Dimensionless Scaling Parameters Research Articles

Particle fracture regimes from impact simulations.

We introduce an approach to particle breakage, wherein the particle is modeled as an aggregate of polyhedral cells with their common surfaces governed by the Griffith criterion of fracture. This model is implemented within a discrete element code to simulate and analyze the breakage behavior of a single particle impacting a rigid plane. We find that fracture dynamics involves three distinct regimes as a function of the normalized impact energy ω. At low values of ω, the particle undergoes elastic rebound and no cracks occur inside the particle. In the intermediate range, the particle is damaged by nucleation and propagation of cracks, and the effective restitution coefficient declines without breakup of the particle. Finally, for values of ω beyond a well-defined threshold, the particle breaks into fragments and the restitution coefficient increases with ω due to kinetic energy carried away by the fragments. We show that particle damage, restitution coefficient, and fracture efficiency (the amount of energy input consumed for particle fracture) collapse well as a function of dimensionless scaling parameters. Our data are also sufficiently accurate to scale fragment size and shape distributions. It is found that fragment masses (volumes) follow a power-law distribution with an exponent decreasing with fracture energy. Interestingly, the average elongation and flatness of fragments are very close to those observed in experiments and lunar samples at the optimal fracture efficiency.

Ionization and current sheet formation in inductive pulsed plasma thrusters

Current sheet formation in inductive pulsed plasma thrusters (IPPTs) is investigated theoretically to determine how non-equilibrium ionization processes influence plasma impermeability to inductive electromagnetic fields and downstream propellant mass. Plasma impermeability to both electromagnetic fields and downstream mass is a prerequisite for efficient IPPT operation. A lumped-element circuit model of an IPPT plasma is modified to include propellant ionization and the electron energy balance under non-equilibrium conditions, neutral gas entrainment via charge exchange collisions, and electromagnetic coupling to a finite skin depth plasma. It is found that current sheets impermeable to both the accelerating fields and downstream mass—presumed to exist in all prior circuit modeling efforts—form only under specific conditions. The dynamics of electron heating during the early portion of the inductive current cycle are identified as the dominant contributors to current sheet formation. A new dimensionless scaling parameter is derived to characterize electron heating relative to inelastic ionization losses, from which it is found that impermeable current sheet formation requires Ohmic heating in the early formation phase to offset ionization losses associated with the entire propellant mass bit. This finding provides a physical explanation and generalization of the semi-empirical requirement on coil current rise rate that is commonly used in the early design phase of IPPTs to ensure current sheet formation.

Distortion of passive scalar structure during suction-based plume sampling

Studies of plume dynamics and olfaction often rely on photoionization detectors (PID) to quantify spatiotemporal distributions of passive scalars (gases, vapors, odors). However, the potential for PID suction to distort filaments and to modify the resulting sensed time record remains unclear. We used computational fluid dynamics to model a widely-used PID to quantify and parameterize suction distortion by considering how sensed concentration records compare to those registered by an ideal probe absent distorting effects. Models cover a range of realistic plume conditions, and we show that PID can modify the peak concentration and pulse shape of sensed records, with peak amplitude reduced by up to 45% and pulse width increased by up to 100% for the cases considered here. We quantified how distortion varies in three key nondimensional parameters describing PID geometry and plume sampling conditions: relative suction rate, relative filament size, and ambient flow Reynolds number. We combined analytical and numerical tools with dimensional analysis and scaling arguments to interpret results and discuss when distortion is likely and what drives it. We built a dimensionless distortion scaling parameter and simple regressions capable of predicting distortion levels, and our results enable PID users to estimate distortion levels and to employ mitigation strategies through suction velocity tuning. We show that an ideal relative suction rate (ratio of suction-to-ambient velocities) near 30 minimizes distortion universally for the Aurora Scientific miniPID. The findings and discussion herein can inform distortion-mitigating design principles and best sampling practices for other suction-based passive scalar sensing schemes.

