We first investigate compressibility effects in backwards-facing step flow and compare with corresponding incompressible flow simulations. The backwards-facing step geometry used in this study is given in figure above along with a typical spectral element discretization. It corresponds to S/h=0.467, where S is the step height and h is the height of the channel. The computational domain is discretized with 52 spectral elements, where each element is further discretized with Nth-order polynomial expansions in each direction. We have performed a resolution study (p-refinement) by increasing N and negligible variations in the solution are observed as N is increased from 6 to 8; in the results presented here we use N=10. The maximum error in mass and momentum global balance is approximately 0.01%.
In the above figure we have also included predictions of mu-Flow for rarefied flows with Kn_o=0.04. Here the value of Knudsen number Kn is based on exit conditions. For the simulations we have used the slip model based on obtaining the slip information one mean-free-path away from the surface. The exit pressure of the channels and the channel thickness were fixed in our simulations. Therefore, the value of Knudsen number Kn_o is constant at the exit of the channel for all of the simulations. However, the distribution of Kn varies from simulation to simulation as the inlet to exit pressure ratio is varied. One limitation in the compressible flow simulations is the possibility of "choking" the flow in the channels, either at the step expansion, where the fluid accelerates significantly, or at the exit of the longer channel. Here we limit the simulations to subsonic cases.
Next, we simulate nitrogen rarefied flow in a backwards-facing step using the DSMC method. For the DSMC simulations 3,000 equally spaced cells are used. The number of cells in the streamwise and cross-flow directions are 100 and 30, respectively. The total number of simulated molecules is 30,000. The DSMC results are sampled for 700,000 time steps. Convergence is verified by monitoring the mass balance with the maximum errors being approximately 0.6%. The inlet channel is also included in the simulation, with the inlet located at x/h=0.86. A uniform flow stream corresponding to M=0.45 approaches the inlet of the channel with a free-stream static temperature of T=335 K. All wall surfaces are maintained at 300 K. In next two figures we plot the streamwise variations of pressure and streamwise velocity, respectively, obtained at five different y/h locations. The values of pressure and velocity are non-dimensionalized with the corresponding free-stream dynamic head and local sound speed, respectively. The specific y/h locations are selected to coincide with the DSMC cell centers to avoid erroneous interpolations or extrapolations. The first location is at y/h=0.01675 near the bottom wall, shown by solid triangles and referred to as "BW". The second location is near the bottom center at y/h=0.25, shown by hollow squares and referred to as "BC". The third location is at y/h=0.48325 near the larger channel center, shown by hollow circles and referred to as "C". The fourth location is shown by solid squares and it is located at y/h=0.75. This location corresponds to the center of entrance section and referred to as "CE". Finally, the fifth location is located at the nearest DSMC cell center to the top wall at y/h=0.9875, and referred to as top-wall "TW" (shown by solid circles).
After the entrance section, the pressure drops very rapidly with corresponding fluid acceleration. The pressure drop in this section (1.2 < x/h < 1.5) is almost uniform across the channel as seen in the figure. In the second figure we see that the velocity increases along the channel similarly to a compressible straight channel flow. In this section the fluid temperature at the center of the entrance region (CE) decreases substantially reaching a minimum at around x/h=2.0. This is accompanied with acceleration of fluid in the streamwise direction, as a result of the transformation of thermal energy into kinetic energy. Around the step expansion we also observe an increase in the cross-flow component of velocity. Therefore, the thermal energy transformation affects both the streamwise and the cross-flow velocity.
The sudden expansion in the geometry creates adverse pressure gradients across the entire cross-section up until x/h=3.25 and x/h=3.65 at the bottom wall (BW) and the top wall (TW), respectively. Due to the adverse pressure gradients the flow at the bottom wall (BW) separates and re-attaches at x/h=2.8 before the pressure gradient at the bottom wall becomes zero (at about x/h=3.25).
Beyond x/h=3.65 favorable pressure gradients are established. The flow goes through a developing region, followed with a typical compressible channel flow behavior far downstream in the channel. This behavior is seen in above figures for x/h > 4.2. In contrast to low Mach number flows, a decrease in the temperature of the fluid near the center of the channel is observed. This shows that the thermal energy of the fluid is converted to kinetic energy, and there is considerable heat transfer from the walls to the fluid.
Next we examine the validity of Maxwell's slip and von Smoluchowski's temperature jump boundary conditions, as well as the boundary conditions based on obtaining the slip information a certain distance away from the surface against the DSMC calculations. This comparison is shown in the previous figures. For a more detailed comparison we present the pressure variation obtained by mu-Flow near the top wall of the step channel (at y/h=0.98325). We have used the first-order slip and temperature jump conditions and our model equations. In employing our model we have obtained the slip information at a distance lambda and 3/2 lambda away from the surface (lambda is mean-free-path). In general, the DSMC and mu-Flow solutions are in good agreement. The slip models predict the variation of top wall pressure including the adverse pressure gradients very well. The pressure distribution near the bottom wall y/h=0.01675 is also given in the figure below. All the continuum slip models show good agreement with the DSMC solution along y/h=0.01675. The model employing information 3/2 lambda away from the surface performs slightly better than the others. Velocity distributions at the top- and bottom-walls are shown below. Also here, the model based on obtaining the slip flow information 3/2 lambda away from the surface predicts the velocity near the walls better than the other models.
To achieve the aforementioned results we had to match the inlet and exit conditions between the DSMC runs and the runs using the mu-Flow code; however some differences were observed due to the numerical diffusion in the DSMC method. In particular, we specified the dynamic viscosity of nitrogen (mu_o=1.76 E-5 [kg/ms] at 293 K as a reference viscosity in the slip flow simulations. This value is corrected for variations in temperature using Sutherland's law of viscosity. Since we do not know precisely the effective viscosity in the DSMC calculations, we had to increase the viscosity in our calculation by a factor 1.69 in order to match the flow rate between the DSMC and the mu-Flow runs. Correspondingly, the mean-free path has to be adjusted proportionally. However, in the evaluation of reference mean-free-path we did not enhance the mean-free-path by a factor of 1.69. This discrepancy can be used to explain the reason for obtaining better comparisons with DSMC results by using the slip information 3/2 lambda away from the surface, effectively enhancing the value of the mean-free path. In other words, if we had used an increased dynamic viscosity for evaluation of the mean-free-path as well, then the our model with slip information obtained at a distance one mean-free-path away from the surface would have given equally good results with the same model based on 3/2 lambda. Since the first-order model and our model based on lambda performed equally well as shown in figure above, we can conclude that both slip models would predict the DSMC solution equally well if the dynamic viscosity was enhanced by a factor of 1.69 in the calculation of mean-free-path.
The reason for obtaining a relatively large viscosity in the DSMC simulations compared to the actual molecular viscosity has been investigated and explained by Breuer, Piekos and Gonzales in the following publication. In the DSMC calculations the relative positions of the molecules within the cells are neglected. Therefore, the momentum of the molecules is uniformly diffused within the cell. If the cell thickness is equal to the mean-free path of the molecules, realistic viscosity values are obtained.
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