Front Velocity Research Articles

Numerical simulation of atmospheric-pressure surface dielectric barrier discharge on a curved dielectric with a curvilinear mesh

A numerical simulation method for atmospheric-pressure surface dielectric barrier discharge (DBD) is presented using a structured curvilinear mesh that is fitted to a curved dielectric surface. The numerical method is based on the plasma fluid model with general coordinate transformation, which is widely used in the field of the computational fluid dynamics. The calculations of a potential distribution formed by a line electrode and a planar surface DBD are performed to confirm the validity of the proposed method. We conducted numerical simulations of the DBD on a wavy surface using the proposed method to investigate the characteristics of the DBD on a curved surface. The propagation velocity of the discharge front and the electric field strength of the discharge front changed depending on the surface shape. The proposed method enables the calculation of the DBD on a complex surface and widens the application range of the numerical simulation in the field of discharge physics.

Comparison of Dynamical and Empirical Simulation Methods of Secondary Dendrite Arm Coarsening

The physical and mechanical properties of an entirely (wrought alloys) or partly (cast alloys) dendritically solidified alloy strongly depend on the secondary dendrite arm spacing (SDAS). The casting practice and the simulation of solidification need a usable but simple method to calculate the SDAS during and at the end of solidification as a function of the cooling rate. Based on many solidification experiments, a simple equation to calculate the SDAS (empirical method) is known to use the local solidification time, which can be obtained from the measured cooling curves (equiaxed solidification), or can be calculated from the temperature gradient and front velocity (directional solidification). This equation is not usable for calculating the SDAS during solidification. Kirkwood developed a semi-empirical method based on the liquid phase’s diffusion, which contains only one geometric factor that seems constant for different alloys. This equation contains some physical parameters that depend on the temperature, so the equation cannot be integral in closed form. In the present work, first, we show the effect of the curvature of the solid/liquid interface on the equilibrium concentrations and then the different processes of SDA coarsening. In our earlier paper, we demonstrated that using the empirical method, the final SDAS can be calculated with acceptable correctness in the case of four unidirectional solidification experiments of Al-7wt%Si alloy. The present work shows that numerically integrated Kirkwood’s equations used the known cooling curve; the SDAS can be calculated at the end and during solidification in good agreement with these experimental results. Compared to the two calculation methods, we stated that the correctness of the methods is similar. Still, the results of the solidification simulation (the microsegregation) will be more correct using the dynamical method. It is also shown that with the dynamical method, the SDAS can be calculated from any type of cooling curve, and using the dynamical method, it is proved that some different SDASs could belong to the same local solidification time.

ANALYSIS OF THE FORM AND EVOLUTION OF THE SOLID PHASE DURING DIRECTIONAL CRYSTALLIZATION OF NON-FERROUS METALS WITH ELECTROMAGNETIC INFLUENCE BY ULTRASONIC AND TEMPERATURE METHODS

The paper is devoted to the experimental study of new mechanisms of controlling the process of directional crystallization of non-ferrous metals. The focus of the study is on the development and testing of measuring techniques applicable both to laboratory modeling and to processes under real operating conditions. The mechanism of controlling the rate and homogeneity of crystallization of a metal melt by changing the phase angles between the supply currents of a traveling magnetic field induction stirrer is proposed. This makes it possible to generate vortex flows of various topologies in the liquid metal, in particular, to change the number of large-scale vortices or to suppress the large-scale flow. It is shown that the hydrodynamic flows have an impact on the crystallization front shape, which allows one to control the homogeneity of metal solidification by changing the characteristics of the power supply of the inductor. It is important to note that a change in the phase angles of the currents while maintaining the supply amplitude does not significantly affect the crystallization rate, which opens up wide possibilities for controlling processes by changing both the current strength and the phase angles. The temperature method for determining the position of the crystallization front was successfully applied and verified by ultrasonic velocimetry measurements. It has been found that, in the presence of developed flows in a liquid medium, thermocouple measurements provide good agreement (up to a few percent) of the measured position and geometric shape of the crystallization front with the ultrasonic measurement data. In the absence of liquid phase stirring, the difference between the thermocouple and ultrasonic measurements increases slightly. Nevertheless, even in this case, the thermocouple method can be used to correctly determine the position and velocity of the crystallization front.

