Three-dimensional Simulations Research Articles

An investigation of force and pressure analysis during the process of equal channel angular pressing to M1 copper using finite element and upper-bound methods

This article presents two methods to determine pressing forces to the M1 copper plate. The first approach employs the upper-bound method, and the second method combines numerical simulation with a central composite design. The upper-bound method uses physical analysis to determine the deformation zone and establish the division of the rigid block model. The findings were identical to the slip-line solution method. The results obtained from the three-dimensional simulation performed using Qform software combined with a central composite design have provided regression equations that help determine the force and punching pressure. The impact of each parameter on the pressing force is similar in both approaches, and the difference between the two methods is insignificant and falls within a dependable range.

Encapsulation structure and in-situ embedding test and model study of a new type of road piezoelectric energy collector

In recent years, research on road piezoelectric energy harvesters (RPEH) has made great progress in the field of road vibration energy harvesting. However, the research on the packaging structure and embedding method of RPEH is not comprehensive. In this paper, a new type of transducer unit structure is developed, which can improve the electrical energy output compared with the drum piezoelectric transducer, and its waterproof performance of the transducer unit is tested under different water depth and duration. Then, a new RPEH package structure is made, and the output performance is compared with the three commonly used package structures in the current research, and the field construction of RPEH with different embedding methods is carried out on the actual road to evaluate the electrical energy output capacity of RPEH under different embedding methods, and the best construction method of RPEH is summarized through durability test. Finally, the three-dimensional simulation model of RPEH with different construction methods is established, and the stress and voltage are analyzed by finite element method to verify the best embedding method. The test results show that the piezoelectric output efficiency of the new transducer unit is increased by 38.9 %, and it still has good electrical energy output after 24 h immersion at a depth of 1.5 m, and its waterproof level reaches IPX8. Under larger excitation frequency and load, the indoor output of RPEH in the new package structure has better output performance in indoor test. The bolt-spring package is used to apply pre-pressure to RPEH, and the output efficiency of RPEH is improved. When the excitation conditions are 10 kN, 10 Hz, the output power is increased by 11.1 %, 127.3 % and 247.4 % compared with the other three package structures. The maximum power of RPEH is 3.41 W by placing concrete on the bottom surface of RPEH sponge washer and using asphalt as the pavement layer to ensure the durability of piezoelectric pavement. The finite element analysis illustrates the strain and voltage variation of piezoelectric pavement with different embedment modes, and the simulation results are consistent with the test results. The research results provide an important basis for the design of RPEH packaging structure and embedding method in road piezoelectric engineering.

VSHPIC: a particle-in-cell algorithm based on vector spherical harmonics expansion

The particle-in-cell (PIC) method has been widely used for studying plasma physics. However, fully three-dimensional PIC simulations always require huge computational resources. For problems with near azimuthal symmetry, recent work (Lifschitz et al 2009 J. Comput. Phys. 228 1803–14, Davidson et al 2015 J. Comput. Phys. 281 1063–77, Li et al 2021 Comput. Phys. Commun. 261 107784, Li et al 2022 J. Comput. Phys. 470 111599) has shown that expanding all the quantities defined on the grid in azimuthal harmonics and truncating the expansion can improve the code efficiency. In this paper, we describe a novel parallel algorithm for efficiently simulating three-dimensional near-spherical symmetry problems. Our approach expands all physical quantities in the θ and ϕ directions in spherical coordinates using vector spherical harmonics. The code is capable of simulating three-dimensional asymmetric scenarios by accurately tracking the evolution of distinct individual modes while preserving the charge conservation law. The fundamental dispersion relation of EM waves in the plasma has been obtained using VSHPIC simulation results. The code also shows a well strong scalability up to more than 1000 cores.

Whistler-mode waves in the tail of Mercury's magnetosphere: A numerical study

Context : Mercury presents a highly dynamic, small magnetosphere in which magnetic reconnection plays a fundamental role. Aim : We aim to model the global characteristics of magnetic reconnection in the Hermean environment. In particular, we focus on waves observed during the third BepiColombo flyby. Method : In this work, we used two fully kinetic three-dimensional (3D) simulations carried out with the iPIC3D code, which models the interaction of the solar wind with the Hermean magnetosphere. For the simulations, we used southward solar wind conditions that allow for a maximum magnetic coupling between the solar wind and the planet. Results : Our simulations show that a significant wave activity, triggered by magnetic reconnection, develops near the diffusion region in the magnetotail and propagates at large scales in the night-side magnetosphere. We see an increase in electron temperature close to the diffusion region and we specifically observe narrowband whistler waves developing near the reconnection region. These waves propagate nearly parallel to the magnetic field at frequency $f ce $. In addition to the electromagnetic component, these waves also exhibit an electrostatic one. Furthermore, we observe a strong electron temperature anisotropy, suggesting it plays a role as the source of these waves.

