Planar Case Research Articles

Aeroelastic Effects in Supersonic Shock-Turbulent Boundary Layer Interaction over Flexible Panels

The dynamic coupling between a Mach 1.94 shock wave/turbulent boundary layer interaction (SBLI) and a flexible panel is investigated. High-order numerical simulations are performed for distinctly different dynamic panel motions and rigid snapshots of their maximum deflected shape. They are compared with a baseline interaction over a rigid planar wall. The panel’s dynamic surface motions were obtained from the Air Force Research Laboratory (AFRL) wind tunnel experiments. The primary aim of the study was to determine whether there were any differences in the flow pressure loading on the compliant panel due to the various rigid and dynamic deformations considered. The results show that the examined panel deformations increase the SBLI size near the panel midpoint, where the deformation amplitude tends to be the largest. Relative to the rigid planar case, the examined surface deformations cause the mean-flow high-pressure surface loading caused by the impinging shock wave to shift downstream along the compliant panel midspan, albeit by a small amount. The spectrogram of the dynamic deformation and the flow surface pressure response suggests that the two are strongly coupled at the dominant (primary) mode but less so at the secondary modes. Although the primary mode frequencies overlap, they do not exactly match, with the pressure response frequency always being slightly higher in all three cases. The rigid deformations did not enhance the pressure power content at the SBLI. However, pre-SBLI and near the panel leading edge, the pressure power spectrum weakly increased throughout the resolved frequency range and overlapped with the onset of the amplification found in the dynamic deformation cases. Post-SBLI, the rigid deformations cause a weak enhancement at frequencies below 1 kHz, which closely match the dominant and secondary pressure response frequencies obtained in the dynamic cases.

АНАЛІЗ ВПЛИВУ КУТА СТРІЛОПОДІБНОСТІ НА ХАРАКТЕРИСТИКИ СТАТИЧНОЇ АЕРОПРУЖНОСТІ КРИЛА НАВЧАЛЬНО-ТРЕНУВАЛЬНОГО ЛІТАКА

The article is dedicated to the analysis of static aeroelastic phenomena in the wing of a trainer aircraft and the general influence of wing sweep on critical speeds. Aeroelasticity studies the interaction between aerodynamic and elastic forces and the impact of this interaction on the aircraft's structure. There are two types of aeroelasticity: static and dynamic. The article reviews and analyzes publications related to this issue. The analysis of the research shows that when designing an aircraft, it is necessary to determine the aeroelastic characteristics of its aerodynamic surfaces. In addition to structural rigidity, the sweep angle also influences these phenomena. The scientific novelty of the study lies in the fact that, unlike the analyzed research, the designed wing structure of a trainer aircraft is considered.For the designed trainer aircraft, the critical speeds of static aeroelastic phenomena were determined. The calculation of the influence of the wing sweep angle on the divergence speeds of the trainer aircraft confirms previously obtained results regarding this influence. When the sweep is increased or decreased for wings with the same span, the torsional rigidity of the wing decreases due to the increase in the length of the line of centers of rigidity. However, the most significant impact on the aeroelastic characteristics is made by the shift of the bending center when the sweep angle changes.With forward sweep, the arm of the aerodynamic forces relative to the bending center increases, thereby reducing the critical divergence speeds. For straight sweep, as the sweep increases, the bending center moves forward, and at a certain point, the bending center is located in front of the aerodynamic center. This means that the wing twist angle decreases as the aerodynamic force increases, which, in the considered planar case, eliminates the occurrence of divergence.The preliminarily determined maximum flight speed does not exceed the critical speeds of divergence and aileron reversal. This result represents the practical value of the study. To achieve higher flight speeds, wings with straight sweep should be used. More precise computational or experimental methods for determining aeroelastic characteristics should be used in the subsequent design stages to prevent aeroelastic phenomena across the entire operational range of speeds and altitudes of the trainer aircraft.

Classification of Solutions to the Anisotropic N-Liouville Equation in ℝN

Abstract Given $N\geq 2$, we completely classify solutions to the anisotropic $N$-Liouville equation $$ \begin{align*} &-\Delta_N^H\,u=e^u \quad\textrm{in}\ \mathbb{R}^N,\end{align*} $$ under the finite mass condition $\int _{\mathbb{R}^{N}} e^{u}\,dx<+\infty $. Here $\Delta _{N}^{H}$ is the so-called Finsler $N$-Laplacian induced by a positively homogeneous function $H$. As a consequence in the planar case $N=2$, we give an affirmative answer to a conjecture made in [ 53].

