We investigate the dynamics of two superposed layers with density contrast flowing countercurrent inside a channel, when the lower layer is much thinner than the wavelength of interfacial waves. We apply a low-dimensional film model to the bottom (heavier) layer and introduce a fast and efficient method to predict the onset of flow reversal in this phase. We study three vertical scenarios with different applied pressure gradients and compare the temporal growth rates of linear and weakly nonlinear waves to the Orr-Sommerfeld problem and to the weakly nonlinear theory, respectively. At the loading point, i.e., when a large wave hump stands at the interface, our spatiotemporal analysis shows that the system is absolutely unstable. We then present profiles of nonlinear saturated waves, pressure field, and streamline distribution in agreement with direct numerical simulation. The reduced model presented here allows us to explore the effect of the upper-layer speed on the wave pattern, showing that the wave profile is very sensitive when the mean film thickness, rather than the liquid flow rate, is maintained constant in the simulation. In addition, we show the strong effect of surface tension on both the maximum wave hump and the crest steepness before the loading point. Finally, we reveal how the nonlinear wave speed affects the vortex distribution within the lower layer by analyzing the stream function under different scenarios.

Proper orthogonal decomposition (POD) is applied to distinct data sets in order to characterize the propagation of error arising from basis truncation in the description of turbulence. Experimental data from stereo particle image velocimetry measurements in a wind turbine array and direct numerical simulation data from a fully developed channel flow are used to illustrate dependence of the anisotropy tensor invariants as a function of POD modes used in low-order descriptions. In all cases, ensembles of snapshots illuminate a variety of anisotropic states of turbulence. In the near wake of a model wind turbine, the turbulence field reflects the periodic interaction between the incoming flow and rotor blade. The far wake of the wind turbine is more homogenous, confirmed by the increased magnitude of the anisotropy factor. By contrast, the channel flow exhibits many anisotropic states of turbulence. In the inner layer of the wall-bounded region, one observes one-component turbulence at the wall; immediately above, the turbulence is dominated by two components, with the outer layer showing fully three-dimensional turbulence, conforming to theory for wall-bounded turbulence. The complexity of flow descriptions resulting from truncated POD bases can be greatly mitigated by severe basis truncations. However, the current work demonstrates that such simplification necessarily exaggerates the anisotropy of the modeled flow and, in extreme cases, can lead to the loss of three-dimensionality. Application of simple corrections to the low-order descriptions of the Reynolds stress tensor significantly reduces the residual root-mean-square error. Similar error reduction is seen in the anisotropy tensor invariants. Corrections of this form reintroduce three-dimensionality to severe truncations of POD bases. A threshold for truncating the POD basis based on the equivalent anisotropy factor for each measurement set required many more modes than a threshold based on energy. The mode requirement to reach the anisotropy threshold after correction is reduced by a full order of magnitude for all example data sets, ensuring that economical low-dimensional models account for the isotropic quality of the turbulence field.

We experimentally investigate the propulsive characteristics of a pitching NACA 0012 airfoil section, with emphasis on thrust and propulsive efficiency, at a Reynolds number of $1.7\times {10}^4$. For the sake of mechanical simplicity, we consider an airfoil restricted to a single actuator in the pitching direction. We examine the effect of changing the airfoil's axis of rotation, finding that contrary to Garrick's linear theory, there exists a pitching axis near the airfoil that maximizes propulsive efficiency. Next, we examine the effect of placing passive springs on the airfoil in the heave (transverse) direction using our Cyber-Physical Fluid Dynamics technique. This elastic heaving motion allows the airfoil to combine pitching and heaving modes while being actuated only in the pitching direction. Two sets of dynamics are considered: one case where the airfoil is weighted unevenly and pitched about its center of mass (so that the resulting heaving motion is independent of inertial forces), and another case where the airfoil's center of mass is fixed at its centroid. For pitching at an amplitude of $8^{\circ }$ and a reduced frequency $k$ of two, we find that elastic heave produces a maximum propulsive efficiency of 35%, compared to 25% without any heave motion. Further, while operating at the same efficiency as the static-pivot case, we find that passive heaving greatly increases the magnitude of the airfoil's thrust. The airfoil configurations with highest propulsive efficiency generally involve pitching near or ahead of the airfoil's leading edge.

The linear response of the wall-shear stress of a turbulent flow to wall waviness is analyzed in the context of a comparison between existing experiments, direct numerical simulations, and analytical approximations. The spectral region where the response is largest is found to be amenable to a simplified quasilaminar analysis. The end result is a parameterless description of this phenomenon that completely captures its physics in a single analytical formula, a Padé approximation of the response function.