Impact Dynamics Simulation for Magnetorheological Fluid Saturated Fabric Barriers

Abstract Experimental research has investigated the non-Newtonian fluid augmentation of fabric barrier materials, aimed at adding impact energy dissipation mechanisms and thereby improving ballistic performance. Published experimental results on the effectiveness of these augmentations are mixed, and numerical models supporting complimentary modeling research are lacking, primarily due to the multiple geometric and material nonlinearities present in the system. The combination of Hamiltonian mechanics with hybrid particle-element kinematics offers a very general modeling approach to impact simulation for these systems, one which includes interstitial fluid–structure interactions, the yarn level dynamics of projectile impacts, and yarn fracture without the introduction of slidelines and without mass or energy discard. Three-dimensional (3D) impact simulations show good agreement with published experiments for magnetorheological (MR) fluid-saturated Kevlar, including fabric tested under bulk field excitation of the target region and magnetomechanically edge-clamped fabric sliding in an excited air gap. The Hamiltonian method employed to develop the system-level model allows for computationally efficient partitioning of the modeled physics while maintaining a thermodynamically consistent formulation.

Attosecond and nano-Coulomb electron bunches via the Zero Vector Potential mechanism

The commissioning of multi-petawatt class laser facilities around the world is gathering pace. One of the primary motivations for these investments is the acceleration of high-quality, low-emittance electron bunches. Here we explore the interaction of a high-intensity femtosecond laser pulse with a mass-limited dense target to produce MeV attosecond electron bunches in transmission and confirm with three-dimensional simulation that such bunches have low emittance and nano-Coulomb charge. We then perform a large parameter scan from non-relativistic laser intensities to the laser-QED regime and from the critical plasma density to beyond solid density to demonstrate that the electron bunch energies and the laser pulse energy absorption into the plasma can be quantitatively described via the Zero Vector Potential mechanism. These results have wide-ranging implications for future particle accelerator science and associated technologies.

Simultaneous influence of contact angle, capillary number, and contraction ratio on droplet dynamics in hydrophobic microchannel

Abstract In droplet-based microfluidic systems, droplet motion and behavior largely depend on fluid properties, flow conditions, and microchannel geometry. This study presents a model to predict droplet dynamics within a contraction microchannel, considering various factors, such as contact angle (θ), capillary number (Ca), and contraction ratio (C), within defined ranges, through the employment of a three-dimensional numerical simulation method. Droplets may undergo trap, squeeze, or breakup as they traverse the contraction microchannel, contingent upon the aforementioned factors. The transition from trap to squeeze is determined by the critical value of Capillary number (〖Ca〗_I). Within the squeezed regime, the contact angle influences droplet deformation and velocity within the contraction microchannel. However, when capillary number (〖Ca〗_II) in contraction microchannel increases to a specific threshold corresponding to the contraction ratio, the droplet no longer in contact with the wall and becomes independent of the contact angle, a phenomenon referred to as detachment. Moreover, the contact angle's impact on the transition of droplets from squeeze to breakup is contingent upon the microchannel's contraction ratio. This study provides a comprehensive overview of droplet behavior within microfluidic systems, offering valuable insights for the optimization and design of microsystems.

