DSMC investigation of rarefied gas flow over a 2D forward-facing step: Effect of Knudsen number
Introduction
Spacecrafts used for re-entry, exploratory, and interplanetary type of deep space exploration, require extensive design studies involving rarefied gas dynamics. These vehicles fly at extremely high (often hypersonic) speeds for a significant part in the rarefied setting [1]. Although the vehicle's streamlined aerodynamic surface form is preferred, minor discontinuities such as protuberances, notches, steps, gaps, or cavities can prevail either during its manufacturing or during its operation. These discontinuities lead to flow separation, circulation, and re-attachment, which is commonly investigated [[2], [3], [4], [5], [6]]. One such discontinuity that can be generally approximated and investigated is the sudden contracting channels, i.e., forward-facing step (FFS). The investigation into the FFS flow has numerous practical uses in the domain of combustors, diffusers, turbine blade cooling, aerospace technology, and other allied areas [7,8]. Furthermore, the space vehicle passes through various rarefaction regimes, atmospheric temperatures, and they experience complex flow behaviors, which include shock interactions [[9], [10], [11]], which are generally investigated using various computational methods. In this article, we primarily discuss the rarefaction effects on flow behavior over the FFS using the Direct Simulation Monte Carlo (DSMC) method, one of the commonly employed methods for simulation of rarefied gas flows [12]. Knudsen number () is used to determine the degree of rarefaction of the gas and is defined as the “ratio of the mean free path () to the characteristic dimension () of the geometry under consideration” [13].
Based on the flows are classified into the following regimes [[14], [15], [16]]: (1) continuum flow regime, where . Here, the rate of collisions between gas molecules is higher than that of collisions between gas molecules and walls. Thus, the fluid can be considered continuous and modeled by the traditional Navier-Stokes equation, without taking into account the rarefaction effect; (2) slip flow regime, where . In this regime, velocity drop happens at the gas-solid boundary interface, and fluid is influenced to some level by the rarefaction effects. Navier-Stokes equations with a slip boundary criterion are used to model the flow [16], (3) transitional flow regime, where . In this regime, the interaction between gas molecules, between walls and gas molecules cannot be ignored, and the continuum model is ineffective as the molecular nature of the fluid is more significant, (4) free-molecular regime, where . In this regime, the colliding frequency between gas molecules and walls is higher than the colliding frequency among gas molecules, and hence detailed molecular motion has to be modeled. The range of slip regime described above is contentious for gas flows in complex 2D/3D geometries and should be accordingly addressed [14].
As the flow enters the slip regime, the continuum hypothesis no longer holds due to predominant rarefaction effects. The study of such rarefied gas flows demands the usage of substitute numerical approaches, namely, Molecular Dynamics (MD) [17], Direct Simulation Monte Carlo (DSMC) [18], and solution of full Boltzmann Equations (BE) [19].
The Knudsen number (, Reynolds number and Mach number for such flows are associated with one another other as follows:where, ρ, , , represent the gas density, free-stream velocity, step height, and gas dynamic viscosity, respectively.
Several studies have been reported by researchers which involve experimental, analytical as well as numerical studies of rarefied gas flow over simple geometries involving protuberances [[19], [20], [21], [22], [23]], notches [24,25], cavities [[25], [26], [27], [28], [29], [30]] and gaps [[31], [32], [33], [34]]. Also, the studies performed involve the subsonic [35,36], supersonic [[37], [38], [39]], and hypersonic flows [[40], [41], [42], [43]] past simple geometries.
Existing literature provides plenty of references to the physics of flow past FFS, but the majority are in the continuum regime, so in this overview, we outline only a few of them. Bogdonoff & Kepler [37] experimentally examined the flow separation and shockwave boundary-layer interaction for the FFS flow and reported flow separation at Mach 3 and pressure ratio of 2, respectively. Rogers & Berry [38] experimentally tested FFS in the supersonic flow of roughly Mach 2, with a thick laminar boundary layer. The study showed that the highest-pressure surge existed on the step face; moreover, this pressure surge was influenced by the ratio of (), i.e., step height ( to the distance from the leading edge of flat-plate to the step position . Similarly, Driftmyer [40] performed an experimental study on FFS for the specific case where step height was below the boundary layer thickness ( for a free-stream Mach number of 4.9 and Reynolds number between per foot. Results showed that the and () influenced the pressure distribution measured in the region of separation before the step.
