Three-dimensional Gas Flow Research Articles

Analysis of the Effect of Sampling Probe Geometry on Measurement Accuracy in Supersonic Gas Flow

The accuracy of sampling of gas components has a significant impact on the measurement of various performance parameters in the combustion chamber of an aero-engine. In order to investigate the effect of the probe geometry of a six-point gas sampling probe on sampling accuracy in supersonic gas flow, a three-dimensional probe gas flow characteristic solution model is established through numerical simulation methods of components of transport and fluid–solid coupling. Probes with three angles of 28°, 30°, and 32° and an optimized conical probe are constructed. The sampling accuracy of the probes with different geometries is compared and evaluated by the deviation of the component volume fraction before and after sampling and the resulting combustion efficiency error. This paper presents a set of calculation methods for solving the relative deviation of volume fraction by an iterative method based on the ideal gas law and the Redlich–Kwong equation (R-K equation). The method is designed to solve the exact component volume fraction problem in the simulation calculation. The study results demonstrate that the 28° and optimized conical probes improve sampling accuracy more effectively than the original 30° structure. The deviation of the volume fractions of the two structures is less than 1.7%, and the combustion efficiency error is less than 0.09%. The developed iterative calculation method can significantly reduce the theoretical calculation error to less than 0.06%. The experimental data of the test bench are in good agreement with the simulation results, thereby demonstrating the reliability and accuracy of the sampling probe following structural optimization.

Physics-Embedded Deep Learning to Predict Real-Time Flow Parameters in Complex Thermodynamic Machinery

Flow through complex thermodynamic machinery is intricate, incorporating turbulence, compressibility effects, combustion, and solid–fluid interactions, posing a challenge to classical physics. For example, it is not currently possible to simulate a three-dimensional full-field gas flow through the propulsion of an aircraft. In this study, a new approach is presented for predicting the real-time fluid properties of complex flow. This perspective is obtained from deep learning, but it is significant in that the physical context is embedded within the deep learning architecture. Cases of excessive working states are analyzed to validate the effectiveness of the given architecture, and the results align with the experimental data. This study introduces a new and appealing method for predicting real-time fluid properties using complex thermomechanical systems.

Analytic description of the gas flow around planets embedded in protoplanetary disks

Context. A growing planet embedded in a protoplanetary disk induces three-dimensional gas flow, which exhibits a midplane outflow that can suppress dust accretion onto the planet and form global dust substructures (rings and gaps). Aims. Because analytic formulae for the planet-induced outflow are useful for modeling its influences on local and global dust surface densities and planet accretion, we derived analytic formulae that describe the morphology and velocity of the planet-induced outflow. Methods. We first performed three-dimensional, nonisothermal hydrodynamical simulations of the gas flow past a planet, which enabled us to introduce a fitting formula that describes the morphology of the outflow. We then derived an analytic formula for the outflow speed using Bernoulli’s theorem. Results. We successfully derived a fitting formula for the midplane outflow morphology (the shape of the streamline), which is valid when the dimensionless thermal mass falls below m ≲ 0.6. The obtained analytic formulae for the outflow, such as the maximum outflow speed and the velocity distributions of the outflow in the radial and vertical directions to the disk, show good agreement with the numerical results. We find the following trends: (1) the maximum outflow speed increases with the planetary mass and has a peak of ~30–40% of the sound speed when the dimensionless thermal mass is m ~ 0.3, corresponding to a super-Earth mass planet at 1 au for the typical steady accretion disk model, and (2) the presence of the headwind (namely, the global pressure force acting in the positive radial direction of the disk) enhances (reduces) the outflow toward the outside (inside) of the planetary orbit. Conclusions. The planet-induced outflow of the gas affects the dust motion when the dimensionless stopping time of dust falls below St ≲ min(10 m2, 0.1), which can be used to model the dust velocity influenced by the outflow.

