# Latest papers in fluid mechanics

### Friction drag reduction based on a proportional-derivative control scheme

Dielectric barrier discharge plasma actuators (DBD-PAs) are deployed experimentally for the first time in a feed-forward proportional-derivative (PD) control system, where the fluctuating wall-pressure Pw is demonstrated to be an effective feed-forward signal, to manipulate a turbulent boundary layer for drag reduction. A floating-element force balance with an area of 50 mm (streamwise length) × 200 mm (spanwise length) is deployed to capture the spatially averaged drag variation behind the DBD-PAs. The DBD-PAs generate streamwise vortices, whose occurrence synchronizes with the output signal of the controller with a predominant frequency of 40 Hz under the optimally tuned PD control. The control system proves to be effective, achieving a spatially averaged drag reduction by 16%, and efficient, cutting down its energy consumption by 30% at a negligibly small expense of drag reduction compared with the open-loop control. It has been found that the optimally tuned PD control aptly increases the voltage applied to the DBD-PAs upon detecting large Pw fluctuations or coherent structures, accounting for the savings in input power, Pinput. The experimental data have been carefully analyzed, which cast light upon the underlying physical mechanism behind the drag reduction. The reason behind the efficient control is also clearly elaborated.

### Flow transition to turbulence and induced acoustics at Mach 6

This paper presents the results of implicit large eddy simulation (iLES) and direct numerical simulation (DNS) for flow and acoustics for transitional and turbulent boundary layer over a flat plate at Mach 6. The DNS was about 50 times more refined grid-wise than iLES. Both DNS and iLES were performed using the same numerical schemes, initial and boundary conditions. We compare the different numerical approaches concerning the shape factor, momentum-thickness-based Reynolds number, heat flux on the wall, Reynolds stress, and near-wall acoustics. We perform pressure fluctuations spectral analysis and propose a predictive model. We show that iLES captures rather accurately the flow and acoustic characteristics in the turbulent region. Differences up to 5 dB occur between iLES and DNS in the transition region. iLES also shifts slightly further downstream the end of the transition and underpredicts the shear stress value peak. The iLES captures the near-wall acoustic spectrum roll-off accurately at low and medium frequencies. It underpredicts high frequencies' content due to grid constraints. Overall, iLES gives excellent results compared to the significantly more refined DNS. The results show that high-order numerical simulations can help adapt and validate semi-empirical models for the engineering design and acoustic loading on hypersonic structures.

### Mechanism of control of the near-wall turbulence using a micro-cavity array

Coherent structures in a turbulent boundary layer have been shown to have an influence on the skin-friction drag acting on surfaces beneath the boundary layer. The use of micro-cavities on a flat surface has recently shown the potential to passively control a turbulent boundary layer by attenuating the sweep events. Previous experiments have determined the design parameters of the cavity array for the optimal boundary-layer control by reducing the sweep events. However, investigating the flow physics behind the interaction of the boundary-layer flow with the cavities is challenging. High near-wall velocity gradients and very small scales and sizes of the cavity holes limit the experiments from investigating the flow characteristics very close to the wall and inside the holes. Therefore, in the present work, direct numerical simulations have been utilized to model the boundary layer flow over a flat surface with a micro-cavity array in order to understand the flow interactions. Detection of coherent structures in the boundary layer shows a reduction in the number of events over the cavity array. Reynolds stresses have been analyzed to determine the effect of micro-cavities. The reduction in the Reynolds shear stress results in a lower skin-friction drag. The flow fluctuations through the holes in the streamwise sequence have been found to be highly correlated using cross correlation. These flow fluctuations interact with the boundary layer to suppress the coherent structures. Overall, the use of the micro-cavity array has resulted in a reduced wall shear stress and approximately 5.6% lower local skin-friction drag.

