fidelity data. In the current study, we present a new engineering wake model that leverages the similarity between the shape of a turbine's wake normal to the streamwise direction and the diffusion of a passive scalar from a disk source. This new wake model provides an analytical expression for a streamwise scaling function that ensures the conservation of linear momentum in the wake region downstream of a turbine. The model also considers the different rates of wake expansion that are known to occur in the near- and far-wake regions. Validation is presented against high-fidelity numerical data and experimental measurements from the literature, confirming a consistent good agreement across a wide range of turbine operating conditions. A comparison is also drawn with several existing engineering wake models, indicating that the diffusion-based model consistently provides more accurate wake predictions. This new unified framework allows for extensions to more complex wake profiles by making adjustments to the diffusion equation. The derivation of the proposed model included the evaluation of analytical solutions to several mathematical integrals that can be useful for other physical applications.

A long-wave asymptotic model is developed for a viscoelastic falling film along the inside of a tube; viscoelasticity is incorporated using an upper convected Maxwell model. The dynamics of the resulting model in the inertialess limit is determined by three parameters: Bond number Bo, Weissenberg number We and a film thickness parameter $a$. The free surface is unstable to long waves due to the Plateau–Rayleigh instability; linear stability analysis of the model equation quantifies the degree to which viscoelasticity increases both the rate and wavenumber of maximum growth of instability. Elasticity also affects the classification of instabilities as absolute or convective, with elasticity promoting absolute instability. Numerical solutions of the nonlinear evolution equation demonstrate that elasticity promotes plug formation by reducing the critical film thickness required for plugs to form. Turning points in travelling wave solution families may be used as a proxy for this critical thickness in the model. By continuation of these turning points, it is demonstrated that in contrast to Newtonian films in the inertialess limit, in which plug formation may be suppressed for a film of any thickness so long as the base flow is strong enough relative to surface tension, elasticity introduces a maximum critical thickness past which plug formation occurs regardless of the base flow strength. Attention is also paid to the trade-off of the competing effects introduced by increasing We (which increases growth rate and promotes plug formation) and increasing Bo (which decreases growth rate and inhibits plug formation) simultaneously.

Ice-crystal icing (ICI) in aircraft engines is a major threat to flight safety. Due to the complex thermodynamic and phase-change conditions involved in ICI, rigorous modelling of the accretion process remains limited. The present study proposes a novel modelling approach based on the physically observed mixed-phase nature of the accretion layers. The mathematical model, which is derived from the enthalpy change after accretion (the enthalpy model), is compared with an existing pure-phase layer model (the three-layer model). Scaling laws and asymptotic solutions are developed for both models. The onset of ice accretion, the icing layer thickness and solid ice fraction within the layer are determined by a set of non-dimensional parameters including the Péclet number, the Stefan number, the Biot number, the melt ratio and the evaporative rate. Thresholds for freezing and non-freezing conditions are developed. The asymptotic solutions present good agreement with numerical solutions at low Péclet numbers. Both the asymptotic and numerical solutions show that, when compared with the three-layer model, the enthalpy model presents a thicker icing layer and a thicker water layer above the substrate due to mixed-phased features and modified Stefan conditions. Modelling in terms of the enthalpy poses significant advantages in the development of numerical methods to complex three-dimensional geometrical and flow configurations. These results improve understanding of the accretion process and provide a novel, rigorous mathematical framework for accurate modelling of ICI.

The spherical Couette system consists of two differentially rotating concentric spheres with the space in between filled with fluid. We study a regime where the outer sphere is rotating rapidly enough so that the Coriolis force is important and the inner sphere is rotating either slower or in the opposite direction with respect to the outer sphere. We numerically study the sudden transition to turbulence at a critical differential rotation seen in experiments at BTU Cottbus-Senftenberg, Germany, and investigate its cause. We find that the source of turbulence is the boundary layer on the inner sphere, which becomes centrifugally unstable. We show that this instability leads to generation of small-scale structures which lead to turbulence in the bulk, dominated by inertial waves, a change in the force balance near the inner boundary, the formation of a mean flow through Reynolds stresses and, consequently, to an efficient angular momentum transport. We compare our findings with axisymmetric simulations and show that there are significant similarities in the nature of the flow in the turbulent regimes of full three-dimensional and axisymmetric simulations but differences in the evolution of the instability that leads to this transition. We find that a heuristic argument based on a Reynolds number defined using the thickness of the boundary layer as a length scale helps explain the scaling law of the variation of critical differential rotation for transition to turbulence with rotation rate observed in the experiments.

