A statistical boundary for 3D rarefied flows through meshes: implementation to a new version of dsmcFoam+ and wind tunnel validation

Abstract

The Direct Simulation Monte Carlo (DSMC) method has become a standard tool for rarefied aerodynamics and microchannel flows. However, the performance benefits of DSMC, such as adaptive grid sizes and number of particles, are constrained by the need to resolve small geometric details of mesh applications within relatively large simulation volumes. The requirement for a sufficient number of particles in even the smallest cells imposes a significant computational burden. A novel set of cyclic statistical boundary conditions is proposed to address the computational bottleneck associated with simulating micrometre-scale structures prevalent in atmospheric and space research under rarefied flow conditions. These conditions account for the geometric parameters of a geometric mesh and the angular dependency of impacting particles, aiming to alleviate the computational challenges posed by conventional approaches. Validation against wind tunnel measurements demonstrates excellent agreement for one of the implemented boundaries, able to simulate fine meshes for conditions of rocket soundings in the Mesosphere. The newly developed boundary conditions are implemented within the advanced DSMC solver, dsmcFoam+ framework. For this study, the solver is ported from OpenFOAM^® version 2.4.0 to the OpenFOAM^® version v2306 to leverage recent code developments, particularly in dynamic meshes, load balancing, and barycentric particle tracking. This advancement enhances the capabilities of DSMC simulations, offering improved fidelity and accuracy in capturing rarefied flow phenomena.

1 Introduction

In rarefied flows, where the interaction between individual molecules begins to determine the nature of a fluid and the continuum approach breaks, the Direct Simulation Monte Carlo (DSMC) [1] has been established as a standard numerical simulation tool for various applications. The rarefication of a flow is described employing the dimensionless Knudsen number that is defined by the ratio of the mean free path of molecules $\lambda$ and a characteristic length scale L:

$$\begin{aligned} Kn = \frac{\lambda }{L} \end{aligned}$$

(1)

DSMC is conventionally used if Kn is above $\sim$0.1. That implies a large field of use, e.g. in hypersonic aerodynamics of spacecraft reentry and sounding rockets (e.g., [2, 3]), that are operated in an environment of large $\lambda$ on the one end, but also in flows, where L becomes microscopic small, e.g., in microtechnology [4]. Recent developments involve coupling the DSMC method with magneto-electric physics, e.g., [5,6,7].

As a microscopic approach, DSMC solves equations of motion for individual particles before deriving macroscopic quantities like density, temperature, and field velocity. Notably, DSMC’s efficiency stems from its requirement for a relatively small number of particles, each representing $F_N$ other particles. The collision process enables both steady-state and transient simulations based on a random approach. A comprehensive overview of the DSMC method is provided in Bird’s textbook [1].

The link between microscopic particles and macroscopic quantities, achieved through averaging over cell volumes, exhibits significant performance potential in external flows. This potential is particularly pronounced in scenarios involving compression upstream of an obstacle and rarefaction downstream, as seen in supersonic aerodynamics, especially with automated cell refinement. However, DSMC simulations with adaptive numerical grids are not commonplace (although an OpenFOAM^® implementation has been reported [8], the source code has not been released), often requiring a priori information for grid setup. Difficulties arise when a flow’s geometry encompasses widely varying scales. For accurate resolution of flow gradients, the numerical grid should ideally at least extend down to the mean free path of particles ($\lambda$). Adaptive mesh refinement (AMR) capabilities in recent OpenFOAM^® versions allow resolution fulfilment without a priori knowledge of flow details. More details of the AMR implementation in OpenFOAM^® can be found in reference [9]. The refinement criterion can be set to maintain a ratio of cell length (x):

$$\begin{aligned} 1 \ge \frac{x}{\lambda }. \end{aligned}$$

(2)

where the mean free path ($\lambda$) can be defined by

$$\begin{aligned} \lambda _{\textrm{HS}} = \frac{1}{\sqrt{2} \pi d^2 N}, \end{aligned}$$

(3)

where subscript HS indicates the application of the Hard-Sphere model (e.g., [1, 10]), N and d denote number density (molecules per unit volume, i.e., m$^{-3}$) and a molecule’s diameter, respectively. Challenges intensify when resolving geometry structures significantly smaller than $\lambda$, necessitating a substantial decrease in the equivalence factor $F_\textrm{N}$ to ensure even the smallest cell contains sufficient particles. This is particularly evident in applications involving micrometre-scale geometries mounted on sounding rockets and satellites for atmospheric investigations, see Fig. 1.

