Gas Flow Equations Research Articles

Influence of thermal buoyancy on the wake dynamics of a heated square cylinder

Direct numerical simulation of the three-dimensional (3-D) wake transition of a heated square cylinder subjected to horizontal cross-flow is performed in the presence of buoyancy. In order to capture the effects of large-scale heating, a non-Oberbeck–Boussinesq model is utilized, which includes the governing equations for compressible gas flow. All computations are performed at low free stream Mach number $M=0.1$ using air (free stream Prandtl number, $Pr=0.71$ ) as the working fluid. The 3-D instability modes A and B, which correspond to free stream Reynolds numbers of 180 and 250, are observed with longer and shorter spanwise wavelengths, respectively, and the onset of three-dimensionality is triggered at a Reynolds number of 173. In the presence of buoyancy, baroclinic vorticity production in the near-wake plays an important role for streamwise vorticity generation. The chaotic wake of the Mode-A instability bifurcates into periodic and quasiperiodic wakes at various heating levels, expressed by the overheat ratio, $\varepsilon =(T_w-T_\infty )/T_\infty$ , where $T_w$ and $T_\infty$ are the temperature of the cylinder surface and the ambient air, respectively. At low heating ( $\varepsilon =0.2$ ), the 3-D Mode-A instability is suppressed leading to a two-dimensional wake flow. Further increase in heating, again brings back the three-dimensionality in the wake through Mode-E instability. The variation of thermophysical properties and the effective Reynolds number with increase in heating level around the cylinder is examined. It is shown that the effect of thermophysical properties competes with the baroclinic streamwise vorticity generation at higher levels of heating ( $\varepsilon \geqslant 0.4$ ) to control the 3-D modes and wake dynamics.

Exploring the Atwood number impact on shock-driven hydrodynamic instability at pentagonal interface using discontinuous Galerkin simulations

Investigation of Atwood number impact is crucial for comprehending the flow dynamics of the evolving hydrodynamic instability. In this paper, the Atwood number impacts on the shock-driven hydrodynamic instability at a backward-facing pentagonal interface in its early stages are explored numerically. The shock-driven pentagonal interface has a unique property in which the incident angle of the shock wave is constant along the interface edge. This property provides the ideal environment for the shock-refraction process. To examine the Atwood number impact, we consider five distinct gases, including sulfur hexafluoride, krypton, argon, neon, and helium, which are filled inside the pentagonal interface and surrounded by nitrogen gas. A mixed modal discontinuous Galerkin method is used to solve the two-dimensional Navier–Stokes–Fourier equations for two-component gas flows for numerical simulations. The simulation results illustrate that the Atwood number plays a crucial role in the flow fields of the shocked-pentagonal gas interface, which have not been reported in the literature. The flow fields are greatly impacted by the Atwood number, leading to complex wave patterns, shock focusing, jet formation, interface deformation, and vorticity generation. Further, the Atwood number effects on the flow fields are also thoroughly examined by a quantitative study based on the integral quantities and the interface features. Finally, a quantitative analysis is performed to examine the intrinsic differences between the evolution of the flow fields at the square and pentagonal interfaces.

Upgrading of the modified Knudsen equation and its verification for calculating the gas flow rate through cylindrical tubes

The modified Knudsen equation was developed to calculate the gas flow rate through a cylindrical tube with arbitrary length-to-diameter ratio in any flow regime including molecular flow, viscous laminar flow, turbulent flow, critical flow, subcritical flow, and their intermediates in our laboratory. Here, it is upgraded for better agreement with literature data, and the upgraded version is compared with 83 literature sources including 31 gas flow equations to verify its reliability. For Kn > 10−5, the modified Knudsen equation mostly agrees with the literature data to within 20%, except for orifice flows with pd/pu ≈ 1.

