2.3.2. CFD Based Correction of Penstock Geometrical Factor

The *NUMECA*/*Hexpress* commercial software [31] was used for generating the computational grid representing the penstock geometry (Figure 5). The unstructured grids consisted of hexahedral elements.

**Figure 5.** Geometry of hydraulic system (calculation domain): head water reservoir (hydraulic diameter of virtual half-cylindrical inlet 30 m) → square pipeline (4.3 × 4.3 m) → cylindrical pipe (4.3 m) → conical pipe (4.3/3.9 m) → cylindrical pipe (3.9 m) → conical pipe (3.9/3.6 m) → cylindrical pipe (3.6 m) → conical pipe (3.6/3.2 m) → cylindrical pipe (3.2 m) → pipe branch for two pump-turbines (2.276 m) → conical pipe (2.276/1.654 m) → outlet cylindrical pipe (1.654 m)).

For flow calculations, *ANSYS*/*Fluent* commercial software was used [32]. The flow was simulated by solving the steady-state Reynolds Average Navier-Stokes (RANS) equations with the *k-*ω *SST* turbulence model. Many studies demonstrate the great usefulness of this turbulence model in the calculation of industrial flow systems [33,34]. It's commonly known that the *k*-ω *SST* model integrates advantages of both *k*- turbulence model and standard *k*-ω turbulence model [35].

The second-order upwind discretization was used with the SIMPLE scheme of pressure-velocity coupling. Non-dimensional distance from wall *Y*<sup>+</sup> was assumed to be in range 1 to 5 according to the used turbulence model. Initialization of calculation was done from all zones limiting the computational domain. The calculations were conducted until all of the residuals (continuity residual, velocity components, turbulent kinetic energy, and specific rate of dissipation) reached values less than 0.001. The parameters for a closure of turbulence model were hydraulic diameter and turbulence intensity. First of them was calculated using formula: *Dh* = 4*A*/*P* [m], in which *A* is the area and *P* is the perimeter (hydraulic diameter was 1.654 m at inlet/outlet of lower part the penstock and 30 m at inlet/outlet of upper part of the penstock). The second parameter was calculated using the formula [32]: *I* = 0.16 *Re*(−1/8) in which *Re* is Reynolds number at inlet or outlet cross-section. At the outlet of the measuring section, constant static pressure was assumed for all calculation cases. The free surface of the reservoir was assumed as a no-slip boundary condition.

The CFD calculations were conducted for four discharge values (20, 25, 30, and 35 m3/s) in the turbine operation modes and for two discharge values (26 and 28 m3/s) in the pump operation modes. The sample of calculation results in the form of water velocity distributions in cross-sections for three chosen flow parts of the penstock were presented in Figures 6–8 for both flow directions, for analyzed discharge of 35 m3/s in turbine regime, and 28 m3/s in pump regime.

**Figure 6.** The water velocity contours in the penstock inlet part with first elbow for discharge of *Q* = 35 m3/s in turbine regime (left view) and for discharge of *Q* = 28 m3/s in pump regime (right view).

**Figure 7.** The water velocity contours in the penstock part containing the cone pipe for discharge of *Q* = 35 m3/s in turbine regime (left view) and for discharge of *Q* = 28 m3/s in pump regime (right view).

**Figure 8.** The water velocity contours in the penstock part containing the pipe branch for discharge of *Q* = 35 m3/s in turbine regime (left view) and for discharge of *Q* = 28 m3/s in pump regime (right view).

The CFD simulation results received for the analyzed penstock flow parts (Figures 6–8) can be characterized as follows:


The CFD results taking account flow irregularities induced in the penstock were used to calculate the equivalent factor *Fe* according to the original procedure presented in Appendix A.

The deviation factor, Δ*f*, representing a relative difference between the equivalent penstock factor, *Fe*, (obtained using CFD calculations) and the penstock geometrical factor, *F,* was included in discharge determination according to the pressure-time method. This factor is calculated as follows:

$$
\Delta f = \frac{F\_\varepsilon - F}{F} \tag{7}
$$

The values of quantity, Δ*f*, determined for chosen discharge values for both flow directions are presented in Table 1. It can be stated that Δ*f* is kept almost constant for both flow directions separately. However, it presents different level for both turbine and pump operational modes: the average value of Δ*f* is about +0.13% and about +0.77% for turbine and pump modes of operation, respectively. These values were used as correction quantities of the geometrical factor *F* calculated based only on the geometry of the entire penstock.


**Table 1.** The relative differences of *F*-factor, Δ*f*, determined for the entire penstock for the assumed discharge values in the both machine operation modes.
