Flat Boundary Research Articles

A universal velocity profile for turbulent wall flows including adverse pressure gradient boundary layers

A recently developed mixing length model of the turbulent shear stress in pipe flow is used to solve the streamwise momentum equation for fully developed channel flow. The solution for the velocity profile takes the form of an integral that is uniformly valid from the wall to the channel centreline at all Reynolds numbers from zero to infinity. The universal velocity profile accurately approximates channel flow direct numerical simulation (DNS) data taken from several sources. The universal velocity profile also provides a remarkably accurate fit to simulated and experimental flat plate turbulent boundary layer data including zero and adverse pressure gradient data. The mixing length model has five free parameters that are selected through an optimization process to provide an accurate fit to data in the range $R_\tau = 550$ to $R_\tau = 17\,207$ . Because the velocity profile is directly related to the Reynolds shear stress, certain statistical properties of the flow can be studied such as turbulent kinetic energy production. The examples presented here include numerically simulated channel flow data from $R_\tau = 550$ to $R_\tau =8016$ , zero pressure gradient (ZPG) boundary layer simulations from $R_\tau =1343$ to $R_\tau = 2571$ , zero pressure gradient turbulent boundary layer experimental data between $R_\tau = 2109$ and $R_\tau = 17\,207$ , and adverse pressure gradient boundary layer data in the range $R_\tau = 912$ to $R_\tau = 3587$ . An important finding is that the model parameters that characterize the near-wall flow do not depend on the pressure gradient. It is suggested that the new velocity profile provides a useful replacement for the classical wall-wake formulation.

Understanding of adopting flat-top laser in laser powder bed fusion processed Inconel 718 alloy: simulation of single-track scanning and experiment

Laser beam intensity profile used in laser powder bed fusion (LPBF) process is generally Gaussian, which may not be the best to deliver site-specific microstructures. In this study, a flat-top laser with large-size laser spot and uniform intensity distribution is adopted in the LPBF process with Inconel 718 powder. Comparison investigations are conducted to the standard Gaussian laser. Firstly, the parameters for the flat-top laser are optimized by multi-physics simulation of single tracks. Then, the thermodynamic behaviors and morphologies of the molten pool are observed combined with experiments. A cubic sample is also fabricated utilizing double-laser system to reveal the different characteristics of the resultant microstructures. The results show that the flat-top laser can work at a relatively wider range of laser power without depression, which often occurs in the case of Gaussian laser. The molten pool induced by the flat-top laser has large width-depth ratio with a flat bottom boundary, and more moderate flow in the half bottom. These features combined with the temperature and flow fields of the flat-top laser provide advantageous conditions for directional epitaxial growth of columnar grains during multi-layer fabrication. The epitaxial growth is retained and can extends to 4 mm from the single-crystal baseplate. This demonstrates great potential of LPBF with the adopted flat-top laser to be applied in fabricating or repairing components with directional crystal structures or single-crystal structures.

Topographically Controlled Marangoni-Rayleigh-Bénard Convection in Liquid Metals

Convection induced in a layer of liquid with a top free surface by a distribution of heating elements at the bottom can be seen as a variant of standard Marangoni–Rayleigh–Bénard Convection where in place of a flat boundary at constant temperature delimiting the system from below, the underlying thermal inhomogeneity reflects the existence of a topography. In the present work, this problem is investigated numerically through solution of the governing equations for mass, momentum and energy in their complete, three-dimensional time-dependent and non-linear form. Emphasis is given to a class of liquids for which thermal diffusion is expected to dominate over viscous effects (liquid metals). Fixing the Rayleigh and Marangoni number to 104 and 5 × 103, respectively, the sensitivity of the problem to the geometrical, kinematic and thermal boundary conditions is investigated parametrically by changing: the number and spacing of heating elements, their vertical extension, the nature of the lateral boundary (solid walls or periodic boundary) and the thermal behavior of the portions of bottom wall between adjoining elements (assumed to be either adiabatic or at the same temperature of the hot blocks). It is shown that, like the parent phenomena, this type of thermal flow is extremely sensitive to the specific conditions considered. The topography can be used to exert a control on the emerging flow in terms of temporal response and patterning behavior.

