Erosion Wear Evaluation Using Nektar ++

Erosion wear is the loss of mass on the surface of a solid body in contact with a moving fluid. When the fluid under study has solid particulate material or sediments, the erosion could be significant (Kulu and Kleis, Solid particle erosion: occurrence, prediction and control. Springer, London, 2008; Karimi and Schmid, Wear 156(1):33–47, 1992). Several studies (Karimi and Schmid, Wear 156(1):33–47, 1992; Neopane and Cervantes, Global J Res Eng Mech Mech Eng 11:17–26, 2011; Zeng et al., Proc IMechE 231(3):182–196, 2017; Padhy and Saini, Energy 39(1):286–293, 2012; Bajracharya et al., Wear 264(3–4):177–184, 2008; Kumar and Saini, Renew Sust Energ Rev 14(1):374–383, 2010; Xiao et al., J Hydrodyn B 19(3):356–364, 2007; Zhu et al., Nucl Eng Des 273:396–411, 2014; Chongji et al., IOP Conf Ser Earth Environ Sci 22(5):052019, 2014; Finnie and Kabil, Wear 8(1):60–69, 1965; Finnie, Wear 3(2):87–103, 1960; Finnie, Wear 19(1): 81–90, 1972) have been developed to investigate this phenomenon. Nevertheless, few attention has been paid to influence of small eddies during the erosion process, limiting the accuracy of the results in the simulation. The use of spectral element methods can allow increasing accuracy in the simulation due to the potential to consider these smaller scales.To the best of the authors’ knowledge, there is no work that uses high order methods to evaluate erosion wear rate. This research aim is to assess the impact of higher resolution methods on the prediction of erosion wear rate and distribution. Nektar+ + is a framework to solve partial differential equations using spectral/hp element method (Cantwell et al., Comput Phys Commun 192:205–219, 2015). This work presents a methodology to model erosion wear in Nektar+ +, with the creation of two modules. The first one is for the particle tracking evaluation and the record of impact on the walls of interest. The second one implements the Hutchings and Finnie models (Finnie, Wear 3(2):87–103, 1960; Finnie, Wear 19(1):81–90, 1972; Padhy and Saini, Renew Sust Energ Rev 12(7):1974–1987, 2008; Dutta, Bulle-effect and its implications for morphodynamics of river diversions. Ph.D. Thesis. University of Illinois at Urbana-Champaign, 2017) to predict the material removal rate as a function of the velocity and the angle of impact of each particle, the particle diameter and density, and the hardness relationship between particles and the wall. At the end, the particles trajectories and erosion distribution on walls are obtained. To assess effectiveness of the methodology a Backward Facing Step was investigate and the erosion patterns were found.


Introduction
Wear is a common phenomenon on many machines and devices, it is characterised by the removal or loss of material. Erosion wear is a particular wear process which occurs when solid particles or droplets, carried by a fluid (liquid or gas), impact on a solid surface [1]. Turbomachinery such as pumps, turbines and pipe accessories (i.e. tees, elbows, nozzles, valves), are examples of elements affected by the erosion wear, decreasing the performance and the lifetime. In many industrial sectors e.g. energy and mining, and oil & gas; massive amounts of resources are used for maintenance and replacement of affected parts [2][3][4]. Despite this phenomenon have been broadly investigated [5][6][7][8][9][10][11][12][13][14] there are still unsolved challenges in establishing the influence of small eddies during the erosion process leading to modest accuracy levels in the simulation results. Due to the microscopic nature of erosion, the smallest scales in the flow play a fundamental role in the complete process. One of the aspects which has not been carefully studied in erosion wear modelling is the effect that the smaller eddies and secondary flows have on the particles interactions with the surface. In general, these secondary flows could not be represented using linear Reynolds Average Navier Stokes (RANS) simulations, this is mainly because the Reynolds stress imbalance is neglected and the secondary flow does not develop. As was mention by Gross and Fasel [15], predictions of the secondary flow require non-linear Reynolds stress, full Reynolds-stress models, Large Eddy Simulations (LES) or Direct Numerical Simulation (DNS). Due to their relatively low computational cost, RANS models often used to predict on erosion using CFD in industrial simulations. The inclusion of smaller eddies and secondary flows in the simulation could be a major breakthrough in the modelling of erosion process. In order to capture in an accurate way the physics related with the small eddies and secondary flows, a numerical technique capable to represent those processes, is needed. As emphasised by Jacobs [16], the use of spectral methods could allow increased accuracy in the simulation due to the potential to simulate a wider range of scales. With this in mind, the purpose of this work is to assess the impact of higher resolution methods on the prediction of erosion wear rate and distribution.

