Flow simulation of a Pelton bucket using finite volume particle method

The objective of the present paper is to perform an accurate numerical simulation of the high-speed water jet impinging on a Pelton bucket. To reach this goal, the Finite Volume Particle Method (FVPM) is used to discretize the governing equations. FVPM is an arbitrary Lagrangian-Eulerian method, which combines attractive features of Smoothed Particle Hydrodynamics and conventional mesh-based Finite Volume Method. This method is able to satisfy free surface and no-slip wall boundary conditions precisely. The fluid flow is assumed weakly compressible and the wall boundary is represented by one layer of particles located on the bucket surface. In the present study, the simulations of the flow in a stationary bucket are investigated for three different impinging angles: 72°, 90° and 108°. The particles resolution is first validated by a convergence study. Then, the FVPM results are validated with available experimental data and conventional grid-based Volume Of Fluid simulations. It is shown that the wall pressure field is in good agreement with the experimental and numerical data. Finally, the torque evolution and water sheet location are presented for a simulation of five rotating Pelton buckets.


Introduction
The deviation of a high-speed water jet by the Pelton buckets is a challenging fluid mechanics problem, which involves complex geometries, moving boundaries, free surface flows and highpressure variations. The ability to simulate accurately the wall pressure field in the Pelton buckets is a key issue for the design of Pelton runners. Mack and Moser [1] and Perrig et al. [2] used grid-based Volume Of Fluid (VOF) computations to simulate the flow in a Pelton bucket. Later, Marongiu et al. [3] used Smooth Particle Hydrodynamics (SPH) with Riemann solver to benefit from its Lagrangian formulation and overcome the mesh difficulties of VOF. The drawback of particle-based simulation compared to grid-based simulation is the significant increase of computing time. Recently, Anagnostopoulos and Papantonis [4] proposed a fast Lagrangian computation to design Pelton buckets. However, this method is only based on the inlet and outlet velocity vector of the particles, which provides an estimation of the integrated pressure. Neither the whole pressure field nor the exact water sheet location can be accurately computed.
The Finite Volume Particle Method (FVPM) is a particle-based solver introduced by Hietel [5]. This method features an Arbitrary Lagrangian-Eulerian (ALE) formulation, which means that the computing nodes can either moves with the material velocity or a user-prescribed velocity. Therefore, FVPM  kernel and a smoothing length to compute the interactions between the particles. Here, we use a method based on the work of Quinlan and Nestor [6] to compute the particle interaction vectors exactly. Like FVM, the interaction vectors are used to weight the fluxes exchanged between the particles. Moreover, FVPM is locally conservative, which enables to perform accurate simulations with variable smoothing length. A variable smoothing length simulation is required to allow an efficient simulation of complex multi-physics phenomenon such as elasto-plastic solid simulation [7] and interactions between fluid and silt particles [8].
The purpose of the present paper is to validate that FVPM simulations are able to capture accurately the deviation of a high-speed water jet by a Pelton bucket. The FVPM simulations are done with the software SPHEROS, developed at EPFL since 2010 [9,10]. The FVPM simulations of the flow in a stationary bucket are validated by VOF numerical simulations and experimental data obtained by Kvicinsky et al. [11]. The wall pressure field in the stationary bucket inner surface is used to compare the FVPM results with pressure sensors and VOF results.
In the following sections, we first introduce the governing equations, FVPM discretization and top-hat kernel. Then, we present the case study including the numerical setup. Finally, we compare the numerical simulations to experimental data and grid-based simulations of the flow in a stationary bucket for three different impinging angles and present the FVPM result of the flow in five rotating buckets.

Governing equations
The water flow is assumed weakly compressible. The flow motion is governed by the mass and linear momentum conservation equations where ρ is the density, C is the velocity vector, g is the gravity vector and σ = s − pI is the stress tensor, which includes s the deviatoric stress contribution and p the static pressure. The latter is computed from the state equation where ρ • is the reference density and a is the sound speed. According to the weakly compressible assumption, the sound speed is set to 10 · C max , C max being the discharge velocity of the water jet. The governing equations (1) can be written as the Partial Differential Equation (PDE) where U = {ρ, ρC} represents the conserved variables and F = Q + P − G is the flux function, which is decomposed in Q = {ρC, ρCC}, P = {0, pI} and G = {0, s}.
In FVPM, the Sheppard interpolating or shape function ψ is used to discretized the conservative form of the PDE where Ω represents the whole computational domain and dV an element of volume. The Sheppard function is zero-order consistent and is defined as is the kernel summation. The spatial resolution of the interpolation is given by the smoothing length h. After some mathematical operations [12], the PDE is simplified as whereẋ is the particle velocity. In order to damp the spurious numerical oscillations, Q ij , U ij and P ij are computed using the AUSM + scheme of Liou [13] and a correction term is applied to the mass flux as described in [12]. The expression of the deviatoric stress G ij is given by where µ is the dynamic viscosity, C ij the averaged velocity and∇ the gradient operator obtained from weighted least square to avoid double summation of gradient operator [12]. In equation (6), ∆ ij represents a weight vector which depends on the interaction vector between particles i and j In the present study, we use a rectangular top-hat kernel to compute the interaction vectors, which reads A 2D example of particles interactions with rectangular support is given in figure 1(a). The tophat kernel is less smooth than a bell-shaped kernel as shown by the contours of the Sheppard shape function given in figure 1(b). However, Quinlan and Nestor [6] demonstrated that tophat kernel allows a fast and exact computation of the interaction vector in 2D with a circular support. In 3D, Jahanbakhsh et al. [7] showed that the use of top-hat kernel with a rectangular support reduces significantly the geometric computations, in order to compute the integral of eq. (9). The latter is simplified as where m is the number of partitioned rectangles, ∆S represents the surface vector of the partitions, σ − and σ + are the summation kernel inside and outside the surfaces respectively. An outline of the 2D computation of eq. (11) is given in figure 1(c), where the rectangular partitions are simplified as lines segments. FVPM is an ALE method, which means thatẋ i , the velocity of the particle i, could be prescribed arbitrarily. In the present study, we set the particle velocity equal to the flow velocity plus a correction vectorẋ where Ω * ij represents the interaction of bisected volume between the kernels of particle i and j. The correction vector is applied to ensure a uniform distribution of particles in the flow and avoid particles clustering. The no-slip wall boundary condition is imposed by setting one layer of fluid particles on the wall surface and fixing their velocities C andẋ equal to the wall velocity. The free-surface boundary condition is given by the particles location. However, the velocity correction applied in eq. (12) is modified to avoid an artificial spreading of particles through the interface. The time integration is performed using a second-order explicit Runge-Kuta scheme and the time step is computed by

