Spherical Case Research Articles

Unsteady Flow Structures and Pressure Pulsations in a Nuclear Reactor Coolant Pump With Spherical Casing

Severe vibrations induced by flow instabilities in the nuclear reactor coolant pump (RCP) are detrimental to the safe operation of the pump. Due to the particular spherical casing in the RCP, the internal flow structures are extremely ambiguity and complicated. The goal of the present work is to shed comprehensive light on the unsteady flow structures and its correlation with the pressure pulsations by using large eddy simulation (LES) method of the RCP. The vorticity distribution and the shedding vortex from the blade trailing edge are depicted in detail. Furthermore, the internal correlations between the flow unsteadiness and pressure pulsation are illustrated in some special regions of the RCP. Evidently, some main excitation components in the pressure spectra are excited by the shedding vortex. Besides, components at blade passing frequency (fBPF) are closely associated with rotor–stator interaction between the wake flow from the impeller outlet and unsteadiness vortexes shedding from the diffuser blade trailing edge. It is thought to be that the pressure pulsations of the RCP are closely associated with the corresponding vorticity distribution and the unsteady vortex shedding effect.

The General Projected Normal Distribution of Arbitrary Dimension: Modeling and Bayesian Inference

The general projected normal distribution is a simple and intuitive model for directional data in any dimension: a multivariate normal random vector divided by its length is the projection of that vector onto the surface of the unit hypersphere. Observed data consist of the projections, but not the lengths. Inference for this model has been restricted to the two-dimensional (circular) case, using Bayesian methods with data augmentation to generate the latent lengths and a Metropolis-within-Gibbs algorithm to sample from the posterior. We describe a new parameterization of the general projected normal distribution that makes inference in any dimension tractable, including the important three-dimensional (spherical) case, which has not previously been considered. Under this new parameterization, the full conditionals of the unknown parameters have closed forms, and we propose a new slice sampler to draw the latent lengths without the need for rejection. Gibbs sampling with this new scheme is fast and easy, leading to improved Bayesian inference; for example, it is now feasible to conduct model selection among complex mixture and regression models for large data sets. Our parameterization also allows straightforward incorporation of covariates into the covariance matrix of the multivariate normal, increasing the ability of the model to explain directional data as a function of independent regressors. Circular and spherical cases are considered in detail and illustrated with scientific applications. For the circular case, seasonal variation in time-of-day departures of anglers from recreational fishing sites is modeled using covariates in both the mean vector and covariance matrix. For the spherical case, we consider paired angles that describe the relative positions of carbon atoms along the backbone chain of a protein. We fit mixtures of general projected normals to these data, with the best-fitting mixture accurately describing biologically meaningful structures including helices, β-sheets, and coils and turns. Finally, we show via simulation that our methodology has satisfactory performance in some 10-dimensional and 50-dimensional problems.

An equivalent system of Einstein Equations

We provide the full set of equations governing the evolution of self-gravitating systems by using a tetrad formalism in General Relativity and the orthogonal splitting of the Riemann tensor. We apply this formalism to spherical case and found that: the only static solution with homogeneous energy density, under the physical reasonable considerations, is the Schwarzchild solution; the conditions under which a non-dissipative fluid is static and finally that shear-free and isotropic pressure conditions are equivalent, for non-dissipative fluids with homogeneous energy density.

Damage response of steel plate to underwater explosion: Effect of shaped charge liner

A shape of charge liners has a great effect on formation of a metal jet and its penetration into a target. In this paper, three different shapes of a charge liner, namely, conical, hemispherical and spherical-segment, are chosen to investigate their effect on damage response of a plate to underwater explosion. A Smooth Particle Hydrodynamic (SPH) method based on mesh-free Lagrange formulation is applied to simulate an entire process of a shaped-charge detonation, formation of a metal jet as well as penetration on a steel plate. Initially, a SPH model of the shaped charge with a spherical-segment liner is developed, and its results are compared with experimental data to verify the effectiveness of this method. Then, numerical simulations of shaped charges with different liners are performed to study the damage characteristics of a steel plate subjected to underwater-explosion shock loading and the metal jet. It was found that for the shock wave the peak value of the radial pressure is larger than that of the axial pressure during the detonation process; the level of pressure in the spherical-segment case was higher than that of the other two cases. After the detonation, the metal jet was gradually produced under the effect of the detonation wave. Three types of the metal jet - a shaped charge jet (SCJ), a jetting projectile charge (JPC) and an explosive formed projectile (EFP) – were formed corresponding to three cases with conical, hemispherical and spherical-segment liners. The obtained results show that the velocity and length of the SCJ in the conical case are greater than that of the other cases, and it therefore may lead to a larger penetration depth. In addition, the EFP has a better motion stability for a velocity difference in the spherical case is lower than that of the other two cases. Subsequently, the shock wave arrives at the plate earlier than the metal jet, which will cause deformation of the plate. Due to higher pressure, the shock wave in the spherical-segment case has a stronger damaging effect on the plate than that in the other two cases. Finally, the metal jet reaches the plate causing a hole. Because of a wider jet head, the EFP results in a more serious damage to the plate. The suggested analysis and its results provide a reference for structural design of shaped charge warheads.