We consider the stability of a binary liquid film flowing down a heated incline. A theoretical model is implemented which captures the Soret effect and the dependence of surface tension on both temperature and solutal concentration. The model also allows for variation in the density of the liquid mixture with thermal and solutal differences. A linear stability analysis is performed with asymptotic and numerical results being obtained. The coupling of the effect of a variable density with the thermosolutal-Marangoni instability and the Soret effect is investigated. Good agreement with previous results for the constant density case is found.

A weighted residual approximation is used to derive a reduced model to analyze the stability of a binary liquid film flowing down a heated incline discussed in Part I. Nonlinear simulations are carried out and reveal a permanent wave structure arising in unstable flow configurations. In addition, a weakly nonlinear analysis is performed and yields a Benney-type equation which has been solved both analytically and numerically. Good agreement between the linear stability analysis, weighted residual model, and weakly nonlinear analysis is found.

Direct numerical simulations are carried out to investigate the effect of the wall temperature on the behavior of oblique shock wave turbulent boundary layer interactions at free-stream Mach number 2.28 and shock angle of the wedge generator $\phi =8^{\circ }$. Five values of the wall-to-recovery-temperature ratio $\left(T_w/T_r\right)$ are considered, corresponding to cold, adiabatic, and hot wall thermal conditions. We show that the main effect of cooling is to decrease the characteristic scales of the interaction in terms of upstream influence and extent of the separation bubble. The opposite behavior is observed in the case of heating, which produces a marked dilatation of the interaction region. The distribution of the Stanton number shows that a strong amplification of the heat transfer occurs across the interaction, with the maximum thermal and dynamic loads found for the case of the cold wall. The analysis reveals that the fluctuating heat flux exhibits a strong intermittent behavior, characterized by scattered spots with extremely high values compared to the mean. Furthermore, the analogy between momentum and heat transfer, typical of compressible, wall-bounded, equilibrium turbulent flows, does not apply for most of the interaction domain. The premultiplied spectra of the wall heat flux do not show any evidence of the influence of the low-frequency shock motion, and the primary mechanism for the generation of peak heating is found to be linked with the turbulence amplification in the interaction region.

The present study attempts to change the regime transitions and heat transfer properties in rotating Rayleigh-Bénard convection by injecting $\sim 100-\mu \mathrm{m}$-diam particles in the flow. The particles start settling out of the fluid immediately after injection and separate entirely from the fluid over a period of several hours. The particles deposit on the top and bottom surfaces, forming porous layers with nonideal thermal properties, and, as expected, decrease the heat flux. The reduction in heat transfer is a result of the inability of the layers to respond rapidly enough to fluid temperature fluctuations. However, in the rotation-dominated geostrophic regime, the heat transfer normalized by its nonrotating value is higher in the presence of the particle layers than without them. Direct numerical simulations with ideal heat transfer walls indicate that the temperature fluctuations in the bulk become slower under the damping effect of rotation, in contrast with those in the boundary layers, which do not show any damping until the flow transitions to the geostrophic regime. In this regime, the dominant time scale of the near-wall fluid temperature fluctuations increases substantially as the rotation rate is increased. It is thus likely that the time response of the particle layers in relation to that of the nearby fluid improves only in the geostrophic regime, which is reflected in a relatively larger heat transfer.

The boundary layer structure of the velocity and temperature fields in turbulent Rayleigh-Bénard flows in closed cylindrical cells of unit aspect ratio is revisited from a transitional and turbulent viscous boundary layer perspective. When the Rayleigh number is large enough, the dynamics at the bottom and top plates can be separated into an impact region of downwelling plumes, an ejection region of upwelling plumes, and an interior region away from the side walls. The latter is dominated by the shear of the large-scale circulation (LSC) roll, which fills the whole cell and continuously varies its orientation. The working fluid is liquid mercury or gallium at a Prandtl number $\text{Pr}=0.021$ for Rayleigh numbers $3\times {10}^5\le \text{Ra}\le 4\times {10}^8$. The generated turbulent momentum transfer corresponds to macroscopic flow Reynolds numbers with $1.8\times {10}^3\le \text{Re}\le 4.6\times {10}^4$. In highly resolved spectral element direct numerical simulations, we present the mean profiles of velocity, Reynolds stress, and temperature in inner viscous units and compare our findings with convection experiments and channel flow data. The complex three-dimensional and time-dependent structure of the LSC in the cell is compensated by a plane-by-plane symmetry transformation which aligns the horizontal velocity components and all its derivatives with the instantaneous orientation of the LSC. As a consequence, the torsion of the LSC is removed, and a streamwise direction in the shear flow can be defined. It is shown that the viscous boundary layers for the largest Rayleigh numbers are highly transitional and obey properties that are directly comparable to transitional channel flows at friction Reynolds numbers ${\text{Re}}_{\tau }\lesssim {10}^2$. The transitional character of the viscous boundary layer is also underlined by the strong enhancement of the fluctuations of the wall stress components with increasing Rayleigh number. An extrapolation of our analysis data suggests that the friction Reynolds number ${\text{Re}}_{\tau }$ in the velocity boundary layer can reach values of 200 for $\text{Ra}\gtrsim {10}^{11}$. Thus the viscous boundary layer in a liquid metal flow would become turbulent at a much lower Rayleigh number than for turbulent convection in gases and gas mixtures.