Space radiation electrostatic shielding scaling laws: Beam-like and isotropic angular distributions

Passive radiation shielding alone cannot provide adequate protection for astronauts on long-term, deep-space missions. High atomic number and energy (HZE) ions, and/or their secondaries, can penetrate any realistic mass shielding for long-term deep-space missions and cause damage to cells via direct energy deposition and/or through the production of secondary particles and fragmented nuclei. Active shielding, or the use of electromagnetic fields to deflect or stop incoming ions before reaching the spacecraft, has gained substantial attention over the last decade as a way to augment passive shielding. Recently, a mathematical relationship between a dimensionless scaling parameter characterizing the active shield and the incoming ion and a protected area formed on a downstream detector due to the deflection of HZE ions by the applied field was validated in Earth-based laboratory conditions. In the present work, the mathematical formulation is extended to relate electrostatic shielding efficacy with the ability to deflect positively charged ions with parameters relevant to space applications. Additionally, a modified scaling parameter is formulated to characterize the shielding efficacy of a “family” of electrostatic active shielding configurations for reducing flux density for a space-relevant isotropic source of energetic protons. The results of this study demonstrate a strong correlation among dimensionless scaling parameters and shielding efficacy metrics for space radiation-relevant HZE ions in beam and isotropic angular distributions. Furthermore, it establishes a framework for optimizing design of three-dimensional electrostatic shielding configurations to improve space radiation protection for astronauts on exploration-class missions.

Stress level effect on mobility of dry granular flows of angular rock fragments

Granular flows of angular rock fragments such as rock avalanches and dense pyroclastic flows are simulated numerically by means of the discrete element method. Since large-scale flows generate stresses that are larger than those generated by small-scale flows, the purpose of these simulations is to understand the effect that the stress level has on flow mobility. The results show that granular flows that slide en mass have a flow mobility that is not influenced by the stress level. On the contrary, the stress level governs flow mobility when granular flow dynamics is affected by clast agitation and collisions. This second case occurs on a relatively rougher subsurface where an increase of the stress level causes an increase of flow mobility. The results show also that as the stress level increases, the effect that an increase of flow volume has on flow mobility switches sign from causing a decrease of mobility at low stress level to causing an increase of mobility at high stress level. This latter volume effect corresponds to the famous Heim’s mobility increase with the increase of the volume of large rock avalanches detected so far only in the field and for this reason considered inexplicable without resorting to extraordinary mechanisms. Granular flow dynamics is described in terms of dimensionless scaling parameters in three different granular flow regimes. This paper illustrates for each regime the functional relationship of flow mobility with stress level, flow volume, grain size, channel width, and basal friction.

Spontaneous and Stimulated Undulator Radiation in Symmetric and Asymmetric Multi-Periodic Magnetic Fields

In this work, the radiation from electrons in multi-periodic undulator fields with symmetric and asymmetric harmonics was analyzed using generalized Bessel functions formalism. The asymmetric, symmetric, and anti-symmetric periodic magnetic fields with harmonics were studied in order to get the enhanced radiation of the high harmonics of undulator radiation (UR). The effect on the spontaneous and stimulated UR was explored. The exact integral forms for the Bessel coefficients were obtained for undulators with general symmetric and asymmetric field harmonics. Spectral properties of the radiation from several configurations of the undulator fields with harmonics were compared with each other. The resulting spontaneous UR spectrum and harmonic intensities were obtained analytically in the form of integrals and compared with the respective results that were obtained numerically with SPECTRA program. The dimensionless scaling parameter of a free electron laser (FEL)—the Pierce parameter (ρ)—was computed and compared for the different considered undulators. We studied the differences in the behavior of the high-gain single pass FEL harmonics and the spontaneous UR harmonics in the same undulators. The undulators with variable deflection parameter (k) were considered. The effect of the k parameter (deflection parameter for a common planar undulator) on the spontaneous UR and on the high-gain FEL radiation was explored. In this context, an experiment with variable strength undulators at FLASH 2 FEL was analyzed; the shorter saturated length in high harmonic self-seeding (HHSS) regime vs. self-amplified spontaneous emission (SASE) is explained.