Some geometrical and topological properties of DNNs' decision boundaries

Geometry and topology of decision regions are closely related with classification performance and robustness against adversarial attacks. In this paper, we use differential geometry to theoretically explore the geometrical and topological properties of decision regions produced by deep neural networks (DNNs). The goal is to obtain some geometrical and topological properties of decision boundaries for given DNN models, and provide some principled guidance to design and regularization of DNNs. First, we present the curvatures of decision boundaries in terms of network parameters, and give sufficient conditions on network parameters for producing flat or developable decision boundaries. Based on the Gauss-Bonnet-Chern theorem in differential geometry, we then propose a method to compute the Euler characteristics of compact decision boundaries, and verify it with experiments.

Error estimation of a homogenized streamwise periodic boundary layer

This work concerns the simulation of a homogenized streamwise periodic boundary layer which reduces the computational cost of direct numerical simulations of incompressible flat plate turbulent boundary layers. It expands upon the understanding of how the transpiration velocity changes dependent on whether a blending function is used to account for inner and outer self-similar scaling effects. With this rescaling correction, lower Reynolds number effects on the transpiration velocity are captured by the homogenized streamwise periodic boundary layer simulation.

High-speed multiview imaging approaching 4pi steradians using conic section mirrors: theoretical and practical considerations

Illuminating or imaging samples from a broad angular range is essential in a wide variety of computational 3D imaging and resolution-enhancement techniques, such as optical projection tomography, optical diffraction tomography, synthetic aperture microscopy, Fourier ptychographic microscopy, structured illumination microscopy, photogrammetry, and optical coherence refraction tomography. The wider the angular coverage, the better the resolution enhancement or 3D-resolving capabilities. However, achieving such angular ranges is a practical challenge, especially when approaching ± 90 ∘ or beyond. Often, researchers resort to expensive, proprietary high numerical aperture (NA) objectives or to rotating the sample or source-detector pair, which sacrifices temporal resolution or perturbs the sample. Here, we propose several new strategies for multiangle imaging approaching 4pi steradians using concave parabolic or ellipsoidal mirrors and fast, low rotational inertia scanners, such as galvanometers. We derive theoretically and empirically relations between a variety of system parameters (e.g., NA, wavelength, focal length, telecentricity) and achievable fields of view (FOVs) and importantly show that intrinsic tilt aberrations do not restrict FOV for many multiview imaging applications, contrary to conventional wisdom. Finally, we present strategies for avoiding spherical aberrations at obliquely illuminated flat boundaries. Our simple designs allow for high-speed multiangle imaging for microscopic, mesoscopic, and macroscopic applications.

Development of a snowdrift model with the lattice Boltzmann method

We developed a snowdrift model to evaluate the snowdrift height around snow fences, which are often installed along roads in snowy, windy locations. The model consisted of the conventional computational fluid dynamics solver that used the lattice Boltzmann method and a module for calculating the snow particles’ motion and accumulation. The calculation domain was a half channel with a flat free-slip boundary on the top and a non-slip boundary on the bottom, and an inflow with artificially generated turbulence from one side to the outlet side was imposed. In addition to the reference experiment with no fence, experiments were set up with a two-dimensional and a three-dimensional fence normal to the dominant wind direction in the channel center. The estimated wind flow over the two-dimensional fence was characterized by a swirling eddy in the cross section, whereas the wind flow in the three-dimensional fence experiment was horizontally diffluent with a dipole vortex pair on the leeward side of the fence. Almost all the snowdrift formed on the windward side of the two-dimensional and three-dimensional fences, although the snowdrift also formed along the split streaks on the leeward side of the three-dimensional fence. Our results suggested that the fence should be as long as possible to avoid snowdrifts on roads.