Fig. 1

Sensor for combined measurements of neutrals and electrons in the atmosphere’s mesosphere and thermosphere. The outer spherical mesh is $\sim$60 mm in diameter. The smallest wire diameter is 0.1 mm. See [11] for more technical details

Resolving such micrometre geometries in larger simulation volumes is impractical, and neglecting their influence on the flow is unphysical and can lead to wrong interpretations of measurement results. Contemporary payload designs for atmospheric research have become increasingly complex and often must be treated as fully three-dimensional due to asymmetries in the design; see, e.g., [12, 13] for recent examples. For such cases, a method is required to allow efficient DSMC to resolve the flow through the fine mesh geometries in a large simulation volume.

To disentangle the numerical resolution required for accurately representing flow features from tiny and complex geometries like metal meshes, Gumbel et al. [2] initially introduced a statistical treatment, collectively treating these mesh structures as a statistical boundary. While an axisymmetric validation test exhibited qualitative agreement with wind tunnel measurements, it underscored the necessity for an angular description of mesh transmissivity. This work implemented and tested new boundary conditions with different levels of complexity to qualitatively describe the flow conditions around transmissive meshes.

In Sect. 2, the underlying models of the new boundary conditions in OpenFOAM^® were described. The implementation of the boundaries in the OpenFOAM^® framework, including key code snippets and demonstration showcases, are presented in Sect. 3. Extensive validation against unique measurements from a supersonic rarefied wind tunnel experiment [2] is shown in Sect. 4. Conclusions are summarized in the final Sect. 5.

2 Model description

The statistical boundaries to mimic the tiny mesh applications (Fig. 1) were implemented to expand the coupled cyclic patch implementation in OpenFOAM^® . If a particle hits such a statistical cyclic patch, the incident angle between the patch normal vector and the particle’s velocity vector, $\omega$ is calculated in a first step. If $\omega$ is in the limits of the transmissivity functions ($\omega _{1,2}$) the transmissivity T is either calculated as a function of a base transmissivity $T_0$ (explained to hereafter), and $\omega$, or otherwise is set $\mbox{T=0}$. The transmissivity (a number between 0 and 1) is subsequently compared to a randomly generated number rand(). If rand() $\ge T$, the particle is rejected and subsequently moved according to its velocity. Otherwise, the particle is reflected. The reflection mechanism applied here is specular, meaning the particle gets the inverse velocity vector after reflection. The procedure is shown in Fig. 2.

Fig. 2

Flow chart of statistical mesh boundary patch

This scheme compares a random number with the calculated transmission probability $T(T_0, \omega )$ following the Monte Carlo principle. The transmission probability describes the probability that a particle crosses the cyclic patch pair. A simple way of an analytical expression of the mesh transmissivity is the two-dimensional description of an infinitely thin mesh. This basic transmission probability $T_0$ is based on the geometric parameters of the mesh only; see Fig. 3:

$$\begin{aligned} T_0 = \frac{(k-b)^2}{k^2} \end{aligned}$$

(4)

where k and b denote the mesh size and the wire width/diameter, respectively. A similar description was also applied to scale meshes for sounding rocket applications while kee** the transmission constant, see [14].