Unified stochastic particle simulation of polyatomic gas flows using SPARTACUS

Numerically simulating multiscale nonequilibrium gas flows that involve internal energy exchange is a challenging problem in engineering applications due to the complexity of the physical processes. The unified stochastic particle (USP) method, based on the Bhatnagar-Gross-Krook (BGK) model, is a promising approach for efficiently simulating multiscale flows. In this paper, we develop a USP method for polyatomic gases that can simulate hypersonic gas flows involving internal energy exchange across all flow regimes. This method is integrated into the open-source SPARTACUS solver, which was previously developed for monatomic gases. The newly developed USP method reassigns the velocity and internal energies of a subset of computational particles based on a precisely designed local target distribution function. A conservation algorithm is applied to ensure that the momentum and energy are conserved. To validate the accuracy and efficiency of the solver, we perform three benchmark cases, including flow past a two-dimensional cylinder, a three-dimensional 70° blunted cone, and an axisymmetric double cone configuration. The results demonstrate that SPARTACUS achieves excellent agreement with the reference results of DSMC and experimental data. Additionally, SPARTACUS provides significant computational efficiency advantages over the DSMC solver SPARTA, particularly in simulating three-dimensional gas flows.

ПРОМИСЛОВИЙ ЦИКЛОННО-РОТАЦІЙНИЙ ПИЛОВЛОВЛЮВАЧ

Cyclone equipment is easy to manufacture and operate, versatile and economical, but its main drawback is the inability to capture finely dispersed dust. The article discusses the most important characteristics of the highly efficient industrial cyclonic rotary dust collector developed by us. The apparatus implements the principle of two-stage separation of heterogeneous dust-gas systems in a centrifugal field. In the first stage, working on the principle of a cyclone, and in the second - working on the principle of a rotary dust collector. During operation of the apparatus, a higher pressure on the periphery of the rotating part returns a concentrated dusty flow with a low gas flow to the cyclone part. Formed by a gap, between the cyclone body and the dividing cone, a closed circuit connects the output and input of the cyclone-rotational dust collector to each other. There is a negative feedback - an effective control mechanism for such complex probabilistic systems as gas heterogeneous curvilinear flows. Negative feedback stabilizes the processes occurring in such flows. The energy of the electric motor is used in the cyclone-rotational dust collector for the movement of dusty gas, concentration, coagulation of fine dust, its return from the rotary to the cyclone zone, dispersion and capture, with the aim of saving energy and material resources. The main task of the theory of centrifugal separation of dusty flows is to find the minimum diameter of a particle that has a practically perceptible relative radial velocity. Known deterministic models of centrifugal separation are either too simple and ignore many factors that determine the process, or in attempts to account for them are too cumbersome and difficult to solve, although they are also not free from various assumptions and limitations. The stochastic model takes into account the integral set of various influences on the radial drift of the Stokes particle from the center to the periphery of the swirling flow. The paper presents the results of the study of the structure of the three-dimensional two-phase gas flow with highly dispersed inclusions in cyclonic-rotary dust collectors using 3D geometric models with the aim of visualizing these flows using computer graphics methods. Numerical studies were carried out using the k-e model of turbulence with scaled wall functions.

SPARTACUS: An open-source unified stochastic particle solver for the simulation of multiscale nonequilibrium gas flows

