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Mesh handling for the CFD simulation of external gear pumps
CHAPTER 12
Mesh handling for the CFD simulation of external gear pumps Javier Martínez Politecnico di Milano, Dipartimento di Energia, Milano, Italy
Contents Introduction EGP working principle Numerical simulation of external gear pumps Theoretical and semiempirical methods 0D and 1D models Computational fluid dynamics Remeshing Mesh superposition Mesh morphing Discussion References
345 346 349 349 349 351 353 357 359 364 366
Introduction External gear pumps (EGP) play a major role in the framework of positive displacement turbomachinery (Ivantysyn & Ivantysynova, 2003). The low cost derived from a simple manufacturing, their robustness, high volumetric efficiency and wide pressure operation range (from low-pressure lubricating pumps to medium-high pressure open and close loop hydraulic circuits), make EGP a popular and reliable choice for a wide variety of applications. In fact, they are commonly found in the petrochemical industry for the pumping of high viscosity fluids, possible thanks to their rigid design, but their tight internal clearances make them also suitable for applications where precise flow control is required, i.e., metering applications of polymers, fuels or chemical additives. Despite their relatively simple manufacturing process, the operation of gear pumps involves complex phenomena both from the mechanical and Positive Displacement Machines https://doi.org/10.1016/B978-0-12-816998-8.00012-1
© 2019 Elsevier Inc. All rights reserved.
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from the fluid motion points of view. As a result, EGP have captured the attention of many researchers both in the experimental and numerical modelling fields. Available investigations focus on several aspects of these pumps, from inter-teeth pressure and forces distribution (Eaton, Keogh, & Edge, 2006) to the fluid leakage (Paltrinieri, Borghi, & Milani, 2004), noise production (Gutes, Gamez-Montero, Castilla, & Codina, 2000), cavitation (Ghazanfarian & Ghanbari, 2014) or the effect of the teeth geometry on the flow ripple (Manring & Kasaragadda, 2003). This chapter will focus on the numerical approach to the problem. While early models were based on semiempirical methods and one-dimensional models, with the advent of low-cost large-scale computing, the use of Computational Fluid Dynamics (CFD) has become a promising solution. In fact, CFD allows for a full three-dimensional description of the fluid region, extremely useful to understand the flow characteristics through the pump and to use such information to improve the efficiency and reduce noise and vibration produced by the flow and pressure pulsations in EGPs. As a general overview, this chapter will first give some basic details of the flow motion through EGPs to describe then the different simulation approaches available in literature, starting with some of the classical simplified 0D or 1D models, to move then into the difficulties that come up in the CFD approach, mainly due to the manipulation of the computational mesh.
EGP working principle External gear pumps belong to the so-called positive displacement pumps. As many other fluid power systems, their function is to supply hydraulic energy to the fluid, obtained from the mechanical energy introduced typically by means of a shaft. The working principle of EGP pumps is not different from any other volumetric displacement machine. An expanding cavity in the suction side brings the fluid towards the pump and drives it to the discharge side, where a collapsing cavity impulses the fluid out of the pump. As a result of the narrow clearances, the volumetric efficiency of these pumps is very high, leading to fluid flow through the pump almost independent of the discharge pressure. For external gear pumps, the reduction of the volume in the suction cavity is obtained by means of two generally identical gearwheels that mesh and rotate towards the suction chamber, as schematized in Fig. 1. On the discharge or pressure side, the fluid is squeezed out as the two gears rotate against each other. The power is externally introduced to the pump through a drive shaft, which is connected to the so-called “driving gear”. While the
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Fig. 1 Fluid motion phases in an EGP.
