Inviscid Regime Research Articles

Probing Dust and Gas Properties Using Ringed Disks

How rapidly a planet grows in mass and how far it may park from the host star depend sensitively on two nondimensional parameters: Stokes number St and turbulent α. Yet these parameters remain highly uncertain, being difficult or impossible to measure directly. Here, we demonstrate how the ringed disks can be leveraged to obtain St and α separately by constructing a simple toy model that combines the dust radial equation of motion under aerodynamic drag and coupling to gas motion with the measured distribution of dust masses in Class 0/I disks. Focusing on known systems with well-resolved dust rings, we find that the ranges of St and α that are consistent with the measured properties of the rings are small: 10−4 ≲ St ≲ 10−2 and 10−5 ≲ α ≲ 10−3. These low St and α ensure the observed rings are stable against clumping. Even in one marginal case where the formation of bound clumps is possible, further mass growth by pebble accretion is inhibited. Furthermore, the derived low α is consistent with the nearly inviscid regime where type I migration can be prematurely halted. Our analysis predicts a minimal planet population beyond ∼tens of au, where we observe dust rings and significantly more vigorous planet formation inside ∼10 au, consistent with current exo-giant statistics. We close with discussions on the implications of our results on small planet statistics at large orbital distances.

The inviscid incompressible limit of Kelvin–Helmholtz instability for plasmas

Introduction: The Kelvin–Helmholtz Instability (KHI) is an interface instability that develops between two fluids or plasmas flowing with a common shear layer. KHI occurs in astrophysical jets, solar atmosphere, solar flows, cometary tails, planetary magnetospheres. Two applications of interest, encompassing both space and fusion applications, drive this study: KHI formation at the outer flanks of the Earth’s magnetosphere and KHI growth from non-uniform laser heating in magnetized direct-drive implosion experiments. Here, we study 2D KHI with or without a magnetic field parallel to the flow. We use both the GAMERA code, which solves the compressible Euler equations, and the STRATOSPEC code, which solves the Navier-Stokes equations under the Boussinesq approximation, coupled with the magnetic field dynamics. GAMERA is a global three-dimensional MHD code with high-order reconstruction in arbitrary nonorthogonal curvilinear coordinates, which is developed for a large range of astrophysical applications. STRATOSPEC is a three-dimensional pseudo-spectral code with an accuracy of infinite order (no numerical diffusion). Magnetized KHI is a canonical case for benchmarking hydrocode simulations with extended MHD options.Methods: An objective is to assess whether or not, and under which conditions, the incompressibility hypothesis allows to describe a dynamic compressible system. For comparing both codes, we reach the inviscid incompressible regime, by decreasing the Mach number in GAMERA, and viscosity and diffusion in STRATOSPEC. Here, we specifically investigate both single-mode and multi-mode initial perturbations, either with or without magnetic field parallel to the flow. The method relies on comparisons of the density fields, 1D profiles of physical quantities averaged along the flow direction, and scale-by-scale spectral densities. We also address the triggering, formation and damping of filamentary structures under varying Mach number or Atwood number, with or without a parallel magnetic field.Results: Comparisons show very satisfactory results between the two codes. The vortices dynamics is well reproduced, along with the breaking or damping of small-scale structures. We end with the extraction of growth rates of magnetized KHI from the compressible regime to the incompressible limit in the linear regime assessing the effects of compressibility under increasing magnetic field.Discussion: The observed differences between the two codes are explained either from diffusion or non-Boussinesq effects.

Maximum Spreading Diameter of Bouncing Droplets at Ultralow Weber Numbers

In the present work, we study the maximum spreading of bouncing droplets in the capillary regime at ultralow Weber numbers with a fixed static contact angle. In the ultralow Weber number region, experiments reveal that existing spreading laws are inapplicable because of gravity exclusion and change in deformation shape. We propose a theoretical scaling law based on energy conservation, modeling the deformed droplet as an ellipsoid with gravity effects. The proposed scaling law indicates the competition between gravity and inertia at ultralow Weber numbers and distinguishes their dominant regimes. By integrating higher-Weber-number regions, we reveal that viscosity is prominent in the previously assumed inviscid regime. Furthermore, we devise a phase diagram to clarify different impact regimes on the basis of energy analysis.