On Generic Singularities of Solutions to the 1D Gas Flow Equations: Chaplygin and Bechert–Stanyukovich Cases

On Generic Singularities of Solutions to the 1D Gas Flow Equations: Chaplygin and Bechert–Stanyukovich Cases

Analyzing Richtmyer–Meshkov Phenomena Triggered by Forward-Triangular Light Gas Bubbles: A Numerical Perspective

In this paper, we present a numerical investigation into elucidating the complex dynamics of Richtmyer–Meshkov (RM) phenomena initiated by the interaction of shock waves with forward-triangular light gas bubbles. The triangular bubble is filled with neon, helium, or hydrogen gas, and is surrounded by nitrogen gas. Three different shock Mach numbers are considered: Ms=1.12,1.21, and 1.41. For the numerical simulations, a two-dimensional system of compressible Euler equations for two-component gas flows is solved by utilizing the high-fidelity explicit modal discontinuous Galerkin technique. For validation, the numerical results are compared with the existing experimental results and are found to be in good agreement. The numerical model explores the impact of the Atwood number on the underlying mechanisms of the shock-induced forward-triangle bubble, encompassing aspects such as flow evolution, wave characteristics, jet formation, generation of vorticity, interface features, and integral diagnostics. Furthermore, the impacts of shock strengths and positive Atwood numbers on the flow evolution are also analyzed. Insights gained from this numerical perspective enhance our understanding of RM phenomena triggered by forward-triangular light gas bubbles, with implications for diverse applications in engineering, astrophysics, and fusion research.

Modeling gas flow in low-permeability formations: An efficient combination of mixed finite elements and high order time integration schemes

Numerical simulation of gas flow in low permeability formations is a challenging task due to the high nonlinearity induced by (i) the compressibility of the gas, (ii) the Klinkenberg slippage effect and (iii) the Langmuir adsorption of the gas on the pore surface. Because of these nonlinearities, modeling gas flow in low permeability formations requires a great deal of computational effort. In this work, we develop an efficient numerical model using advanced spatial and temporal discretization methods for a simultaneous solution of the coupled equations of gas flow, cubic Peng-Robinson equation of state, slippage effect and Langmuir adsorption. The spatial discretization is performed with the lumped hybrid formulation of the mixed finite element method which is well adapted for fluid flow in heterogeneous porous media. The time integration is performed with high-order methods via the method of lines (MOL) which allows large time steps and efficient solution of the nonlinear system of equations. Numerical experiments, performed for gas extraction in a heterogeneous domain, point out the high efficiency of the new model since it can be until 10 times more efficient than the classical first-order time discretization method. Results of global sensitivity analysis show that the intrinsic gas-phase permeability and the Klinkenberg factor are the most influential parameters controlling the pumping rate in the case of gas extraction in a homogeneous domain.

Multiscale modeling of gas-induced fracturing in anisotropic clayey rocks

In the context of repositories for nuclear waste, understanding the behavior of gas migration through clayey rocks with inherent anisotropy is crucial for assessing the safety of geological disposal facilities. The primary mechanism for gas breakthrough is the opening of micro-fractures due to high gas pressure. This occurs at gas pressures lower than the combined strength of the rock and its minimum principal stress under external loading conditions. To investigate the mechanism of microscale mode-I ruptures, it is essential to incorporate a multiscale approach that includes subcritical microcracks in the modeling framework. In this contribution, we derive the model from microstructures that contain periodically distributed microcracks within a porous material. The damage evolution law is coupled with the macroscopic poroelastic system by employing the asymptotic homogenization method and considering the inherent hydro-mechanical (HM) anisotropy at the microscale. The resulting permeability change induced by fracture opening is implicitly integrated into the gas flow equation. Verification examples are presented to validate the developed model step by step. An analysis of local macroscopic response is undertaken to underscore the influence of factors such as strain rate, initial damage, and applied stress, on the gas migration process. Numerical examples of direct tension tests are used to demonstrate the model's efficacy in describing localized failure characteristics. Finally, the simulation results for preferential gas flow reveal the robustness of the two-scale model in explicitly depicting gas-induced fracturing in anisotropic clayey rocks. The model successfully captures the common behaviors observed in laboratory experiments, such as a sudden drop in gas injection pressure, rapid build-up of downstream gas pressure, and steady-state gas flow following gas breakthrough.