other is dragged by meshing with the former and is referred to as “driven gear”. Considering the small number of parts involved in the system, this simplicity leads to an increase in the reliability of this kind of pumps, as compared to other more complex designs. As a result, gear pumps can operate at high speeds and up to moderate pressure heads. Following the fluid motion shown in Fig. 1, the energy transformation in the pump occurs in three phases: • Suction: The motion of both gears generates an opening volume that is continuously filled by the fluid. Inlet pressure, typically atmospheric, pushes the liquid towards the vacuum created by the opening volume. • Displacement: Volumes of fluids are trapped in the region between gears and the casing and are transported towards the discharge side. • Impulsion: The volume containing the liquid is here reduced, forcing the fluid to move towards the outlet pump. As the fluid is pushed into the chambers and squeezed out in the pressure side, the pressure field continuously varies. The motion it experiments leads to an oscillatory behaviour generating the so-called flow and pressure ripple. The pressure distribution is of major importance. In fact, the minimum value determines the possible presence of cavitation. Furthermore, the pressure distribution on the teeth can be used to determine the chances of mechanical failure of the gears and pump casing, typically related to a fatigue process. In order to reduce the stress suffered by gears and casing, indentation of relief grooves is a common choice. In fact, decompression slots allow improving the fatigue life of the different elements of the pump. However, they
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also increase the liquid leakage, reducing the suction capability of the pump by decreasing its volumetric efficiency. Nonetheless, leakage is just one of the factors that can influence the volumetric efficiency of the pump (del Campo, 2012). • Leakage: As shown in Fig. 2, the fluid can find its way from the pressure side to the suction side through several paths, that lead to the definition of: ▪ Relief groove leakage: Fluid from the meshing inter-teeth volume that leaks through the volume trapped between contact points, due to the effect of the relief grooves. ▪ Axial leakage: Fluid passage between the gears and the side plates. ▪ Radial leakage: Movement of fluid through the teeth and the casing towards adjacent inter-teeth volumes. • Compressibility: In a real pump, the volumetric efficiency also depends on the fluid working conditions. • Cavitation: When cavitation occurs, volumetric efficiency can be drastically reduced.
Fig. 2 Fluid leakage: relief groove leakage (grey); axial leakage (dark grey); radial leakage (black).
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As it will be detailed in the next sections, different simulation approaches are more or less suitable for the correct prediction of volumetric efficiency, flow and pressure ripple, teeth pressure distribution. A complete description of all the phenomena involved is only possible through a 3D CFD simulation of the flow and therefore will be analyzed in detailed.
Numerical simulation of external gear pumps This section describes the state of the art for the available methods in the numerical study of EGP.
Theoretical and semiempirical methods Different degrees of accuracy can be obtained with simplified theoretical and semiempirical methods. To cite some examples, Manring and Kasaragadda (2003) studied the theoretical mass flow through external gear pumps neglecting leakages or cavitation effects. Simple techniques were however proposed to study cavitation phenomena too, including for instance the semiempirical methods presented by Myllykyl€a (1999), which focused on the suction capabilities of these pumps before cavitation occurs. Marginal suction conditions were also studied by Khalaf (1989), who paid attention to the effect on pump performance. However, most of the existing models try to replicate the pressure and flow pulsation occurring in these pumps, given the importance of this phenomena on the deterioration of the gearwheels, affecting the lifetime of the pump. Eaton et al. (2006) used an analytical model to predict the pressure ripple based on an equivalent hydraulic circuit, while Edge and Johnston (1990a, 1990b) worked on the estimation of the flow ripple from pressure ripple measurements. While theoretical and semiempirical methods can provide accurate prediction in some cases, their application is typically restricted to very particular designs and difficult to generalize without the appropriate calibration.
0D and 1D models The next degree of generalization consists in the use of more sophisticated 0D and 1D models. Generally speaking, 1D codes model gear pumps by considering the fluid region to be divided into control volumes, each of which represents the volume of the fluid trapped between two teeth and
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the casing, or between the teeth of both gears in the gearing region. Properties within the chamber are uniform and the evolution of these properties depend on the interaction with the adjoining chambers by variable orifices (Rundo, 2017). In 0D models, also known as lumped parameter models, each control volume is simulated by a capacitive element, which calculates the pressure as a function of the net ingoing flow rate. Volumes are connected by resistive components calculating the flow rate as a function of the pressure drop. 0D models can be further divided (see Fig. 3). In single-chamber models (Wang, Sakura, & Kasarekar, 2011), a control volume is associated to each volume chamber, whereas in multichamber models (Rundo & Corvaglia, 2016), all chambers are connected to the same volume and are lumped together, in order to obtain two main volumes: inlet and outlet. In 1D models, also known as distributed parameter models, the discretization continues further than in 0D models. Suction and delivery ducts for instance, which are simulated with a single capacitive component in 0D models, are here split into many volumes connected in series along the pump axis allowing, for instance, the simulation of pressure waves in the pipes. The simplification, despite the sometimes-extreme assumptions, has been successfully applied to the description of simple straight-cut gears (Casoli, Vacca, & Franzoni, 2005; Liping, Yan, Fanili, Jianjun, & Xianzhao, 2011; Mucchi, 2012). The main drawbacks of such methods are, among others: the difficult description of the geometrical volumes and throat areas as a function of the shaft angular position for a general realistic profile, and the lack of a detail description of the pressure distribution in the teeth. Besides, the consideration of three-dimensional effects and the extension to helical gears becomes critical.