Experimental study on the rise of aqueous foams in vertical pipes

In this paper we report experiments on the growth of dry foams and their rise in vertical pipes of different circular cross-sectional radii with length over diameter ratios in the interval 25 ≤ h/D ≤ 80 for applications in the study of fracture stimulation in enhanced oil recovery processes. Air injection at the bottom of the pipes is performed at a constant flow rate by means of a single capillary tube. The formation and rising of the foam was investigated for two different cases: 1) when the top cap of the vertical pipes is open and 2) when it is closed. We find that the position and velocity of the foam front as well as the foam dispersivity are both dependent on the pipe diameter and on whether its top end is open or capped. When the top is open, the foam column grows faster compared to the case when it is sealed. In pipes with h/D ≥ 30, the growth rate is non-linear and faster than in pipes with h/D < 30 in which cases the foam rises at an almost constant rate. As the diameter of the pipe increases, the size of the produced bubbles also increases. In closed-top pipes the foams tend to be more homogeneous than in open top pipes. The experimental observations indicate that under foam drainage driven by gravity, the liquid flow velocity across the Plateau borders is indicative of a drainage model based on a plug-like flow in channels with fully mobile interfaces, where viscous dissipation occurs only in the nodes.

In uence of numerical diffusion on the growth rate of viscous ngers in the numerical implementation of the Peaceman model by the finite volume method

A numerical model of oil displacement by a mixture of water and polymer based on the Peaceman model is considered. Numerical experiments were carried out using the DuMux package, which is a software library designed for modeling nonstationary hydrodynamic problems in porous media. The software package uses the vertex-centered variant of finite volume method. The effect of diffusion on the growth rate of ''viscous fingers'' has been studied. The dependencies of the leading front velocity on the value of model diffusion are obtained for three viscosity models. It is shown that the effect of numerical diffusion on the growth rate of ''viscous fingers'' imposes limitations on calculations for small values of model diffusion.

Gradient-controlled freeze casting of preceramic polymers

Solidification is the foundation upon which freeze casting is built, and this work seeks to understand the solidification process in freeze casting, especially with respect to the growth of dendrites. To this end, two solidification parameters, freezing front velocity and temperature gradient, were independently controlled using a gradient-controlled freeze casting setup to investigate the effects of each parameter. Changes in dendritic pore size and morphology with solidification parameters were investigated by scanning electron microscopy and mercury intrusion porosimetry. The results agree with dendrite growth theory. The theory of constitutional supercooling is used to describe the morphological changes of dendritic pores to cellular pores by the control of freezing front velocity, temperature gradient, and preceramic polymer concentrations.

Application of capacitive sensors and controlled injection pressure to minimize void formation in resin transfer molding

AbstractVoid formation as a result of irregular resin flow at the flow front is discussed and a practical method for reducing void formation during resin transfer molding (RTM) is introduced. In this study, a sensor system is developed for in situ measurement of resin velocity inside a closed cavity. Assisted by the acquired data, a resin injection system is augmented to automatically adjust the injection pressure and achieve a uniform flow front velocity. It is proven, that the developed system is suited to monitor the resin flow front and is able to sufficiently control flow velocity of a linear flow front. Test specimen produced by this method show significantly reduced void contents in comparison to a common resin transfer molding process.

Effect of passive pre-chamber orifice diameter on the methane combustion process in an optically accessible SI engine

Due to its potential to extend the lean limit or overcome knock-limited conditions on spark-ignited (SI) engines, the pre-chamber spark-ignition (PCSI) concept has gained attention as an efficient alternative to the conventional ignition architecture. Nevertheless, especially in engine-like conditions, the effect of the main geometrical parameters on combustion development is still not fully understood and might be a hindrance to the use of this technology. In this sense, the current study aims to evaluate the effect of the orifice diameter on the jet characteristics and main chamber combustion of a novel passive pre-chamber in an optically accessible single-cylinder SI engine, as well as to compare the PCSI with the typical SI ignition method. High-speed broadband chemiluminescence imaging was used to track the jet penetration and flame front propagation of three mixture compositions at various spark timings (ST). Additionally, based on the in-cylinder pressure, a heat-released analysis was performed to assess the overall combustion process. When compared with the typical SI, the pre-chamber concept presented a higher equivalent flame front velocity, which is also corroborated by the higher in-cylinder pressure rise rate and rate of heat released. For all mixtures tested, the 1.2 mm orifice diameter PC presented the shortest flame development angle (crank angle interval between ST and 10 % of mass fraction burned in the MC), followed by 1.5 and lastly 1.0 mm. When comparing different STs, the shortest flame development angle occurred at MBT condition. Despite the highest flame development angle, the PC 1.0 mm case presented the lowest main chamber combustion duration (CA10-90), which increases as the orifice diameter is increased.