Numerical homogenization of the gas filtration problem

The non-stationary gas flow equation in homogeneous and heterogeneous media is considered. Areas of this type are often encountered when considering gas production processes from a gas-bearing reservoir. The heterogeneous structure of the reservoir can strongly affect the extraction processes. For the numerical solution of the problem under consideration, we construct an approximation of the equations on a coarse grid using the numerical averaging method. The method allows us to solve the problem using less computational power and in a shorter time. A numerical comparison of the results of solving the model problem is carried out in a two-dimensional domain for the cases of linear and nonlinear variants of the equations, as well as in a homogeneous and heterogeneous medium. The finite element solution on a fine mesh was taken as a reference solution. The computational realization was performed using the FEniCS library. The construction of the geometric computational domain was performed in the program Gmsh. Paraview and the Matplotlib library in Python were used for data visualization.

A NUMERICAL ANALYSIS ON THE SUBMICRON- AND MICRON-SIZED PARTICLE SEDIMENTATION IN A WIRE-TO-PLATE ELECTROSTATIC PRECIPITATOR

Electrostatic precipitators (ESPs) are frequently utilized in collecting fine organic and inorganic materials from continuous liquid with few moving parts and high efficiency using electrically charging the particles. In this study, cross-sectional 2D geometry of a wire-to-plate electrostatic precipitator the parametric data of which originally published elsewhere was numerically modeled and validated to investigate submicron-micron particle charging in terms of diffusion and field charging mechanisms and precipitation behavior of particles with detailed electric field properties. Electric field, gas flow, and particle trajectory equations are coupled and solved in a multiphysics solver. Particle tracking is realized with the Lagrangian approach. Results indicate variations in electric field strength and space charge density between corona electrodes, with space charge present in the entire precipitation channel. Between two different charging mechanisms, diffusion charging prevails for charge accumulated on submicron particles, whereas field charging becomes dominant for particles larger than 1μm diameter. However, for the ESP configuration considered in this study, particles reach a charge saturation in less than 0.7 seconds, regardless of their size. Although calculated precipitation efficiencies for micron-sized particles can reach to 100%, efficiencies for submicron particle range drop with increasing particle size, as diffusion charging rapidly loses its effectiveness, in 50-250nm range.

MODELING THE DYNAMICS OF DISSOLVED OXYGEN CONCENTRATION IN WATERS OF THE ANIVA BAY (THE SEA OF OKHOTSK)

To determine the risks for mariculture farms, the intra-annual change of dissolved О2 concentration was simulated for five zones in the Aniva Bay using the CNPSi model. Zone 1 differed sharply from other zones as the most shallow and freshened. Zone 2 is characterized by a pronounced water exchange with Zone 3 and Zone 4: during spring two layers were formed and stood out in these zones, in the summer the water column was homogeneous. Zone 3 has free water exchange with the open waters of the La Perouse Strait. An outstanding feature of Zone 4, in the deep-water part of the bay, was a distinctive subsidence of waters in the centre of the anticyclonic circulation and the maximum thermocline depth (up to 60-70 m). Zone 5 extends along the western coast of the Tonino-Aniva Peninsula and is characterized by the constant upwelling of waters during the icefree period, which is clearly expressed by lower water temperatures. The calculation showed that in the areas suitable for mariculture farms coastal waters were provided with oxygen throughout the year. Anaerobic conditions developed in spring only in the deepest parts of the bay. An additional source of oxygen in the Aniva Bay is natural thickets of macrophytes, among which the Japanese saccharin (Saccharina japonica) dominates in terms of biomass and area. Annually, Japanese saccharin itself absorbed at least 1200 tons of C in its biomass and supplied at least 3100 tons of О2. Unlike the artificially grown biomass, the biomass of all macrophytes would remain in the system and be destroyed during the life cycle, and the oxygen would be consumed for oxidation. The carbon accumulated in the biomass would again return to the rapid cycle, with the exception of the amount transported to the deep central part of the bay, where it would slowly decompose under nearly anaerobic conditions. It would be possible to place additional algae plantations in the bay, which could absorb up to 49 500 tons of C annually, while supplying up to 132 000 tons of О2. The obtained model estimates could be a starting point for determining the “baseline” of the content of dissolved oxygen and compiling balance equations for gas flows in the ocean-atmosphere system in the Aniva Bay before the development of seaweed plantations, which simultaneously act as carbon farms.