Quasi-symmetric mappings and their generalizations on Riemannian manifolds

A relation between 𝜂-quasi-symmetric homomorphisms and K-quasiconformal mappings on n-dimensional smooth connected Riemannian manifolds has been studied. The main results of the research are presented in Theorems 2.6 and 2.7. Several conditions for the boundary behavior of 𝜂-quasi-symmetric homomorphisms between two arbitrary domains with weakly flat boundaries and compact closures, QED and uniform domains on the Riemannian manifolds, which satisfy the obtained results, were also formulated. In addition, quasiballs, c-locally connected domains, and the corresponding results were also considered.

An evaporation flux of pure vapor in the method of lattice Boltzmann equations

The regularities of the evaporation flux of pure vapor in the method of lattice Boltzmann equations (LBE) are investigated. The simulations show that the mass flux during the evaporation of a flat surface is proportional to the difference in the densities of the saturated vapor at the surface temperature and surrounding vapor, which is in good agreement with the Hertz–Knudsen law. A simple method is proposed for setting the vapor flow at the flat boundary of the computational domain for the LBE method.

Perturbative estimates for the one-phase Stefan problem

We provide perturbative estimates for the one-phase Stefan free boundary problem and obtain the regularity of flat free boundaries via a linearization technique in the spirit of the elliptic counterpart established in De Silva (IFB 13, 223–238, 2011).

Normal modes with boundary dynamics in geophysical fluids

Three-dimensional geophysical fluids support both internal and boundary-trapped waves. To obtain the normal modes in such fluids, we must solve a differential eigenvalue problem for the vertical structure (for simplicity, we only consider horizontally periodic domains). If the boundaries are dynamically inert (e.g., rigid boundaries in the Boussinesq internal wave problem and flat boundaries in the quasigeostrophic Rossby wave problem), the resulting eigenvalue problem typically has a Sturm–Liouville form and the properties of such problems are well-known. However, when restoring forces are also present at the boundaries, then the equations of motion contain a time-derivative in the boundary conditions, and this leads to an eigenvalue problem where the eigenvalue correspondingly appears in the boundary conditions. In certain cases, the eigenvalue problem can be formulated as an eigenvalue problem in the Hilbert space L2⊕C and this theory is well-developed. Less explored is the case when the eigenvalue problem takes place in a Pontryagin space, as in the Rossby wave problem over sloping topography. This article develops the theory of such problems and explores the properties of wave problems with dynamically active boundaries. The theory allows us to solve the initial value problem for quasigeostrophic Rossby waves in a region with sloping bottom (we also apply the theory to two Boussinesq problems with a free-surface). For a step-function perturbation at a dynamically active boundary, we find that the resulting time-evolution consists of waves present in proportion to their projection onto the dynamically active boundary.

Microstructure refinement mechanism of undercooled Cu55Ni45 alloys

Abstract Different undercooling degrees of Cu55Ni45 alloy were obtained by the combination of molten glass purification and cyclic superheating, and the maximum undercooling degree reached 284 K. The microstructure of the alloy was observed by metallographic microscope, and the evolution of microstructure was studied systematically. There are two occasions of grain refinement in the solidification structure of the alloy: one occurs in the case of low undercooling, and the other occurs in the case of high undercooling. Electron backscatter diffraction (EBSD) technology was used to analyze the rapid solidification structure under high undercooling. The features of flat polygonal grain boundary, high proportion of twin boundary, and large proportion of large angle grain boundary indicate recrystallization. The change in microhardness of the alloy under different undercooling degrees was studied by microhardness tester. It was found that the average microhardness decreased sharply at high undercooling degrees, which further confirmed the recrystallization of the solidified structure at high undercooling degrees.