Pullin & Harvey [43] computationally evaluated a 2D, rarefied hypersonic FFS flow with as the fluid and free-stream Mach number of 22. Their results indicated rapidly decelerating flow, accompanied by compression near the vicinity of the step. Furthermore, their numerical findings matched well with the experimental results. Grotowsky & Ballmann [44] also numerically investigated laminar hypersonic flow over both FFS and BFS using the Navier-Stokes equations for a free-stream Mach number of 8, Reynolds number of the order of 108, at an altitude of 30 km. Although their numerical results matched well with experimental data from the literature, they indicated that there were significant differences in the analysis of the wall heat flux, perhaps because of its imprecise measurement. Mahdavi et al. [45] studied the nano/micro BFS flow using the slip/jump boundary conditions with N2 as working fluid and compared the CFD and DSMC solutions. Their results showed that the hybrid slip/jump boundary conditions had a better prediction of flow and thermal properties compared to the DSMC results. Darbandi and Roohi [46] studied the subsonic flow through micro/nano BFS using the DSMC method for different rarefaction regimes. It was observed that the rarefaction reduces the length of separation considerably. In a recent investigation, Leite et al. [47] performed a numerical study using DSMC for rarefied flow over FFS for different step heights. Their study showed that a formidable compression region in front of the frontal face influenced the flow characteristics upstream of the step. Furthermore, high-pressure regions and high heating loads occurred on the upstream and frontal face.
Based on the above discussion, it is understandable that the majority of the research on FFS focused mainly on laminar or turbulent flow in the continuum regime, most of which were experimental. However, the aerothermodynamic conditions encountered by space vehicles and the physics behind such extreme hypersonic flows remain poorly understood. They are challenging to analyze experimentally and can be designed using numerical techniques. Furthermore, to the best of the authors’ knowledge, very few prior studies thoroughly analyzed the different rarefaction regimes which the space vehicles experience while entering into the atmosphere. Therefore, in the present investigation, to bring an extensive and representative insight, we apply the DSMC method to numerically investigate the influence of Knudsen number into the physics of the rarefied hypersonic flow past FFS.
The rest of the paper is organized as follows. Sec.2. describes the computational methodology adopted to study the problem; Sec.3. describes the computational parameters along with the assumption made, Sec.4. describes the validation and independence studies performed. Sec.5. describes the effect of the Knudsen number on the flow and surface properties, respectively, and Sec.6. describes the principal conclusions.
Section snippets
Direct Simulation Monte Carlo
The DSMC approach, established by Bird [18], is one of the most widely used methods for modeling the dynamics of the rarefied gas flows. Over the past few decades, this approach has been commonly used in various flow regimes and has been validated experimentally. The DSMC method is based on Boltzmann's Equation employed with certain restrictions. As is established, the air density declines gradually with altitude, and the rarefaction effects become more apparent near the outer space as the flow
Geometry, boundary conditions, and grid
Schematic of the flow over a 2D forward-facing step is shown in Fig. 1, where the flow traverses from left to right. The cartesian coordinates along and represent the streamwise and transverse directions, respectively. The step height () was fixed to 3 mm for all the cases. Various geometric parameters of the FFS considered in this study are shown in Table 1.
The boundary conditions applied for different surfaces are given in Table 2. Surface-I includes the step, lower and upper surfaces
Validation
The dsmcFoam solver has been used to perform the numerical simulations, whose validity has been extensively documented in our recent works [[75], [76], [77], [78]]. However, in the present work, a benchmark case of hypersonic flow past a flat plate is included to evaluate the applicability of the DSMC method. The flat plate has a relatively simple geometry with both experimental results of Becker et al. [79] and numerical results of Hermina et al. [80], thus making it a useful case for
Results and discussion
To evaluate the effects of Knudsen number on the flow and surface properties over the FFS, five different Knudsen numbers in various rarefaction regimes, i.e., (Slip), (Transitional), (Transitional), (Free-molecular), and (Free-molecular), were considered. The Mach number was fixed to 25, step height = 3 mm), and is used for all instances in this section. Henceforth, the variation of sampled results in the transverse direction to the flow is
Conclusion
In this paper, the Direct Simulation Monte Carlo method was used to investigate the effect of Knudsen number on the non-reacting rarefied hypersonic gas flow over a forward-facing step. We assume the typical atmospheric conditions encountered by a space vehicle at 55.02 Km, 60.5 Km, 77 Km, 91.5 Km, 95 Km altitudes, which result in a of 0.05, 0.1, 1.06, 10.33 and 21.33 respectively, covering the various rarefaction regimes. The rarefaction effects on flow-field properties, i.e., velocity,
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors gratefully acknowledge the High Performance Computing (HPC) Laboratory, BITS Pilani, Hyderabad Campus, for their technical assistance and support in providing the computing facilities to perform the numerical simulations. The authors express their sincere thanks to the editor and anonymous reviewers for their constructive suggestions, which helped us improve the manuscript.
References (91)
- et al.
Hypersonic nonequilibrium flow simulations of a hemispherical nose with a counterflowing jet
Acta Astronaut.