Spectral Methods
Several numerical techniques are used to solve Navier Stokes (NS) equations. Some of them are finite differences, finite volumes and finite elements. Nevertheless, when high accuracy is required the use of a lot of elements is needed in the modelling, which significantly increase the computational cost [17,18]. Hence novel methods are subject of research to offer a better rate accuracy and computational cost.
Among novel numerical methods considered nowadays are spectral methods, which have shown to be a powerful tool with high level of accuracy for solving large problems in computational fluid dynamics (CFD), according to the available literature, especially in the studies developed by Boyd [19], Canuto et al. [20][21][22], Trefethen [23,24] and Sherwin [25,26]. Nektar++ is an open-source software framework designed to support the development of high-performance scalable solvers for partial differential equations using the spectral/hp element method [27]. High Order CFD methods have been receiving considerable attention in the past two decades. Traditional CFD software could be replaced by high order code in many applications in few years [28].

Particles Tracking
To the best of the authors' knowledge, there is no work that uses high order methods to evaluate erosion wear rate. This research aims to assess the impact of higher resolution methods on the prediction of erosion wear rate and distribution. It comprises the solution of fluid flow using incompressible NS solver with implicit LES modelling, the implementation of a Lagrangian particle tracking model and the later data processing through traditional erosion rate models but using the available high order information. This could allow the evaluation of traditional rate models with more spatial resolution and accuracy.
The Lagrangian particle tracking model is based on one-way coupling approach, that is the most simple case when just the iteration between the fluid and each particle is taking into account in just one way. That means that the particles are moved by the fluid but the fluid flow is not perturbed by the particles. Moreover, the effects of the collision between particles are also neglected. The one-way coupling model is valid for volume concentrations of particles lower than 10 −6 [29,30]. The problem of predicting particle motion in a fluid flow can be predicted by solving an evolution equation in time: where v p and x p are the particle velocity and position and F is a function of the velocity of the fluid u, the density of the fluid ρ and particle density ρ p , among others.
To start, it is necessary to obtain the velocity on a certain point from the eulerian velocity field. This process consists of finding the element containing the particle and interpolating the velocity with the element information. In a higher-order velocity element field, the use of linear interpolation is inaccurate and could vanish the advantage won with the use of high order methods. On the other hand, using high order interpolation could be computationally expensive. Therefore, special attention to this procedure is required [16,31,32].

Erosion Wear Evaluation
Once the information about the collisions is complete, the erosion wear model is used to predict the pattern of material removed. The general erosion equation, based on the work of Finnie [33][34][35][36][37][38] can be presented as W is the erosion rate or material removed by collision, k is a wall material dependent constant, F s is the particle geometric factor, V p and n are the collision velocity and the velocity exponent, and f (θ) is a function of collision angle. Several authors define these values for different materials configurations and test cases. Three of the most used models, which include experimental results are the jet impingement test [39][40][41][42][43][44][45], elbow erosion [46][47][48][49], and the works of the Wong et al. [4,46,50,51].

Implementation
This section describes the implementation of erosion wear in Nektar++. To achieve this objective is important to have in mind the partition of the problems into two parts. The first one is the particle tracking as a filter within the Nektar++ incompressible Navier Stokes solver. A filter in Nektar++ is a module for calculating a variety of useful quantities from the field variables as the solution evolves in time [27].
The second one is implemented as a FieldConvert module to evaluate the erosion of each collision and generate the fields on the boundaries walls. FieldConvert is a utility embedded in Nektar++ with the primary aim of allowing the user to work with the Nektar++ output files, some of the modules within FieldConvert allow the user to postprocess the output data [27].