Asymptotic Growth of the Local Ground-State Entropy of the Ideal Fermi Gas in a Constant Magnetic Field

We consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane {{mathbb {R}}}^2 perpendicular to an external constant magnetic field of strength B>0. We assume this (infinite) quantum gas to be in thermal equilibrium at zero temperature, that is, in its ground state with chemical potential mu ge B (in suitable physical units). For this (pure) state we define its local entropy S(Lambda ) associated with a bounded (sub)region Lambda subset {{mathbb {R}}}^2 as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region Lambda of finite area |Lambda |. In this setting we prove that the leading asymptotic growth of S(LLambda ), as the dimensionless scaling parameter L>0 tends to infinity, has the form Lsqrt{B}|partial Lambda | up to a precisely given (positive multiplicative) coefficient which is independent of Lambda and dependent on B and mu only through the integer part of (mu /B-1)/2. Here we have assumed the boundary curve partial Lambda of Lambda to be sufficiently smooth which, in particular, ensures that its arc length |partial Lambda | is well-defined. This result is in agreement with a so-called area-law scaling (for two spatial dimensions). It contrasts the zero-field case B=0, where an additional logarithmic factor ln (L) is known to be present. We also have a similar result, with a slightly more explicit coefficient, for the simpler situation where the underlying single-particle Hamiltonian, known as the Landau Hamiltonian, is restricted from its natural Hilbert space text{ L}^2({{mathbb {R}}}^2) to the eigenspace of a single but arbitrary Landau level. Both results extend to the whole one-parameter family of quantum Rényi entropies. As opposed to the case B=0, the corresponding asymptotic coefficients depend on the Rényi index in a non-trivial way.

Transverse isotropy in magnetoactive elastomers

Magnetoactive elastomers (MAEs) are field-controllable materials with magnetically switchable properties. The presence of an external magnetic field results in the change in the macroscopic shape of the MAE. The magnetic field also introduces a mechanical anisotropy with an axis of symmetry along the magnetic field. We aim to derive an effective material model which considers the initial shape of the MAE from the free energy of an isotropic MAE in the dipole approximation. Primarily, we consider uniaxial deformations parallel and perpendicular to the applied field direction. The stress–strain behaviour of the MAE yields direction-dependent tensile modulus. It indicates that the MAE behaves as a transversely isotropic material in the presence of an external magnetic field. Finally, we compare our predictions with the conventional transversely isotropic material model and find a good agreement. This comparison allows us to extract the dimensionless scaling parameter of a transversely isotropic MAE which characterizes the influence of the magnitude of a magnetic field as well as the initial shape of an MAE sample.

Testing the energy diffusion approximation for the escape of a Brownian particle from a potential pocket

For the first time, the energy diffusion approximation is confronted at the percent level with the exact numerical modeling of thermal decay of a metastable state. This model is useful in many branches of natural sciences: e.g. in biology, nuclear physics, chemistry, etc. The exact (within the statistical errors about 2%) quasistationary decay rates result from the Langevin equations for the coordinate and conjugated momentum. For the energy (or action) diffusion approach, a Langevin-type equation for the action is constructed, validated, and solved numerically. The comparison of these two approaches is performed for four potentials (two of which are anharmonic) in a wide range of two dimensionless scaling parameters: i) the governing parameter G reflecting how high is the barrier with respect to the temperature and ii) the damping parameter φ expressing the friction strength. It turns out that the energy diffusion approach produces the rate which comes into 50%-agreement with the exact one only at φ < 0.02. Thus, we quantify, for the first time by our knowledge, the condition φ ≪ 1 known in the literature.

Evolution of wet agglomerates inside inertial shear flow of dry granular materials.

We use particle dynamics simulations to investigate the evolution of a wet agglomerate inside homogeneous shear flows of dry particles. The agglomerate is modeled by introducing approximate analytical expressions of capillary and viscous forces between particles in addition to frictional contacts. During shear flow, the agglomerate may elongate, break, or be eroded by loss of its capillary bonds and primary particles. By systematically varying the shear rate and surface tension of the binding liquid, we characterize the rates of these dispersion modes. All the rates increase with increasing inertial number of the flow and decreasing cohesion index of the agglomerate. We show that the data points for each mode collapse on a master curve for a dimensionless scaling parameter that combines the inertial number and the cohesion index. The erosion rate vanishes below a cutoff value of the scaling parameter. This leads to a power-law borderline between the vanishing erosion states and erosion states in the phase space defined by the inertial number and the cohesion index.