(2019) - et al.
The effect of fin oscillation in heat transfer enhancement in separated flow over a backward facing step
Int. J. Heat Mass Tran.
(Jan. 2019) - et al.
A review of backward-facing step (BFS) flow mechanisms, heat transfer and control
Thermal Science and Engineering Progress
(2018) - et al.
Laminar separation on a forward-facing step
Eur. J. Mech. B Fluid
(Jul. 1999) Hypersonic environment assessment of the CIRA FTB-X re-entry vehicle
Aero. Sci. Technol.
(2013)- et al.
Hypersonic flow over Stardust Re-entry Capsule using ab-initio based chemical reaction model
Acta Astronaut.
(Sep. 2019) - et al.
Effects of solar panels on Aerodynamics of a small satellite with deployable aero-brake
Acta Astronaut.
(Oct. 2018) - et al.
Flowfield structure characteristics of the hypersonic flow over a cavity: from the continuum to the transition flow regimes
Acta Astronaut.
(2019) - et al.
Study of rarefied gas flows in backward facing micro-step using Direct Simulation Monte Carlo
Vacuum
(Sep. 2018) - et al.
DSMC simulation of subsonic flow through nanochannels and micro/nano backward-facing steps
Int. Commun. Heat Mass Tran.
(Dec. 2011)
An asymptotic-preserving Monte Carlo method for the Boltzmann equation
J. Comput. Phys.
An asymptotic preserving Monte Carlo method for the multispecies Boltzmann equation
J. Comput. Phys.
DSMC and R13 modeling of the adiabatic surface
Int. J. Therm. Sci.
Generalized scheme of the no-time-counter scheme for the DSMC in rarefied gas flow analysis
Comput. Fluid
Collision partner selection schemes in DSMC: from micro/nano flows to hypersonic flows
Phys. Rep.
On the vortical characteristics and cold-to-hot transfer of rarefied gas flow in a lid driven isosceles orthogonal triangular cavity with isothermal walls
Int. J. Therm. Sci.
Molecular models for simulation of rarefied gas flows using direct simulation Monte Carlo method
Fluid Dynam. Res.
Statistical collision model for Monte Carlo simulation of polyatomic gas mixture
J. Comput. Phys.
Improved sampling techniques for the direct simulation Monte Carlo method
Comput. Fluids
Proper cell dimension and number of particles per cell for DSMC
Comput. Fluid
An open source, parallel DSMC code for rarefied gas flows in arbitrary geometries
Comput. Fluid
dsmcFoam+: an OpenFOAM based direct simulation Monte Carlo solver
Comput. Phys. Commun.
Monte Carlo simulation for aerodynamic coefficients of satellites in Low-Earth Orbit
Acta Astronaut.
The impact of the length-to-depth ratio on aerodynamic surface quantities of a rarefied hypersonic cavity flow
Aero. Sci. Technol.
The effects of Maxwellian accommodation coefficient and free-stream Knudsen number on rarefied hypersonic cavity flows
Aero. Sci. Technol.
“Simulation of rarefied gas flow in a microchannel with backward facing step by two relaxation times using Lattice Boltzmann method–Slip and transient flow regimes
Int. J. Mech. Sci.
Some considerations on thermal boundary condition of slip flow
Int. J. Heat Mass Tran.
Analysis of Laminar Flow over a Backward Facing Step
Turbulent boundary-layer separation in front of a forward-facing step
AIAA J.
Experimental and theoretical investigation of backward-facing step flow
J. Fluid Mech.
Computational analysis of a rarefied hypersonic flow over backward-facing steps
J. Thermophys. Heat Tran.
Nonequilibrium Hypersonic Aerothermodynamics
Thermal and second-law analysis of a micro- or nanocavity using direct-simulation Monte Carlo
Phys. Rev. E - Stat. Nonlinear Soft Matter Phys.
The Art of Molecular Dynamics Simulation by D. C. Rapaport
Molecular Gas Dynamics and the Direct Simulation of Gas Flows
The Boltzmann Equation and its Applications
Heat transfer to wavy wall in hypersonic flow
AIAA J.
Heating augmentation for short hypersonic protuberances
J. Spacecraft Rockets
Hypersonic interference heating in the vicinity of surface protuberances
Exp. Fluid
Experimental measurement of aerodynamic heating about complex shapes at supersonic Mach numbers
J. Spacecraft Rockets
Separation of a supersonic accelerated flow over notches
AIAA J.
Separation controlled transonic drag-rise modification for V-shaped notches
AIAA J.
An investigation of separated flows - Part I: the pressure field
J. Aero. Sci.
An investigation of separated flows- Part I I : flow in the cavity and heat transfer
J. Aero. Sci.
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