The simulation of rarefied gas flow based on the Boltzmann equation is challenging, especially when the gas mixtures have disparate molecular masses. In this paper, a computationally tractable kinetic model is proposed for monatomic gas mixtures, to mimic the Boltzmann collision operator as closely as possible. The intra- and inter-collisions are modelled separately using relaxation approximations, to correctly recover the relaxation time scales that could span several orders of magnitude. The proposed kinetic model preserves the accuracy of the Boltzmann equation in the continuum regime by recovering four critical transport properties of a gas mixture: the shear viscosity, the thermal conductivity, the coefficients of diffusion and the thermal diffusion. While in the rarefied flow regimes, the kinetic model is found to be accurate when comparing its solutions with those from the direct simulation Monte Carlo method in several representative cases (e.g. one-dimensional normal shock wave, Fourier flow and Couette flow, two-dimensional supersonic flow passing a cylinder and nozzle flow into a vacuum), for binary mixtures with a wide range of mass ratios, species concentrations and different intermolecular potentials. Pronounced separations in species properties have been observed, and the flow characteristics of gas mixtures in shock waves are found to change as the molecular mass ratio increases from 10 to 1000.

In this study, direct numerical simulation of the particle dispersion and turbulence modulation in a sonic transverse jet injected into a supersonic cross-flow with a Mach number of 2 was carried out with the Eulerian–Lagrangian point-particle method. One single-phase case and two particle-laden cases with different particle diameters were simulated. The jet and particle trajectories, the dispersion characteristics of particles, and the modulation effect of particles on the flow were investigated systematically. It was found that large particles primarily accumulate around shear layer structures situated on the windward side of the jet trajectory. In contrast, small particles exhibit radial transport, accessing both upstream and downstream recirculation zones. Moreover, small particles disperse extensively within the boundary layer and large-scale shear layers, evidently influenced by the streamwise vortices. The particles increase the mean wall-normal velocity near the wall in the wake region of the transverse jet, while reducing the mean streamwise and wall-normal velocities in outer regions. Particles significantly alter the flow velocity adjacent to shock fronts. In particular, the turbulent fluctuations near the windward barrel shock and bow shock are reduced, while those around the leeward barrel shock are increased. An upward displacement of the bow shock in the wall-normal direction is also observed due to particles. In the regions away from the shocks, small particles tend to amplify the Reynolds stress, while large particles attenuate the turbulent kinetic energy.

This work aims to perform a parametric study on a round supersonic jet with a design Mach number Md = 1.8, which is manipulated using a single steady radial minijet with a view to enhancing its mixing. Four control parameters are examined, i.e. the mass flow rate ratio Cm and diameter ratio d/D of the minijet to main jet, and exit pressure ratio Pe/Pa and fully expanded jet Mach number Mj, where Pe and Pa are the nozzle exit and atmospheric pressures, respectively. Extensive pressure and schlieren flow visualization measurements are conducted on the natural and manipulated jets. The supersonic jet core length Lc/D exhibits a strong dependence on the four control parameters. Careful scaling analysis of experimental data reveals that Lc/D = f1(Cm, d/D, Pe/Pa, Mj) may be reduced to Lc/D = f2(ξ), where f1 and f2 are different functions. The scaling factor $\xi = J({d_i}/{D_j})/(\gamma M_j^2{P_e}/{P_a})$ is physically the penetration depth of the minijet into the main jet, where $J({d_i}/{D_j})$ is the square root of the momentum ratio of the minijet to main jet (di and Dj are the fully expanded diameters of d and D, respectively), γ is the specific heat ratio and $\gamma M_j^2{P_e}/{P_a}$ is the non-dimensional exit pressure ratio. Important physical insight may be gained from this scaling law into the optimal choice of control parameters such as d/D and Pe/Pa for practical applications. It has been found for the first time that the minijet may induce a street of quasi-periodical coherent structures once Cm exceeds a certain level for a given ${P_e}/{P_a}$. Its predominant dimensionless frequency Ste (≡ feDj/Uj) scales with a factor $\zeta = J({d_i}/{D_j})\; \sqrt {\gamma M_j^2{P_e}/{P_a}} $, which is physically the ratio of the minijet momentum thrust to the ambient pressure thrust. The formation mechanism of the street and its role in enhancing jet mixing are also discussed.