Case study
In the present study, a high-speed water jet impinges on: first one stationary bucket and second five rotating buckets. An outline of the case study is given in figure 2.  Figure 2. Outline of the case study. The bucket can tilt of an angle θ around the Z axis. The inlet of the water jet has a diameter D 2 and its axis is in the −X direction at a distance Y = −D 1 /2. The 32 pressure samples are located on the bucket inner surface, their location fits the position of lines X1 to X7 and Z1 to Z5. The bucket geometry and the 32 pressure samples are taken from Kvicinsky et al. [11]. The bucket width is B 2 = 0.09 m and its reference diameter is D 1 = 0.315 m. The location of the pressure samples fits the position of lines X1 to X7 and Z1 to Z5. In the stationary analysis, the bucket is tilted of an angle θ = 72 • , 90 • or 108 • around the Z axis. In the rotating analysis, five buckets rotate around the Z axis with a rotation speed N = 1280 rpm. The five buckets are spaced with an angle of 18 • . The jet diameter is D 2 = 0.03 m and its axis is in the −X direction at a distance Y = −D 1 /2. The discharge velocity of the water jet is C max = 28.5 m s −1 and C max = 38.056 m s −1 for the stationary and rotating analysis respectively.
The FVPM simulations are run on two Intel Xeon CPUs E5 2670 at 2.6 GHz with 32 cores (hyper-threading 2 × 16) and 32 Gb of memory. The domain decomposition is 4 × 4 × 4 and the domains size is adapted according to the particles load using the adaptive domain decomposition strategy from Vessaz et al. [14].

stationary bucket
The influence of the reference particle spacing X ref on the FVPM results is analyzed for an impinging angle of θ = 90 • . Figure 3 shows the time history of F/F * , where F is the magnitude of the force applied on the bucket and F * = 2ρπ(D 2 /2) 2 C 2 max is the maximum force of the water jet. A mean value of the converged force, the computing time as well as the number of particles at the end of the simulation are given in table 1.    According to these results, the convergence of the FVPM results with the refinement of the spatial discretization is highlighted. However, the computing cost increases significantly with the spatial discretization. Figure 4 shows a free surface reconstruction of the water sheet for a discretization of D 2 /X ref = 30. The free surface location is less influenced by the spatial discretization compared to the force or pressure measurements.
The pressure coefficient C p = (p − p ref )/(0.5ρC 2 max ) is compared to the VOF computations and measurements from Kvicinsky et al. [11]. The averaged C p profile along the lines X1 to X7 and Z1 to Z5 are given in figures 5 and 6 respectively.  In these figures, the FVPM pressure profiles fit qualitatively well the VOF and measurements from Kvicinsky et al. [11] despite some quantitative differences. Moreover, the convergence of the FVPM results according to the spatial discretization is highlighted. Finally, we present in figure 7 the wall pressure field comparisons between the finest FVPM results of D 2 /X ref = 50 for the three different impinging angles θ = 72 • , θ = 90 • and θ = 108 • . Once again, the FVPM results for the three different impinging angles fit qualitatively well the VOF computations and measurements.

Rotating buckets
The operating point simulated corresponds to a discharge coefficient φ B 2 = 0.20 and an energy coefficient ψ 1 = 3.38. The simulated time corresponds to the rotation of the buckets of 230 • , which allows the passage of the five buckets through the high-speed water jet. The spatial discretization selected for the FVPM simulation is D 2 /X ref = 30 because it provides a good compromise between accuracy and computing time. This simulation of 224'150 wall particles and up to 279'152 fluid particles lasted 5 days on 64 cores. Figure 8 shows the evolution of torque in each bucket as well as the total torque applying on the five buckets in function of the angular position. Figure 9 shows the evolution of torque in one bucket. This averaged torque is obtained by averaging the torque in the buckets 2, 3 and 4. A free surface reconstruction of the water sheet is given in figure 10. These results are promising by comparing them to the experimental measurements of Perrig et al. [2]. However, further FVPM simulations have to be investigated in order to obtain a proper comparison.

Conclusion
The FVPM method with rectangular top-hat kernel allowed us to compute exactly and efficiently the particle interaction vectors. The convergence of the method according to the spatial discretization was highlighted in the stationary bucket analysis. Moreover, the FVPM results were qualitatively validated with the VOF computations and measurements from Kvicinsky et al. [11]. The FVPM results of the five rotating buckets were satisfactory. However, further simulations will be investigated to compare the FVPM simulation of a rotating Pelton runner to experimental data.  Figure 10. FVPM simulation of five rotating buckets: free surface reconstruction.