Lagrangian statistics of concentrated emulsions

The dynamics of stabilised concentrated emulsions presents a rich phenomenology including chaotic emulsification, non-Newtonian rheology and ageing dynamics at rest. Macroscopic rheology results from the complex droplet microdynamics and, in turn, droplet dynamics is influenced by macroscopic flows via the competing action of hydrodynamic and interfacial stresses, giving rise to a complex tangle of elastoplastic effects, diffusion, breakups and coalescence events. This tight multiscale coupling, together with the daunting challenge of experimentally investigating droplets under flow, has hindered the understanding of concentrated emulsions dynamics. We present results from three-dimensional numerical simulations of emulsions that resolve the shape and dynamics of individual droplets, along with the macroscopic flows. We investigate droplet dispersion statistics, measuring probability density functions (p.d.f.s) of droplet displacements and velocities, changing the concentration, in the stirred and ageing regimes. We provide the first measurements, in concentrated emulsions, of the relative droplet–droplet separations p.d.f. and of the droplet acceleration p.d.f., which becomes strongly non-Gaussian as the volume fraction is increased above the jamming point. Cooperative effects, arising when droplets are in contact, are argued to be responsible of the anomalous superdiffusive behaviour of the mean square displacement and of the pair separation at long times, in both the stirred and in the ageing regimes. This superdiffusive behaviour is reflected in a non-Gaussian pair separation p.d.f., whose analytical form is investigated, in the ageing regime, by means of theoretical arguments. This work paves the way to developing a connection between Lagrangian dynamics and rheology in concentrated emulsions.

Study on the Application of Water Resource Management Based on GF-7 Stereo Mapping Satellite

Abstract. GF-7 satellite was successfully launched on 3 Nov 2019. The two-line stereo camera can effectively obtain 20 km-width panchromatic stereo images with resolution better than 0.8m and 3.2m-resolution multispectral images. Through the composite mapping mode of stereo camera and laser altimeter, the satellite realizes 1:10,000-scale stereo mapping. This article selected the Qiafuqihai Reservoir and its upstream of the Yili River Basin as the study area. Hydrological landforms and vegetation coverage were monitored and three-dimensional dynamic simulation software was developed to verify the potential application of GF-7 stereo mapping satellite data in supporting drainage basin water resource allocation and management in future scenarios. 3 meters DSM derived from GF-7 surveying and mapping satellite finely portrayed the characteristics of hydroponic landforms of small watersheds. The upstream of Qiafuqihai was divided into eleven small watersheds. The water density was 8.5 times more than which was produced from STRM 90m. The elevation of the water-level fluctuation zone of the Qiafuqihai Reservoir ranged from 970 to 998 meters with the water area changed from 28.3 to 57.6 square kilometres. The terrain in the northwest and northeast was flat, with the main vegetation types being natural grasslands (64.9%) and arid lands (5.03%). The three-dimensional dynamic visualization software was developed and demonstrated hydrology information, vegetation coverage information and dynamic changes in the landform of the basin in three-dimensional environment. GF-7, the stereo surveying and mapping satellite, could be perfectly able to support the application of water resources deployment and management in the future.

Investigation of Sidewall and Reynolds Number Effects in a Ribbed Square Duct

AbstractThis paper presents a comprehensive three-dimensional Direct Numerical Simulation (DNS) investigation of flow characteristics over a roughened wall in a duct, exploring bulk Reynolds number variations from 4400 to 13,200 and considering the influence of side walls. Notably, a significant increase in friction factor highlights heightened resistance to flow due to surface roughness, emphasizing the importance of considering surface conditions in engineering applications. The study reveals three recirculation zones in the streamwise direction, indicating intricate separation phenomena caused by the interaction between the ribbed wall and the flow. Furthermore, the presence of sidewalls creates two major vortices perpendicular to the flow direction, highlighting the complexity of the vortical structures in this configuration, crucial for predicting flow behavior and optimizing system performance. The study demonstrates that the influence of the Reynolds number on these vortices is not well-scalable with respect to outer units, in contrast with respect to smooth ducts. Investigating side-wall effects, increased turbulent production rate, and non-equilibrium turbulence along the side walls highlight the sensitivity of turbulent stresses to Reynolds number and side-wall proximity. In conclusion, the paper provides novel insights into the interplay between Reynolds numbers, roughened surfaces, and boundary conditions in turbulent flows, laying a foundation for a deeper understanding of the flow in duct with high roughness.