Fig. 3

Transmission description of a two-dimensional (i.e. infinite thin) mesh grid. The transmissivity is the ratio of the overall grid area $A_\textrm{grid}$ to the permeable area $A_\textrm{perm}$. The grid size is given by k and the wire width/diameter by b, see [14]

Real mesh geometries are often manufactured from wires, so the simple formulation of Eq. 4 is of limited accuracy. To account for the angular dependency of a real mesh, a modified mathematical expression needs to be formulated based on Fig. 4:

$$\begin{aligned} \begin{aligned} x = \frac{(1 - \sqrt{T_0})}{2} \left( \frac{1}{\cos (\omega )} -1 \right) ,\\ T(\omega ) = \left( 1 - \left[ ( 1 - \sqrt{T_0}) + 2x\right] \right) \sqrt{T_0}. \end{aligned} \end{aligned}$$

(5)

where $\omega$ denotes the angle between the face normal vector and a particle trajectory through the mesh, x denotes additional blocking due to the wire thickness. Note that $T_0$ denotes the constant basic transmission of Eq. 4, whereas $T(\omega )$ is the angular dependent transmission probability. The transmission $T(\omega )$ is the base transmission $T_0$, if $\omega\mbox{=0}$ and $T(\omega )\mbox{=1}$ (i.e. total blocking) for the limit denoted by:

$$\begin{aligned} \omega _{1} = \pm \cos ^{-1}\left( 1 - \sqrt{T_0}\right) , \end{aligned}$$

(6)

see also vertical dashed lines of Fig. 5, panel A.

Fig. 4

Geometric description of three-dimensional grid mesh with finite extension. For explanation of variables k and b refer to Fig. 3. Additional blocking x appears if the angle between mesh/face normal and particle vector is not zero

In former DSMC simulations of geometric mesh applications, it was speculated that the real transmissivity of a mesh should be higher than those calculated by, e.g. Equation 4 because some particles do not pass the mesh directly, but are reflected by the wires and scattered in the direction of the flow (hereafter denoted as forward scatter), ultimately passing the mesh. For an in-depth study of this forward scattering on a mesh segment, a ray tracing tool (with the use of basic features of the pyoctree software package [15]) was set up, able to handle arbitrary STL files (i.e., tessellated CAD files). The results for a mesh segment (similar to Fig. 3) for 24 different geometries with base transmissions ($T_0$) from 0.1%$-$99.0% are shown in Fig. 5.

Fig. 5

Ray tracing results (12 of 24) for various mesh base transmissions $T_0$. A Geometric transmission only: black lines show the analytical description of Eq. 5. B Forward scattered (ray tracing): vertical lines indicate the calculated blocking angle for pure transmission and include forward scattering (dashed and dashed-dotted lines, respectively). C Overall ray tracing transmission (direct and forward scattered)

For this study, 10,000 randomly distributed ray vectors with angles $\omega$ starting from 0° up to 89.9° and a resolution of 0.1° were used, where $\omega$ again denotes the angle between the particle velocity vector and face normal. Each run was repeated ten times to minimize statistical errors in the random placement of ray starting points. The ray tracing results indicate that the forward scatter (Fig. 5B) has a sufficient effect on the overall transmission function (Fig. 5C) and essentially affects the transmission even on small deviations from $\omega$=0°, especially for meshes with lower base transmissions $T_0$. The additional forward scattering ultimately shifts the angle of total blocking to higher values of $\omega$. The limiting angle for the forward scattering $\omega _{2}$ can be directly derived from ray tracing results and is given by

$$\begin{aligned} \omega _{2} = \pm \cos ^{-1}\left( 1 - \root 4 \of {T_0}\right) , \end{aligned}$$

(7)

see also dashed-dotted lines in Fig. 5B and C. The angles $\omega _{1,2}$ (Eq. 6 and 7) define the angle where total blocking appears. These transmissivity function limits are shown in Fig. 5 for pure transmission (Fig. 5A) and transmission that includes forward scattering (Fig. 5C). If the incident angle between patch normal and particle velocity vector $\omega$ is larger than the limit (either $\omega _{1}$ or $\omega _{2}$), the mesh blocks the particle. The blocking of pure transmission $\omega _1$ additionally denoted the angle where the population of forward scattered rays peak, e.g., $\sim$40 % of particles hitting a mesh with $T_0$=85 % (base transmissivity) are scattered in a forward direction at $\omega$=85°. To estimate the transmission probability for a particle with direction $\omega$ through a mesh with base transmission $T_0$, a 2D linear interpolation scheme is used. A consequence of higher limits of $\omega _2$ derived involving a forward scatter mechanism compared to $\omega _1$ is, that generally lower density increase is expected upstream the modelled mesh.