(A)
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Fig. 3 0D models: single chamber (left and centre); multichamber (right).
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Computational fluid dynamics CFD is probably the only approach that allows us to treat the problem in its full complexity. A detailed CFD simulation can take into account not only the 3-dimensionality of the problem but also the fluid compressibility, turbulence, cavitation and the fluid–structure interaction (FSI), where even a two-way coupling with the gear-wheels vibration can be considered. In CFD the fluid region (and the solid too for the cases of FSI) is divided into discrete cells (computational mesh) where the governing equations are solved (Ferziger & Peric, 2002). Depending on the approach to solve these equations and the discretization methods, several options are common nowadays, including among others: spectral-element method (SEM), finite element method (FEM), finite volume method (FVM) and finite-difference method (FDM). Finite volume methods are the preferred choice for most commercial codes dealing with the solution of the fluid motion conservation equations (Navier– Stokes NS equations). FVM is robust, simple and a natural choice when the partial differential equations to be solved are conservation laws. FVM is based on the application of the integral form of the NS equations on each of the small control volumes defined by the computational mesh. The critical aspect when CFD is used to simulate EGP lies in the meshing process. As the gears rotate the fluid domain changes and the computational mesh should adapt to the gear motion. In order to deal with a mesh that is continuously changing, most methods rely on the so-called Arbitrary Lagrangian–Eulerian (ALE) formulation (Donea, Huerta, Ponthot, & Rodrı´guez-Ferra´n, 2004), of extensive applicability in the literature (Castilla, Wojciechowsky, Gamez-Montero, Vernet, & Codina, 2008; Houzeaux & Codina, 2007; Strasser, 2006; Wilson, 1948). In the usual Eulerian method, equations, including convective terms, are solved in a steady mesh. With pure Lagrangian methods, the mesh moves with the fluid particles and the convective term vanishes. ALE approach tries to adopt an intermediate solution, the mesh can be independently moved, leading to additional terms when the NS equations are solved. Nonetheless, the generation and motion of the computational mesh are not straight forward. The accuracy and convergence of the solution depend primarily on the quality of the computational mesh. Several parameters are typically used to determine the quality of a mesh to be used in an FV CFD code, including for instance: nonorthogonality, skewness, aspect ratio and growth ratio. Nonorthogonality: For every face of a computational cell (see Fig. 4), refers to the angle γ between the surface normal Sf and the line that joins the
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P o Sf
Q
no g
Fig. 4 Definition of nonorthogonality.
two centroids sharing the face PQ. Depending on the method used for the solution of NS equations, the computation of surface gradients at the cell boundaries might be required. For instance, in the calculation of diffusion terms in a collocated grid for FV methods, the required surface gradients are typically decomposed in an orthogonal component o, taken implicitly, and an explicit correction for the nonorthogonal component no, that is sometimes limited since it can lead to instabilities. Ideally, γ should be kept as close to 0 as possible. Convergence speed and stability close depend on how low γ is kept in the mesh. Skewness: for a given cell, the skewness expresses the vicinity of the cell to the optimal shape for the same volume. Several definitions can be found in literature depending on the type of cell. For instance, for a 2D triangular mesh (see Fig. 5), the skewness can be obtained as the relative difference d and an ideal triangular cell with in area between the current cell ABC 0 0 0 d the same circumradius A B C , following equation: s¼
Ad Ad 0 0 0 ABC
ABC
Ad 0 0 0
(1)
ABC
Alternatively, it can also be obtained following equation: γ γ e γ e γ min , s ¼ max max 180 γ e γe
(2)
where γ max and γ min are respectively the maximum and minimum angles in the current cell, and γ e is the angle for the equilateral optimal cell (i.e., 60 degrees for a triangular cell or 90 degrees for a quadrilateral cell).