The influence of magnetic field on the physical explosion of a heavy gas cloud

ABSTRACTRichtmyer–Meshkov (RM) instability arises during the eruption of heavy gas cloud. In this study, we numerically study the effects of magnetic fields on the RM instability induced by the ionised cylindrical and spherical heavy gas cloud eruption using corner transport upwind + constrained transport algorithm. Our numerical results show that magnetic fields can suppress the formation of spike and bubble structures induced by the eruption in both cylindrical and spherical cases. The magnetic pressure of the interface along the perpendicular direction of magnetic field is the main factor to control the distortion of the interface. Even weak magnetic fields can drastically alter the evolution of the cloud and result in different distributions and amplifications of the magnetic pressure, which will affect further transformation of RM instability during the ionised gas eruption. Meanwhile, the magnetic pressure on the interface decreases gradually when the initial magnetic field is relatively large; when the initial magnetic field is small enough, the opposite results will occur.

The one-dimensional Riemann problem on an elliptic curve

A one-dimensional generalization of the Riemann–Hilbert problem from the Riemann sphere to an elliptic curve is considered. A criterion for its positive solvability is obtained and the explicit form of all possible solutions is found. As in the spherical case, the solutions turn out to be isomonodromic.

Effective friction of granular flows made of non-spherical particles

Understanding the rheology of dense granular matter is a long standing problem and is important both from the fundamental and the applied point of view. As the basic building blocks of granular materials are macroscopic particles, the nature of both the response to deformations and the dissipation is very different from that of molecular materials. In the absence of large gradients, the best approach formulates the constitutive equation as an effective friction: for sheared granular matter the ratio of the off-diagonal and the diagonal elements of the stress tensor depends only on dynamical parameters, in particular the inertial number. In this work we employ numerical simulations to extend this formalism to granular packings made of frictionless elongated particles. We measured how the shape of the particles affects the effective friction, volume fraction and first normal stress difference, and compared it to the spherical particle case. We had to introduce polydispersity in particle size in order to keep the systems of the more elongated particles disordered.

Numerical investigation on heat transfer performance and flow characteristics in a rectangular air cooling channel (AR = 2) with ridged dimples

As an innovative type of structure, ridged dimples exhibit great potential for heat transfer enhancement. In this study, the effects of ridged dimples on heat transfer, flow characteristics and entropy generation in a rectangular air cooling channel with aspect ratio of 2:1 are numerically investigated. A comparative and comprehensive analysis is accomplished, considering the influences of extending ratio r and Reynolds number Re, as well as a comparison with the results of spherical dimple case. It is shown that heat transfer is further enhanced when ridged dimples are employed compared with spherical dimple case, and the effect is markedly promoted when r increases. The spherical dimple primarily affects the dimpled wall, while the influence of ridged dimple on heat transfer enhancement is mainly reflected on the side walls and top wall. In addition, the areas of the high temperature regions at the side boundaries of all walls significantly decrease with an increasing r. Besides, the overall turbulent kinetic energy TKE in the domain of spherical dimple case is in a relatively low level. When ridged dimples are adopted, a considerably large area of high TKE is detected in the domain above the dimpled wall. And the proportion of the high TKE region in the domain rises with the increase in r. In ridged dimple cases, the low TKE regions found in spherical dimple case at the side boundaries are not detected. The overall TKE level in ridged dimple case is higher than that of spherical dimple case. Furthermore, the average heat transfer entropy generation rate and average total entropy generation rate are prominently reduced with the employment of ridged dimples, while a slight increase in average friction entropy generation rate is detected. The employment of ridged dimple significantly reduces the maximum wall temperature and distinctly improves the wall temperature uniformity, with a further enhanced effect when r ascends, greatly beneficial for safe and stable operations of engineering devices.