### Geostrophic adjustment on the f-plane: Symmetric versus anti-symmetric initial height distributions

We study the fundamental process of geostrophic adjustment in infinitely long zonal and meridional channels of widths Ly and Lx, respectively, by deriving analytic solutions and simulating the linearized rotating shallow water equations (LRSWE). All LRSWE's variables are divided into time-independent (geostrophic) and time-dependent components. The latter includes Kelvin waves, Poincaré waves, and inertial oscillations. Explicit expressions are derived for both components, which are confirmed numerically. Anti-symmetric and symmetric initial height distributions, [math], are considered, both of which introduce a length scale, D, into the problem. We show that for an anti-symmetric [math], (i) the rate of approach to geostrophy is D-independent; (ii) the decay rate of inertial oscillations is [math]; and (iii) for [math], the energy of the final state in any finite sub-domain of the channel exceeds that of the initial state, while for [math] the energy in the final state is smaller than in the initial state. In contrast, for a symmetric [math]: (i) the rate of approach to geostrophy increases with D; (ii) the decay rate of inertial oscillations is [math], and (iii) the energy of the final state is always smaller than that of the initial state. In meridional channels, the effect of boundaries is to (i) block the waves, propagation to infinity; (ii) alter the spatial structure of the geostrophic flow; and (iii) discretize the frequency spectrum, thus eliminating the inertial oscillations. The ratio of wave energy to initial energy decreases with Lx for anti-symmetric [math] and increases with Lx for symmetric [math].

### Development of generalized bubble growth model for cavitation and flash boiling

Cavitation occurs in a wide range of applications, such as in marine propellers, diesel injectors, supercavitating projectiles, etc. Currently, the available cavitation models rely on expressions derived from inertial bubble growth models and fine-tuned using a few experiments. Revisiting the literature on bubble growth models indicates that there is scope for improvement in the bubble growth expressions presently employed. The previous studies in this subject have assumed that the vapor in the bubble remains saturated. Detailed numerical studies using one-dimensional saturated vapor model reveals over-prediction of the bubble radius when compared with a wide range of experimental data. To overcome this, a coupled mass, momentum, and energy model, termed full model, is then developed and the analysis suggests that this model gives good agreement over the entire experimental data. Parametric studies carried out to generate non-dimensional bubble growth rate expressions indicate that the growth rate climbs linearly on a log –log plot during initial stages of bubble growth which is function Jakob number [math] and finally settles into an asymptotic non-linear curve which is independent of [math]. The bubble growth rate expressions when integrated to obtain bubble radius as function of time is able to predict the experimental data with mean relative error of 1.2% and root mean square relative error of 8% for [math] varying from 13.53 to 2745.

### Energetic structures in the turbulent boundary layer over a spanwise-heterogeneous converging/diverging riblets wall

Time-invariant (or mean) secondary flows, in terms of large-scale counter-rotating roll modes filling boundary layer, have been observed in turbulent boundary layer (TBL) flows over various spanwise-heterogeneous rough walls. Recent studies show that these mean secondary flows are inherently connected with instantaneous large-scale structures in TBL flows over such spanwise-heterogeneous rough walls. In this work, the technique of proper orthogonal decomposition (POD) is used to extract dominant energetic (and thus large-scale) structures from TBL flows over the surface with a spanwise-periodic converging/diverging riblets pattern, one of the spanwise-heterogeneous rough walls adopted previously. POD analyses are conducted on the three fluctuating components of velocities, measured by stereoscopic particle image velocimetry at [math] = 13 000, in a cross-stream plane of the TBL flows over a spanwise section of the converging riblets pattern, where low-momentum eruptions consistently occur. It is found that first two POD modes with large temporal coefficients are linked to large-scale structures oscillating vigorously in the transverse direction within one section of the converging riblets. Superposition of these instantaneous large-scale structures reveals the observed pattern of mean secondary flows. Furthermore, these large-scale structures are associated with enhanced streamwise vortices and spanwise gradients of the streamwise velocity, compared with that in the smooth wall flow, thus rendering profound effects on the spatial correlations of velocities as well as the distributions of Reynolds stresses.

### Simultaneous velocity and density measurements of fully developed Rayleigh-Taylor mixing

Author(s): Mark Mikhaeil, Prasoon Suchandra, Devesh Ranjan, and Gokul Pathikonda

The dynamics of molecular mixing and the energy transfer process in the Rayleigh-Taylor instability (RTI) are studied through the collection of simultaneous velocity-density measurements using particle image velocimetry (PIV) and laser induced fluorescence (LIF). Statistically stationary experiments are performed in a convective-type gas tunnel facility which allows long experimental times and enables collection of statistically important turbulence data. The data and analyses presented in this paper are expected to help validate variable-density turbulence models and further our understanding of instability-driven flows.