Estimation of the Propagation Time of the Solar Wind Pressure Pulse between a Shock Wave and the Magnetopause Based on Simultaneous Satellite and Ground-Based Observations

The propagation of solar wind inhomogeneities through the transition region between the bow shock front and the daytime magnetopause is an important link in the chain of solar-terrestrial connections. We analyzed a unique situation when two THEMIS satellites were near the bow shock during the solar wind pressure pulse, and the favorable location of the ground-based magnetometric equipment made it possible to determine the time of the solar wind impact on the magnetopause with greater accuracy than it was done previously. Based on the measurements, the propagation time of sudden impulse (SI) in the magnetosheath was estimated. The obtained value (~1 min) agrees with model calculations, but turns out to be 2–3 times less than that previously obtained by other authors also from satellite and ground-based observations. The propagation velocity of the solar wind in the magnetosheath (~280 km/s), the displacement velocity of the bow shock front to the Earth (~80 km/s), and the thickness of the magnetosheath at the location of the satellites (2.6–3.9RE) are determined.

Effects of elevated temperature steam oxidation and subsequent quenching on light water reactor clad

Following a postulated severe accident, the temperature of fuel pins of light water reactor escalates because of decay heat and devoid of coolant. The presence of steam inside the core causes exothermic oxidation of the cladding surface at elevated temperatures. The in-vessel injection system of severe accident management guidelines injects the cold water inside the hot uncovered core to arrest the escalation of temperature rise and terminate the accident progression. However, the heated clad may be subjected to thermal shock during the quenching process, which inevitably assists in the failure and fragmentation of the severely oxidised fuel pins column.In this paper, the effect of thermal quenching shock on the severely oxidised fuel pins column during the reflooding of the hot uncovered core is studied experimentally. A 2ˣ2 square lattice arrangement of zircaloy-4 clads is used for experiments. First, the clads are oxidised in the presence of steam in the temperature range of 1100–1250 °C, and then the test-section is flooded with water at ambient temperature from its bottom. The violent heat transfer between the clad and cold water has been observed during the reflooding. Furthermore, the clad has failed with a circumferential crack at multiple locations along the clad length during this period; however, longitudinal cracks are not observed on the clad surface. The scanning electron micrographs of the clad samples show a significant oxide layer over the clad surface. At the same time, energy dispersive x-ray spectroscopy reveals significant oxygen diffusion inside the zirconium metal matrix, particularly near the outer tube surface.Additional experiments are performed to study the clad failure mechanism during the reflooding of the severely oxidised clad. The set of experiments reveals that the clad fails under the thermal quench shock (>7.41 °C/s) for severely oxidised cladding (ECR>25%) at an average temperature above 1100 °C. The failed clad is likely to create debris. The average quench front velocity is 0.6 cm/s while the reflood velocity is 1.13 cm/s. Quench velocity below the reflood velocity indicates excessive water evaporation and dispersed type flow ahead of the quench front. The total analytical H2 generated before reflooding is 3.75 g.

Study on flame propagation of H2/LPG premixed gas in a tube

This work aims to is to study the flame propagation characteristics of the hydrogen/liquefied petroleum gas (LPG) premixed gas in a closed tube. With the aid of a self-developed square glass explosion tube and a high-speed camera, the characteristic parameters (i.e., flame structure, flame tip speed, maximum explosion overpressure, maximum rate of explosion pressure rise, deflagration index) were obtained and analyzed the explosion characteristics of the premixed gas. The results show that the time for the flame to reach the end of the pipe decreased, and the maximum explosion overpressure increases as the hydrogen addition ratio increases. However, the explosion overpressure and flame propagation velocity increased slowly when the hydrogen addition ratio was less than 40%, While the explosion overpressure and flame propagation velocity increased drastically in the range of hydrogen addition ratio 40–80%. Therefore, the key turning point for LPG hydrogenation is when the hydrogen volume fraction is 40%. The peak explosion flame propagation velocity and the peak explosion pressure showed the change law of increasing first and then decreasing with the equivalent ratio increasing. The Pmax and Vmax are 6.4 bar and 71.2 m/s respectively. The explosive flame structure of the premixed gas formed a "multiple tulip" structure after the typical tulip flame structure due to the influence of hydrodynamic. In addition, the propagation law of H2-LPG-Air premixed flame in the pipeline was influenced by the interaction of pressure wave and flame front velocity, and Pmax trajectory and Vmax curve showed similar oscillation law. Hence, the study provides crucial data for the prevention and control of H2-LPG-Air gas mixture combustion and explosion accidents.