Physical constraints on the non-dimensional absorption coefficients of compressional and shear waves for viscoelastic cylinders

BackgroundNormalized absorption coefficients for the longitudinal and shear waves in viscoelastic (polymer-type) materials, extracted from non-fictional experimental data showed anomalous effects, such as the generation of a negative radiation force (NRF) in plane progressive waves, negative energy absorption and extinction efficiencies and a scattering enhancement, not in agreement with energy conservation. ObjectiveThe objective of this work is directed towards analyzing those anomalies from the standpoint of energy conservation. Physical conditions which demonstrate that the ratio of the normalized absorption coefficients cannot be of arbitrary value but depends on the ratio of the square of the compressional and shear wave speeds, are established and discussed. MethodThe necessary physical condition for the validity of the linear viscoelastic (VE) model for any passive (i.e. that does not generate energy) polymeric cylinder with an ultrasonic absorption of hysteresis-type submerged in a non-viscous fluid requires that the absorption efficiency be positive (Qabs>0) since there are no active radiating sources inside the core material. This condition imposes restrictions on the values attributed to the normalized absorption coefficients for the compressional and shear-wave wavenumbers for each partial-wave mode n. The forbidden values produce anomalous/unphysical NRF, negative absorption and extinction efficiencies, as well as an enhancement of the scattering efficiency using plane progressive waves, not in agreement with energy conservation. ResultsBased on the partial wave series expansion method in cylindrical coordinates, numerical results for the radiation force, extinction, absorption and scattering energy efficiencies assuming plane progressive wave incidence are performed for three VE polymer cylinders immersed in a non-viscous host liquid (i.e. water) with particular emphasis on the shear-wave absorption coefficient, the dimensionless size parameter ka (where k is the wavenumber and a is the radius of the cylinder) and the partial-wave mode number n. Physical and mathematical conditions are established for the non-dimensional absorption coefficients of the longitudinal and shear waves for a cylinder (i.e. the 2D case) in terms of the sound velocities in the VE material. The physical condition for the spherical 3D case is also noted. ConclusionFor passive materials, the physical conditions must be always satisfied to allow accurate computations of the acoustic radiation force, torque, and energy absorption, extinction and scattering efficiencies for VE cylinders having a hysteresis type of absorption (such as polymers and plastics), and submerged in a non-viscous fluid. The physical conditions must be always satisfied regardless of the shape of the incident field. They also serve to validate and verify experimental data for VE materials and test the accuracy of related numerical computations.

Spatio-temporal Modeling of Lasing Action in Core-Shell Metallic Nanoparticles.

Nanoscale laser sources based on single metallic nanoparticles (spasers) have attracted significant interest for their fundamental implications and technological potential. Here we theoretically investigate the spatio-temporal dynamics of lasing action in core–shell metallic nanoparticles that include optically pumped four-level gain media. By using detailed semiclassical simulations based on a time-domain generalization of the finite-element method, we study the evolution of the lasing dynamics when going from a spherical case to an elongated nanorod configuration. Our calculations show that there exists an optimal nanoparticle elongation that exhibits significantly improved lasing threshold and slope efficiency over those obtained for its spherical counterpart. These results are accounted for in terms of a coupled-mode theory analysis of the variation with elongation of the light confinement properties of localized surface plasmons. This work could be of importance for further development of nanoscale light sources based on localized surface plasmon resonances.

Impact of topology in causal dynamical triangulations quantum gravity

We investigate the impact of spatial topology in 3+1 dimensional causal dynamical triangulations (CDT) by performing numerical simulations with toroidal spatial topology instead of the previously used spherical topology. In the case of spherical spatial topology we observed in the so-called phase C an average spatial volume distribution n(t) which after a suitable time redefinition could be identified as the spatial volume distribution of the four-sphere. Imposing toroidal spatial topology we find that the average spatial volume distribution n(t) is constant. By measuring the covariance matrix of spatial volume fluctuations we determine the form of the effective action. The difference compared to the spherical case is that the effective potential has changed such that it allows a constant average n(t). This is what we observe and this is what one would expect from a minisuperspace GR action where only the scale factor is kept as dynamical variable. Although no background geometry is put in by hand, the full quantum theory of CDT is also with toroidal spatial toplogy able to identify a classical background geometry around which there are well defined quantum fluctuations.