We report details of the fluid motion formed within and above grooves when a laminar continuous phase fluid flows over a second immiscible fluid enclosed in a grooved microchannel. Vortical structures within the transverse grooves were caused by a slip velocity at the fluid-fluid interface and act as “fluid bearings” on the boundary to lubricate the flow of the continuous phase. We investigated the drag reduction in the laminar flow in the microchannel by measuring slip at the boundaries and calculating an effective slip length, taking into account the influence of the effect of the viscosity ratio of the two fluids on the effective slip length. The “fluid bearing” effect can be used to transport high viscosity fluids using low viscosity fluids trapped in cavities to reduce drag.

Singular solutions to the Euler equations could provide essential insight into the formation of very small scales in highly turbulent flows. Previous attempts to find singular flow structures have proven inconclusive. We reconsider the problem of interacting vortex tubes, for which it has long been observed that the flattening of the vortices inhibits sustained self-amplification of velocity gradients. Here we consider an iterative mechanism, based on the transformation of vortex filaments into sheets and their subsequent instability back into filaments. Elementary fluid mechanical arguments are provided to support the formation of a singular structure via this iterated mechanism, which we analyze based on a simplified model of filament interactions.

The present work investigates the effectiveness of the control strategy introduced by Schoppa and Hussain [Phys. Fluids 10, 1049 (1998)] as a function of Reynolds number ($\text{Re}$). The skin-friction drag reduction method proposed by these authors, consisting of streamwise-invariant, counter-rotating vortices, was analyzed by Canton et al. [Flow, Turbul. Combust. 97, 811 (2016)] in turbulent channel flows for friction Reynolds numbers (${\text{Re}}_{\tau }$) corresponding to the value of the original study (i.e., 104) and 180. For these $\text{Re}$, a slightly modified version of the method proved to be successful and was capable of providing a drag reduction of up to 18%. The present study analyzes the Reynolds number dependence of this drag-reducing strategy by performing two sets of direct numerical simulations (DNS) for ${\text{Re}}_{\tau }=360$ and 550. A detailed analysis of the method as a function of the control parameters (amplitude and wavelength) and $\text{Re}$ confirms, on the one hand, the effectiveness of the large-scale vortices at low $\text{Re}$ and, on the other hand, the decreasing and finally vanishing effectiveness of this method for higher $\text{Re}$. In particular, no drag reduction can be achieved for ${\text{Re}}_{\tau }=550$ for any combination of the parameters controlling the vortices. For low Reynolds numbers, the large-scale vortices are able to affect the near-wall cycle and alter the wall-shear-stress distribution to cause an overall drag reduction effect, in accordance with most control strategies. For higher $\text{Re}$, instead, the present method fails to penetrate the near-wall region and cannot induce the spanwise velocity variation observed in other more established control strategies, which focus on the near-wall cycle. Despite the negative outcome, the present results demonstrate the shortcomings of the control strategy and show that future focus should be on methods that directly target the near-wall region or other suitable alternatives.

In this paper, we present results from an experimental study into turbulent Rayleigh-Bénard convection forced externally by periodically modulated unidirectional rotation rates. We find that the azimuthal rotation velocity $\dot{\theta }\left(t\right)$ and thermal amplitude $\delta \left(t\right)$ of the large-scale circulation (LSC) are modulated by the forcing, exhibiting a variety of dynamics including increasing phase delays and a resonant peak in the amplitude of $\dot{\theta }\left(t\right)$. We also focus on the influence of modulated rotation rates on the frequency of occurrence $\eta$ of stochastic cessation or reorientation events, and on the interplay between such events and the periodically modulated response of $\dot{\theta }\left(t\right)$. Here we identify a mechanism by which $\eta$ can be amplified by the modulated response, and these normally stochastic events can occur with high regularity. We provide a modeling framework that explains the observed amplitude and phase responses, and we extend this approach to make predictions for the occurrence of cessation events and the probability distributions of $\dot{\theta }\left(t\right)$ and $\delta \left(t\right)$ during different phases of a modulation cycle, based on an adiabatic approach that treats each phase separately. Last, we show that such periodic forcing has consequences beyond influencing LSC dynamics, by investigating how it can modify the heat transport even under conditions where the Ekman pumping effect is predominant and strong enhancement of heat transport occurs. We identify phase and amplitude responses of the heat transport, and we show how increased modulations influence the average Nusselt number.