Breakup of a low-viscosity liquid thread

We examine the breakup of low-viscosity ($\ensuremath{\mu}$) liquid filaments. When $\ensuremath{\mu}$ is small, the filament initially thins as if it were inviscid and its minimum radius ${h}_{m\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}n}$ obeys a universal scaling law. Here, we use simulations to show that for fluids of sufficiently small $\ensuremath{\mu}$, a coefficient value in the scaling law predicted from computations agrees with theory to three decimal places and inviscid power-law behavior can be observed over 2-3 decades in ${h}_{m\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}n}$ as ${t}_{b}\ensuremath{-}t\ensuremath{\rightarrow}0$ where ${t}_{b}$ is the filament break up time. Transition from the inviscid regime to a viscous one is also demonstrated from simulations.

Nozzle design optimization for supersonic wind tunnel by using surrogate-assisted evolutionary algorithms

For high-precision measurement in supersonic wind tunnel experiments, it is of crucial importance to produce a uniform flow in the measurement section downstream of the nozzle. This paper proposes and verifies a new design methodology for supersonic wind tunnel nozzles that can generate highly uniform airstream at a design Mach number at the nozzle exit. Shape design optimization has been conducted by employing surrogate-assisted evolutionary algorithms coupled with computational fluid dynamics. This approach has yielded higher flow uniformity than that of the nozzle designed by using the method of characteristics in the inviscid regime. By applying boundary layer correction to the nozzle contour obtained from the inviscid optimization, nearly uniform core flow of Mach 2.5 has been achieved at the nozzle exit in the presence of viscosity. In the viscous optimization, nozzle shape optimization has been performed by incorporating viscous simulations without using boundary layer correction to evaluate its efficacy in comparison with the former approach. It has been found that the deviation from the design Mach number and the flow deflection from the horizontal direction cannot be minimized simultaneously. This has been attributed to the constraint associated with the nozzle length. It has also been found that the result of the former approach combining the inviscid optimization and boundary layer correction can be regarded as one of the results of the viscous optimization.

Physical insights into multi-point global optimum design of scramjet intakes for ascent flight

Scramjet propulsion offers promise for flexible and sustainable space transportation. Hypersonic airflow compression through the intake plays a major role in successful scramjet-powered ascent. It is thus of crucial importance to design high-performance intakes that can work efficiently and robustly throughout the ascent trajectory. The present research is conducted to gain physical insight into multi-point global optimum scramjet intake design and associated requirements by means of multi-point design optimization and analytical investigation in inviscid and viscous regimes. It has been found and verified theoretically that at any operating conditions, compression efficiency, drag, compression ratio, and mean exit temperature can theoretically be determined uniquely in an interrelated manner under calorically perfect and adiabatic flow assumptions, provided one of them is given or known except for compression efficiency. The multi-point optimization study in the inviscid regime has verified that the multi-point global optimum design can exist with respect to compression efficiency and drag. On the other hand, while the theoretical analysis also prescribes that the global optimum design can exist for a single operating condition in viscous flow, trade-off relations have been found to exist between two operating conditions in the viscous regime, where boundary layer is responsible for counteracting behaviors between compression efficiency and drag at different operating conditions. The insights gained contribute to the guiding principle for intake design of scramjet engines for access-to-space.

Physical insight into axisymmetric scramjet intake design via multi-objective design optimization using surrogate-assisted evolutionary algorithms

Scramjet engines are a hypersonic airbreathing technology that offers a potential for economical and flexible space transportation in lieu of traditional rocket-based systems. Accurate prediction of inviscid flowfields is of particular importance for high-performance intake design, prior to consideration of viscous effects in the design process. Further, inviscid axisymmetric intakes serve as a base for streamline tracing, one of the most promising design methodologies for scramjet intakes. Multi-objective optimization studies have been conducted via surrogate-assisted evolutionary algorithm to gain physical insights into axisymmetric intake design in this study. The results indicate the existence of global optimum solutions that can simultaneously achieve maximum compression efficiency and minimum drag for any degree of compression in case the outflow is supersonic at the intake exit, which has been verified by theory. In addition, a correlation between compression efficiency and flow uniformity has been found and discussed quantitatively. This assures the optimality of the Busemann intakes in that they simultaneously offer high compression efficiency and uniform flow at the intake exit in the inviscid regime.