3 Implementation and demonstration showcase

3.1 Implementation of cyclic statistical patch boundaries

Two different versions of statistical boundaries were implemented, differing in the estimation of the transmission probability only. The first, named dsmcCyclicSpecularPatch applies specular physics only by applying Eq. 5. The second, denoted as dsmcCyclicForwardScatterSpecularPatch inhibits both, a specular description for backscattered particles and the possibility of forward scatter. This is done by interpolation of the ray tracing data. The statistical boundaries are designed as coupled cyclic patches, i.e., transmitted particles are placed in a position according to the former calculus of their trajectory. Particles that are reflected, instead, are released from the patch it originally hit.

A code example for the main functionality of the statistical boundary implementation is provided in Appendix A and consists of two parts. First, a modified class is called if a particle hits a cyclic patch, and then the actual controlling method is called if the statistical boundary is hit. The hole source code is publicly available, see Sect. 5.

3.2 Demonstration showcase

To test the general performance of the statistical boundary conditions, simple simulations have been set up for different angles of attack $\Phi$ for one base transmission probability $T_0$=0.5, density $n_{Ar,\infty }$=1$\cdot 10^{19}$ m$^{-3}$, and a free stream temperature of 273.15 K. These cases are available within the repository [16] and can be run on ordinary Laptops, (e.g., a couple of hours (varying with case) on a single  CPU with at least 4 GB ram). Results for both the dsmcCyclicSpecularPatch and the dsmcCyclicForwardScatterSpecularPatch are shown in Figs. 6 and 7 respectively.

Fig. 6

Test simulations for the dsmcCyclicSpecularPatch for a grid placed at x=50 mm extending from y=-20 to y=20 mm. Various angles of attack (given above the sub-figures) with and $T_0$=0.5. Glyphs indicate the velocity field, and the background colour reflects the relative density. The lower left inset gives the effective transmission $T_{\textrm{sim}}$

Fig. 7

Same as Fig. 6 but for dsmcCyclicForwardScatterSpecularPatch

Both Figs. 6 and 7 conclusively show that the transmission decreases with increasing angle $\Phi$ between the velocity field in the free stream and the face normal. The lower (higher) transmissions $T_{\textrm{sim}}$ at small (high) angles $\omega$ should be interpreted as the manifestation of the molecular flow character: An analysis of the angles $\omega$ for particles that hit the boundary revealed a mean value of $\sim$22 (for dsmcCyclicSpecularPatch) even when the free flow was directly perpendicular to the mesh boundary. This is because the particles’ velocity component in the x-direction becomes comparably low as the z and y-components upstream of the mesh boundary due to collisions with reflected particles. Considering this physical effect, the boundaries perform as expected from the analytical function (Eq. 5) and the ray tracing results, both shown in Fig. 5: I.e., with a larger transmission probability for the dsmcCyclicForwardScatterSpecularPatch and a decrease at larger angles $\omega$, compared to dsmcCyclicSpecularPatch.

4 Validation with wind tunnel measurements

4.1 Wind tunnel measurements

The experiments were conducted in the SR3 low-density, Mach adaptable wind tunnel at CNES, now located in Orléans and called MARHY (fr. Soufflerie á Mach Adaptable Raréfié HYpersonique). This facility consists of a setting chamber with a gas inlet, a two-meter diameter test chamber, and a diffuser, connected by a heatable nozzle. Density fields were measured using electron beam fluorescence (EBF), where an electron gun emits an 25 kV electron beam exciting $N_2^+$ to emit light at 391.4 nm, detected perpendicularly by a photomultiplier, see [17, 18].

A mesh, with an application diameter of 28 mm and a 76 % geometric transmission at 0.1 mm wire width, was placed on a 1 mm thick metal ring within the test chamber, see Fig. 8. The wire width (0.1 mm) is the characteristic length for calculating dimensionless Re, Kn, and Ma numbers.