Thermal conductance of one-dimensional disordered harmonic chains

We study heat conduction mediated by longitudinal phonons in one dimensional disordered harmonic chains. Using scaling properties of the phonon density of states and localization in disordered systems, we find non-trivial scaling of the thermal conductance with the system size. Our findings are corroborated by extensive numerical analysis. We show that a system with strong disorder, characterized by a `heavy-tailed' probability distribution, and with large impedance mismatch between the bath and the system satisfies Fourier's law. We identify a dimensionless scaling parameter, related to the temperature scale and the localization length of the phonons, through which the thermal conductance for different models of disorder and different temperatures follows a universal behavior.

Unified theory for excited-state, fragmented, and equilibriumlike Bose condensation in pumped photonic many-body systems

We derive a theory for Bose condensation in nonequilibrium steady states of bosonic quantum gases that are coupled both to a thermal heat bath and to a pumped reservoir (or gain medium), while suffering from loss. Such a scenario describes photonic many-body systems such as exciton-polariton gases. Our analysis is based on a set of kinetic equations for a gas of noninteracting bosons. By identifying a dimensionless scaling parameter controlling the boson density, we derive a sharp criterion for which system states become selected to host a macroscopic occupation. We show that with increasing pump power, the system generically undergoes a sequence of nonequilibrum phase transitions. At each transition a state either becomes or ceases to be Bose selected (i.e. to host a condensate): The state which first acquires a condensate when the pumping exceeds a threshold is the one with the largest ratio of pumping to loss. This intuitive behavior resembles simple lasing. In the limit of strong pumping, the coupling to the heat bath becomes dominant so that eventually the ground state is selected, corresponding to equilibrium(-like) Bose condensation. For intermediate pumping strengths, several states become selected giving rise to fragmented nonequilibrium Bose condensation. We compare these predictions to experimental results obtained for excitons polaritons in a double-pillar structure [Phys. Rev. Lett. 108, 126403 (2012)] and find good agreement. Our theory, moreover, predicts that the reservoir occupation is clamped at a constant value whenever the system hosts an odd number of Bose condensates.

Dimensionless Scaling Parameters During Thermal Flooding Process in Porous Media

The scaling concept is important, effective, and consistent in any application of science and engineering. Scaled physical models have inimitable advantages of finding all physical phenomena occurring in a specific process by transforming parameters into dimensionless numbers. This concept is applicable to thermal enhanced oil recovery (EOR) processes where continuous alteration (i.e., memory) of reservoir properties can be characterized by various dimensionless numbers. Memory is defined as the continuous time function or history dependency which leads to the nonlinearity and multiple solutions during modeling of the process. This study critically analyzed sets of dimensionless numbers proposed by Hossain and Abu-Khamsin in addition to Nusselt and Prandtl numbers. The numbers are also derived using inspectional and dimensional analysis (DA), while memory concept is used to develop some groups. In addition, this article presents relationships between different dimensionless numbers. Results show that proposed numbers are measures of thermal diffusivity and hydraulic diffusivity of a fluid in a porous media. This research confirms that the influence of total absolute thermal conductivities of the fluid and rock on the effective thermal conductivity of the fluid-saturated porous medium diminishes after a certain local Nusselt number of the system. Finally, the result confirms that the convective ability of the fluid-saturated porous medium is apparently more pronounced than its conductive ability. This study will help to better understand the modeling of the EOR process thus improving process design and performance prediction.

Generalized Knudsen Number for Unsteady Fluid Flow.

We explore the scaling behavior of an unsteady flow that is generated by an oscillating body of finite size in a gas. If the gas is gradually rarefied, the Navier-Stokes equations begin to fail and a kinetic description of the flow becomes more appropriate. The failure of the Navier-Stokes equations can be thought to take place via two different physical mechanisms: either the continuum hypothesis breaks down as a result of a finite size effect or local equilibrium is violated due to the high rate of strain. By independently tuning the relevant linear dimension and the frequency of the oscillating body, we can experimentally observe these two different physical mechanisms. All the experimental data, however, can be collapsed using a single dimensionless scaling parameter that combines the relevant linear dimension and the frequency of the body. This proposed Knudsen number for an unsteady flow is rooted in a fundamental symmetry principle, namely, Galilean invariance.