Electrified cone formation in perfectly conducting viscous liquids: Self-similar growth irrespective of Reynolds number

Above a critical field strength, the free surface of an electrified, perfectly conducting viscous liquid, such as a liquid metal, is known to develop an accelerating protrusion resembling a cusp with a conic tip. Field self-enhancement from tip sharpening is reported to generate divergent power law growth in finite time of the forces acting in that region. Previous studies have established that tip sharpening proceeds via a self-similar process in two distinguished limits—the Stokes regime and the inviscid regime. Using finite element simulations to track the shape and forces acting at the tip of an electrified protrusion in a perfectly conducting Newtonian liquid, we demonstrate that the conic tip always undergoes self-similar growth irrespective of the Reynolds number. The blowup exponents at the conic apex for all terms in the Navier-Stokes equation and the normal stress boundary condition at the moving interface reveal the dominant forces at play as the Reynolds number increases. Rescaling of the tip shape by the power law representing the divergence in capillary stress at the apex yields an excellent collapse onto a universal cone shape with an interior half-angle dependent on the Maxwell stress. The rapid acceleration of the liquid interface also generates a thin interfacial boundary layer characterized by a significant rate of strain. Additional details of the modeled flow, applicable to cone growth in systems such as liquid metal ion sources, help dispel prevailing misconceptions that dynamic cones resemble conventional Taylor cones or that viscous stresses at a finite Reynolds number can be neglected.

Self-focusing dynamics of patches of ripples

The dynamics of focussing of extended patches of nonlinear capillary–gravity waves within the primitive fluid dynamic equations is presented. It is found that, when the envelope has certain properties, the patch focusses initially in accordance to predictions from nonlinear Schrödinger equation, and focussing can concentrate energy to the vicinity of a point or a curve on the fluid surface. After initial focussing, other effects dominate and the patch breaks up into a complex set of localised structures–lumps and breathers–plus dispersive radiation. We perform simulations both in the inviscid regime and for small viscosities. Lastly we discuss throughout the similarities and differences between the dynamics of ripple patches and self-focussing light beams.

Super free fall of an inviscid liquid through interconnected vertical pipes

In this work the super free fall motion of the interface of a liquid column flowing downwards, suddenly from the rest in a system of vertical concentrically interconnected tubes, is studied theoretically, using a one-dimensional model in the inviscid regime. The ratio of the surface area of the upper tube to that of the lower one, , is assumed to be smaller than unity. Experiments show a good agreement with the main results of the developed theory.

High-order pole-treatment in cylindrical coordinates for incompressible flow simulations with finite-difference collocated schemes

A finite-difference collocated scheme is proposed for the simulation of unsteady incompressible flows in cylindrical coordinates. The Rhie–Chow flux-interpolation technique is used to avoid the odd–even decoupling of the pressure field. It is shown that the calculation of the Coriolis and centrifugal acceleration terms based on the cell-center values lead to numerical instabilities near the pole in the inviscid regime. To avoid this issue, a new formulation based on the combination of cell-center and cell-face velocity components is proposed for the calculation of these terms. Furthermore, a new pole treatment methodology is introduced to calculate the viscous terms, resulting single-valued properties at the pole. As regards the extensions of the present scheme to higher orders of accuracy, a fourth-order spatial discretization is also introduced and verified. Moreover, it is shown that for higher-order extensions, the present pole treatment is able to preserve the order of accuracy and the L2-norm and L∞-norm errors are greatly improved in comparison with the previous schemes. The robustness of the present method is assessed via five different numerical tests including laminar flows in a polar cavity, vortex-wall interaction, evolution of a monopole vortex-system to a tripole vortex-system, advection of the lamb dipole, and large eddy simulation (LES) of turbulent pipe flows. Compared to the previous LES results for turbulent pipe flows, significant improvement is obtained for the mean and fluctuating velocities especially at the core region of the channel. Similarly, at the wall region, the fluctuating axial and azimuthal velocities and the skewnesses of the axial and radial velocities are very close to the previous direct numerical simulation and experimental data.