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Fig. 5 Definition of skewness.
Aspect ratio: for a given cell, is a measurement of the stretching of the cell. Low aspect ratio should be kept in the bulk regions, while high aspect ratio cells might be useful for highly anisotropic regions, where it will allow a reduction in the number of cells (consider for instance boundary layers). Again, several definitions can be found in the literature. A common one is the ratio between the largest and the smallest edges, as defined in Eq. (3) and Fig. 6. AR ¼
dmax dmin
(3)
Cell size growth: The influence of this parameter depends a lot on the method used to solve NS equations. However, the change in the cell or element size should be gradual as sudden changes in the cell/element sizes can typically lead to low accuracy and stability issues. In order to obtain a satisfactory mesh for fast convergence of the equations, several mesh motion strategies have been applied in literature for the simulation of EGP and will be commented next. For the different strategies, mesh quality is not the only issue to consider, but the simplicity of the method, the speed of the process, possible requirement of additional modelling effort, and the parallelizability are also major factors to determine the most appropriate method for the intended simulation.
Remeshing A mesh replacement strategy is probably the most common approach (Castilla, Gamez-Montero, del Campo, Raush, & Garcia-Vilchez, 2015; Castilla, Gamez-Montero, Vernet, Coussirat, & Codina, 2010; Kim,
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dmin
dmax
Fig. 6 Definition of aspect ratio.
Marie, & Patil, 2007). Typically, an unstructured mesh is created for the initial position of the gears. As the gears rotate, mesh points are displaced following the gears motion, and the quality of the resulting mesh is controlled. When the quality lies below a given threshold, a new mesh is generated for that particular position of the gears. Variables are then interpolated from one mesh into the other and the simulation proceeds with the new mesh repeating the process. The approach is frequently used mainly due to its simplicity. However, it is clear that the mesh generation process can consume a significant amount of computational power since it needs to be performed every small angle of rotation (the frequency depends obviously on the degree of refinement of the mesh and the quality thresholds). Besides, interpolation of variables between meshes generates numerical errors which are difficult to control. Furthermore, a good mesh quality in the regions near the contact points or the circumferential clearance gaps is hard to achieve. This strategy is generally used with unstructured hexahedral dominant meshes (see Fig. 7). For these, the quality is high in internal regions where a perfectly orthogonal uniform mesh can be considered and slightly reduced when increasing the refinement level (nonorthogonality is increased), and near the mesh boundaries, in the connection between the boundary layer and internal cells. It is a common method, and available in a number of open-source CFD codes, and it has been developed for several years, increasing its robustness and resulting mesh quality. However, the computational cost to generate each mesh is not negligible in the simulation solution process. When considering the parallelizability of the method, while these algorithms work properly in parallel, redistribution of generated cells between processors is generally required as the mesh generation process continues every time a new mesh is needed. This obviously increases the computational cost. Regarding the mesh-to-mesh interpolation errors, the effect
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Fig. 7 Example of mesh-replacement strategy with a hex-dominant mesh.
will not be very significant far from the gearwheels’ boundaries, where the mesh is almost unaffected, but it will near the gearing region, which might be the most important when considering the computation of the volumetric efficiency. In fact, with such a method, the contact region is hardly resolved properly without significantly increase the computational cost, and additional modelling is generally required. Since the actual solution of the contact point as a real contact (separation of the chambers connected by this point in two regions) increases the complexity of the method, the distance between gears centers is artificially increased and contact does not exist anymore (upper and lower gear wall boundaries are not in contact so that the fluid could freely move through the contact point), and a model is applied to mimic the behaviour of physical contact. This modelling is typically based on an artificial increase of the fluid viscosity in the surroundings of the contact point (Castilla et al., 2015; del Campo, Castilla, Raush, GamezMonter, & Codina, 2012; Gamez-Montero et al., 2012), or by introducing a porous media, which acts as an additional force to impede the flux between the regions connected by the contact point. A sudden increase in the viscosity or a too large porous explicit source term can typically lead to instabilities in the solution. Therefore, both methods are blended so that the additional terms are smoothly added to the equations (both in time and space). Depending on the artificially increased distance between gears centers, and how sudden is the addition of the extra terms, the solution might be affected, particularly when there is interest in the determination of the volumetric efficiency since it directly affects the computation of the leakage.