A vortical impulse theory is used to compute the thrust force of a plunging and pitching airfoil in forward flight at high Reynolds numbers within the framework of linear potential flow theory. The result is significantly different from the classical one of Garrick, which considered only two effects, the leading-edge suction and the projection in the flight direction of the pressure force on the airfoil. By taking into account the complete vorticity distribution on the airfoil and the wake the mean thrust coefficient contains, in addition to the pressure force projection term, a new term that generalizes the leading-edge suction term in Garrick's theory. This term depends on Theodorsen function $C\left(k\right)$ and on a new complex function $C_1\left(k\right)$ of the reduced frequency $k$. The main qualitative difference with Garrick's theory is that the propulsive efficiency, or ratio of the mean thrust power and the mean input power required to drive the airfoil, tends to zero as the reduced frequency increases to infinity (as $k^{-1}$), in contrast to Garrick's propulsive efficiency that tends to a constant $\left(1/2\right)$. Consequently, for pure pitching and combined pitching and plunging motions, the maximum of the propulsive efficiency is not reached as $k\to \infty$ like in Garrick's theory, but at a finite value of the reduced frequency that depends on the remaining nondimensional parameters. The present analytical results are in good agreement, for small amplitude oscillations, with numerical results from unsteady panel methods, and with experimental data and numerical results from the Navier-Stokes equations, except for small reduced frequencies where viscous effects are obviously important.

Intraventricular pressure gradients were clinically demonstrated to represent one useful indicator of the left ventricle (LV) function during the development of heart failure. We analyze the fluid dynamics inside a model LV to improve the understanding of the development of hemodynamic forces (i.e., mean pressure gradient) in normal conditions and their modification in the presence of alterations of LV tissue motion. To this aim, the problem is solved numerically and the global force exchanged between blood flow and LV boundaries is computed by volume integration. We also introduce a simplified analytical model, based on global conservation laws, to estimate hemodynamic forces from the knowledge of LV tissue information commonly available in cardiac imaging. Numerical results show that the normal intraventricular gradients feature a deep brief suction at early diastolic filling and a persistent thrust during systolic ejection. In presence of abnormalities of the wall motion, the loss of time synchrony is more relevant than the loss of spatial uniformity in modifying the normal pressure gradient spatiotemporal pattern. The main findings are reproduced in the integral model, which represents a possible easy approach for integrating fluid dynamics evaluations in the clinical examination.

In this paper we propose equations of motion for the dynamics of liquid films of surfactant suspensions that consist of a general gradient dynamics framework based on an underlying energy functional. This extends the gradient dynamics approach to dissipative nonequilibrium thin-film systems with several variables and casts their dynamic equations into a form that reproduces Onsager's reciprocity relations. We first discuss the general form of gradient dynamics models for an arbitrary number of fields and discuss simple well-known examples with one or two fields. Next we develop the three-field gradient dynamics model for a thin liquid film covered by soluble surfactant and discuss how it automatically results in consistent convective (driven by pressure gradients, Marangoni forces, and Korteweg stresses), diffusive, adsorption or desorption, and evaporation fluxes. We then show that in the dilute limit, the model reduces to the well-known hydrodynamic form that includes Marangoni fluxes due to a linear equation of state. In this case the energy functional incorporates wetting energy, surface energy of the free interface (constant contribution plus an entropic term), and bulk mixing entropy. Subsequently, as an example, we show how various extensions of the energy functional result in consistent dynamical models that account for nonlinear equations of state, concentration-dependent wettability, and surfactant and film bulk decomposition phase transitions. We conclude with a discussion of further possible extensions towards systems with micelles, surfactant adsorption at the solid substrate, and bioactive behavior.