Fig. 8

Mesh that is used for wind tunnel measurements. A mesh size of 0.8 mm, a wire width of 0.1 mm, resulting in 76 % transmissivity. The mesh is mounted on a ring of 28 mm diameter and 1 mm wire thickness, (J. Gumbel, personal communication). The centre-line of the ring is indicated in red. The contact point between ring and rod, that places the mesh in the wind tunnel, is marked by a green circle

The tunnel operated at free stream conditions detailed in Table 1, simulating conditions similar to rocket sounding in the Earth’s mesosphere when scaled to full size [18]. Results of these measurements are presented in Fig. 9.

Table 1 Wind tunnel conditions: free stream ($\infty$), application (wall)

Fig. 9

Results from the MARHY wind tunnel measurements. Right panel shows a tricomntour plot of the raw measurements (converted to relative values only). Black dots indicate positions where actual measurements were taken. The middle panel shows corrected data to match with data presented in reference [2] (see text for more information). The right panel reveals corrected results suggested in this manuscript

The raw data, provided by the author of reference [2] (personal communication), indicated the mesh was not perfectly aligned vertically due to drag or manufacturing tolerances and has been corrected. Reproducing this correction, reveals that the mesh had been rotated by approximately 4.15° around the contact point of the rod and the mesh at (x=0, z=−13.5) mm, see green circle in Fig. 8. Subsequently, data has been cropped at the suspected symmetry line. The anticipated mesh position is shown in green at the leftmost panel of Fig. 9. The resulting field is shown in the middle panel.

Despite potential parasitic reflections near the reflecting surface [19], measurements close to the object are included here for qualitative analysis. The author’s interpretation is, that the correction of the miss alignment is reasonable and only medium influence of miss alignment on the overall transmission (its angular dependency) is suspected for small angles for values along the symmetry line.

The correction appears reasonable, with minor alignment impact on overall transmission for small angles along the symmetry line. Asymmetry around the symmetry line (z=0) likely stems from the rod placing the mesh at the Laval nozzle’s centre. Raw (i.e. uncorrected) measurements suggests that the coordinate system of measurements was centred at the centre-line and not the rod, which was not sampled at all. Supersonic conditions induced shear forces in the model, peaking at the short wire connecting the rod to the mesh ring (see Fig. 8, green circle). Consequently, the rotation point in reference [2] should be at the mesh’s centre (x=0, y=0) mm, as indicated by the red patch/line in Fig. 9.

This correction reveals a denser field upstream and a decreased density downstream of the mesh, see right panel of Fig. 9. The low density inside the mesh ring (red patch) is explained by the EBF setup, as the ring blocks most emitted light, leading to lower measured densities, (see reference [17] for a description of the measurement setup.) This new correction also shows expected molecule concentration enhancement along the undisturbed flow direction, starting from the shock (Ma=1) to the mesh’s surface.

4.2 Validation test cases set-up

Simulations were conducted on one-quarter of the mesh using x-y and x-z planes as symmetry planes. All other outer boundaries were set as free stream boundaries, with various statistical boundary conditions: dsmcCyclicSpecularPatch, dsmcCyclicForwardScatterSpecularPatch, and dsmcCyclicForwardScatterDiffusivePatch, to model angular-dependent specular and diffusive reflections, with and without forward scattering physics.

Fig. 10

Simulation domain setup, visualized with ParaView [20]

The hexagonal numerical grid was generated using OpenFOAM^® ’s blockMesh and snappyHexMesh utilities, with the coarsest grid resolution with a cell size of 1 mm, well above the expected mean free path in the free stream (Fig. 10).

The simulation time step was set at 50 ns., significantly smaller than the mean collision time of 3.6 $\mu$s for free stream parameters. Adaptive mesh refinement (AMR) was performed every 500 steps. A FunctionObject was used to define a volScalarField named refineField, which marked cells for refinement if its value exceeded 1, and for unrefinement if below 0.45. The refineField was set to the ratio of the cell dimension (in the main flow direction, x) to the mean free path, provided the number of simulation particles was sufficiently high. A minimum of 400 particles was maintained to ensure numerical stability and robust statistics.