Recently, a unified stochastic particle (USP) method [14] has been developed for the simulation of multiscale nonequilibrium gas flows. Compared with the conventional stochastic particle methods, such as the direct simulation Monte Carlo (DSMC) method, USP can be applied with much larger time steps and cell sizes as it couples the effects of particle movements and collisions. To extend the applicability of USP to complex nonequilibrium gas flows, we develop a USP solver referred to as SPARTACUS within the framework of SPARTA, which is a widely used software based on the DSMC method for the simulation of rarefied gas flows. Inheriting from SPARTA, SPARTACUS is parallelized with the message passing interface (MPI), and it is open-source under the GNU General Public License (GPL) and released in an online repository with essential documentation, tutorial examples, and benchmark cases. To evaluate the performance of SPARTACUS, the test cases cover from one to three dimensional flows with a wide range of the Knudsen number and Mach number, including Couette flow, lid-driven cavity flow, hypersonic flow past a cylinder, and supersonic flow around a blunt body. The results obtained by SPARTACUS in the rarefied and continuum regimes are in good agreement with DSMC and computational fluid dynamics (CFD) results, respectively. Moreover, SPARTACUS has significant computational efficiency advantage over traditional DSMC solvers, especially for three-dimensional nonequilibrium gas flows. In addition, SPARTACUS shows comparable parallel efficiency with SPARTA, so it is promising to be applied with high performance computing for the simulation of complex multiscale gas flows encountered in engineering problems. Program summaryProgram Title: SPARTACUSCPC Library link to program files:https://doi.org/10.17632/pyb343w7zv.1Developer's repository link:https://github.com/KKFeng/spartacusLicensing provisions: GNU General Public License version 2Programming language: C++External routines/libraries: SPARTA (http://sparta.sandia.gov/)Nature of problem: Multiscale simulation of nonequilibrium gas flows using unified stochastic particle method.Solution method: Unified Stochastic Particle (USP) method.

CFD simulation of hot spot in PEM fuel cell with diverging and converging flow channels

Hot spot formation is one of the most critical factors contributing to the membrane layer deterioration and, hence, fuel cell performance degradation. When local hot spot patches develop, the long-term endurance of a PEM fuel cell catalyst and membrane layer is dramatically reduced. The reactive flow gas distribution inside a PEM fuel cell with a 3-D gas diffusion layer has been investigated using divergent and convergent gas flow channels. COMSOL Multiphysics 4.4a was used for simulation studies. A three-dimensional gas flow channel for a single-phase, a non-isothermal simulation model of cathode side PEM fuel cells was created to investigate the impact of the local hot spot patch on the cathode side PEM fuel cells. The effect of local hot spot patches on the catalyst and membrane layer is investigated in this work, using a transient-state modeling approach. The conclusions have important implications for evaluating the internal reaction uniformity in the PEM fuel cell using divergent and convergent flow channels and developing a theoretical basis for evaluating local hot spot patches on the catalyst and membrane layer surface. Maximum surface temperature of the divergent flow channel was found to be 4% lower than that of the convergent flow channel. The studies aim to examine the possibility of altered variation of local hot spots on the catalyst and membrane layer due to changes in depth-dependent flow fields.

Flutter of Fan Blades in the Aircraft Engine in the Three-Dimensional Subsonic Gas Flow

Aeroelasticity problems arise in the different fields of technology. The accident-free operation of the airborne machines is one of the most important factors that should be taken into account during their designing and upgrading. The solution of this problem involves the implementation of many measures to provide the system reliability on the whole, including its individual elements, in particular aircraft engine, its fan whose wide-chord blades can be exposed to the wreckage due to different reasons including the aeroelastic effects, i.e. self-excited vibrations. As a result, the origination of the aeroelastic phenomenon (flutter) in design and off-design modes should be eliminated at the stage of the design and operational development of the rotor wheel that would result in a considerable increase of the level of reliability of the aircraft engine. Based on the analysis of the available methods used for the flutter prediction we can draw a conclusion that the most promising approach to the analysis of the aeroelastic behavior of the blade ring of fan is the use of the method based on the three-dimensional model of the aerodynamics and dynamics (the method used for the solution of the coupled aeroelastic problem). By solving the coupled aeroelastic problem of the nonstationary aerodynamics and elastic vibrations of the blades we can get the amplitude –frequency blade vibration spectrum for the three-dimensional gas flow, including forced vibrations and self-excided vibrations in order to increase the reliability of the blade row of turbine machines. The developed numerical method was used for the analysis of the aeroelastic behavior of the blade ring of the fan mounted in the airborne engine for the operation mode of 3520 rmp with appropriate boundary conditions at the inlet and outlet behind the ring. The computation data confirmed the origination of self-vibrations for the given fan operation mode.