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It must be mentioned that the mesh-replacement method becomes even more interesting when applied to spur gears (straight-cut) where the efficiency of this strategy can be maximized (Concli, Gorla, Della Torre, & Montenegro, 2016; Concli & Gorla, 2016). This is not only applicable to the mesh-replacement strategy but also to the other methods that will be described next. In such cases, the generation of the 3D mesh can be simplified to the extrusion of a 2D mesh. The generation of a 2D mesh and its extrusion is much faster, reducing one of the drawbacks of this method, the computational cost. This will however not apply to the case of helical gears, which cannot be generated through direct extrusion. Dynamic mesh refinement-agglomeration The mesh substitution does not need to be performed in the entire mesh, or even in the entire gearing region. In fact, the mesh can be locally refined or coarsened according to the requirements given by the gears motion. It is the case of the method used in (Strasser, 2006; Gorla et al., 2013; Concli & Gorla, 2012a, 2012b). Fig. 8 shows an example of such a method, of common use in commercial codes. A 2D triangular mesh is slowly deformed following the rotation. Due to the continuous deformation, distorted elements appear and must be substituted. This is taken care of by proceeding with a local adjustment of the triangle sizes in those regions, which can be quickly performed using the available algorithms that generate a tetrahedral grid patch. New elements are however not regularly distributed compared to the initial mesh size. To reduce these issues, and maintained lower aspect ratio and mesh skewness, triangular/tetrahedral cells are created/agglomerated using spring-based smoothing to maintain grid quality. New cells are created on the trailing tooth edge, while cells are destroyed in the leading tooth edge. In the gearing region, this creation/destruction process must
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Fig. 8 Example of the dynamic mesh refinement strategy in a 2D case: mesh overview (left); zoom in the gear meshing region (right).
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proceed aggressively but smoothly. Mesh skewness is the main concern in this method. In order to reduce this, other ideas have been proposed, such as reducing the motion of smaller cells (Khurram & Masud, 2006), that adjust relatively less than larger cells from time step to time step. The main drawback of this method is that the mesh type is restricted to the triangular cells in two-dimensions or tetrahedral cells in 3D. These meshes are quick to generate and are very useful when very complex geometries are intended. However, the average nonorthogonality and skewness are much higher than the ones that can be obtained with a structured grid. As a result, convergence issues can appear, and more dissipative schemes are required, affecting the quality of the results. Besides, the restriction of using triangular-tetrahedral cells impedes the resolution of the boundary layer. Cell-shape is therefore not oriented with the flow near solid boundaries, necessary if we intend to resolve the large gradients near the wall in a wall-resolved simulation. Regarding the parallelizability and computational cost of this method, even if the replacement and agglomeration occur only in some regions of the mesh, the number of cells and faces continuously changes. The change in the topology forces the recalculation of the mesh connectivity, which increases the computational time, and difficulties the proper balancing of the processor cells distributions. One of the benefits that should be mentioned for his method is that, given the continuous mesh modification that involves the addition and subtraction of cells and faces, the consideration of internal baffles and wall boundary conditions for the contact point does not produce a significant increment in the computational cost. Therefore, a realistic approximation of the contact point, completely impeding the flow, can be easily considered for these algorithms (Castilla et al., 2010).
Mesh superposition Most of the problems presented in the previous methods are linked to the fact that the continuous motion of the gears forces the deformation/replacement of the computational mesh. An alternative method to avoid these problems is to consider overlapping meshes, also known as overset meshing (Bruce, Wilson, & Generalis, 1997). If the volumes of stationary and rotational components are constructed and gridded separately, so that grid cells are free to overlap, mesh quality is preserved in each of the separate domains (see Fig. 9). In the solution procedure, for a given time step, some of the
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(A)
(B)
Fig. 9 Example of overlapping grids.