Dynamic load balancing was disabled to avoid issues with reconstruction and redistribution for a large number of simulation particles (approximately 7$\cdot$10$^{8}$ for these simulations). The simulations were executed in densityOnly mode to reduce computational load, calculating mean field properties. Thus, densities of simulated and physical particles, along with the refinement parameter, were derived as macroscopic mean quantities. Convergence criteria included mean kinetic energy and mean collision acceptance rate. Once steady state was achieved, fields were averaged to enhance statistical accuracy.

4.3 Validation test cases results and comparison

The main parameters of the DSMC simulations are summarized in Table 2.

Table 2 Simulation parameters

Figure 11 shows the numerical grid at the final state, refined up to a level of 4 (0.0625 mm) in regions with the largest gradients, such as the shock. The simulations run on 40–80 CPUs, on $\sim$650GB ram for 8–12 days, applying a variable time step (i.e., larger time steps at the beginning of a simulation).

Fig. 11

Numerical grid at final states. Grids have been refined according to the mean free path of the flow. For a dsmcCyclicSpecularPatch, b dsmcCyclicForwardScatterSpecularPatch and c dsmcCyclicForwardScatterSpecularPatch

Three simulations were performed: a simple dsmcCyclicSpecularPatch simulation without forward scatter, and two, applying advanced transmission models with forward scatter (dsmcCyclicForwardScatterSpecularPatch and dsmcCyclicForwardScatterDiffusivePatch). The interaction between molecules and the surface was modelled as specular in the first advanced model and as thermal-diffusive for full thermal accommodation in the second.

The final results for both simulations are shown in Fig. 12.

Fig. 12

Relative density field derived by simulation (coloured contours) and wind tunnel measurements (red contour lines), see Fig. 9 (right panel) for a full-colour plot. Same colour bar and contour levels for all plots

All three simulations exhibit a steep density increase upstream of the mesh, propagating to the mesh itself. The shock region’s shape at the mesh edges is consistent across simulations, validating the applied correction. However, the exact shock position and relative density values differ from measurements. The dsmcCyclicSpecularPatch simulation shows a thicker shock region with a larger relative density increase compared to measurements.

For a quantitative comparison, Fig. 13 presents the relative number density along the mesh symmetry axis from wind tunnel measurements and DSMC simulations. This includes results from a previous DSMC study [2] with a constant, non-angular dependent transmissivity factor (T=T$_0$), the original and reconstructed values with data correction from reference [2], raw data, and the suggested correction based on Sect. 4.1. Note, that the low density value of the first raw data point leeward the mesh most likely is a result of (partial) blocking of the EBF detector (discussed before). The low values for the reproduced and corrected data result from rotation and additional interpolation between the raw data points.

The dsmcCyclicForwardScatterSpecularPatch simulation aligns well with wind tunnel data, within error bars of both raw and corrected data. The AMR technique effectively resolved the steep density increase at the shock boundary, allowing precise prediction of the shock’s position. Downstream of the mesh, this simulation mostly agrees with the wind tunnel data.

The dsmcCyclicForwardScatterDiffusivePatch simulation underestimates shock thickness and overestimates density behind the mesh, but shows closer agreement with measurements than the non-forward scatter, specular setup. The peak densities of both forward scatter setups are similar, whereas the specular setup overestimates the density increase. Interestingly, the relative density downstream of the mesh surpasses 1, which may seem counter-intuitive. This effect, not observed in Sect. 3, is attributed to the solid metal ring supporting the mesh, acting as an "aerodynamic lens" with a leeward "focal point" in supersonic flow, as discussed in reference [2] (their Fig. 5).

Previous DSMC simulations [2] overestimated shock thickness and underestimated downstream density. The primary differences are that the previous simulations were axisymmetric 2D and used a constant transmission probability.

Fig. 13

Normalized densities derived from DSMC and wind tunnel measurements. The wind tunnel data processing is discussed in Sect. 4.1

Instruments like ion-gauges (Fig. 1) are particularly affected by density downstream of such meshes, while sensors like meteor smoke particle detectors are affected by the shock ahead of the sensor [3, 21]).