A Novel Numerical Model Considering Unsaturated Soil Properties and Computational Study on Heat and Moisture Transfer Characteristics of Helix Ground Heat Exchanger

Abstract A novel three-dimensional numerical simulation model for the helix ground heat exchanger was proposed in this paper, which takes into account unsaturated soil properties. This model is more suitable for real working conditions. To validate its accuracy, a miniature model heat transfer experimental platform was constructed. Additionally, the study conducted simulation research using three types of soil with significantly different thermal and moisture characteristics. Moreover, the comprehensive thermal conductivity and water diffusion coefficient of these three soil types were determined by relevant literature and experimental tests. The aim was to comprehensively explore the impact of different soil types on the heat and mass transfer of the helix ground heat exchanger. The results indicate that the numerical model developed in this paper accurately captures the heat and mass transfer characteristics of the helix ground heat exchanger to a certain extent. Increasing the comprehensive thermal conductivity and water diffusion coefficient of the soil can significantly enhance the heat exchange capacity of the exchanger. For instance, under sandy loam conditions, the heat exchange capacity is approximately 20.73% higher compared to clay loam conditions. The study also identifies two distinct areas around the helix ground heat exchanger: the severe change region and the soft change region. In the severe change region, there is a notable decrease in soil water content near the exchanger, which inevitably weakens the thermal conductivity of the soil. It is advised to minimize this effect through measures like active water spraying.

Effect of horizontal magnetic field position on oxygen distribution in CZ silicon crystal growth

The horizontal magnetic field-assisted Czochralski(CZ) method is gaining significant popularity in the semiconductor industry for producing 200 mm semiconductor-grade silicon wafers. In this study, the impact of the horizontal magnetic field position on the oxygen content and radial uniformity of semiconductor-grade silicon crystals produced by the CZ method was investigated. Three-dimensional simulations were conducted using finite element software to calculate the melt convection and oxygen content before and after adjustment of the magnetic field position, respectively. The simulations results predicted a decrease in the oxygen content of the crystals following the adjustment of the magnetic field position from 0 to 75 mm below the free surface, and this was experimentally verified. The velocity distribution of the free melt surface exhibits better uniformity when lowering the magnetic field position. The influence of the magnetic field position on the temperature distribution and Marangoni flow in the melt was investigated, and the result of oxygen concentration difference identified the adjustment of the magnetic field position promoted the greater Marangoni flow below the free melt surface, which promoted the volatilization of SiO gas and thus reduced the oxygen concentration in the silicon ingot. Therefore, adjusting the position of the magnetic field is an effective way to increase the Marangoni flow and to obtain semiconductor-grade silicon crystal with lower oxygen content and better oxygen radial homogeneity.

A low loss silicon waveguide bend based on width and curvature variations

A simple, low loss and compact 90° SOI waveguide bend constructed by using an inner circular curve and an outer Euler-circular curve was proposed, optimal designed and fabricated. By adopting this simple structure, the width and curvature of the waveguide bend can be varied simultaneously, which can effectively suppress both the mode mismatch loss and the radiation loss. Three-dimensional finite difference time domain simulation results indicate a very low excess loss of about 0.0056 dB can be achieved for equivalent bending radius of 2 μm. By using the cut-back method, a low loss of about 0.0175 dB at 1550 nm was measured, which is less than one quarter of the traditional circular bend. In addition, the wavelength dependence for the proposed SOI waveguide bend was investigated and 0.0088 dB–0.0247 dB was measured in the wavelength range of 1480∼1580 nm.

A three-dimensional numerical model for computing the actual voltage induced in a Barkhausen surface coil

This investigation presents a novel three-dimensional simulation model designed for the analysis of Magnetic Barkhausen Noise (MBN). The model’s formulation is directly deduced from Maxwell’s equations, establishing a Boundary Value Problem (BVP) with boundary conditions tailored to a specific configuration involving a lengthy steel rod encased within a solenoid, subjected to sinusoidal current excitation, all contained within an air enclosure. This chosen configuration holds physical equivalence to the majority of setups commonly utilized in Barkhausen experiments. Utilizing the proposed three-dimensional model, it became feasible, for the first time, to computationally assess the voltage induced in the coil by the flux emanating from all spatial directions. The findings indicate that the MBN signal is primarily generated by the flux rate aligned with the direction of the applied field. Nevertheless, simulations confirm additional contributions to the signal originating from magnetic flux in directions significantly deviating from the applied field direction. It is demonstrated that these contributions from lateral flux significantly influence the angular dependence of the Barkhausen signal energy, showcasing a notable alignment between the proposed model and experimental results. The results underscore the capability of the proposed model to facilitate the simulation of Barkhausen Noise.