Entraining gravity currents

AbstractEntrainment of ambient fluid into a gravity current, while often negligible in laboratory-scale flows, may become increasingly significant in large-scale natural flows. We present a theoretical study of the effect of this entrainment by augmenting a shallow water model for gravity currents under a deep ambient with a simple empirical model for entrainment, based on experimental measurements of the fluid entrainment rate as a function of the bulk Richardson number. By analysing long-time similarity solutions of the model, we find that the decrease in entrainment coefficient at large Richardson number, due to the suppression of turbulent mixing by stable stratification, qualitatively affects the structure and growth rate of the solutions, compared to currents in which the entrainment is taken to be constant or negligible. In particular, mixing is most significant close to the front of the currents, leading to flows that are more dilute, deeper and slower than their non-entraining counterparts. The long-time solution of an inviscid entraining gravity current generated by a lock-release of dense fluid is a similarity solution of the second kind, in which the current grows as a power of time that is dependent on the form of the entrainment law. With an entrainment law that fits the experimental measurements well, the length of currents in this entraining inviscid regime grows with time approximately as ${t}^{0. 447} $. For currents instigated by a constant buoyancy flux, a different solution structure exists in which the current length grows as ${t}^{4/ 5} $. In both cases, entrainment is most significant close to the current front.

Early regimes of capillary filling

In this paper we analyze the inviscid regime (for which viscosity is unimportant and the flow occurs due to the balance between the capillary and the inertial effects) that invariably precedes the classical century-old Washburn regime during capillary filling. We demonstrate that a new nondimensional number, namely, the product of the Ohnesorge number and the ratio between the filling length (ℓ) and the radius of the capillary (R), dictates the occurrence of this regime and the other well-known regimes in a capillary filling problem. We also identify that this inviscid regime occurs for the time that is of the order of the capillary time scale and, as has been quantified before [Quere, Eur. Phys. Lett. 39, 533 (1997); Joly, J. Chem. Phys. 135, 214705 (2011)], is characterized by the filling length being linearly proportional to the filling time. We establish the universality of this regime by pinpointing the existence of this regime (showing appropriate dependencies of the capillary radii and density) from existing experimental and Molecular Dynamics Simulation results that signify disparate ranges of length and time scales.

Compressible inviscid instability of rapidly expanding spherical material interfaces

A high-order weighted essentially non-oscillatory scheme is employed to investigate the stability of a rapidly expanding material interface produced by a spherical shock tube. The flow structure is characterized by a forward moving primary shock, a backward moving secondary shock, and a spherical contact interface in-between. We consider herein the linear inviscid regime and focus on the development of the three-dimensional perturbations around the contact interface by solving a one-dimensional system of partial differential equations. Numerical simulations are performed to illustrate the effects of the contact interface's density discontinuity on the growth of the disturbances for various spherical wave numbers. In a spherical shock tube the instability is influenced by various mechanisms which include classical Rayleigh-Taylor (RT) effects, Bell-Plesset or geometry/curvature effects, the effects of impulsively accelerating the interface, and compressibility effects. Henceforth, the present instability will be referred to as non-classical RT instability to distinguish it from classical RT instability. For an extended intermediate time period, it can be shown that the small disturbances grow exponentially as in the classical RT instability. During this stage, the exponential growth rate increases with the spherical wave number, until it saturates for very large wave numbers due to the finite thickness limitation of the numerical representation of the contact interface. The results compare favorably with previous theoretical models; but indicate that in addition to compressibility, the space-time evolution of the contact interface's thickness plays a significant role. A parametric study is performed that varies the pressure and density ratios of the initial spherical container. The characteristics of the contact interface and the applicability of various instability theories is investigated for these regimes. Furthermore, varying the pressure and density ratios aids in understanding significance of compressibility effects on the instability at different operating conditions.

On the interaction of an oblique shock with a flexible panel

In this communication, the dynamics of a flexible two-dimensional panel subjected to an impinging oblique shock is considered numerically in the inviscid regime. Computations have been performed for a Mach number equal to two and for several shock strengths employing a previously validated fluid–structural approach. Results show for the first time that for a sufficiently strong shock self-excited oscillations arise in the coupled system. Depending on the shock strength, either supercritical or subcritical bifurcations emerge at a value of dynamic pressure which can be considerably lower than that corresponding to standard panel flutter (i.e., in the absence of a shock). For fixed dynamic pressure, both the amplitude and frequency of the limit-cycle oscillations increase with the impinging shock strength. In addition to the incident and reflected oblique shocks, the unsteady flows are characterized by a shock at the plate leading edge and by a strong recompression and subsequent expansion near the trailing edge. The significant unsteadiness generated by the fluttering panel propagates along the reflected shock and trailing-edge recompression/expansion wave system with potential structural implications not only on the panel but on opposing surfaces in internal flow applications.