Optimal scheduling of hydrogen blended integrated electricity–gas system considering gas linepack via a sequential second-order cone programming methodology

In the context of increasing global environmental pollution and constant increase of carbon emission, hydrogen production from surplus renewable energy and hydrogen transportation using existing natural gas pipelines are effective means to mitigate renewable energy fluctuation, build a decarbonized gas network, and achieve the goal of “carbon peak and carbon neutral” in China. Existing gas network modeling with hydrogen blended is divided into steady-state and dynamic. The steady-state modeling oversimplifies the physical process, while the dynamic modeling is computationally intensive. Thus, a quasi-steady-state modeling method of a hydrogen blended integrated electricity–gas system (HBIEGS) considering gas linepack is developed in this paper. The gas linepack refers to the gas stored in natural gas pipelines due to the compressibility of the gas. As a form of gas energy storage, linepack can enhance system flexibility by coping with wind power uncertainty. In the face of nonlinear gas flow equations in HBIEGS, conventional relaxation or approximate methods cannot provide a sufficiently feasible optimal solution. Therefore, a sequential second-order cone programming (S-SOCP) method is proposed to solve the developed model. It can effectively strike a balance between solution speed and the quality of the solution obtained. Finally, the modified IEEE-24-node power system and 12-node natural gas system are used to validate the effectiveness of the developed model and the proposed method. The optimization results show that the proposed method improves the computational efficiency by 91% compared with a general nonlinear solver.

Investigation on interaction effects of multiple factors on clean gas extraction: implementation of response surface methodology

Abstract Gas-related disasters have become a major threat to mining safety of coal resources. Investigation into the optimization approach for inseam borehole gas extraction is important for improving extraction efficiency. By establishing control equations for coal seam deformation, porosity-permeability properties, and gas flow, effects of extraction time, initial gas pressure, initial coal permeability, and borehole diameter on gas flow pattern and effective extraction radius are analyzed. Based on response surface methodology and orthogonal experimental design, a regression analysis model of effective extraction radius and multiple influencing factors is established. Meanwhile, the interaction mechanism between effects of multiple factors is analyzed, thus identifying optimal parameters for gas extraction. Results show that effective extraction radius is positively correlated with initial permeability, extraction time, and borehole diameter, and negatively associated with original gas pressure. In terms of effect on effective extraction radius, those parameters could be ranked as follows: initial gas pressure, extraction time, initial coal permeability, then borehole diameter. Interaction between multiple factors could inhibit the impact of individual factors on extraction radius. An increase in initial gas pressure reduces the positive effect of permeability on extraction radius. Growing initial gas pressure inhibits the contribution of extraction time to effective drainage radius. Prolonged duration of extraction suppresses the positive effect of permeability on extraction radius. Interaction between initial gas pressure and initial permeability has the most significant influence on extraction radius. Results from orthogonal experimental design correspond to those of response surface methodology. These findings could provide guidance for dynamically adjusting drilling extraction parameters and improving gas-extraction outcomes.