Celestial IR divergences and the effective action of supertranslation modes

Infrared divergences in perturbative gravitational scattering amplitudes have been recently argued to be governed by the two-point function of the supertranslation Goldstone mode on the celestial sphere. We show that the form of this celestial two-point function simply derives from an effective action that also controls infrared divergences in the symplectic structure of General Relativity with asymptotically flat boundary conditions. This effective action finds its natural place in a path integral formulation of a celestial conformal field theory, as we illustrate by re-deriving the infrared soft factors in terms of celestial correlators. Our analysis relies on a well-posed action principle close to spatial infinity introduced by Compère and Dehouck.

Geometry and entanglement in the scattering matrix

A formulation of nucleon–nucleon scattering is developed in which the S-matrix, rather than an effective-field theory (EFT) action, is the fundamental object. Spacetime plays no role in this description: the S-matrix is a trajectory that moves between RG fixed points in a compact theory space defined by unitarity. This theory space has a natural operator definition, and a geometric embedding of the unitarity constraints in four-dimensional Euclidean space yields a flat torus, which serves as the stage on which the S-matrix propagates. Trajectories with vanishing entanglement are special geodesics between RG fixed points on the flat torus, while entanglement is driven by an external potential. The system of equations describing S-matrix trajectories is in general complicated, however the very-low-energy S-matrix –that appears at leading-order in the EFT description– possesses a UV/IR conformal invariance which renders the system of equations integrable, and completely determines the potential. In this geometric viewpoint, inelasticity is in correspondence with the radius of a three-dimensional hyperbolic space whose two-dimensional boundary is the flat torus. This space has a singularity at vanishing radius, corresponding to maximal violation of unitarity. The trajectory on the flat torus boundary can be explicitly constructed from a bulk trajectory with a quantifiable error, providing a simple example of a holographic quantum error correcting code.

Effect of various combinations of Ti and Zr interlayers on the tensile properties of laser welded joints of molybdenum

Molybdenum (Mo) is of great potential in the nuclear energy field, however, the embrittlement and the lower tensile strength limit the application of the fusion welded joints. Titanium (Ti) and zirconium (Zr) were synchronously added into the fusion zone (FZ) to improve the tensile strength and certain ductility of laser beam welding joints. By utilizing high-resolution scanning electron microscopy (SEM), electron backscatter diffraction (EBSD) and nanoindentation, the microstructures and mechanical properties of FZs in the joints under different alloying strategies with Ti and Zr were analyzed. Zr is mainly present in Mo as Mo2Zr, playing the second-phase strengthening effect; Ti is mainly present in Mo in an atomic state, playing a role of the solid - solution strengthening. The addition of Ti and Zr with different mass proportions can control the relative contents of Mo2Zr phase and solid - solution atoms in the FZ, control the distribution of zigzag and flat grain boundaries and influence the relative distributions of low - angle grain boundaries (LAGBs) and high - angle grain boundaries (HAGBs). In the end, combined addition of Ti and Zr is possible to improve the tensile strength while decrease the loss of ductility of the joint.

Non-Equilibrium Crystallization of Monotectic Zn-25%Bi Alloy under 600 g.

This study investigated the influence of supergravity on the segregation of components in the Zn–Bi monotectic system and consequently, the creation of an interface of the separation zone of both phases. The observation showed that near the separation boundary, in a very narrow area of the order of several hundred microns, all types of structures characteristic for the concentration range from 0 to 100% bismuth occurred. An additional effect of crystallization in high gravity is a high degree of structural order and an almost perfectly flat separation boundary. This is the case for both the zinc-rich zone and the bismuth-rich zone. Texture analysis revealed the existence of two privileged orientations in the zinc zone. Gravitational segregation also resulted in a strong rearrangement of the heavier bismuth to the outer end of the sample, leaving only very fine precipitates in the zinc region. For comparison, the results obtained for the crystallization under normal gravity are given. The effect of high orderliness of the structure was then absent. Despite segregation, a significant part of bismuth remained in the form of precipitates in the zinc matrix, and the separation border was shaped like a lens. The described method can be used for the production of massive bimaterials with a directed orientation of both components and a flat interface between them, such as thermo-generator elements or bimetallic electric cell parts, where the parameters (thickness) of the junction can be precisely defined at the manufacturing stage.