[Phys. Rev. Fluids 6, 073902] Published Mon Jul 19, 2021

### Asymptotic theory of hydrodynamic interactions between slender filaments

Author(s): Maria Tătulea-Codrean and Eric Lauga

Hydrodynamic interactions are important in biophysics because they influence the collective behaviour of microorganisms and active particles, and also play a key role in the emergence of swimming gaits. We determine the hydrodynamic interactions between slender filaments by means of asymptotic calculations and numerical simulations, for the case when two filaments are separated by a distance greater than their contour length (d > L). We then show how our theory explains the collective dynamics of two rigid helices rotating side-by-side.

[Phys. Rev. Fluids 6, 074103] Published Mon Jul 19, 2021

### Large eddy simulation of transitional channel flow using a machine learning classifier to distinguish laminar and turbulent regions

Author(s): Ghanesh Narasimhan, Charles Meneveau, and Tamer A. Zaki

Breakdown to turbulence in wall-bounded flows takes place through sporadic bursts of turbulent spots. Wall-modelled large-eddy simulations (LES) of transition to turbulence must dynamically identify the nascent turbulent regions, track their evolution, and apply the appropriate wall stress within and outside the turbulent/non-turbulent (T-NT) interface. Self-organized maps (SOM), a machine learning classifier, objectively and efficiently captures the T-NT interface. Wall-modeled LES with SOM interface identification predicts both orderly and bypass transition.

[Phys. Rev. Fluids 6, 074608] Published Mon Jul 19, 2021

### Wake structure and evolution of flow over a finned circular cylinder

This paper reports the results of a numerical investigation into the flow over a circular cylinder evenly attached with rectangular ribs around its circumference and the associated near-wall vortex structure as well as the evolution of wake flow. The effect of the number of ribs (n) ranging from 1 to 12 is examined in a low Reynolds number range of 60–180. Five kinds of near-wall vortices are identified with the presence of rectangular ribs, including quasi-stagnation vortices, subordinate vortices, inter-rib vortices, inter-rib quasi-stagnation vortices, and dynamic inter-rib vortices. These near-wall vortices and their evolution are sensitive to Re as well as n. With the introduction of ribs, each boundary layer experiences multiple separations that increase from 2 to 4 as n increases from 1 to 12. The intermittent emergency of the last separation is attributed to the switching of the final separation between two adjacent ribs. Furthermore, the duration of the last separation depends on Re and the evolution of dynamic inter-rib vortices. The hydrodynamic coefficients and vortex shedding frequency are closely related to the vortex formation length (Lf*) and wake width (W*). The different variation of Lf* and W* with Re is possibly associated with the appearance and number of subordinate vortices. In addition, the fluctuation of W* is mainly attributed to the switching of boundary layer separation point. The wake flow experience two evolutions as the vortices are convected downstream: the transition from one-row primary vortex street to the two-layered vortices, and the transition from two-layered vortices to the secondary vortex street. The latter is only observed in the wake of a finned cylinder with 3–5 ribs at Re = 180, while the two-layered vortices are decayed in the far wake in other cases.

### Editorial: Machine Learning and <i>Physical Review Fluids</i>: An Editorial Perspective

Author(s): Michael P. Brenner and Petros Koumoutsakos

[Phys. Rev. Fluids 6, 070001] Published Fri Jul 16, 2021

### Scaling of hypersonic shock/turbulent boundary layer interactions

Author(s): Clara M. Helm and M. P. Martín

A large database of shock/turbulent boundary layer interactions is compiled to study the separation length scaling over the range of flow conditions including hypersonic interactions. Experimental and computational data of two-dimensional and axisymmetric geometries are included with Mach number from 2 to 10 and ratio of wall to adiabatic recovery temperature from 0.3 to 1.9. A new scaling shows weak interactions collapse by the upstream boundary layer properties, strong interactions do not, and strong separation cases depend on the structure of the downstream flow.