Multi-year three-dimensional simulation of seasonal variation in phytoplankton species composition in a large shallow lake

Lake Erie has been negatively impacted by multiple stressors, including nutrient enrichment and climate change, that have exacerbated eutrophication and harmful algal blooms. Management of these long-term water quality problems requires numerical models that can be run over years to decades. The three-dimensional hydrodynamics and biogeochemistry models applied to date, however, have not been tested for continuous runs longer than one year and have not been shown to accurately reproduce seasonal variation in phytoplankton species composition (e.g., the development of harmful algal blooms) over decadal timescales. We simulated the three-dimensional nutrient and phytoplankton concentrations in western Lake Erie continuously from 2002 to 2014. Using a single parameter set, we were able to reproduce both seasonal and inter-annual variation in phytoplankton species composition. The model qualitatively reproduced the observed seasonal succession (i.e., variation in phytoplankton species composition), including the spring diatom bloom and late summer cyanobacterial growth. This study demonstrates that three-dimensional models can be applied for multi-year simulations of nutrients and phytoplankton to inform large lake research and management.

Precise execution of personalized surgical planning using three-dimensional printed guide template in severe and complex adult spinal deformity patients requiring three-column osteotomy: a retrospective, comparative matched-cohort study

BackgroundThe surgical treatment of severe and complex adult spinal deformity (ASD) commonly required three-column osteotomy (3-CO), which was technically demanding with high risk of neurological deficit. Personalized three dimensional (3D)-printed guide template based on preoperative planning has been gradually applied in 3-CO procedure. The purpose of this study was to compare the efficacy, safety, and precision of 3D-printed osteotomy guide template and free-hand technique in the treatment of severe and complex ASD patients requiring 3-CO.MethodsThis was a single-centre retrospective comparative cohort study of patients with severe and complex ASD (Cobb angle of scoliosis > 80° with flexibility < 25% or focal kyphosis > 90°) who underwent posterior spinal fusion and 3-CO between January 2020 to January 2023, with a minimum 12 months follow-up. Personalized computer-assisted three-dimensional osteotomy simulation was performed for all recruited patients, who were further divided into template and non-template groups based on the application of 3D-printed osteotomy guide template according to the surgical planning. Patients in the two groups were age- and gender- propensity-matched. The radiographic parameters, postoperative neurological deficit, and precision of osteotomy execution were compared between groups.ResultsA total of 40 patients (age 36.53 ± 11.98 years) were retrospectively recruited, with 20 patients in each group. The preoperative focal kyphosis (FK) was 92.72° ± 36.77° in the template group and 93.47° ± 33.91° in the non-template group, with a main curve Cobb angle of 63.35° (15.00°, 92.25°) and 64.00° (20.25°, 99.20°), respectively. Following the correction surgery, there were no significant differences in postoperative FK, postoperative main curve Cobb angle, correction rate of FK (54.20% vs. 51.94%, P = 0.738), and correction rate of main curve Cobb angle (72.41% vs. 61.33%, P = 0.101) between the groups. However, the match ratio of execution to simulation osteotomy angle was significantly greater in the template group than the non-template group (coronal: 89.90% vs. 74.50%, P < 0.001; sagittal: 90.45% vs. 80.35%, P < 0.001). The operating time (ORT) was significantly shorter (359.25 ± 57.79 min vs. 398.90 ± 59.48 min, P = 0.039) and the incidence of postoperative neurological deficit (5.0% vs. 35.0%, P = 0.018) was significantly lower in the template group than the non-template group.ConclusionPerforming 3-CO with the assistance of personalized 3D-printed guide template could increase the precision of execution, decrease the risk of postoperative neurological deficit, and shorten the ORT in the correction surgery for severe and complex ASD. The personalized osteotomy guide had the advantages of 3D insight of the case-specific anatomy, identification of osteotomy location, and translation of the surgical planning or simulation to the real surgical site.