Viscous-Fingering Function for Unstable Immiscible Flows

Summary Displacement of viscous oils often involves unstable immiscible flow. Viscous instability and its influence on relative permeability were studied in this work at different viscosity ratios, injection rates, and domain widths. Micromodels and pore-scale models were used to visually inspect the interplay of viscous and capillary forces in the viscous-dominated regime. A new dimensionless scaling parameter, NI=(vwμwσow)(μoμw)2(D2/K), was developed that is useful in predicting the recoveries of unstable displacements at various viscosity ratios and injection rates. The scaling parameter showed excellent fit with experimental data of 68 corefloods. A lumped finger model was developed to modify multiphase flow equations and to yield pseudorelative permeability functions that account for viscous fingering. The parameters of the lumped model can be estimated from the new dimensionless number, NI. This pseudorelative permeability function could be applied at each gridblock on the basis of the local NI to simulate large-scale unstable floods in water-wet porous media.

Wave-dominated, mesotidal headland-bay beach morphodynamic classsfications of the Shuidong Bay in South China

Beach morphodynamic classifications have achieved extensive acceptance in foreign coastal geomorphological studies. Three beaches located in different zones of a headland-bay coast are classified according to a dimensionless fall parameter, a relative tide range parameter and a dimensionless embayment scaling parameter. Synchronous data, including wave, tide, sediment and beach morphology, are respectively collected from the tangential beach, the transitional beach and the shadow beach of the Shuidong Bay during each spring tide for 16 successive months. The research results indicate that (1) the beach in the tangential zone falls between two major categories which are low tide terrace beaches with rips and barred beaches; the beach in the transitional zone exhibits two main types which are low tide bar/rip beaches and barred dissipative beaches; and the beach in the shadow zone mainly mirrors dissipative states with presence or absence of bars; and (2) the sequential changes and differences of beach states in different coastal zones reflect spatial and temporal variabilities of the headland-bay coast, totally meeting the actual measured beach morphology changes, showing that studies on wave-dominated, meso-macrotidal beaches need to consider the influences of the tides. Meanwhile, the research mainly provides a framework about beach state studies, due to different beach states with different erosion patterns, which requires the need to strengthen the researches in this respect, in order to further enrich theoretical basis for a beach topography evolution, beach morphodynamic processes and beach erosion prevention in China.

Coupling of Surface-Plasmon-Polariton-Hybridized Cavity Modes between Submicron Slits in a Thin Gold Film

Electron energy loss spectroscopy (EELS) in a monochromated transmission electron microscope is applied to probe standing-wave-like cavity modes hybridized with surface plasmon polaritons (SPP) in rectangular submicron slits in a thin gold film. Coupling of hybridized SPP-cavity modes between two adjacent slits is studied by systematically varying the width of the metal bar d that separates the identical slits in a two-slit system. Measurements on two-slit systems with different slit lengths L and fixed width reveal energy shifts and mode splitting of the fundamental SPP cavity mode which can be generally described as a function of a dimensionless scaling parameter L/d. Numerical simulations with the Discontinuous Galerkin Time-Domain (DGTD) method confirm the experimental data and reveal insights into the underlying complex coupling mechanisms.