[Phys. Rev. Fluids 6, 074607] Published Fri Jul 16, 2021

### Acoustic signatures and bubble entrainment mechanisms of a drop impacting a water surface with surfactant

The acoustics of a water drop impact on a bath of water and sodium dodecyl sulfate (SDS) is studied close to the irregular entrainment regime. In particular, acoustic events, corresponding to bubble vibration, are observed for several SDS concentrations. These acoustic events are induced by five different kinds of hydrodynamic events, including four different bubble entrainment mechanisms and one bubble excitation mechanism. These families of events appear to have their own acoustic signature. The different mechanisms are described in detail and typical signals belonging to these families are presented. Their main features are highlighted and linked to the hydrodynamics of the corresponding event.

### Energy stable modeling of two-phase flow in porous media with fluid–fluid friction force using a Maxwell–Stefan–Darcy approach

Two-phase incompressible flow in porous media plays an important role in various fields including subsurface flow and oil reservoir engineering. Due to the interaction between two phases flowing through the pores, the fluid–fluid friction force may have a significant effect on each phase velocity. In this paper, we propose an energy stable (thermodynamically consistent) Maxwell–Stefan–Darcy model for two-phase flow in porous media, which accounts for the fluid–fluid friction. Different from the classical models of two-phase flow in porous media, the proposed model uses the free energy to characterize the capillarity effect. This allows us to employ the Maxwell–Stefan model to describe the relationships between the driving forces and the friction forces. The driving forces include the pressure gradient and chemical potential gradients, while both fluid–solid and fluid–fluid friction forces are taken into consideration. Thermodynamical consistency is the other interesting merit of the proposed model; that is, it satisfies an energy dissipation law and also obeys the famous Onsager's reciprocal principle. A linear semi-implicit numerical method is also developed to simulate the model. Numerical simulation results are provided to show that the fluid–fluid friction force can improve the oil recovery substantially during the oil displacement process.

### A length scale approach to the highest standing water wave

The highest standing surface wave at infinite depth is a classical hydrodynamic problem, illuminated by Taylor's excellent experiments [G. I. Taylor, “An experimental study of standing waves,” Proc. R. Soc. London, Ser. A 218, 44–59 (1953)]. Based on length scale arguments, we present a compact analytical approach to the highest standing wave. Our physical postulate is that the highest deep-water wave has a single length scale, i.e., its wavelength. The single-scale postulate for standing periodic deep-water waves is confronted with two distinctly different cases where zero and two length scales are postulated as follows: (i) No physical length scale for an isolated rogue-wave peak at deep water suggests a similarity solution. (ii) Two length scales for the periodic peaked surface at constant depth suggest a one-parameter family of standing waves. Moreover, the two length scales are the wavelength and average fluid depth. The deep-water limit with its single-length scale postulate confirms Grant's theory [M. A. Grant, “Standing Stokes waves of maximum height,” J. Fluid Mech. 60, 593–604 (1973)], taking the highest standing wave as a state of zero kinetic energy. The reversible motion is irrotational according to Lord Kelvin's theorem. The acceleration field for the highest deep-water wave has a single Fourier component according to our single length scale postulate. The resulting free-surface shape follows from the exact nonlinear dynamic condition. Our analytical theory confirms the ratio 0.203 for maximal wave height to wavelength, found by Grant. We test its robustness by extending the theory to a moderate spatial quasi-periodicity. Appendix A provides a simple proof for the right-angle peak, representing a regular extremal point of a locally quadratic complex function. Appendix B presents a quadrupole solution for an isolated peak of stagnant deep-water rogue waves.

### Numerical study of the shock wave and pressure induced by single bubble collapse near planar solid wall

Bubble collapse is one of the leading causes for the cavitation erosion of submerged structures. For better understanding of the destructive mechanism of cavitation, high-fidelity simulation is performed to simulate the complete process of single bubble collapse near a planar solid wall. The wave propagation method with the approximate Riemann solver Harten Lax and van Leer Contact is adopted to solve the compressible two-phase five-equation model. We implement fifth-order weighted essentially non-oscillatory scheme with the block-structured adaptive mesh method to resolve shock waves and moving interface with high-resolution. We simulate single bubble collapsing in free-field to validate the present numerical methods and solver. Our results (e.g., averaged bubble-interior pressure and the radius variation) are found in excellent agreement with the theoretical Keller–Miksis solutions. In this study, the shock wave transmitted inside the bubble and the water-hammer shock formed in the liquid are under quantitative investigation. Numerical results reveal that the interactions between the shock wave and bubble interface give rise to peak pressures of liquid phase, and the initial stand-off distances have important influence on shock wave pattern, wall peak pressure, and bubble dynamics.

