Investigation on the Interface Smoothing of Coupled N–S/DSMC Method Using Image Processing Filters
Gas flow problems involving both continuum and rarefied regimes are common in many scientific and engineering applications. Coupled N–S/DSMC method is a major solution to simulate the continuum–rarefied transitional flow. Interface determination is one of the key aspects in developing coupled N–S/DSMC solver, which is often implemented using continuum breakdown parameters to indicate the rarefication level. Due to the statistics characteristic of DSMC method, distribution of the continuum breakdown parameters usually fluctuates, resulting in difficulty in locating the interface. Considering the similarity between continuum breakdown parameter smoothing and noise reduction in image processing, a general smoothing method is proposed in this paper, which employs the image filters to smoothen the distribution of continuum breakdown parameters. Two commonly used image filters, the mean filter and the median filter, are investigated. Several comparison studies are implemented by simulating a typical vacuum plume impingement flow problem. The median filter with 5 × 5 mask provides the best performance. The flow problem of flow over a 2D cylinder is used to validate the coupled N–S/DSMC solver using median filter to smoothen continuum breakdown parameter, which proves the accuracy of the coupled solver and validity of the smoothing method.
KeywordsCoupled N–S/DSMC method Continuum breakdown parameter Interface smoothing Mean filter Median filter
In many scientific and engineering applications, gas flows containing both continuum and rarefied regimes need to be studied, such as in the vacuum plume flow problem of space thrusters, inside the thruster nozzle, the gas density is high, while outside the nozzle, it drops rapidly, approaching to the vacuum state and the degree of the rarefaction of gas changes extremely, from continuous gas flow to rarefied gas flow. It is difficult to adopt one single method to simulate this kind of problem and the coupled Navier–Stokes (N–S) and Direct Simulation Monte Carlo (DSMC)  method is a preferable solution, which handles the continuous and rarefied gas regions using Computational Fluid Dynamics (CFD) method and DSMC method, respectively. There are two important issues in developing an N–S/DSMC solver, which are interface location and information exchange between the continuums and rarefied regions, and current paper pays special attention to the former one.
To measure the extent of rarefaction quantitatively, researchers have proposed different continuous breakdown parameters. Because the Kn parameter  is a global parameter, it isn’t suitable for coupled N–S/DSMC method as a continuous breakdown parameter. Based on the Kn number, several continuous breakdown parameters have been developed, including P parameter , KnGLL parameter , KnGL parameter , B parameter [5, 6, 7], and the Ptne parameter .
Previous studies have been carried out on the formation of continuous breakdown parameters, but no matter what kind of continuous breakdown parameters are employed, an extra smoothing operation must be performed before a satisfactory interface location result can be obtained. The reason is that, the continuum breakdown parameters within the DSMC region are calculated from variables those are got from particle statistics, and thus the fluctuation of the continuum breakdown parameters is inevitable. After a threshold value is set, the interface will not be an ideal curve or curved surface in 2D and 3D coordinate, respectively, but a complicated situation will appear. Many scattered isolated partitions will appear near the interface, making the further coupled calculation hard to implement.
Former coupled N–S/DSMC solvers [9, 10, 11] must have had considered this problem in some manner, but no special research has focused on it. In the present paper, investigations are performed on this subject, aiming to find a general solution for coupled N–S/DSMC method to locate the interface more reliably and flexibly.
The interface smoothing problem and the noise reduction problem in the image processing are considered analogous in the present paper. Thus two common filtering methods in image processing, the mean filter and median filter, are introduced to deal with the partition smoothing in coupled N–S/DSMC calculation.
2 Coupled N–S/DSMC Method
The DSMC method, which was proposed by G. A. Bird [1, 12], uses a small number of simulation particles instead of a large number of gas molecules in the real flowfield to obtain the macroscopic flow parameters by statistical means. In the actual gas flow, the movement and collision of gas molecules are always carried out simultaneously, meaning that they are coupled. DSMC method uses probabilistic (Monte Carlo) means to decouple the motion and the collision of gas molecules, simplifying the process of the algorithm and saving much computational time. But the consumption of computational resources is still unacceptable to employ DSMC method in the continuum region, while the CFD method, which solves the gas dynamic equations using finite difference schemes, is quite mature and efficient in modeling the continuum flow. The N–S solver is only available in the continuum region because of the hypothesis that the equations based on. Since DSMC method and N–S solver have their own proper applications, respectively, the concept to couple these two methods to form a continuum–rarefied transitional flow solver is straightforward.