Dynamic Instability of Shallow Shells Interacting with a Three-Dimensional Potential Gas Flow

To study the interaction of a vibrating shallow shell with a three-dimensional subsonic gas flow, we deduce a system of hypersingular integral equations for the aerodynamic derivatives of the pressure drop. This system of equations is convenient for the solution of the problems of aeroelasticity. We solve the system of hypersingular integral equations by using a numerical approach based on the discrete vortex method. To model vibrations of shallow shells, we deduce a system of ordinary differential equations with the help of the assumed-mode method. We also perform the numerical investigation of dynamic instability of the equilibrium state of a shallow shell in the subsonic gas flow.

Pore-scale gas flow simulations by the DSBGK and DVM methods

Shale gas flow at the pore scale is very challenging to simulate due to high Knudsen numbers (Kn), low speeds and complicated pore geometries. The direct simulation BGK (DSBGK) method and the discrete velocity method (DVM) are promising methods to simulate three-dimensional (3D) gas flows in the shale rock. As the grid-convergence accuracy and computational cost highly depend on the spatial and molecular velocity grids, here we present a rigorous study of the computational performance of the two methods by simulating gas flows through the standard 3D Sierpinski carpet. In this study, we found reasonable agreement between the DSBGK and DVM simulations when comparing the velocity profiles and permeability curves obtained by using fine spatial and velocity grids. However, the grid convergence of the DSBGK simulation is faster than that of the DVM, which requires a very large number of spatial and velocity grids for convergence. Consequently, the DSBGK simulation is orders of magnitude cheaper than the DVM simulation in terms of the CPU time and memory usage. While the DVM requires a high-performance computing facility to meet the memory demand of several hundreds of gigabyte for simulating this 3D flow problem, the DSBGK simulation can run on an ordinary laptop. This work therefore demonstrates the DSBGK method as a practical simulation tool for 3D pore-scale gas flows in the shale rock. Converged results generated in this study can be used as comparison data by other simulation methods.

The Influence of Mesh Resolution on 3D RANS Flow Simulations in Turbomachinery Flow Parts

The question of the difference mesh refinement degree influence on the results of calculation of the three-dimensional viscous gas flows in the flow parts of turbomachines using the RANS flow models and second order numerical methods is considered. Calculations of flows for a number of turbine and compressor grids on successively refining grids have been performed. We used H-type grids with approximate orthogonalization of cells in the boundary layer. The calculations were carried out using a CFD solver F with the use of an implicit ENO scheme of the second order, a local time step, and a simplified multigrid algorithm. When calculating the flow on fine grids, the following were used: convergence acceleration tools implemented in the solver; truncation of the computational domain with subsequent distribution of the results based on the symmetry property; the computational domain splitting into parts and computations parallelizing. Comparison of the obtained results is carried out, both in terms of qualitative resolution of the complex structure of three-dimensional flows, and in terms of quantitative assessment of losses. Grid convergence was estimated in two ways. In the first, the characteristic two-dimensional distributions of parameters obtained on different grids were visually compared. The purpose of such comparisons was to evaluate the sufficient degree of solution of both the general structure of the flow in grids and its features, namely, shock waves, contact discontinuities, separation zones, wakes, etc. The second estimation method is based on the grid convergence index (GCI). The GCI calculated from the three-dimensional density field was considered in this paper. It is concluded that for scientific research requiring high accuracy of calculations and detailing of the structure of a three-dimensional flow, very fine difference meshes with the number of cells from 106 to 108 in one blade-to-blade channel are needed, while for engineering calculations, under certain conditions, it is sufficient to use meshes with the number of cells less than 1 million in one blade-to-blade channel.