Observation of Pattern Formation during Electromagnetic Levitation Using High-Speed Thermography

Electromagnetic levitation (EML) was employed for studying the velocity and morphology of the solidification front as a function of undercooling of metallic materials. The limitation of the EML technique with respect to low melting alloys that emit outside the visible light spectrum was overcome by employing state-of-the-art high-speed mid-wavelength infrared cameras (MWIR cameras) with a photon detector. Due to the additional thermography contrast provided by the emission contrast of the solid and liquid phases, conductor, and semi-conductor, the pattern formation of Al-based alloys was studied in detail, revealing information on the nucleation, phase selection during solidification, and the influence of convection.

Macrosegregation Evolution in Eutectic Al-Si Alloy under the Influence of a Rotational Magnetic Field

Using magnetic stirring during solidification provides a good opportunity to control the microstructure of alloys, thus controlling their physical properties. However, magnetic stirring is often accompanied by a change in local concentrations, and new structures form which could harm the physical properties. This research paper investigated the effect of forced melt flow by a rotating magnetic field (RMF) on the macrostructure of an Al-Si eutectic alloy. To serve this purpose, Al-12.6 wt% Si alloy samples were solidified in a vertical Bridgman-type furnace equipped with a rotating magnetic inductor to induce the flow in the melt. The diameter and length of the sample are 8 mm and 120 mm, respectively. The solidification parameters are a temperature gradient (G) of 6 K/m, and the solid/liquid front velocity (v) of 0.1 mm/s. These samples were divided into parts during the solidification process, where some of these parts are solidified under the effect of RMF stirring while others are solidified without stirring. The structure obtained after solidification showed a distinct impact of stirring by RMF; new phases have been solidified which were not originally present in the structure before stirring. Besides the eutectic structure, the new phases are the primary aluminum and the primary silicon. The Si concentration and the volume fraction of each phase were measured using Energy-Dispersive Spectroscope (EDS)and new image processing techniques. The experimental results reveal that applying the RMF during the solidification has a distinct effect on the macrostructure of Al-Si eutectic alloys. Indeed, the RMF provokes macro-segregation, reduces the amount of eutectic structure, and changes the sample’s Si concentration distribution.

Investigation of dynamic impact responses of layered polymer-graphene nanocomposite films using coarse-grained molecular dynamics simulations

Polymer nanocomposite films have recently shown superior energy dissipation capability by using the micro-projectile impact testing method. However, how stress waves interact with nanointerfaces and the underlying deformation mechanisms have remained largely elusive. This paper investigates the detailed stress wave propagation process and dynamic failure mechanisms of layered poly(methyl methacrylate) (PMMA)-graphene nanocomposite films during piston impact through coarse-grained molecular dynamics simulations. The spatiotemporal contours of stress and local density clearly demonstrate shock front, reflected wave, and release wave. By plotting shock front velocity (Us) against piston velocity (Up), we find that the linear Hugoniot Us−Up relationship generally observed for bulk polymer systems also applies to the layered nanocomposite system. When the piston reaches a critical velocity, PMMA crazing can emerge at the location where the major reflected wave and release wave meet. We show that the activation of PMMA crazing significantly enhances the energy dissipation ratio of the nanocomposite films, defined as the ratio between the dissipated energy through irreversible deformation and the input energy. The ratio maximizes at the critical Up when the PMMA crazing emerges and then decreases as Up further increases. We also find that a critical PMMA-graphene interfacial strength is required to activate PMMA crazing instead of interfacial separation. Additionally, layer thickness affects the amounts of input energy and dissipated energy for nanocomposite films under impact. This study provides important insights into the detailed dynamic deformation mechanisms and how nanointerfaces/nanostructures affect the energy dissipation capability of layered nanocomposite films.