Efficiency Assessment on Roof Geometry and Trombe Wall Shape for Improving Buildings’ Heating Performance

It is crucial to consider structural design issues in Trombe wall (T-wall) buildings to promote more suitable indoor climates and thermal comfort standards. Therefore, the present study examined the impact of two different T-wall designs and six different roof types on the energy and operational efficiency of a building located in a low-temperature and high-humidity winter climate. Ansys-CFX 15.0 software was employed to simulate the thermal and fluid dynamics behavior of the T-wall system, and flow, thermal comfort, energy, and exergy analyses were conducted. Three-dimensional simulation results and the pertinent literature data showed a good level of agreement, and the accuracy of the model was ensured. Outcomes revealed an average air velocity variation of 0.186 m/s and maximum average indoor air temperature variation of 3.3 °C between the six roof geometries. The highest air speed (0.988 m/s) was recorded for the gambrel roof while the lowest one (0.802 m/s) was recorded for the typical flat roof. The shed roof right with a rounded T-wall was more comfortable for standing and sitting activity than the others for the two T-wall shapes, and, at Y = 0.6 m and Y = 1.1 m, the average predicted percentages of dissatisfied (PPD) values were 31 and 28%, respectively. Furthermore, it was determined in the study that solar radiation intensity and T-wall and roof geometries had a significant effect on energy and exergy efficiency, and high energy and exergy efficiencies were achieved at higher solar intensity values. The best energy and exergy efficiencies were obtained for the butterfly and shed roof configurations. This study can serve as a reference for the thermal environment design of buildings with T-walls.

Three-dimensional numerical modeling of the temporal evolution of backward erosion piping

Backward erosion piping (BEP) is a complex degradation mechanism in geotechnical flood protection infrastructure (GFPI) that is still relatively less understood, particularly when considering its time-dependent features. This manuscript presents a novel dual random lattice modeling approach for three-dimensional simulation of BEP, with a focus on its evolution over time. The key novelty of this presented framework is twofold: (1) we propose and incorporate a novel constitutive relationship for computation of time-dependent soil erosion based on the theory of rate processes, and (2) we devise an algorithm for calculation of coupled degradation of the dual lattices for accurate computation of 3-D hydraulic gradients. The constitutive relationship was developed from fundamental granular physics, and brings the potential to provide deeper fundamental physical understanding of the phenomenon. The capabilities of the modeling framework are investigated by comparison with available laboratory experiments which illustrates good agreement in the spatial advancement of piping erosion, pipe progression speeds, as well as the evolution of local gradients. To the best knowledge of the authors, the presented model is the first to be able to capture all of the aforementioned features when simulating BEP.

CFD modeling of a fluidized bed with volatiles distributor for biomass chemical looping combustion

Achieving high volatiles conversion is crucial to biomass chemical looping combustion. Challenges arise from rapid devolatilization of biomass and limited biomass injection ports, resulting in volatiles with insufficient contact with oxygen carriers in fluidized beds. A concept called volatiles distributor (VD) has recently been proposed and investigated in a cold-flow fluidized bed, which shows excellent performance in achieving an even distribution of volatiles over the cross section. To deeply understand VD's impact on hydrodynamics behaviors, pioneering three-dimensional full-loop cold-flow CFD simulations were conducted using an Eulerian multiphase granular model. Three drag models, i.e., Gidaspow, Filtered, and two-step EMMS/bubbling, were evaluated against experimental data. While all models perform well in bubbling fluidization, the two-step EMMS/bubbling model excels in turbulent fluidization. Additionally, CFD simulations reveal improved mixing between volatiles and bed materials with VD, highlighting its efficiency in addressing incomplete conversion of high-volatile fuels like biomass in fluidized bed systems.

Decretion disc size in Be/X-ray binaries depends upon the disc aspect ratio

ABSTRACT With three-dimensional hydrodynamical simulations we show that the size of the decretion disc and the structure of the accretion flow on to the neutron star in a Be/X-ray binary strongly depends upon the disc aspect ratio, H/R. We simulate a Be star disc that is coplanar to the orbit of a circularly or moderately eccentric neutron star companion, thereby maximizing the effects of tidal truncation. For low-disc aspect ratio, H/R ≲ 0.1, the disc is efficiently tidally truncated by the neutron star. Most material that escapes the Roche lobe of the Be star is accreted by the neutron star through tidal streams. For larger disc aspect ratio, the outflow rate through the Be star disc is higher, tidal truncation becomes inefficient, the disc fills the Roche lobe and extends to the orbit of the companion. Some material escapes the binary as a gas stream that begins near the L2 point. While the accretion rate on to the neutron star is higher, the fraction of the outflow that is accreted by the neutron star is smaller. Low-density Be star discs are expected to be approximately isothermal, such that H/R increases with radius. Tidal truncation is therefore weaker for larger separation binaries, and lower mass primaries.