Low-Emission, Liquid Fuel Combustion System for Conventional and Alternative Fuels Developed by the Scaling Analysis

The objective of this study is to develop a theoretical basis for scalability considerations and design of a large-scale combustor utilizing flow blurring (FB) atomization. FB atomization is a recently discovered twin-fluid atomization concept, reported to produce fine spray of liquids with wide range of viscosities. Previously, we have developed and investigated a small-scale swirl-stabilized combustor of 7-kWth capacity. Spray measurements have shown that the FB injector's atomization capability is superior when compared to other techniques, such as air blast atomization. However, despite these favorable results, scalability of the FB injector and associated combustor design has never been explored for large capacity; for example, for gas turbine applications. In this study, a number of dimensionless scaling parameters that affect the processes of atomization, fuel–air mixing, and combustion are analyzed, and scaling criteria for the different components of the combustion system are selected. Constant velocity criterion is used to scale key geometric components of the system. Scaling of the nonlinear dimensions and complex geometries, such as swirler vanes and internal parts of the injector is undertaken through phenomenological analysis of the flow processes associated with the scaled component. A scaled-up 60-kWth capacity combustor with FB injector is developed and investigated for combustion performance using diesel and vegetable oil (VO) (soybean oil) as fuels. Results show that the scaled-up injector's performance is comparable to the smaller scale system in terms of flame quality, emission levels, and static flame stability. Visual flame images at different atomizing air-to-liquid ratio by mass (ALR) show mainly blue flames, especially for ALR &gt; 2.8. Emission measurements show a general trend of lower CO and NOx levels at higher ALRs, replicating the performance of the small-scale combustion system. Flame liftoff height at different ALRs is similar for both scales. The scaled-up combustor with FB injector preformed robustly with uncompromised stability for the range of firing rates (FRs) above 50% of the design capacity. Experimental results corroborate with the scaling methodology developed in this research.

Effects of permeability heterogeneity on CO2 injectivity and storage efficiency coefficient

AbstractWe study the dependency of CO2 storage efficiency coefficient (E) and injectivity index (Iinj) on the geostatistical parameters of the permeability field. CO2 injection simulations are conducted for multiple realizations of log‐normally distributed permeability fields parameterized by log permeability standard deviation (σ) and dimensionless horizontal correlation length (λ). Results show that the injectivity index increases with increasing λ, the magnitude of the effect depending on σ. Increasing σ leads to poorer injectivity for cases with small λ, but improves injectivity when λ is large. Further analysis indicates that the enhancing effect of σ on injectivity can be attributed to cases with channelized flow, while the decrease effect of σ is seen in more dispersive flow regime. The dependence of injectivity on both λ and σ is captured with a linear correlation between Iinj and a parameter group , where ξ is a dimensionless scaling parameter. The storage efficiency coefficient, on the other hand, decreases with both increasing σ and λ, and a simple linear fit is found between E and the parameter group λσ2, a well‐established heterogeneity parameter group describing e.g. macro‐dispersivity in solute transport studies. These relationships provide potentially useful tools for the preliminary evaluation of a site. Future studies should address the validity of the relationships in alternative injection scenarios and domain geometries. © 2015 Society of Chemical Industry and John Wiley &amp; Sons, Ltd

A Dimensionless Scaling Parameter for Thermal Effects on Cavitation in Turbopump Inducers

To characterize cryogenic pump performance, at least one parameter in addition to flow coefficient and cavitation number is required. This parameter arises because the heat of vaporization and other physical parameters change along the saturation line of liquids and results in a thermal effect on cavitation that has been observed and studied by previous researchers over a range of operating conditions and working fluids. These previous efforts have defined both dimensionless and dimensional parameters governing thermal effects in pumps. In the present work, a dimensionless parameter (DB) scaling thermal effects in a cavitating pump across different tip diameters, rotational speeds, and working fluids is derived using a model of bubble growth in a time-varying pressure field. Although the derivation is somewhat different than others have used, the result is similar and in some cases identical to that of others. Careful testing is carried out to experimentally validate this parameter with (deaerated) variable temperature water and with variable pump speed. The results show that within the accuracy of the data, the same head fall-off curve is obtained when either the water temperature or the pump speed is used to set the DBs. This suggests the proposed parameter can thermally scale cryogenic pumping conditions for suction performance when testing in hot water. Also examined is the effect of thermodynamics on inducer cavitation instabilities. The quasi-steady, dynamic environments at the pump inlet are compared in cold and superheated water. The cavitation instabilities of the test inducer are found to be dramatically changed by thermal effects. These findings emphasize the importance of considering both fluid mechanical and thermal scaling when designing a test program to evaluate the suction performance of a cryogenic pump.