Predicting the Flutter of the Fan Row in the Aircraft Engine

The aeroelasticity of the cascades of blades is considered to be the most burning and important problem, because it effects the reliability of the flow channels of turbine machines. The main complexity of the investigation of aeroelastic phenomena that occur in turbine machines consists in the need of the simulation of the interaction of the two physical environments (liquid and elastic). One of the approaches to the computation of aeroelastic phenomena (the flutter) is the nonstationary aerodynamic analysis that studies nonstationary flows in turbine machines independently of their origin and over the past decades it covered the way from the linear theory to the solution of Navier -Stokes equations. The second approach to the investigation of the aeroelasticity of the cascades of blades of turbine machines is related to the studies of the motion of blades under the action of aerodynamic forces induced by the nonuniformity of the main flow and independent of the motion of the blades. These approaches ignore the influence of vibrating blades on the main gas flow. Within the framework of individual problems considered in scientific papers the problem of self-excited vibrations (the flutter) that are the most complicated and least studied phenomena falls out of consideration. Based on the analysis of the contemporary state of the problem dealing with the turbine machine aeroelasticity and available methods of the flutter prediction we can draw a conclusion that the most promising approach to the investigation of the aeroelastic behavior of the blade row of the turbine machine is the approach that is based on the three-dimensional model of the nonstationary dynamics and the modal analysis of the blade motion ( the method of the solution of the coupled aeroelastic problem). This method of the solution of the coupled problem of the nonstationary aerodynamics and elastic vibrations of the blades will enable the prediction of the amplitude-&-frequency spectrum of blade vibrations in the three-dimensional gas flow, including forced self-excited vibrations in order to increase the reliability of the cascades of blades of turbine machines. Using the developed numerical method, we analyzed the aeroelastic behavior of the fan row of the aircraft engine in different modes. The computation data confirmed the appearance of nonstationary modes in the fan operation. The research done allows us to give recommendations on the elimination of the aeroelastic phenomena, i.e. the flutter (by changing the fan blade geometry).

Numerical modelling of a gas-dust sphere scattering by SPH-IDIC method

The paper is devoted to the first application of the SPH-IDIC method to simulation of three-dimensional dusty gas flow. The expansion of a sphere consisting of gas and monodisperse fine dust into vacuum is considered as a model problem. Computing was carried out using graphic accelerators based on nVidia CUDA technology. The numerical simulations show ability of the method to deal with three-dimensional dynamics. We found that in 3D case the method kept all the advantages previously demonstrated on one-dimensional problems: robustness, efficiency and simplicity in implementation.

Diffusion Deposition of Nanoparticles in a Layer of Granules

The deposition of aerosol nanoparticles has been studied in a model granular filter composed of parallel monolayers of contacting spherical granules with a square packing, where each even monolayer is shifted in its plane relative to an odd layer by the granule radius in two directions normal to one another. The monolayers of the granules are oriented normal to an incident flow. The efficiencies of diffusion collection of point particles by the granules have been calculated as depending on the distance between the layers in the regime of a three-dimensional gas flow at small Reynolds numbers $$\operatorname{Re} $$ $$ \ll $$ 1 within a Peclet number range of $${\text{Pe}}$$ = 102−104. The calculation results have agreed with the published data and the results of measuring the deposition of monodisperse nanoparticles from nitrogen and helium upon filtration through layers of monodisperse metal spheres. The possibility of using a layer of granules as a diffusion battery for measuring the sizes of aerosol particles has been discussed.