Numerical Solution of the Steady-State Network Flow Equations for a Nonideal Gas

Numerical Solution of the Steady-State Network Flow Equations for a Nonideal Gas

Similarity rules for inductive radio frequency plasmas with thermohydrodynamic coupling effects

We demonstrate similarity rules for inductively coupled plasmas with thermohydrodynamic coupling effects using two-dimensional fluid simulations and theoretical analyses of the gas flow and heat transfer equations. The results confirm the validity of conventional similarity laws, e.g., the similarity relation for electron density, which can be violated by the nonlinear gas heating effects from exothermic and endothermic reactions. The nonlinear gas heating can obviously perturb the invariance of spatial distributions of the gas flow velocity, resulting in the electron density decreasing nonproportionally with different scaling factors. Adding an external heat source can mitigate the violation of the gas temperature scaling law, thus maintaining the validity of similarity relations to some extent. In addition, two kinds of scaling relations for excited-state argon atoms are identified with and without the consideration of nonlinear collisions.

Plasma behavior of a low current orificed hollow cathode using krypton propellant for electric thruster applications

As a fundamental component, hollow cathodes have been widely used in electric thruster applications. Krypton has become one of the ideal alternative propellants for hollow cathode due to its economy, thus it is vital to understand the basic physical process of the discharge of krypton-fed hollow cathode. This study sets out to establish a two-dimensional fluid model coupled with the equations for gas flow and heat transfer allowing to obtain a self-consistent description of the cathode discharge. The anomalous collision term based on the formulations of Sagdeev and Galeev is considered in this model. The model is validated by comparing the simulated and measured plasma characteristic parameters and reasonable agreement is reached. The findings show that the discharge of the krypton-fed hollow cathode is characterized by the lower plasma density, higher electron temperature, and higher spatial electric potential compared to that of the xenon-fed hollow cathode and tends to present a higher plasma density in the cathode plume region when it is operating with the diode configuration. The double layer formed at the keeper orifice is an important factor for the low electron temperature distribution, which is greatly affected by the keeper current. These findings add to our understanding of plasma behavior in the discharge of a low current krypton-fed hollow cathode and support further development of credible krypton-fed hollow cathode.

Numerical investigation of shock Mach number effects on Richtmyer–Meshkov instability in a heavy square bubble

This study conducts a two-dimensional numerical investigation of incident shock Mach number effects on the Richtmyer–Meshkov (RM) instability after a shock wave impulsively drives a heavy square bubble. The square bubble is composed of SF6 gas, which is surrounded by nitrogen gas. Five different shock Mach numbers are considered: Ms=1.12, 1.22, 1.4, 1.7, and 2.1. Two-dimensional compressible Euler equations for two-component gas flows are simulated with a high-order modal discontinuous Galerkin solver. For validation, the numerical results are compared with the existing experimental results and are found to be in good agreement. The present results reveal that the incident shock Mach number is critical in the growth of RM instability in a heavy square bubble. The shock Mach number affects flow morphology significantly, resulting in complicated wave patterns, shock focusing, jet creation, bubble deformation, and vorticity generation. The convergent shape of the bubble causes shock focusing, which creates a local high-pressure zone and, as a result, an outward jet generation. It is found that the bubble deforms differently with increasing shock Mach number, and the different shock Mach numbers alter the distance between the shock focusing position and the downstream bubble interface. The effects of shock Mach numbers are investigated in depth through physical phenomena such as vorticity production, kinetic energy, and enstrophy. Finally, the shock Mach number impacts on the time-variations of the shock trajectories and interface structure are thoroughly investigated.