Instability development of a viscous liquid drop impacting a smooth substrate

We study the instability development during a viscous liquid drop impacting a smooth substrate, using high-speed photography. The onset time of the instability highly depends on the surrounding air pressure and the liquid viscosity: it decreases with air pressure with the power of minus two, and increases linearly with the liquid viscosity. From the real-time dynamics measurements, we construct a model which compares the destabilizing stress from air with the stabilizing stress from liquid viscosity. Under this model, our experimental results indicate that at the instability onset time, the two stresses balance each other. This model also illustrates the different mechanisms for the inviscid and viscous regimes previously observed: the inviscid regime is stabilized by the surface tension and the viscous regime is stabilized by the liquid viscosity.

Ultrafast X-Ray Phase-Contrast Imaging of the Initial Coalescence Phase of Two Water Droplets

We report an ultrafast x-ray phase-contrast imaging study of the early merging dynamics of two water drops in air. Owing to the edge-enhancement capability, the high penetrability, and the unprecedented temporal and spatial resolutions offered by this new x-ray technique, the coalescence singularity of two water drops was revisited. A finite initial contact radius was identified and the evolvement of the trapped toroidal air bubble was studied for the first time. Despite the existence of this finite initial contact radius, the subsequent meniscus radius followed power laws which agree with theoretical predictions for the inviscid regime.

Microjetting from wave focusing on oscillating drops

We present experimental observations of microjetting from an oscillating drop. The jet is generated by the focusing of axisymmetric capillary waves that overturn and collide at an apex of the drop. These jets are up to two orders of magnitude smaller than the original drops. We present two widely different configurations that produce such microjets. The first occurs on a satellite drop, produced by the pinch-off of a water drop from a vertical nozzle. The large oscillations following the contraction of the satellite bridge focus waves at the bottom, sending out a 30μm jet at 9.9m∕s. The second jet arises when a water drop, containing surfactants, falls onto and passes through a hemispherical soap film. The gentle deformation of the drop creates a surface wave that focuses at its top, shooting out a tiny jet and entrapping a small bubble inside the drop. This jet is 16±5μm in diameter and emerges at 6.3m∕s. In this configuration, the soap film wraps around the drop and acts as a sensor of the air flow, revealing that the liquid jet is preceded by a localized faster-moving air jet. The jetting in both configurations is quite robust and occurs even for slightly asymmetric conditions. These microjets appear for much lower values of the Reynolds and Weber numbers than previously observed, suggesting that free-surface jetting is not limited to the inviscid capillary-inertial regime, which has been the focus of much of the theoretical work.

Nearly Inviscid Faraday Waves in Slightly Rectangular Containers

In the weakly inviscid regime parametrically driven surface gravity-capillary waves generate oscillatory viscous boundary layers along the container walls and the free surface. Through nonlinear rectification these generate Reynolds stresses which drive a streaming flow in the nominally inviscid bulk; this flow in turn advects the waves responsible for the boundary layers. The resulting system is described by amplitude equations coupled to a Navier-Stokes-like equation for the bulk streaming flow, with boundary conditions obtained by matching to the boundary layers, and represents a novel type of pattern-forming system. The coupling to the streaming flow is responsible for new types of secondary instabilities of standing waves leading to chaotic dynamics, and in appropriate regimes can lead to the presence of relaxations oscillations. These are present because in the nearly inviscid regime the streaming flow decays much more slowly than the waves, and resemble a class of oscillations discovered by Simonelli and Gollub [J. Fluid Mech. 199 (1989), 349] in a domain with an almost square cross-section.

Dynamics of nearly inviscid Faraday waves in almost circular containers

Parametrically driven surface gravity-capillary waves in an elliptically distorted circular cylinder are studied. In the nearly inviscid regime, the waves couple to a streaming flow driven in oscillatory viscous boundary layers. In a cylindrical container, the streaming flow couples to the spatial phase of the waves, but in a distorted cylinder, it couples to their amplitudes as well. This coupling may destabilize pure standing oscillations, and lead to complex time-dependent dynamics at onset. Among the new dynamical behavior that results are relaxation oscillations involving abrupt transitions between standing and quasiperiodic oscillations, and exhibiting ‘canards’.