*2.3. Assumptions for CFD Calculations*

CFD calculations were performed using Ansys Fluent 19. The investigated geometry has inflow section of the length of 6H (Figure 6), where *H* denotes the height of the inflow channel (Figure 2). The length of the outflow section was assumed as 11*H*. At the inlet to the seal, the total pressure *pin* and temperature *Tin* are known, while downstream the segment the static pressure *pout* is known. Based on the above boundary conditions (Table 1.) CFD calculations are performed.

**Figure 6.** Computation domain.

**Table 1.** Boundary conditions.


In these calculations air was treated as the compressible ideal gas. RANS type calculations were performed for 2D axisymmetric geometry. Equations (8)–(10) were taken into account in the CFD calculations. The continuity equation

$$
\nabla \left( \rho \stackrel{\rightarrow}{v} \right) = 0,\tag{8}
$$

and the momentum conservation equation

$$\nabla \left( \rho \vec{v} \,\stackrel{\rightarrow}{v} \vec{v} \right) = -\nabla p + \nabla \cdot \stackrel{\rightarrow}{\tau}\_{\prime} \tag{9}$$

where <sup>→</sup> *τ* is the stress tensor, was analyzed. The energy conservation equation [39] was considered

$$\nabla \left( \stackrel{\rightarrow}{\upsilon} (\rho E\_t + p) \right) = \nabla \left( k\_{eff} \nabla T + \left( \stackrel{\rightarrow}{\tau}\_{eff} \cdot \stackrel{\rightarrow}{\upsilon} \right) \right) \tag{10}$$

where *Et* is the total energy, keff is the effective thermal conductivity, <sup>→</sup> *τ eff* is the deviatoric stress tensor.

The next part of the paper deals with the method of selection of the grid for CFD calculations according to [40]. The grid convergence method was used, which is based on the Richardson extrapolation method. In order to select the mesh, a representative mesh size *h* was used

$$h = \left[\frac{1}{N} \sum\_{i=1}^{N} (\Delta A\_i) \right]^{1/2}. \tag{11}$$

where Δ*Ai* is area of the *i*-th cell, and *N* is total number of cells. Three significantly different mesh sets were selected for the analysis. In order to determine the mass flux change calculations were performed for the boundary conditions given in the Table 1 (*pin* = <sup>2</sup>·10<sup>5</sup> *Pa*). Let *<sup>h</sup>*<sup>1</sup> < *<sup>h</sup>*<sup>2</sup> < *<sup>h</sup>*<sup>3</sup> and *<sup>r</sup>*<sup>21</sup> = *<sup>h</sup>*2/*h*1, *<sup>r</sup>*<sup>32</sup> = *<sup>h</sup>*3/*h*<sup>2</sup> then calculate the apparent order *p* of the method using the expression

$$p = \frac{1}{\ln(r\_{21})} |\ln|\varepsilon\_{32}/\varepsilon\_{21}| + q(p)|\tag{12a}$$

$$q(p) = \ln\left(\frac{r\_{21}^p - s}{r\_{32}^p - s}\right) \tag{12b}$$

$$s = 1 \cdot \text{sgn}(\varepsilon\_{32}/\varepsilon\_{21}) \tag{12c}$$

where *<sup>ε</sup>*<sup>21</sup> <sup>=</sup> . *<sup>m</sup>*<sup>2</sup> <sup>−</sup> . *<sup>m</sup>*<sup>1</sup> and *<sup>ε</sup>*<sup>32</sup> <sup>=</sup> . *<sup>m</sup>*<sup>3</sup> <sup>−</sup> . *m*2.

Extrapolated mass flow was calculated from expression

$$
\dot{m}\_{ext}^{21} = \left(r\_{21}^p \dot{m}\_1 - \dot{m}\_2\right) / \left(r\_{21}^p - 1\right). \tag{13}
$$

Then the following parameters were calculated and reported, along with the apparent order *p*:


$$e\_a^{21} = \left(\dot{m}\_1 - \dot{m}\_2\right) / \dot{m}\_{1\prime} \tag{14}$$


$$e\_{\rm ext}^{21} = \left(\dot{m}\_{\rm ext}^{21} - \dot{m}\_1\right) / \dot{m}\_{\rm ext}^{21} \tag{15}$$


$$\text{GCI}\_{fine}^{21} = \frac{1.25e\_a^{21}}{\left(r\_{21}^p - 1\right)}\,. \tag{16}$$

The analysis was performed for the initial geometry with dimensions given in point 3 and exact geometry presented in Figure 10a (t = 8, LP = 4 mm, RC = 0.315 mm). The results are summarized in Table 2.

**Table 2.** Parameters used in the grid convergence method [40].


According to Table 2, the obtained numerical uncertainty for the fine grid solution N1 is 0.86%. This is a satisfactory value. The grid with the number of approximately N1 = 354,000 elements was used for further analysis. In boundary layers, approx. 20 grid cells were assumed. Illustrative grid taken for calculations is shown in Figure 7. The *k-ω* SST turbulence model [41] was included in calculations. To obtain the grid of appropriate quality for the *k-ω* SST model, the condition that y<sup>+</sup> < 2 in the boundary layer was assumed.

**Figure 7.** Illustrative grid used for calculations.

Stationary calculations were performed using the pressure-based coupled solver. Convergence tolerance of 4 × <sup>10</sup>−<sup>6</sup> was assumed for calculations. To obtain the required convergence criterion, for each case approx. 150 iterations of calculations were made.

For the model research presented in this paper, CFD simulations were performed for similar grid parameters and the same solver settings as those described in papers [15,24]. The same working medium and similar boundary conditions as in [15,24] were applied. Relative mass-flux error between the values obtained experimentally and from CFD calculations for the segment of the straight-through seal [15] was within the range from 0.3 to 0.7%.

This paper presents initial analysis of the new design method. It was decided to perform two-dimensional calculations due to reduced calculation time for many geometries and various boundary conditions. Data shown in the paper [42] indicate that the rotational speed has a little impact on the gas leakage value. Therefore, the impact of the rotational speed was not included in the general CFD analysis.