Size and velocity of jet drops produced by bursting bubbles at the interface of a water jet

Bursting bubbles at the free surface of aerated faucet water jets may spread pathogens through the released droplets. Many studies focused on the production of jet drops from bursting bubbles at a planar interface, particularly for the first jet drop. The extent to which previous findings apply to bubbles in aerated jets remains unknown. In this study, we produce a wide range of bubble size distributions within different jet diameters and characterize the diameter and velocity of jet drops released from individually bursting bubbles. Several similarities with the planar case are recovered, such as the overall dependence of the jet drop diameter and bursting dynamics on the bubble diameters and the formation of secondary jet drops. However, we observe asymmetries in the ejection of the droplets, and droplets ejected horizontally appear to have the highest ejection velocity among all jet drops. By modeling the evolution of the ejected drops for the different bubble size distributions, we find that for a mean Laplace number Labub=ρwσwRbubμw2≲6×104, a fraction of the drops ejected can become airborne. Droplets deposit within 9 cm for a mean Labub≲2.1×104 and within 33 cm for a mean 2.1×104≲Labub≲1.8×105 from a faucet jet, assuming a countertop situated 20 cm below the faucet outlet. A bubble size distribution with a mean Labub of 6×104 would minimize both the risk of airborne pathogen transmission and that resulting from surface contamination.

Local dimer dynamics in higher dimensions

We consider local dynamics of the dimer model (perfect matchings) on hypercubic boxes [n]^{d} . These consist of successively switching the dimers along alternating cycles of prescribed (small) lengths. We study the connectivity properties of the dimer configuration space equipped with these transitions. Answering a question of Freire, Klivans, Milet, and Saldanha, we show that in three dimensions any configuration admits an alternating cycle of length at most 6. We further establish that any configuration on [n]^{d} features order n^{d-2} alternating cycles of length at most 4d-2 . We also prove that the dynamics of dimer configurations on the unit hypercube of dimension d is ergodic when switching alternating cycles of length at most 4d-4 . Finally, in the planar but non-bipartite case, we show that parallelogram-shaped boxes in the triangular lattice are ergodic for switching alternating cycles of lengths 4 and 6 only, thus improving a result of Kenyon and Rémila, which also uses 8-cycles. None of our proofs make reference to height functions.

(Invited) Developing Periodically-Aligned Fibrous Structures to Suppress Lithium Dendrites for Anode-Free Batteries

We have developed electrospinning-based manufacturing capability for dendrite-free anode-free and anode-less batteries. Electrospinning is a widely used, feasible manufacturing approach to create nano- and micro-porous layers of functional fibers. By tailoring fiber compositions, the electrospun layers can afford a wide variety of functionalities applicable to biomedical templates, separation membranes, and energy storage. Its manufacturing capability is, however, limited to producing randomly oriented fibrous structures. Topology and tortuosity of the electrospun fibrous layers are poorly controlled, making it difficult to systematically investigate structure-property relationships for any given applications, especially electrochemical systems.In this presentation, we will demonstrate how to improve the controllability of fiber construction geometry by electrospinning. Unlike conventional electrospinning, our approach employs a mobile stage that allows precise alignment of fibers. Produced fibrous structures will be used to construct current collectors of the anode-free batteries, one of the proposed beyond-lithium (Li)-ion battery systems for high energy density. While direct, yet reversible, Li plating and stripping on and from the current collector has proven difficult due to uncontrolled dendrite growth, we found that the flat copper foil reinforced by the three-dimensional fibrous structure can enhance Li storage efficiency over an extended number of cycles, outperforming the planar case. We will discuss the effect of fiber compositions and controlled geometrical configurations on stabilizing Li plating and stripping morphologies to suppress Li dendrite growth and provide a fundamental design principle to construct anode-free and anode-less battery cells. Battery manufacturing advanced by this work will offer a systematic strategy to develop next-generation energy storage systems for a sustainable energy future.

Shape optimization for a nonlinear elliptic problem related to thermal insulation

In this paper, we consider a minimization problem of a nonlinear functional I_{\beta,p}(D,\Omega) related to a thermal insulation problem with a convection term, where \Omega is a bounded connected open set in \R^{n} and D\subset\overline{\Omega} is a compact set. The Euler–Lagrange equation relative to I_{\beta,p} is a p -Laplace equation, 1<p<\infty , with a Robin boundary condition with parameter \beta>0 . The main aim is to study extremum problems for I_{\beta,p}(D,\Omega) , among domains D with given geometrical constraints and \Omega\setminus D of fixed thickness. In the planar case, we show that under perimeter constraint the disk maximizes I_{\beta,p} . In the n -dimensional case we restrict our analysis to convex sets showing that the same is true for the ball but under different geometrical constraints.