Reflection of coupled waves from the flat boundary surface of a nonlocal micropolar thermoelastic half-space containing voids

Reflection behavior of a set of coupled longitudinal/transverse waves incident obliquely at the flat boundary surface of a nonlocal micropolar thermoelastic half-space containing voids is investigated. Appropriate boundary conditions are setup for mechanically stress-free and thermally insulated boundary surface of the half-space. The equations yielding the reflection coefficients and the expressions of corresponding energy ratios of various reflected waves are presented. The nature of dependence of the reflection coefficients and their corresponding energy ratios against the angle of incidence is studied numerically for a specific model. Effect of several parameters on the reflection coefficients as well as on the corresponding energy ratios is noticed and depicted graphically. It is found that the sum of the energy ratios is approximately equal to unity at each angle of incidence. Some earlier known results are also recovered as special cases of the present formulation.

Effect of Manufacturing on the Transverse Response of Polymer Matrix Composites.

The effect of residual stress build-up on the transverse properties of thermoset composites is studied through direct and inverse process modeling approaches. Progressive damage analysis is implemented to characterize composite stiffness and strength of cured composites microstructures. A size effect study is proposed to define the appropriate dimensions of Representative Volume Elements (RVEs). A comparison between periodic (PBCs) and flat (FBCs) boundary conditions during curing is performed on converged RVEs to establish computationally efficient methodologies. Transverse properties are analyzed as a function of the fiber packing through the nearest fiber distance statistical descriptor. A reasonable mechanical equivalence is achieved for RVEs consisting of 40 fibers. It has been found that process-induced residual stresses and fiber packing significantly contribute to the scatter in composites transverse strength. Variation of ±5% in average strength and 18% in standard deviation are observed with respect to ideally cured RVEs that neglect residual stresses. It is established that process modeling is needed to optimize the residual stress state and improve composite performance.

Intensity of scattered wave energy in the slab having a discontinuity

The main purpose of this study is to explore the intensity of the scattered wave energy and the total intensity scattered around a discontinuity in a slab, considering strain and kinetic energy in the case of SH waves. The problem is very intriguing because of the multiple scattering and reflections from both the failure and the traction-free boundaries of the slab. The lowest mode of the wave is considered to decrease the complexity of the problem. Increase in frequency results in the growth of chaotic patterns with additional side lobes appearing along with the backward and forward peaks. Reflected waves of the flat boundaries in the slab disappear on the discontinuity when the thickness of the slab increases, and the results approach the infinite medium solution. However, when the size of the failure is comparable to the thickness of the slab, their effect dominates the intensity. In addition, resonance and focusing occur depending on the wavelengths and the thickness of the slab. The biggest increase in the intensity of scattered energy is especially detected on the upper and lower side of the discontinuity. All these effects of SH waves on the slab with a singularity are calculated and presented for various frequency ranges.

The role of dimensionality and geometry in quench-induced nonequilibrium forces

We present an analytical formalism, supported by numerical simulations, for studying forces that act on curved walls following temperature quenches of the surrounding ideal Brownian fluid. We show that, for curved surfaces, the post-quench forces initially evolve rapidly to an extremal value, whereafter they approach their steady state value algebraically in time. In contrast to the previously-studied case of flat boundaries (lines or planes), the algebraic decay for curved geometries depends on the dimension of the system. Specifically, steady-state values of the force are approached in time as t −d/2 in d-dimensional spherical (curved) geometries. For systems consisting of concentric circles or spheres, the exponent does not change for the force on the outer circle or sphere. However, the force exerted on the inner circles or sphere experiences an overshoot and, as a result, does not evolve to the steady state in a simple algebraic manner. The extremal value of the force also depends on the dimension of the system, and originates from curved boundaries and the fact that particles inside a sphere or circle are locally more confined, and diffuse less freely than particles outside the circle or sphere.