A DSMC code named PWS  (Plume Work Station) is used as the DSMC solver in the current coupled N–S/DSMC research. The PWS is developed by the authors based on the Cartesian grid system. The surfaces in complex geometries of flowfield are divided into several simple convex units and embedded in the grid as described in Ref. . HS (Hard Sphere), VHS (Variable Hard Sphere), VSS (Variable Soft Sphere), CLL (Cercignani–Lampis–Lord) molecular models are implemented in PWS, and RSF (Random Sampling Frequency) collision sampling method is adopted .
3 Problem in Interface Determination
In the coupled method, the continuum/rarefied interface is decided by continuous breakdown parameters. From the definitions of KnGL, it can be figured out that they are all calculated from the derivation of macroscopic gas parameters. When calculating the KnGL within the rarefied regions, the related gas parameters are all obtained by the statistics of particles, and thus the fluctuation is inevitable. What is even worse is that before the interface reaches to a steady state, there will not be enough steps of DSMC calculations in each coupled iteration to increase the convergence speed, and thus there will not be enough number of particles to obtain smooth macroscopic gas parameters by statistics.
This demonstration case shows that, without continuum breakdown parameter smoothing technique, further coupled N–S/DSMC calculation is difficult to perform.
4 Two Commonly Used Spatial Image Filters
4.1 Mean Filter
The smoothing effect of mean filter is related to the radius of the mask. The larger the radius, the greater the fuzzy degree, the more smooth the image is. Due to the very small area occupied by the mask in an image, it is difficult to see the difference between the images after smoothing by various masks in Fig. 6 .
4.2 Median Filter
In practical, the larger the masking, the more obvious the de-noising effect is. If the mask is too large, it will lead to changes in contour lines. It is necessary to select the size of the masking carefully.
5 Image Filters Applied to Kn GL Smoothing
In image filtering, the data to be operated is the pixel values. Correspondingly in N–S/DSMC-coupled computation, the filtering method is employed, while the target data to be operated is replaced with continuum breakdown parameter, which is KnGL parameter in the present study. A cell in the coupled calculation mesh corresponds to a pixel position in an image, and the KnGL is regarded as the pixel gray value in an image.
There are several configuration parameters for each image filter, such as mask radius and using weight or not in mean filter. To investigate which filter and what kind of filter configuration is more preferable, several simulations are performed using different filters and different configurations.
5.1 Mean Filter
It can be figured out that, when the radius is 3 × 3, KnGL distribution is more disordered, the 0.05 contour lines surround several isolated zones, and some cells within the shock wave have KnGL larger than 0.05, which means it will be determined as DSMC cells. The 5 × 5 mask includes more pixels to average, increases the fuzzy level, and thus achieves a more smooth KnGL distribution result, at the same time reserves the overall shape. Within the shock-wave cells, all the KnGL values are below 0.05, and thus can be considered as continuum partition.
Among the three configurations, comparison results show that the 5 × 5 standard average mean filter mask provides an acceptable smoothing effect.
5.2 Median Filter
6 2D Cylinder Flow for Validation
The comparison validates the accuracy of the coupled solver employing median filter to smoothen KnGL and demonstrate the smoothing effect again.
The coupled N–S/DSMC method is an efficient and accurate solution for continuum–rarefied gas flow simulation. Determination of the continuum–rarefied interface is one of the important aspects in coupled N–S/DSMC computation. Although several continuum breakdown parameters have been proposed by former researchers to indicate the rarefication level, the continuum breakdown parameter would fluctuate in practice, making the interface hard to locate. The present paper proposed a new method to pre-process the continuum breakdown parameter, which borrows the ideas form image processing. Two commonly used image filters, the mean filter and the median filter, are employed to smoothen the KnGL continuum breakdown parameter in the coupled N–S/DSMC calculation. Several comparison studies were implemented by simulating a typical vacuum plume impingement flow problem. The median filter with 5 × 5 mask was proved to be more preferable. A benchmark flow problem of flow over a 2D cylinder was simulated with the coupled N–S/DSMC method, using median filter to smoothen continuum breakdown parameter, and was compared to the full DSMC results, which validated the accuracy of the coupled solver and demonstrated the median filter smoothing effect. The image filter investigated in this paper is also suitable for other coupled solver and continuum breakdown parameters.
8 Future Work
The interface determination technique contains more or less some empirical factors. To ensure the accuracy, the median filter with strong smoothing property is employed in the present investigation. The reviewers inspired us a new interesting idea that the interface can be defined using an edge detector operator. In this case, the image is smoothed first by a Gaussian Kernel and then the interface is determined using the Canny operator. This idea will be studied in the future.
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