Reactive and Hydraulic Behavior of Granular Mixtures Composed of Zero Valent Iron

Zero valent iron (ZVI) is widely used in permeable reactive barriers (PRBs) for the remediation of contaminated groundwater. The hydraulic conductivity of ZVI can be reduced due to iron corrosion processes activated by water and its constituents including pollutants. To overcome this issue, ZVI particles can be mixed with granular materials that avoid a drastic reduction in the hydraulic conductivity over time. In light of the most recent studies concerning iron corrosion processes and recalling the basic principles of century-old chemistry of iron corrosion, we have revised the results of 24 long-term column tests investigating the hydraulic and reactive behavior of granular mixtures composed of ZVI and pumice or lapillus. From this analysis, we found a clear correlation between the reactive behavior, described by the retardation factor (i.e., the ratio between flow velocity and propagation velocity of the contamination front), and the hydraulic behavior, described by means of the permeability ratio of the reactive medium (i.e., the ratio between the final and initial value of hydraulic conductivity). In particular, the permeability ratio decreased with the increase in the retardation factor. Moreover, it was found that the retardation factor is a useful parameter to evaluate the influence of flow rate, contaminant concentration, and ZVI content on the reactive behavior of the granular medium.

Density-driven exchange flow propagating over an array of densified obstacles

The evolution of bottom-propagating gravity currents with the presence of an array of densified obstacles submerged in a channel is investigated using large-eddy simulations. Our attention is particularly focused on the flow transition of gravity currents over rough surfaces with extra resistance that provokes significant dissipative processes. Two geometric parameters of the roughness elements, namely, the submergence ratio of the obstacle D/H and the gap-spacing ratio λ/D between obstacles, govern their kinematic and dynamic effects on the propagation of gravity currents. Physically, D/H plays a significant role in the control of the current diversion, and λ/D regulates the flow pathway of gravity current propagation. The integrated measures show that two distinct flow morphologies are identified. For a low submergence ratio (D/H<0.15), an overtopping flow is formed in which the gravity current travels on the top of the array and undergoes an inconspicuous loss of buoyancy, subject to minimal vertical convective instability interacting with the underlying ambient fluid within the gap regions. For a sufficiently high submergence ratio (D/H≥0.15) and a certain gap spacing (2≤λ/D<4), an overrunning flow is formed in which the current rapidly decelerates to a buoyancy–inertia state and then transitions to a drag-dominated state with a gain in excessive drag, in which the front velocity is proportional to t−0.5. However, the simulation results show a turning point toward an increase in the gap spacing as λ/D≥4, that the maximum drag acting on the gravity current is measured when it impinges on the second obstacle of an array, and that the drag coefficient goes up by 10%–40%, depending on D/H. The propagation of the gravity current does not show a higher sensitivity to the retarding effect instead. Meanwhile, the promotion of energy conversion occurs because of the gravity current encountering the continuous climbing and plunging flow behavior between two adjacent obstacles in regular motions.

Permeation-diffusion characteristics and air-leakage blocking mechanism for the fire-extinguishing inorganic gel flows in loose broken coal particles

• Three various stages of gel slurry transports in loose coal particles were acquired experimentally and numerically. • Investigated the influence of grouting pressure and coal voids on permeation, diffusion, and filtration of gel. • Theoretically revealed the mechanisms for permeation-diffusion and wind-plugging of gel. Fire prevention with inorganic gel is the most efficient approach to the prevention and control for spontaneous fires of loose coal piles. This work explores the impact of grouting pressures on the seepage, diffusion, and filtration of fire-extinguishing inorganic gel in loose broken coal particles with different voids. Firstly, an experimental system for investigating the filtering effect, permeation behaviors, and diffusion modes for the permeation grouting of inorganic gel in loose coal particles was developed. A 2D discrete element numerical modeling was then performed to probe the intuitive features and underlying mechanisms in the experimental observation, with the numerical modeling matching fairly well with the experimental findings. Results showed that the entire permeation-diffusion process presents obvious stage characteristics with permeation-compaction, permeation-splitting flow, compacting-splitting flow-permeation, and the diffusion distance of gel slurry front increased exponentially while the diffusion velocity decreased initially and then retained stable gradually. Meanwhile, the permeability coefficient of gel slurry showed spatial–temporal properties affected by the filtration effect. Besides, the increasing grouting pressure would promote the filtration effect of gel slurry in loose coal particles, while the decreasing coal sample voids pose an opposite tendency. Under lower grouting pressures (P ≤ 0.3 MPa), the bubble radius increased slightly, and the variation in diffusion distance was faint with the increase in grouting pressure. Under higher grouting pressures (P ≥ 0.3 MPa), the bubble radius increased obviously, and the variation in diffusion distance increased obviously with the increase in grouting pressure, demonstrating that increasing grouting pressure would increase the diffusion distance and diffusion velocity significantly. In addition, increasing the gel grouting pressure and coal voids would result in a non-linear increase in the diffusion distance and diffusion velocity of the gel slurry front, which would also shorten the time required for coal particles to reach the saturation state. Moreover, the mechanisms of seepage-diffusion and wind-plugging of gel slurry in loose coal particles were discussed theoretically. This work would assist us in interpreting and illuminating the seepage-diffusion mechanism and macro–micro characteristics of inorganic gel for high-temperature loose broken coal particles.