Influences of protoplanet-induced three-dimensional gas flow on pebble accretion

Context.Pebble accretion is among the major theories of planet formation. Aerodynamically small particles, called pebbles, are highly affected by the gas flow. A growing planet embedded in a protoplanetary disk induces three-dimensional (3D) gas flow. In our previous work, Paper I, we focused on the shear regime of pebble accretion and investigated the influence of planet-induced gas flow on pebble accretion. In Paper I, we found that pebble accretion is inefficient in the planet-induced gas flow compared to that of the unperturbed flow, particularly when St ≲ 10−3, where St is the Stokes number.Aims.Following on the findings of Paper I, we investigate the influence of planet-induced gas flow on pebble accretion. We did not consider the headwind of the gas in Paper I. Here, we extend our study to the headwind regime of pebble accretion.Methods.Assuming a nonisothermal, inviscid sub-Keplerian gas disk, we performed 3D hydrodynamical simulations on the spherical polar grid hosting a planet with the dimensionless mass,m=RBondi∕H, located at its center, whereRBondiandHare the Bondi radius and the disk scale height, respectively. We then numerically integrated the equation of motion for pebbles in 3D using hydrodynamical simulation data.Results.We first divided the planet-induced gas flow into two regimes: flow-shear and flow-headwind. In the flow-shear regime, where the planet-induced gas flow has a vertically rotational symmetric structure, we find that the outcome is identical to what we obtained in Paper I. In the flow-headwind regime, the strong headwind of the gas breaks the symmetric structure of the planet-induced gas flow. In the flow-headwind regime, we find that the trajectories of pebbles with St ≲ 10−3in the planet-induced gas flow differ significantly from those of the unperturbed flow. The recycling flow, where gas from the disk enters the gravitational sphere at low latitudes and exits at high latitudes, gathers pebbles around the planet. We derive the flow transition mass analytically,mt, flow, which discriminates between the flow-headwind and flow-shear regimes. From the relation betweenm,mt, flowandmt, peb, wheremt, pebis the transition mass of the accretion regime of pebbles, we classify the results obtained in both Paper I and this study into four groups. In particular, only when the Stokes gas drag law is adopted andm< min(mt, peb,mt, flow), where the accretion and flow regime are both in the headwind regime, the accretion probability of pebbles with St ≲ 10−3is enhanced in the planet-induced gas flow compared to that of the unperturbed flow.Conclusions.Combining our results with the spacial variety of turbulence strength and pebble size in a disk, we conclude that the planet-induced gas flow still allows for pebble accretion in the early stage of planet formation. The suppression of pebble accretion due to the planet-induced gas flow occurs only in the late stage of planet formation, more specifically, in the inner region of the disk. This may be helpful for explaining the distribution of exoplanets and the architecture of the Solar System, both of which have small inner and large outer planets.

System of Orthogonal Curvilinear Coordinates on the Isentropic Surface behind a Detached Bow Shock Wave

Three-dimensional ideal gas flows behind a detached bow shock wave near the nonsymmetrical nose part of a bluff body in a uniform supersonic flow are investigated. The vector lines of the vector product of the velocity and entropy gradient are considered on isentropic surfaces, which are stream surfaces starting on closed lines located on the shock wave and covering the leading point of the shock wave. It is shown that if these vector lines encircle the isentropic surface they are closed. This means that an orthogonal curvilinear coordinate system in which the coordinate lines coincide with the streamlines may be constructed on each isentropic surface.

Numerical investigation of particles characteristics on cyclone performance for sustainable environment

In this paper, a three-dimensional turbulent gas flow with solid particles was numerically simulated to optimize the performance of a cyclone preheater. The numerical approach for the flow development was based on unsteady simulations using the Reynolds Stress Model (RSM). The Discrete Phase Model (DPM) was used for the dispersion of particles due to turbulence in the fluid phase. The size of raw material solid particles ranged from 1 µm to 30 µm and the inlet gas velocity ranged from 10 m/s to 20 m/s were considered for parametric study. The main objective of this study consists of the performance evaluation of the cyclone pre-heater in terms of collection efficiency, pressure drop across the cyclone, heat transfer rate. The results indicate that the increase of particle size leads to increase collection efficiency of the particles up to 99.89% and temperature difference of about 212 K with the slight variation of pressure drop. The heat transfer rate decreases with increase in inlet air velocity due to less particles residence time inside the cyclone pre-heater. For a given cyclone design and working conditions, the maximum heat transfer was 87 W for 30 µm particle size with hot air velocity of 10 m/s.

A Numerical Method Used for the Simulation of Self-Excited Vibrations that Arise in the Aircraft Engine Fan Blade Ring