Schrödinger–Poisson systems with zero mass in the Sobolev limiting case

AbstractWe study the existence of positive solutions for a class of systems which strongly couple a quasilinear Schrödinger equation driven by a weighted ‐Laplace operator and without the mass term, and a higher‐order fractional Poisson equation. Since the system is set in , the limiting case for the Sobolev embedding, we consider nonlinearities with exponential growth. Existence is proved relying on the study of a corresponding Choquard equation in which the Riesz kernel is a logarithm, hence sign‐changing and unbounded from above and below. This is in turn solved by means of a variational approximating procedure for an auxiliary Choquard equation, where the logarithm is uniformly approximated by polynomial kernels. Our results are new even in the planar case .

Weak solvability of one model of a nonlinearly retarded fluid in a thermal field

For the initial-boundary value problem of the dynamics of a thermoviscoelastic medium of Oldroyd type in the planar case, a nonlocal theorem regarding the existence of a weak solution is established.

Random walks and the “Euclidean” association scheme in finite vector spaces

Abstract In this paper, we provide an application to the random distance-t walk in finite planes and derive asymptotic formulas (as $q \to \infty $ ) for the probability of return to start point after $\ell $ steps based on the “vertical” equidistribution of Kloosterman sums established by N. Katz. This work relies on a “Euclidean” association scheme studied in prior work of W. M. Kwok, E. Bannai, O. Shimabukuro, and H. Tanaka. We also provide a self-contained computation of the P-matrix and intersection numbers of this scheme for convenience in our application as well as a more explicit form for the intersection numbers in the planar case.

An innovative approach to planar mixed-mode fatigue crack growth study

Determining fatigue crack growth parameters under mixed-mode loading is crucial for damage tolerance design, particularly in assessing a component's residual life with defects to prevent disasters. This study introduces a novel methodology for determining fatigue crack growth rate. A digital microscope is utilized to track the fatigue crack tip with respect to local coordinate systems established on the specimen surface. A grid is drawn and subdivided into units with proposed nomenclature. Compact Tension and Shear (CTS) specimens, extracted from AA-7085 plates, undergo fatigue loading in a servo-hydraulic universal testing machine. A fixture proposed by Richard et al. induces planar mixed-mode loading cases in the specimen. During experiments, the digital microscope tracks the crack tip using grid unit corners as local origins. The local coordinates of the crack tip are transformed into the crack tip coordinates with respect to the pre-crack tip. Crack evolution data are used to determine the fatigue crack growth rate by analyzing tangents of a three-parameter fitted curve. The equivalent SIF ranges are computed at different crack lengths in an XFEM script written in Ansys. The modified Paris parameters are found and experimentally observed crack deflection angles are compared with the Maximum Tangential stress, Strain energy density, and Richard’s criterion.

Hadamard’s inequality in the mean

Let Q be a Lipschitz domain in Rn and let f∈L∞(Q). We investigate conditions under which the functional In(φ)=∫Q|∇φ|n+f(x)det∇φdx obeys In≥0 for all φ∈W01,n(Q,Rn), an inequality that we refer to as Hadamard-in-the-mean, or (HIM). We prove that there are piecewise constant f such that (HIM) holds and is strictly stronger than the best possible inequality that can be derived using the Hadamard inequality nn2|detA|≤|A|n alone. When f takes just two values, we find that (HIM) holds if and only if the variation of f in Q is at most 2nn2. For more general f, we show that (i) it is both the geometry of the ‘jump sets’ as well as the sizes of the ‘jumps’ that determine whether (HIM) holds and (ii) the variation of f can be made to exceed 2nn2, provided f is suitably chosen. Specifically, in the planar case n=2 we divide Q into three regions {f=0} and {f=±c}, and prove that as long as {f=0} ‘insulates’ {f=c} from {f=−c} sufficiently, there is c>2 such that (HIM) holds. Perhaps surprisingly, (HIM) can hold even when the insulation region {f=0} enables the sets {f=±c} to meet in a point. As part of our analysis, and in the spirit of the work of Mielke and Sprenger (1998), we give new examples of functions that are quasiconvex at the boundary.

Periodic partitions with minimal perimeter

We show existence of fundamental domains which minimize a general perimeter functional in a homogeneous metric measure space. In some cases, which include the usual perimeter in the universal cover of a closed Riemannian manifold, and the fractional perimeter in Rn, we can prove regularity of the minimal domains. As a byproduct of our analysis we obtain that a countable partition which is minimal for the fractional perimeter is locally finite and regular, extending a result previously known for the local perimeter. Finally, in the planar case we provide a detailed description of the fundamental domains which are minimal for a general anisotropic perimeter.