4.4 Discussion

Real molecule-surface interactions may be modelled as a combination of idealised approaches: specular scatter, assuming a smooth and rigid surface; isotropic scatter, for perfectly rough and rigid surfaces; and full thermal accommodation, where molecules are in perfect equilibrium with the surface and are diffusively scattered according to the Maxwell distribution (see e.g. [22]).

Describing molecule-surface interaction on a small mesh geometry probabilistically, as done here, requires differentiating between molecules transmitted through the mesh and those reflected. In an ideal case of an infinitely thin, two-dimensional mesh (see Fig. 3), this differentiation is straightforward, with only two possible states: the molecule is either reflected or interacts with the surface upon collision. The advancement in transmission probability calculation, which accounts for the three-dimensional nature of most meshes and the higher transmission probability due to forward scattering, necessitates incorporating molecule-surface interaction into the transmission probability.

For specular reflections of molecules (dsmcCyclicForwardScatterSpecularPatch), the probability function, including forward scatter derived in Sect. 2, remains valid and consistent, accurately reproducing the density field observed in wind tunnel measurements. The ray tracing applied uses a specular scatter mechanism, resulting in the same physical process as specular molecule reflection.

The dsmcCyclicForwardScatterDiffusivePatch boundary in the current implementation combines the thermal-diffusive approach, triggered upon molecule reflection, with the specular approach, as part of the transmission probability function. The thinner shock front (compared to measurements), indicates either an overestimation of the transmission probability or a density decrease in front of the mesh due to thermal heating (see [23]). Thermal heating of the flow, entering at only 71 K, seems plausible but does not explain the increased density leeward of the mesh. The temperature of molecules increases at the stagnation point of the flow to $\approx$380 K. This is moderately higher compared to the mesh’s surface temperature. This may explain the relatively close agreement with the dsmcCyclicForwardScatterSpecularPatch, as thermal accommodation plays a minor role when stagnation point temperature and surface temperature spread is moderate. A closer look at the statistics of the velocity vector deflection from x-direction upstream the mesh of both forward scatter boundary cases reveals, that more particles interact with the mesh under small impact angles, see Fig. 14 Consequently, the transmission probability is higher compared to the larger impact angles of the specular scattered molecules. This contributes to the comparably lower (higher) densities in front (behind) the mesh.

Fig. 14

Probability density of molecule velocity vector deflection angles for the forward scatter simulations. Deflection is relative to x-direction, i.e., $\overline{n}$=(1, 0, 0). The sampled volume is: -2<x<0; 0<y|z<10 [mm]. The black line indicates absolute deviations

The results also indicate that, at least for the wind tunnel conditions, the molecule-surface mechanism’s influence is much smaller than the influence of the transmission probability. However, a more physical wall interaction model becomes more significant at higher Ma numbers with higher temperature spreads between particles and surfaces, such as those encountered during sounding rocket missions in the thermosphere.

Using advanced molecule-surface interaction models like the multi-coefficient Cercignani-Lampis-Lord (CCL) model [24] for both transmissivity calculation and molecule-surface interaction of reflected particles, with appropriate coefficient values, is expected to yield better results under such high Ma conditions. To justify the implementation and utilisation effort of such an advanced (e.g., CCL) boundary, additionally, proper validation measurements are necessary.

5 Conclusion

In this study, an implementation of cyclic boundaries designed to handle fine meshes with scales much smaller than the required computational grid resolution (i.e., $\ll$ $\lambda$) has been introduced. Utilizing a stochastic approach through a Monte Carlo technique, this implementation enables more efficient simulation setups, reducing overall computational effort significantly. Recognizing that constant transmissivity parameters ($\mbox{T=}T_0$, independent of incident angle) may lead to unrealistic behaviour, particularly in oblique flows, new boundaries were developed that account for angular dependencies at various levels of complexity. The performance of these models was demonstrated and validated against physical expectations.

Simulations were conducted and compared with wind tunnel measurements under mesosphere-like conditions. The detailed process of wind tunnel data processing was outlined, providing a comprehensive understanding of the methods used in reference [2], leading to suggestions for proper data processing.