Machine learning for enhanced semiconductor characterization from time-resolved photoluminescence

<h2>Summary</h2> To enhance the accuracy and effectiveness of optoelectronic characterization, Bayesian inference has been applied to the statistical analysis of time-resolved photoluminescence (TRPL) data with large-scale graphics-processing unit (GPU)-based simulations of electron dynamics. A simulated TRPL dataset, derived from a CH₃NH₃PbI_3-xCl_x perovskite absorber, was used as a case study. From a power scan, Bayesian inference extracts values of the (ambipolar) carrier mobility, free-carrier density, radiative recombination rate, and the overall lifetime of the nonradiative recombination processes. Analysis of the experimental TRPL data yields similar parameter values. The independent contributions of bulk and surface recombination can be distinguished via the introduction of an additional sample thickness. Further experiments separate the front and back surface recombination velocities and can distinguish the electron and hole mobilities. Ultimately, Bayesian inference enables a significant increase in the information yield from TRPL measurements.

Land cover changes the soil moisture response to rainfall on the Loess Plateau

AbstractInsight into the rainfall‐soil moisture (SM) response to land cover is critical for soil hydrological process modelling and management. In this study, five typical land‐cover types (forest, shrub, grass, crop and bare land) and four rainfall patterns (heavy, intermediate, light and continuous rains) were selected to assess the effects of SM response characteristics on the Loess Plateau of China. We monitored SM at five depths on each land‐cover type at 1‐h intervals over the growing season of 2019. The results showed that rainfall patterns and land‐cover types together determined the SM response process and infiltration efficiency. A minimum accumulated rainfall amount of 5 mm was the threshold required to trigger a 10‐cm SM response. Rain events with a higher intensity and smaller total rainfall amount triggered a quick surface SM response, while larger amounts could percolate deeper and faster. Land‐cover change significantly altered the rainfall‐SM response dynamics and rainwater utilization efficiency after 20 years of ecological construction. Revegetation sites (mean values of forest, shrub and grass) increased the soil wetting depth by 14.7%, shortened the SM response time by 27.3%, and accelerated the SM wetting front velocity by 67.2%, which promoted a 35.2% rainfall transformation rate (RTR) across the 1‐m profile over all rainfall events (R1–13). Moreover, planted forest showed the highest RTR of R1–13 and the maximal increase in soil water storage, which did not aggravate the soil water deficit across the 1‐m profile over the growing season. Therefore, we present evidence that planted forests, instead of shrubs, may be beneficial for water conservation if precipitation is greater than 550 mm. The findings of this study prove the role of revegetation on rainwater infiltration capacity and efficiency and can help improve the management of afforestation in arid and semiarid regions.

The Value of the Early-Time Lieb-Robinson Correlation Function for Qubit Arrays

The Lieb-Robinson correlation function is one way to capture the propagation of quantum entanglement and correlations in many-body systems. We consider arrays of qubits described by the tranverse-field Ising model and examine correlations as the expanding front of entanglement first reaches a particular qubit. Rather than a new bound for the correlation function, we calculate its value, both numerically and analytically. A general analytical result is obtained that enables us to analyze very large arrays of qubits. The velocity of the entanglement front saturates to a constant value, for which an analytic expression is derived. At the leading edge of entanglement, the correlation function is well-described by an exponential reduced by the square root of the distance. This analysis is extended to arbitrary arrays with general coupling and topologies. For regular two and three dimensional qubit arrays with near-neighbor coupling we find the saturation values for the direction-dependent Lieb-Robinson velocity. The symmetry of the underlying 2D or 3D lattice is evident in the shape of surfaces of constant entanglement, even as the correlations front expands over hundreds of qubits.