In the present DSMC analysis, something similar to the Rayleigh-Taylor instability has been applied as a disturbance near the inner wall of the circular tube, and the transition of the flow velocity in the tube from a laminar distribution to a turbulent distribution has been calculated. Since this calculation uses artificial approximation, sufficient consideration will be required in the future to determine how close it can be to the actual turbulence in the tube. The newly improved DSMC calculation method for tube flow velocity (turbulent and laminar) as of 2026 is described in Section 3.3(e).
This time, I will first explain the introductory part of this analysis. The DSMC method has been developed for rarefied gas dynamics and it reproduces the flow based on the motion of a large number of molecules. Therefore, the lower the density (lower the pressure if the temperature is constant) and/or the smaller the characteristic length, the easier the analysis is. Furthermore, if the average speed of molecules is assumed to be 350 m/s, it is easy for the DSMC method to analyze phenomena with flow velocities close to or larger than that, but difficult to solve phenomena caused by winds of 10 m/s or less in daily life. Based on this, we first considered the following conditions when simulating the flow in a circular tube with Re=5000. The axisymmetric flow field is assumed, which consists of a cylindrical tube with a diameter d = 2.5 mm (this is the characteristic length) and a length equal to or longer than d in the tube axis direction. Periodic boundary condition is also assumed at the inlet and outlet of the tube. Naturally, since there is friction on the tube wall, the momentum lost on the wall is compensated equally for all molecules present in the flow field, so that the momentum of the entire flow field, that is, the average flow velocity, is kept constant. The idea that the lost momentum is equally imparted to all molecules is reasonable if you consider that a flow falling at a constant flow rate in a vertical tube is maintained by balancing the gravity acting on the molecules and the wall friction. It seems that the longer the tube length, the more interesting the results will be, but you need to be prepared that the calculation time will also be longer. If a simple argon gas (viscosity coefficient 2.2e-5 Pa_s) with a pressure of 0.5 atm (380 mmHg) and a temperature of 288 K flows at an average velocity of 50 m/s, Re will be approximately 5000. The boundary condition at the wall surface used in the DSMC method is generally such that molecules are diffusely reflected by the wall surface (diffuse reflection).
If the molecular velocity is generated from a uniform distribution at the average flow velocity as an initial condition and the calculations are continued, even if Re=5000, it will converge to a parabolic distribution indicating laminar flow (Poiseuille flow). This result is not unnatural since no disturbance was applied. From then on, the reflection velocities are determined by artificially manipulating them of the molecules from the wall (the author proposed this from the perspective of undulating at long positional intervals in surface roughness, but strictly speaking, this is not accurate). As a result, if a disturbance similar to Rayleigh-Taylor instability is generated near the inner wall of the tube, this causes distortion in the flow velocity distribution (Usami’s pseudo disturbance for DSMC). Note that the axial momenta of the reflected molecules cancel each other so that their total sum becomes zero. Two parameters are used to determine this disturbance, and the final shape of the flow velocity distribution in the circular tube is determined depending on the selection of their values. The horizontal axis of each figure is the distance from the center of the tube, which is normalized by the tube diameter. The vertical axis is the flow velocity normalized by the average flow velocity. Figures 1 and 2 (animation) indicate the results by averaging the entire area in the axial direction, on the other hand, figures 3 and 4 (animation) show the results by dividing the axial direction into several sections so that it is easy to investigate how the flow velocity changes near the wall surface. Figures 5 and 6 (animation) are those in which both are displayed alternately. In addition, figures 7 and 8 (animation) show the undulations of the flow that occur near the wall using streamlines and contour lines of the axial flow velocity. Note that the temperature and density changes in the entire flow field are approximately 1% at most, so the results are shown in terms of flow velocity and intentionally not in mass flux (rate of mass flow). In addition, only the Usys method (U_system) has been used in this calculation, and the Bird method (B_original) is not used. Using the cell structure adopted here, the Bird method could not completely obtain even the parabolic distribution of laminar flow. Since the Bird method has excellent calculation speed than the Usys method, it would be fair to use a more fine cell structure for comparison, but this is not done here. You may reproduce these results without permission, but please cite the source, ‘DSMC calculation by M.Usami (https://usamimas.net)’. The same applies to the descriptions in other chapters.