We consider the initial ‘slumping phase’ of a lock-release gravity current (GC) on a down slope with focus on particle-driven (turbidity) flows, in the inertia–buoyancy (large Reynolds number) and Boussinesq regime. We use a two-layer shallow-water (SW) model for the depth-averaged variables, and compare the predictions with previously published experimental data. In particular, we analyse the empirical conclusion of Gadal et al. (J. Fluid Mech., vol. 974, 2023, A4) that the slumping displays a constant speed for a significant range of slopes and particle-sedimentation speeds. We emphasize the physical definition of the slumping phase (stage): the adjustment process during which (a) the fluid in the lock is set into motion by the dam break, then (b) forms a tail from the backwall to the nose. We focus on the question of if and when the propagation speed $u_N$ of the nose (front) of the GC is constant during this process (there is consensus that a significant deceleration of $u_N$ appears in the post-slumping stage.) The SW theory predicts correctly the adjustment of the flow field during the slumping stage, but indicates that a constant $u_N$ appears only for the classical case ($\gamma =E=c_D=\beta =0$) where $\gamma, E, c_D, \beta$ are the slope, entrainment and drag coefficients, and the scaled particle settling speed for a particle-driven GC. However, since $\gamma, E, c_D, \beta$ are typically small, the change of $u_N$ during the slumping phase is also small in many cases of interest. The interaction between the various driving and hindering mechanisms is elucidated. We show that, in a system with a horizontal (open) top (typical laboratory experiments), the height of the ambient increases along the slope, and this compensates for buoyancy loss due to particle sedimentation. We point out the need for further experimental and simulation studies for a better understanding of the slumping phase and transition to the next phases, and further assessment/improvement of the SW predictions.

This paper investigates the amplification and propagation of swirl fluctuations in turbulent swirling flows using resolvent analysis. Swirl fluctuations have been repeatedly observed in acoustically excited swirl flows and play a significant role in triggering thermoacoustic instabilities in swirl-stabilized flames. While recent research on simplified rotating laminar base flows suggests that the linear inertial-wave mechanism is a key driver of swirl fluctuations, it remains unclear whether this applies to the fully turbulent regime and whether a linear method is sufficient for modelling. To address this issue, a turbulent swirling pipe flow is considered using large-eddy simulations and phase-locked particle image velocimetry, which are combined with mean-field resolvent analysis. A sound agreement between the empirical and physics-based modes is found in terms of shape and propagation velocity. The latter is particularly important for thermoacoustic time-lag models. The comparison with a generic rotating pipe flow shows that the observed swirl fluctuations are indeed driven by a linear inertial wave mechanism. The resolvent framework is, then, exploited to further investigate the coupling and amplification mechanisms in detail. It is discovered that the combined effects of inertia and strong shear lead to very high amplification rates of the swirl fluctuations, explaining the high potential of these structures to trigger combustion instabilities. The study further demonstrates the capability of the resolvent to reveal the driving mechanisms of flow response structures in highly complex turbulent flows, and it opens the path for efficient physics-based optimization to prevent combustion instabilities.

This paper presents a numerical study on the flow around two tandem circular cylinders beneath a free surface at a Reynolds number of $180$. The free-surface effects on the wake dynamics and hydrodynamic forces are investigated through a parametric study, covering a parameter space of gap ratios from $0.20$ to $2.00$, spacing ratios from $1.50$ to $4.00$ and Froude numbers from $0.2$ to $0.8$. A jet-like flow accompanied by a shear layer of positive vorticity separating from the free surface is formed in the wake at small gap ratios, which significantly alters the wake pattern through its dynamic behaviours. At shallow submergence depths, the three-dimensional wake transitions from mode B to mode A as the distance between the cylinders increases. As submergence depth increases, the wavy deformation of the primary vortex cores disappears in the wake, and the flow transitions to a two-dimensional state. Higher Froude numbers can extend the effect of the free surface to deeper submergence depths. The critical spacing ratio tends to be larger at higher Froude numbers. Furthermore, the free-surface deformation is examined. The free-surface profile typically comprises a hydraulic jump immediately ahead of the upstream cylinder, trapped waves in the vicinity of the two tandem cylinders and well-defined travelling waves on the downstream side. The frequencies of the waves cluster around the vortex shedding frequency, indicating a close association between the generation of waves and the vortex shedding process.