### Vortex flow evolution in a growing microdroplet during co-flow in coaxial capillaries

Using micro-particle image velocimetry (μPIV), the convective flow inside a silicone oil droplet was investigated in detail during its formation in coaxial capillaries under co-flow in a water/glycerol mixture continuous phase. The analysis of μPIV measured flow field revealed that two characteristic flow areas exist in the droplet in formation: an inflow zone and a circulation zone. The intensity of vortex flow in these zones was estimated by calculating the average angular velocity of these vortices under the condition of no shear for different dispersed phase and continuous phase flow rates and for different viscosity ratios between the two phases. The evolution of the vortex flow pattern inside the droplet was investigated thoroughly all the way from the step of their formation to the step of the free-moving droplet. The results of this study are important for understanding the mixing processes inside the droplet at different stages of its formation.

### Sheltering electroconvective instability in a weak electrolyte

Sheltering vortices by shear flow is a common method in plasma and neutral fluids and has recently been successfully applied to control ionic fluids. This work proposes a new chemical sheltering vortex method for electroconvective instability (ECI) based on the Onsager chemical effect. We reveal unique ECI behaviors in a weak electrolyte with the Onsager effect, including the vortex height selection, overlimiting transport, and vortex structure. Due to the strong electric field strength in the electric double layer, the Onsager effect in a weak electrolyte causes neutral molecules to generate additional free ions, which weakens the thickness of the extended space charge layer and causes the fluid to transition from a chaotic ECI to a steady ECI. Consequently, the Onsager effect shelters ECI without an oblique vortex, which is significantly different from the shear flow effect [Kwak et al., “Sheltering the perturbed vortical layer of electroconvection under shear flow,” J. Fluid Mech. 813, 799 (2017)]. We believe that the proposed chemical control strategy can be an alternative candidate for ionic fluids.

### Flow dynamics of shear-coaxial cryogenic nitrogen jets under supercritical conditions with and without acoustic excitations

The three-dimensional flow structures and dynamics of shear-coaxial nitrogen jets under supercritical conditions are comprehensively investigated using the large-eddy simulation technique. The theoretical framework is based on the full conservation laws and accommodates real-fluid thermodynamics and transport theories over the entire range of fluid states. Cryogenic liquid nitrogen (132 K) is delivered through the inner tube of a shear-coaxial nozzle, and gaseous nitrogen (191 K) is injected through the outer annulus, into a high-pressure environment at 233 K. Particular attention is paid to the influence of operating parameters on the flow evolution of the coaxial jets. Two supercritical pressure conditions, 4.94 and 10.0 MPa, and three outer-to-inner velocity ratios in the range of 2.03–3.75, are considered. Results reveal that the flowfield downstream of the coaxial jets is characterized by two potential cores and three shear layers. Two counter-rotating recirculation zones are formed next to the central post-tip. The characteristic frequencies of flow recirculation and shear-layer instability are analyzed using power spectral analysis. Increasing the pressure and the velocity ratio enhances turbulent mixing, promotes the entrainment of the outer stream into the inner region, and advances the pinching point of vortical motion to the centerline, thereby leading to a decreased length of the inner dark core. The predicted dark core length and outer spreading angle show good agreement with experimental results. The introduction of transverse acoustic excitation induces large sinusoidal oscillations of the inner dense-fluid stream along the direction of acoustic motion and promotes the mixing of the inner and outer streams.

### Some general solutions for linear Bragg–Hawthorne equation

Linear cases of Bragg–Hawthorne equation for steady axisymmetric incompressible ideal flows are systematically discussed. The equation is converted to a more convenient form in a spherical coordinate system. A new vorticity decomposition is derived. General solutions for 16 linear cases of the equation are obtained. These solutions can be specified to gain new analytical vortex flows, as examples in the paper demonstrate. A lot of well-known solutions like potential flow past a sphere, Hill's vortex with and without swirl, are included and extended in these solutions. Special relations between some vortex flows are also revealed when exploring or comparing related solutions.