The dsmcCyclicForwardScatterSpecularPatch boundary condition, which models the angular dependency of transmission probability and mimics the process of forward scattering of molecules through a mesh with the smallest dimensions not resolvable in real scale DSMC, showed excellent quantitative agreement with unique wind tunnel measurements. Conversely, the dsmcCyclicSpecularPatch, were shown to underestimate the transmission probability by neglecting forward scattering contributions. The closer agreement with measurements for the dsmcCyclicForwardScatterDiffusivePatch indicates that a proper transmission probability function for thermal diffusive scattering should have lower transmission probabilities. These findings suggest that the dsmcCyclicForwardScatterSpecularPatch is a suitable model for small mesh applications in atmospheric rocket soundings. The new boundary models have been implemented to a ported version of the dsmcFoam+ package that was originally released by White et al. [25] in OpenFOAM version 2.4.0 (https://github.com/MicroNanoFlows/OpenFOAM-2.4.0-MNF/), has been successfully benchmarked [26] and now is modified to run on v2306. This new version, now compatible with v2306, combines the advanced features of dsmcFoam+ with recent developments in grid generation, particle handling, and compatibility with modern Linux systems.

Future work should focus on further refining the transmission probability functions and validating them against additional experimental data, particularly under high Ma number conditions relevant to thermosphere soundings. Implementing advanced molecule-surface interaction models, alongside appropriate transmission probability functions, that account for the influence of respective scattering mechanism is expected to yield better results for higher Ma conditions.

Appendices

Appendix A Code for cyclic statistical boundary

To implement the cyclic boundary, a new function was added to the dsmcParcel class of the advanced dsmcFoam+ (which corresponds to the DSMCParcel) in the original DSMC that comes with OpenFOAM-v2306.

The main functionality of List 1, is to call the controlMol method if a patch is being hit, that is registered in the cyclicBoundaryModels class. After this procedure, the reflected method is called to check whether the molecule should be placed according to the prior calculated trajectory or is being reflected. This is achieved by using hitCyclicPatch in dsmcParcel.C of the dsmcFoam+ source code directory. The general procedure of the particle handling of the statistical boundary condition is given in List 2.

Depending on the model, the method calculateTransmission() either calculates the transmission using Eq. 5, or calls an interpolation routine regularGridInterpolator() with a data grid derived by the ray tracing results described earlier. The function reflected() (see List 3) is a simple wrapper to call the final reflection state in List 1:

Appendix B Quality measuremets

The quality of Direct Simulation Monte Carlo results heavily depends on a thoughtful simulation setup. The main quality parameters are: (1) A sufficient number of sampled particles inside a single cell volume, which is used to derive macroscopic quantities through averaging.

Fig. 15

Number of simulated particles inside the simulation volume. (Colour figure online)

Most practical guidelines, recommend at least 10 particles per cell for reliable statistics, see e.g., [27]. The least populated cell for the presented simulations contains a mean of 43 particles, see Fig. 15. (2) A sufficiently small cell volume, to resolve even steep gradients in the flow. This plays an important role, when studying e.g., supersonic flows, or heat transfers. For the presented simulation, the AMR technique is applied to justify that the cells sizes with the mean free path, see Eq. 2. Figure 16 shows the refinement parameter, that is equal to the cell-size to mean free path ratio.

Fig. 16

Number of simulated particles inside the simulation volume. (Colour figure online)

Exceptions are possible for cells that contain less than 400 particles, where the refinement value is set to a constant value of 0.7. (3) The decoupling of collisions and translation of particles, requires a sufficiently small simulation time step. In good practice the simulation time $\Delta t$ to mean collision time (mct) ratio should be (much) smaller compared to unity:

$$\begin{aligned} \frac{\Delta t}{mct}<< 1. \end{aligned}$$

The mean collision time for the free stream (see Table 2) is 0.013 and even for regions with density and temperature increase by one order of magnitude, i.e. far beyond values that can be expected in the flow investigated here, increases to a value of 0.4 only.