Aeroelastic effects that are caused by blade vibrations under the action of perturbation forces are characterized by the energy exchange between the gas flow and the vibrating blades and form the basis of the physical mechanism of self-excited vibrations that can either decay (aerodamping) or be in unstable flutter mode that can result in the structure failure in a short period of time. Based on the developed mathematical model and the numerical method of aeroelastic behavior of the blade ring in the transonic gas flow (the coupled problem of nonstationary dynamics and elastic blade vibrations) the numerical analysis of aeroelasic behavior shown by the fan blade ring in the three-dimensional ideal gas flow was performed. The three-dimensional transonic ideal gas flow is described by the complete system of Euler equations represented in the integral form of the laws of conservation of mass, pulse and energy in the relative system of Cartesian coordinates. Using a modal approach, the dynamic model of a vibrating blade was reduced to the system of ordinary differential equations in terms of the modal coefficients of eigenmodes. The values of blade displacements and its speed were derived from the solution of a dynamic problem and these values are used as boundary conditions for aerodynamic problem during each of the iteration. This method of the solution of the coupled aeroelastic problem enables the prediction of the amplitude and frequency spectra for blade vibrations in the three-dimensional ideal gas flow, including forced and self-excided vibrations in order to increase the reliability of the blade systems of turbine machines. The investigations carried out provided the data on the nonstationary loads (peripheral and axial loads and the aerodynamic moment) that have an effect on the blades and also the amplitude and frequency spectra of blade vibrations.

Problem on Conjugate Nonstationary Heat Exchange in Supersonic Flow over a Blunt-Nosed Cone at an Angle of Attack

The flow of a gas over a conic body, having a nose blunted over a sphere, with a supersonic velocity at different angles of attack was considered. The possibility of control of the temperature regime of such a body in the case of three-dimensional gas flow over it was investigated. The dependence of the temperature of the surface of the indicated body on the material of which it is made, the angle of attack of the body, and other constitutive parameters of the problem on conjugate nonstationary heat exchange between the body and the gas fl ow over it was determined for standard conditions of an aerodynamic experiment. Results of numerical solution of this problem for different angles of attack of the body were processed in the criteria form. The dependences obtained allow one to estimate the degree of decrease in the maximum temperature at the surface of a blunt-nosed conic body depending on the thermophysical parameters of the material of which this surface is made, including in the case of nonstationary gas fl ow over it. It is shown that a uniform heating of the peripheral part of such a body and a marked decrease in the temperature of its windward part at large angles of attack can be provided with the use of highly heat-conducting materials for these parts of the body.

Influences of protoplanet-induced three-dimensional gas flow on pebble accretion

Context. The pebble accretion model has the potential to explain the formation of various types of planets. The main difference between this and the planetesimal accretion model is that pebbles not only experience the gravitational interaction with the growing planet but also a gas drag force from the surrounding protoplanetary disk gas. Aims. A growing planet embedded in a disk induces three-dimensional (3D) gas flow, which may influence pebble accretion. However, so far the conventional pebble accretion model has only been discussed in the unperturbed (sub-)Keplerian shear flow. In this study, we investigate the influence of 3D planet-induced gas flow on pebble accretion. Methods. Assuming a nonisothermal, inviscid gas disk, we perform 3D hydrodynamical simulations on the spherical polar grid, which has a planet located at its center. We then numerically integrate the equation of motion of pebbles in 3D using hydrodynamical simulation data. Results. We find that the trajectories of pebbles in the planet-induced gas flow differ significantly from those in the unperturbed shear flow for a wide range of investigated pebble sizes (St = 10−3–100, where St is the Stokes number). The horseshoe flow and outflow of the gas alter the motion of the pebbles, which leads to a reduction of the width of the accretion window, wacc, and the accretion cross section, Aacc. On the other hand, the changes in trajectories also cause an increase in the relative velocity of pebbles to the planet, which offsets the reduction of wacc and Aacc. As a consequence, in the Stokes regime, the accretion probability of pebbles, Pacc, in the planet-induced gas flow is comparable to that in the unperturbed shear flow except when the Stokes number is small, St ~ 10−3, in 2D accretion, or when the thermal mass of the planet is small, m = 0.03, in 3D accretion. In contrast, in the Epstein regime, Pacc in the planet-induced gas flow becomes smaller than that in the shear flow in the Stokes regime in both 2D and 3D accretion, regardless of assumed St and m. Conclusions. Our results combined with the spacial variety of turbulence strength and pebble size in a disk, suggest that the 3D planet-induced gas flow may be helpful to explain the distribution of exoplanets and the architecture of the Solar System.