Particles Tracking
The first step was the implementation of a ODE time solver. Several options are available, but having into account the discrete time flow fields calculated with the Navier Stokes incompressible solver, and to avoid the use of temporal interpolation, the selected option was the Adams-Bashforth (AB) and Adams-Moulton (AM) schemes.
The implementation was tested with a benchmark case presented in [31]. In this model, the particle velocity is the fluid velocity at certain point and the evolution equation is reduced to one equation; Eq. 1 is reduced to: To solve this system a Time-Marching Method was implemented, meaning that the future values are evaluated using the present and past values of the variables. Explicit AB and Implicit AM methods were implemented using first to fourth integration order. The error values obtaining using AB and AM with different order presents features from this kind of methods. The next step was the implementation of the solid particles. In this case, the momentum equation is evaluated on each particle, resulting:  (7) where F d is the drag force, Re p is the Reynolds Number based on the diameter of the particle, g is the gravity acceleration, and C d is the drag force evaluated on each particle. Figure 1 shows a diagram of the evolution equation. Current position, velocities and forces are evaluated to get the future positions (BP, OP) until the next position is located outside of the domain (NP). When this happens, the evolution algorithm stop and a function is used to evaluate the collision point (CP) and the position after of collision (NP') using the high order information about the walls.

Erosion Wear Evaluation
Erosion rate per collision (Eq. 2) has to be integrated over each element of the eroded surface. For each particle collision, more material is removed from the surface, the elemental erosion rate has to take into account this cumulative effect over the surface.
As mentioned before, the set of parameters used in this work, has been based on experimental data. One of the most used parameter set is the one proposed by Erosion group of the University of Tulsa [38,48,52]. The erosion rate takes the form of Eq. 2, F s = 1 for sharp (angular), 0.53 for semi-rounded, or 0.2 for fully rounded sand particles. V p is the impact velocity and n = 1.73. The angle function has the form:

Test Case
To test the new feature in Nektar++, a Backward Facing Step (BFS) model was developed based on the experimental setup of [30,32,54] showed in Fig. 2. In the model developed in this work, the simulations were done with the addition of gravitational effects on the −y direction. In original experiments the air at the inlet is a well development turbulent flow (ū = 10.5 m/s), this is used a inlet condition and, to complete the model, a zero pressure condition at the output. The additional boundaries were set as walls. A zero velocity field was set as initial condition. The particles used have a 70 µm diameter and 8808 kg/m 3 density. Figure 3 (top) shows a snapshot of the velocity field when the statistically stationary regime is reached (t = 8 s), next the particles are released and were convected by the flow. Particular trajectories are shown in grey lines in Fig. 3 (bottom). In the same figure, results of the particle collision with the walls, computed with Eq. 2, are also shown.
From the results presented Fig. 3, the typical BFS velocity profiles can be recognised. It is important to note the details behind the step, the main flow originates the secondary eddies and defines the limit of the recirculation zone (x/H = 7 from the step) where backflow occurs. Additionally, interesting details appear in between each main velocity flow ripple and the walls along the x-direction. It is noteworthy that particle tracking is evaluated using a steady velocity field, therefore the existence of several irregularities is expected, for instance, particle trajectories inside recirculation zone. Erosion rate depends on the number of collisions at specific points. It is a localised phenomenon that does not occur continuously in the domain. Its distribution shows a strong dependence on the flow dynamics.

Conclusion and Future Work
This work presented a method developed to asses the erosion wear rate using a high-order (spectral) element based technique on a modified test case implemented in Nektar++. The methodology proposed in this study have a potential to increase the accuracy when solving this kind of problems. Future research activities are going to be focused on the determination of accuracy improvements and optimisation of the proposed methodology. Several more cases have to be tested to produce solid conclusions about the implemented methodology, as well as a detailed comparison with experimental test cases.
Despite the methodology implemented had several important simplifications, as the use of one-way coupling and the few forces taken into account, allowed quicker implementations and results. This work would be an interesting starting to implement this kind of simulations using Nektar++. However, to run more realistic cases, additional research efforts are required for the implementation of two-way and four-way coupling and the effects of other forces over each particle.
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