A study is made about the energy spectrum $E\left(k\right)$ of turbulence on the basis of high-resolution direct numerical simulations (DNSs) of forced incompressible turbulence in a periodic box using a Fourier spectral method with the number of grid points and the Taylor scale Reynolds number $R_{\lambda }$ up to $12{288}^3$ and approximately 2300, respectively. The DNS data show that there is a wave-number range (approximately $5\times {10}^{-3}<k\eta <2\times {10}^{-2}$) in which $E\left(k\right)$ fits approximately well to Kolmogorov's $k^{-5/3}$ scaling, where $\eta$ is the Kolmogorov length scale. However, a close inspection shows that the exponent is a little smaller than $-5/3$, and $E\left(k\right)$ in the range fits to $E\left(k\right)/\left[{\langle \epsilon \rangle }^{2/3}{k}^{-5/3}\right]=c{\left(kL\right)}^m$, where $\langle \epsilon \rangle$ is the mean energy dissipation rate per unit mass; $L$ is the integral length scale; and $m\approx -0.12$. The coefficient $c$ is independent of $k$, but has a $R_{\lambda }$ dependence, such as $c=C{R}_{\lambda }^{\zeta }$, where $C\approx 0.9$ and $\zeta \approx 0.14$.

We analyze the angular velocity of a small neutrally buoyant spheroid log rolling in a simple shear. When the effect of fluid inertia is negligible the angular velocity $\mathbf{\omega }$ equals half the fluid vorticity. We compute by singular perturbation theory how weak fluid inertia reduces the angular velocity in an unbounded shear, and how this reduction depends upon the shape of the spheroid (on its aspect ratio). In addition we determine the angular velocity by direct numerical simulations. The results are in excellent agreement with the theory at small but not too small values of the shear Reynolds number ${\mathrm{Re}}_{\mathrm{s}}$, for all aspect ratios considered. For the special case of a sphere we find $\omega /s=-1/2+0.0540{\mathrm{Re}}_{\mathrm{s}}^{3/2}$ where $s$ is the shear rate. The $O\left({\mathrm{Re}}_{\mathrm{s}}^{3/2}\right)$ correction differs from that derived by Lin et al. [J. Fluid Mech. 44, 1 (1970)], who obtained a numerical coefficient roughly three times larger.

Selective noise amplifiers are characterized by large linear amplification to external perturbations in a particular frequency range despite their global linear stability. Applying a stochastic forcing with increasing amplitude, the response undergoes a strong nonlinear saturation when compared to the linear estimation. Building upon our previous work, we introduce a predictive model that describes this nonlinear dynamics, and we apply it to a canonical example of selective noise amplifiers: the backward-facing step flow. Rewriting conveniently the stochastic forcing and response in the frequency domain, the model consists in a mean flow equation coupled to the linear response to forcing at each frequency. This coupling is attained by the Reynolds stress, which is constructed by the integral in frequency of the independent responses. We generalize the model for a response to a white noise forcing $\delta$-correlated in space and time restricting the flow dynamics to its most energetic patterns calculated from the optimal harmonic forcing and response of the flow. The model estimates accurately the response saturation when compared to direct numerical simulations, and it correctly approximates the structure of the response and the mean flow modification. It also shows that the response undergoes a selective process governed by the nonlinear gain, which promotes a response structure with an approximately single frequency and wavelength in the whole domain. These results suggest that the mean flow modification by the Reynolds stress is the key nonlinearity in the saturation process of the response to white noise.

We perform a linearized reduced-order modeling of the aeroacoustic sound sources in supersonic reacting mixing layers to explore their sensitivities to some of the flow parameters in radiating sound. Specifically, we investigate the role of outer modes as the effective flow compressibility is raised, when some of these are expected to dominate over the traditional Kelvin-Helmholtz (K-H) -type central mode. Although the outer modes are known to be of lesser importance in the near-field mixing, how these radiate to the far-field is uncertain, on which we focus. On keeping the flow compressibility fixed, the outer modes are realized via biasing the respective mean densities of the fast (oxidizer) or slow (fuel) side. Here the mean flows are laminar solutions of two-dimensional compressible boundary layers with an imposed composite (turbulent) spreading rate, which we show to significantly alter the growth of instability waves by saturating them earlier, similar to in nonlinear calculations, achieved here via solving the linear parabolized stability equations. As the flow parameters are varied, instability of the slow modes is shown to be more sensitive to heat release, potentially exceeding equivalent central modes, as these modes yield relatively compact sound sources with lesser spreading of the mixing layer, when compared to the corresponding fast modes. In contrast, the radiated sound seems to be relatively unaffected when the mixture equivalence ratio is varied, except for a lean mixture which is shown to yield a pronounced effect on the slow mode radiation by reducing its modal growth.