Global classical solutions of the full compressible Navier–Stokes equations with cylindrical or spherical symmetry

In this paper, we consider the full compressible Navier–Stokes equations in N(N≥2) space dimension with cylindrical or spherical symmetric initial data. The global existence of strong and classical solutions is established. The analysis is based on some delicate a priori estimates which depend on the assumption κ(θ)=θq where q⩾0 and (ρ0,θ0)∈H2,(u0,v0,w0)∈H01∩H2. Compared with the results in Wen and Zhu (2014) and Qin, Yang, Yao and Zhou (2015), our results relax the restriction q>0, when there is no initial vacuum and include the global existence of classical solutions for both the cylindrical or spherical symmetric cases, respectively. It should point out that we obtain the global classical solutions with the help of weighted H3 estimates of (u,v,w,θ).

Numerical investigation on particle behavior in a bubbling fluidized bed with non-spherical particles using discrete hard sphere method

In this work, a hard sphere particle method for non-spherical particles is applied to study particle behaviors in a bubbling fluidized bed. The influence of the non-spherical particle method and the restitution coefficient are examined. The particle behaviors, including the overall fluidization state (distributions of particle translational and rotational velocities, void fraction, etc.), bubble and particle fluctuations (particle and bubble granular temperatures), are studied. Specific variation rules which are difficult to measure experimentally are presented. As the bubble boundaries in the non-spherical particle system are not as clear as in the spherical case, there is a stronger fluctuation movement of bubbles, which also dominates the particle turbulent movement. The absolute values of particle translational and rotational velocities increase when the restitution coefficient decreases. And it affects the periodic fluctuation characteristics of gas-solid interactions. The variation rules for bubbles and particles as a function of the restitution coefficient are different.

Anomalous recurrence properties of many-dimensional zero-drift random walks

AbstractFamously, a d-dimensional, spatially homogeneous random walk whose increments are nondegenerate, have finite second moments, and have zero mean is recurrent if d∈{1,2}, but transient if d≥3. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially nonhomogeneous random walks (Markov processes) whose increment covariance matrix is asymptotically constant along rays from the origin, and which, in any ambient dimension d≥2, can be adjusted so that the walk is either transient or recurrent. Natural examples are provided by random walks whose increments are supported on ellipsoids that are symmetric about the ray from the origin through the walk's current position; these elliptic random walks generalize the classical homogeneous Pearson‒Rayleigh walk (the spherical case). Our proof of the recurrence classification is based on fundamental work of Lamperti.

Functional summary statistics for point processes on the sphere with an application to determinantal point processes

We study point processes on Sd, the d-dimensional unit sphere Sd, considering both the isotropic and the anisotropic case, and focusing mostly on the spherical case d=2. The first part studies reduced Palm distributions and functional summary statistics, including nearest neighbour functions, empty space functions, and Ripley’s and inhomogeneous K-functions. The second part partly discusses the appealing properties of determinantal point process (DPP) models on the sphere and partly considers the application of functional summary statistics to DPPs. In fact DPPs exhibit repulsiveness, but we also use them together with certain dependent thinnings when constructing point process models on the sphere with aggregation on the large scale and regularity on the small scale. We conclude with a discussion on future work on statistics for spatial point processes on the sphere.

A remark on geometric desingularization of a non-hyperbolic point using hyperbolic space

A steady state (or equilibrium point) of a dynamical system is hyperbolic if the Jacobian at the steady state has no eigenvalues with zero real parts. In this case, the linearized system does qualitatively capture the dynamics in a small neighborhood of the hyperbolic steady state. However, one is often forced to consider non-hyperbolic steady states, for example in the context of bifurcation theory. A geometric technique to desingularize non-hyperbolic points is the blow-up method. The classical case of the method is motivated by desingularization techniques arising in algebraic geometry. The idea is to blow up the steady state to a sphere or a cylinder. In the blown-up space, one is then often able to gain additional hyperbolicity at steady states. The method has also turned out to be a key tool to desingularize multiple time scale dynamical systems with singularities. In this paper, we discuss an explicit example of the blow-up method where we replace the sphere in the blow-up by hyperbolic space. It is shown that the calculations work in the hyperbolic space case as for the spherical case. This approach may be even slightly more convenient if one wants to work with directional charts. Hence, it is demonstrated that the sphere should be viewed as an auxiliary object in the blow-up construction. Other smooth manifolds are also natural candidates to be inserted at steady states. Furthermore, we conjecture several problems where replacing the sphere could be particularly useful, i.e., in the context of singularities of geometric flows, for avoiding compactification, and generating 'interior' steady states.