Not all the information in a turbulent field is relevant for understanding particular regions or variables in the flow. Here, we present a method for decomposing a source field into its informative $\boldsymbol {\varPhi }_{I}(\boldsymbol {x},t)$ and residual $\boldsymbol {\varPhi }_{R}(\boldsymbol {x},t)$ components relative to another target field. The method is referred to as informative and non-informative decomposition (IND). All the necessary information for physical understanding, reduced-order modelling and control of the target variable is contained in $\boldsymbol {\varPhi }_{I}(\boldsymbol {x},t)$, whereas $\boldsymbol {\varPhi }_{R}(\boldsymbol {x},t)$ offers no substantial utility in these contexts. The decomposition is formulated as an optimisation problem that seeks to maximise the time-lagged mutual information of the informative component with the target variable while minimising the mutual information with the residual component. The method is applied to extract the informative and residual components of the velocity field in a turbulent channel flow, using the wall shear stress as the target variable. We demonstrate the utility of IND in three scenarios: (i) physical insight into the effect of the velocity fluctuations on the wall shear stress; (ii) prediction of the wall shear stress using velocities far from the wall; and (iii) development of control strategies for drag reduction in a turbulent channel flow using opposition control. In case (i), IND reveals that the informative velocity related to wall shear stress consists of wall-attached high- and low-velocity streaks, collocated with regions of vertical motions and weak spanwise velocity. This informative structure is embedded within a larger-scale streak–roll structure of residual velocity, which bears no information about the wall shear stress. In case (ii), the best-performing model for predicting wall shear stress is a convolutional neural network that uses the informative component of the velocity as input, while the residual velocity component provides no predictive capabilities. Finally, in case (iii), we demonstrate that the informative component of the wall-normal velocity is closely linked to the observability of the target variable and holds the essential information needed to develop successful control strategies.

We analyse the motion of a flagellated bacterium in a two-fluid medium using slender body theory. The two-fluid model is useful for describing a body moving through a complex fluid with a microstructure whose length scale is comparable to the characteristic scale of the body. This is true for bacterial motion in biological fluids (entangled polymer solutions), where the entanglement results in a porous microstructure with typical pore diameters comparable to or larger than the flagellar bundle diameter, but smaller than the diameter of the bacterial head. Thus, the polymer and solvent satisfy different boundary conditions on the flagellar bundle and move with different velocities close to it. This gives rise to a screening length $L_B$ within which the fluids exchange momentum and the relative velocity between the two fluids decays. In this work, both the solvent and polymer of the two-fluid medium are modelled as Newtonian fluids with different viscosities $\mu _s$ and $\mu _p$ (viscosity ratio $\lambda = \mu _p/\mu _s$), thereby capturing the effects solely introduced by the microstructure of the complex fluid. From our calculations, we observe an increased drag anisotropy for a rigid, slender flagellar bundle moving through this two-fluid medium, resulting in an enhanced swimming velocity of the organism. The results are sensitive to the interaction between the bundle and the polymer, and we discuss two physical scenarios corresponding to two types of interaction. Our model provides an explanation for the experimentally observed enhancement of swimming velocity of bacteria in entangled polymer solutions and motivates further experimental investigations.

Metastructures composed of a closely spaced plate array have been widely used in bespoke manipulation of waves in contexts of acoustics, electromagnetics, elasticity and water waves. This paper focuses on wave scattering by discrete plate array metastructures of arbitrary cross-sections, including isolated vertical metacylinders, periodic arrays and horizontal surface-piercing metacylinders. A suitable transform-based method has been applied to each problem to reduce the influence of barriers in a two-dimensional problem to a set of points in a one-dimensional wave equation wherein the solution is constructed using a corresponding Green's function. A key difference from the existing work is the use of an exact description of the plate array rather than an effective medium approximation, enabling the exploration of wave frequencies above resonance where homogenisation models fail but where the most intriguing physical findings are unravelled. The new findings are particularly notable for graded plate array metastructures that produce a dense spectrum of resonant frequencies, leading to broadband ‘rainbow reflection’ effects. This study provides new ideas for the design of structures for the bespoke control of waves with the potential for innovative solutions to coastal protection schemes or wave energy converters.