Non-integrability of the restricted three-body problem

Abstract The problem of non-integrability of the circular restricted three-body problem is very classical and important in the theory of dynamical systems. It was partially solved by Poincaré in the nineteenth century: he showed that there exists no real-analytic first integral which depends analytically on the mass ratio of the second body to the total and is functionally independent of the Hamiltonian. When the mass of the second body becomes zero, the restricted three-body problem reduces to the two-body Kepler problem. We prove the non-integrability of the restricted three-body problem both in the planar and spatial cases for any non-zero mass of the second body. Our basic tool of the proofs is a technique developed here for determining whether perturbations of integrable systems which may be non-Hamiltonian are not meromorphically integrable near resonant periodic orbits such that the first integrals and commutative vector fields also depend meromorphically on the perturbation parameter. The technique is based on generalized versions due to Ayoul and Zung of the Morales–Ramis and Morales–Ramis–Simó theories. We emphasize that our results are not just applications of the theories.

Active smectics on a sphere

The dynamics of active smectic liquid crystals confined on a spherical surface is explored through an active phase field crystal model. Starting from an initially randomly perturbed isotropic phase, several types of topological defects are spontaneously formed, and then annihilate during a coarsening process until a steady state is achieved. The coarsening process is highly complex involving several scaling laws of defect densities as a function of time where different dynamical exponents can be identified. In general the exponent for the final stage towards the steady state is significantly larger than that in the passive and in the planar case, i.e. the coarsening is getting accelerated both by activity and by the topological and geometrical properties of the sphere. A defect type characteristic for this active system is a rotating spiral of evolving smectic layering lines. On a sphere this defect type also determines the steady state. Our results can in principle be confirmed by dense systems of synthetic or biological active particles.

On the stability of a pair of vortex rings

The growth of perturbations subject to the Crow instability along two vortex rings of equal and opposite circulation undergoing a head-on collision is examined. Unlike the planar case for semi-infinite line vortices, the zero-order geometry of the flow (i.e. the ring radius, core thickness and separation distance) and by extension the growth rates of perturbations vary in time. The governing equations are therefore temporally integrated to characterize the perturbation spectrum. The analysis, which considers the effects of ring curvature and the distribution of vorticity within the vortex cores, explains several key flow features observed in experiments. First, the zero-order motion of the rings is accurately reproduced. Next, the predicted emergent wavenumber, which sets the number of secondary vortex structures emerging after the cores come into contact, agrees with experiments, including the observed increase in the number of secondary structures with increasing Reynolds number. Finally, the analysis predicts an abrupt transition at a critical Reynolds number to a regime dominated by a higher-frequency, faster-growing instability mode that may be consistent with the experimentally observed rapid generation of a turbulent puff following the collision of rings at high Reynolds numbers.

Existence and approximation of densities of chord length- and cross section area distributions

In various stereological problems an $n$-dimensional convex body is intersected with an $(n-1)$-dimensional Isotropic Uniformly Random (IUR) hyperplane. In this paper the cumulative distribution function associated with the $(n-1)$-dimensional volume of such a random section is studied. This distribution is also known as chord length distribution and cross section area distribution in the planar and spatial case respectively. For various classes of convex bodies it is shown that these distribution functions are absolutely continuous with respect to Lebesgue measure. A Monte Carlo simulation scheme is proposed for approximating the corresponding probability density functions.

Coloring lines and Delaunay graphs with respect to boxes

AbstractThe goal of this paper is to show the existence (using probabilistic tools) of configurations of lines, boxes, and points with certain interesting combinatorial properties. (i) First, we construct a family of lines in whose intersection graph is triangle‐free of chromatic number . This improves the previously best known bound by Norin, and is also the first construction of a triangle‐free intersection graph of simple geometric objects with polynomial chromatic number. (ii) Second, we construct a set of points in , whose Delaunay graph with respect to axis‐parallel boxes has independence number at most . This extends the planar case considered by Chen, Pach, Szegedy, and Tardos.

Approximating Maximum Integral Multiflows on Bounded Genus Graphs

We devise the first constant-factor approximation algorithm for finding an integral multi-commodity flow of maximum total value for instances where the supply graph together with the demand edges can be embedded on an orientable surface of bounded genus. This extends recent results for planar instances. Our techniques include an uncrossing algorithm, which is significantly more difficult than in the planar case, a partition of the cycles in the support of an LP solution into free homotopy classes, and a new rounding procedure for freely homotopic non-separating cycles.

A few words about maps

In this paper, we survey some properties, encoding, and bijections involving combinatorial maps, double occurrence words, and chord diagrams. We particularly study quasi-trees from a purely combinatorial point of view and derive a topological representation of maps with a given spanning quasi-tree using two fundamental polygons, which extends the representation of planar maps based on the equivalence with bipartite circle graphs. Then, we focus on Depth-First Search trees and their connection with a poset we define on the spanning quasi-trees of a map. We apply the bijections obtained in the first section to the problem of enumerating loopless rooted maps. Finally, we return to the planar case and discuss a decomposition of planar rooted loopless maps and its consequences on planar rooted loopless map enumeration.