Eigenmodes of brain activity: Neural field theory predictions and comparison with experiment

Neural field theory of the corticothalamic system is applied to predict and analyze the activity eigenmodes of the bihemispheric brain, focusing particularly on their spatial structure. The eigenmodes of a single brain hemisphere are found to be close analogs of spherical harmonics, which are the natural modes of the sphere. Instead of multiple eigenvalues being equal, as in the spherical case, cortical folding splits them to have distinct values. Inclusion of interhemispheric connections between homologous regions via the corpus callosum leads to further splitting that depends on symmetry or antisymmetry of activity between brain hemispheres, and the strength and sign of the interhemispheric connections. Symmetry properties of the lowest observed eigenmodes strongly constrain the interhemispheric connectivity strengths and unihemispheric mode spectra, and it is predicted that most spontaneous brain activity will be symmetric between hemispheres, consistent with observations. Comparison with the eigenmodes of an experimental anatomical connectivity matrix confirms these results, permits the relative strengths of intrahemispheric and interhemispheric connectivities to be approximately inferred from their eigenvalues, and lays the foundation for further experimental tests. The results are consistent with brain activity being in corticothalamic eigenmodes, rather than discrete “networks” and open the way to new approaches to brain analysis.

Virial series for inhomogeneous fluids applied to the Lennard-Jones wall-fluid surface tension at planar and curved walls.

We formulate a straightforward scheme of statistical mechanics for inhomogeneous systems that includes the virial series in powers of the activity for the grand free energy and density distributions. There, cluster integrals formulated for inhomogeneous systems play a main role. We center on second order terms that were analyzed in the case of hard-wall confinement, focusing in planar, spherical, and cylindrical walls. Further analysis was devoted to the Lennard-Jones system and its generalization, the 2k-k potential. For these interaction potentials, the second cluster integral was evaluated analytically. We obtained the fluid-substrate surface tension at second order for the planar, spherical, and cylindrical confinement. Spherical and cylindrical cases were analyzed using a series expansion in the radius including higher order terms. We detected a lnR/R(2) dependence of the surface tension for the standard Lennard-Jones system confined by spherical and cylindrical walls, no matter if particles are inside or outside of the hard walls. The analysis was extended to bending and Gaussian curvatures, where exact expressions were also obtained.

Flow Unsteadiness and Pressure Pulsations in a Nuclear Reactor Coolant Pump

Unsteady flow induced by rotor-stator interaction is detrimental to the safe operation of the nuclear reactor coolant pump, so it is essential to clarify flow structures and pressure pulsation in such pumps, especially within the spherical casing. In this paper, unsteady flow characteristics in a mixed-flow nuclear reactor coolant model pump were investigated using large-eddy simulation (LES) method. Results show that at the nominal flow rate, in two particular diffuser channels near the spherical casing discharge nozzle, the flow structures are uneven compared with that in the other flow channels. The reason is associated with the position of the flow channel with respect to the spherical casing nozzle. Large- scale flow separation and backflow structures easily occur at the regions near these two channels. In the right and the middle region of the casing nozzle, due to the large-scale separate flow and high vorticity magnitude, unsteady flow structures are more complicated in comparison with the other regions. It has been found that the vorticity spectra and the pressure spectra almost have the same main excitation frequencies. Therefore, it has been confirmed that for particular regions pressure pulsations are determined by the shedding vortex wake from the diffuser blade trailing edge of the nuclear reactor coolant model pump.

AGN coronal emission models – I. The predicted radio emission

Accretion discs in AGN may be associated with coronal gas, as suggested by their X-ray emission. Stellar coronal emission includes radio emission, and AGN corona may also be a significant source for radio emission in radio quiet (RQ) AGN. We calculate the coronal properties required to produce the observed radio emission in RQ AGN, either from synchrotron emission of power-law (PL) electrons, or from cyclo-synchrotron emission of hot mildly-relativistic thermal electrons. We find that a flat spectrum, as observed in about half of RQ AGN, can be produced by corona with a disc or a spherical configuration, which extends from the innermost regions out to a pc scale. A spectral break to an optically thin power-law emission is expected around 300-1000 GHz, as the innermost corona becomes optically thin. In case of thermal electrons, a sharp spectral cutoff is expected above the break. The position of the break can be measured with VLBI observations, which exclude the cold dust emission, and it can be used to probe the properties of the innermost corona. Assuming equipartition of the coronal thermal energy density, the PL electrons energy density, and the magnetic field, we find that the energy density in a disc corona should scale as $\sim R^{-1.3}$, to get a flat spectrum. In the spherical case the energy density scales as $\sim R^{-2}$, and is $\sim 4\times 10^{-4}$ of the AGN radiation energy density. In paper II we derive additional constraints on the coronal parameters from the Gudel-Benz relation, $L_{\rm Radio}/L_{\rm X-ray}\sim 10^{-5}$, which RQ AGN follow.