A second-order slip/jump boundary condition modified by nonlinear Rayleigh–Onsager dissipation factor

A newly heuristic form of second-order slip/jump boundary conditions (BCs) for the Navier–Stokes–Fourier (NSF) equations is proposed from the viewpoint of generalized hydrodynamic equations (GHE) to extend the capability of the NSF equations for moderately rarefied gas flows. The nonlinear Rayleigh–Onsager dissipation function appearing in the GHE, which contains useful information about the nonequilibrium flow fields of interest, is introduced into the proposed BCs named the simplified generalized hydrodynamic (SGH) BCs as a correction parameter. Compared with the classical Maxwell/Smoluchowski (MS) BCs, the SGH BCs may be more sensitive to capture the nonequilibrium information of flows adaptively and produce physically consistent solutions near the wall. Subsequently, the SGH BCs are implemented in the NSF equations for planar micro-Couette gas flows over a wide range of Knudsen numbers. The results indicate that the SGH BCs make impressive improvements against the MS BCs for diatomic and monatomic gases at the slip region and early transition regime, particularly in terms of capturing precisely the temperature and normal heat flux profiles in the flow and the temperature jump on the wall. More importantly, the SGH BCs conducted in NSF equations with less computational cost still can obtain well-pleased results comparable to the non-Newton–Fourier equations, such as several Burnett-type equations and regularized 13-moment equations, and even perform better than these models near the wall compared with direct simulation Monte Carlo data for the Couette flows to some extent.

Pressure fields produced by single-bubble collapse near a corner

Damage from repeated bubble collapse to neighboring rigid objects is a consequence of cavitation. Few studies exist on the dynamics of bubbles collapsing near a corner, i.e., two flat rigid surfaces making a right angle. Here we solve the three-dimensional compressible Navier-Stokes equations for gas and liquid flows to quantify the pressure fields. The second wall affects the collapse by (i) causing the re-entrant jet to no longer point in the direction normal to the closest wall but at an angle toward the corner and (ii) causing the maximum wall pressure to be observed in the corner when the bubble is sufficiently close to equidistant from each boundary due to shock reflections.

Mathematical Model for Flow of High-pressure Gas in Microscale

Fused silica micro-tubes, whose size is similar to the pore throat in a low permeability reservoir, simplify the complex and tortuous of channels and throats and are used in this study to stimulate the flow pattern with micro-scale methods. A high-pressure micro-tube holder (designed and developed independently with a patent for invention) has been used to study gas flow patterns within the pressure range of 0.1MPa-30MPa. Moreover, we establish a gas flow equation and analysis the applicability of different inlet pressure based on the experiment result. The result shows: First, the pressure drop in the micro-tube is predominantly induced by friction resistance at low inlet pressure. While under relatively high inlet pressure, the impact of kinetic energy conversion and throttling effect on pressure drop shall be considered. Second, we build up a gas-flow equation in a micro-tube considering friction resistance, kinetic energy loss, and throttling effect. We also analyze the applicability for different diameters and inlet pressure. These research achievements were enriched in microtubule flow; different conditions have been identified for a different equation which has significant meaning to gas flow regulation in tight sandstorm reservoirs.

A New Interpretation of Gas Viscosity for Flow through Micro-Capillaries and Pores.

The Hagen-Poiseuille equation for gas flowhad never been derived theoretically; it is rather a simple analogy of the same for liquid flow, and "gas viscosity" is a measure for overall resistance to flow. In this work, experimental flow data for different gases through capillaries and porous media, reported in literature by different groups, including those measured and treated by Knudsen are treated with Hagen-Poiseuille equation, but taking "gas viscosity" as an adjustable parameter. It is found that, at constant temperature, there exists an unambiguous relation between the viscosity (µ) of a given gas, and the product of average pressure (Pav ) and capillary diameter (D). In addition, for Pav * D<0.01, a universal linear relation exists between µ/M0.5 (where M is molecular mass) for different gases and the parameter Pav * D. The new interpretation of gas viscosity avoids the differentiation of regimes into "Knudsen" and "viscous" flow as it is frequently done in literature. The concept can be applied to obtain a reliable data base for gas viscosities in different fields of applications, for example in microfluidic systems or the analysis of pore size distributions of filters and membranes by gas flow porometry.