**2. Method of Aerodynamic Design**

The leakage rate in sealing is strongly affected by the gas expansion in clearances and by the gas kinetic energy dissipation in chambers. Gas flowing through subsequent clearances expands, which results in the density reduction. The relation between the clearance height and the pressure distribution in the straight-through seal was presented in the paper [34]. In each subsequent clearance the gas velocity is higher and higher (Figure 1). Widely used labyrinth seals are characterized by constant chamber size.

**Figure 1.** Example vectors of velocity fields in a representative segment for *pin/pout* = 2 and *Tin* = 300 K, *pout* = 105 Pa.

In the seal chambers of constant dimensions one can observe the unequal degree of gas kinetic energy dissipation. The measure of the gas kinetic energy dissipation in chambers is the kinetic energy carry-over coefficient, which was analyzed in the paper [35]. The phenomenon of the gas kinetic energy carry-over has a great impact on the leakage rate [36–38]. The idea of a novel method of aerodynamic design is to adjust the seal geometry to flow conditions so that the gas kinetic energy dissipation as even as possible in the seal chambers is obtained.

The novel design method is discussed based on the geometry of a staggered labyrinth seal (Figure 2) of the outer diameter *D*, segment height *H*, radial clearance *RC*, segment length *LS*, pitch *LP*, and tooth thickness *B*.

**Figure 2.** Illustrative geometry of a staggered labyrinth seal.

The design method presented in this paper is based on CFD calculations pertaining to gas flow in the labyrinth seal.

Labyrinth seals have axisymmetric geometries, therefore the flow can be described along the axis *X*. In the method presented herein, calculations were based on the local non-dimensional coordinate x of the seal length being parallel to the seal axis. The origin of the coordinate system is determined at the place being the beginning of the first seal tooth on the gas inflow side. The end of the non-dimensional coordinate system is located at the end of the segment (Figure 2). Hence,

$$\mathbf{x} = \frac{\mathbf{X}}{LS}.\tag{1}$$

The method consists in the analysis of the kinetic energy *E*(*x*) distribution in the axial direction (the main flow) within the seal area. The gas kinetic energy in the axial direction was described by the dependency

$$E(\mathbf{x}) = \frac{u^2}{2},\tag{2}$$

where *u* denotes the gas velocity in the axial direction. Non-dimensional kinetic energy was defined as follows:

$$e = \frac{E(\mathbf{x}) - E\_{\rm min}}{E\_{\rm max} - E\_{\rm min}} \text{.} \tag{3}$$

where *Emin* and *Emax* denote the minimal and maximal gas kinetic energy, respectively, in the axial direction in the seal. For calculations it was assumed that *Emin* = 0.

Illustrative fields of non-dimensional gas kinetic energy *e* are shown in Figure 3. To improve the readability of the Figure 3, the scale was set within the range from 0 to 0.6.

**Figure 3.** Example distribution of non-dimensional kinetic energy e in the axial direction in the staggered seal for *pin/pout* = 2, *Tin* = 300 K, *pout* = 105 Pa.

The initial stage of the method of adjusting the seal geometry to flow conditions is searching for local maximal non-dimensional values of the gas kinetic energy in the axial direction. In staggered labyrinth seals local maxima of the kinetic energy occur in the area of clearances and in seal chambers just behind clearances (Figure 3).

Illustrative distribution of local maxima of non-dimensional kinetic energy *e*max(*i*) occurring in the staggered seal with the approximation function *e*(*x*) is shown in Figure 4.

**Figure 4.** Local maxima of non-dimensional gas kinetic energy *emax*(*i*) in the staggered seal with the approximating line *e*(*x*); discontinuous green line shows the location of tooth sidewalls on the inflow side (the distance between them is *LP*).

In non-dimensional coordinates, the field below the approximating line, hereinafter designated as *e*(*x*), (Figure 4), is divided into n areas with beginnings and ends included between points (*xi*, *xi*+1) for *i* = 1, 2, ... , *n* + 1. The number n of areas is equal to the number of chambers in the seal segment, and their length corresponds with the length of pitch *xi+*<sup>1</sup> − *xi* = *LP* of the initial geometry.

The area below the approximating curve can be described as follows:

$$A\_{\mathfrak{E}} = \int\_0^1 e(\mathfrak{x})d\mathfrak{x} = \sum\_{i=1}^n A\_{\mathfrak{E}}(i),\tag{4}$$

where fields of respective areas are as follows:

$$A\_{\varepsilon}(i) = \int\_{\chi\_{i}}^{\chi\_{i+1}} e(\mathbf{x})d\mathbf{x}, \text{ where } i = 1, 2, \dots, n. \tag{5}$$

Fields below the curve *e*(*x*) of respective areas are proportional by weight to the length of respective pitches of the new geometry.

The length of respective pitches of the improved geometry results from the following equation:

$$LP(i) = LS \frac{A\_{\mathcal{E}}(i)}{A\_{\mathcal{E}}}.\tag{6}$$

As a result of the applied method, a variable pitch length of the seal adapted to the flow conditions was obtained. Depending on the possibility of changing the inner diameter and the external geometry of the seal, two variants of the method were specified.