(A)
(B)
Fig. 10 Active cells in the overlapping grid method.
computational cells are temporarily or permanently de-activated (passive cells). The active cells along the interface between overlapping meshes refer to donor cells at another grid, instead of to the passive neighbours on the same grid. Grids are therefore implicitly coupled, and the solution is computed on all grids simultaneously. As an example, Fig. 10 shows only the active cells in an overset mesh calculation performed with commercial software. The problem arises when considering how to interpolate the information accordingly. As the gears rotate, the fluid area covered by the different mesh regions changes. If interpolation is performed between a very refined area in one of the meshes and a very coarse one in other mesh, the diffusive effect of this interpolation is clear. As a result, the quality of the solution is significantly affected. Furthermore, the required resolution in the vicinity of the contact region is much higher than the necessary far from it. When overlapping grids are used, the grids are not deformed, therefore the resolution is fixed, and might not be sufficient if the contact point needs to be properly studied. Despite the complexity in the handling of several meshes, the absence of the need of any mesh topological modification (the total number of cells and faces remains constant) typically increases the speed with respect to the
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remeshing and mesh refinement/agglomeration algorithms. However, the detection of active and passive cells must still be done every time step, as well as the interpolation scheme allocation between grids.
Mesh morphing The last method worth mentioning in this chapter is the so-called mesh morphing. The described issues related to the changes in the topology of the mesh in the remeshing and mesh refinement/agglomeration methods (which increases the computational cost), and the problems of the meshto-mesh interpolation in the overlapping algorithms, would be both eliminated if we were able to handle a deforming mesh that adapts to the motion of the gears, but still maintains a constant number of cells and faces. Furthermore, this is the only method that allows studying the leakage gaps (which can be considered in the order of the micron) accurately without an extreme increase in the computational cost. This is the main reason behind the extensive use of mesh deformation algorithms in the recent years (Martı´nez, 2018; Qi et al., 2016; Vande Voorde, Vierendeels, & Dick, 2004). In the mesh morphing method, the complex motion of the gears in the gearing region is solved by continuously deforming a block-structured grid. While the use of structured grids generally satisfies very high-quality grids, particularly in the vicinity of the leakage gaps, and a very simple datastructured (the grid can be easily defined just with the position of the nodes). the problem lies now on the determination of the field of displacement of the grids to simulate the gears motion. While the method is used by several commercial codes, some algorithms performing this kind of mesh motion strategy are also available in the open literature. In particular, two of the existing methods will be exposed here. Vande Voorde, Vierendeels & Dick, (2004) recently proposed an innovative method. In order to determine the field of displacements of a structured grid, they decided to use an additional unstructured grid that is moved according to the mesh refinement/agglomeration method described earlier but is only sued to solve for a Laplace equation (see Fig. 11). The constrains on the unstructured mesh quality are therefore relaxed (NS equations will not be solved in this mesh), and the complex motion algorithms of the structured grid points is simplified since it is just directly exported from the solution of the Laplace equation in the unstructured grid. The method however also presents some clear disadvantages. First of all, it forces the use of two simultaneous meshes, doubling the problems linked to
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(A)
(B) Fig. 11 Unstructured grid for the solution of the Laplace equation (left); Structured morphing mesh (right).
the motion algorithm. While the mesh in the unstructured grid continuously changes its topology (the number of cells and faces changes as the mesh adapts to the gears motion), the deformation of the structured grid also implies possible changes in the processor decomposition of the mesh, since the processor-to-processor boundaries might need to be modified. Secondly, the structured grid nodes displacement must be obtained from interpolation from the unstructured grid, which might still incur in a nonnegligible computational cost. Recently, an example of an algorithm to directly determine the field of displacements of a block-structured grid was proposed in Martı´nez (2018). The objective of that work was to minimize the computational cost of the mesh motion algorithm, while still maintaining a good quality mesh, using a general method that could be applied to a given gear profile both in spur-cut and helical gears. The process will be now described for a two-dimensional case, while the extension to the 3D simulations will be commented later. The motion algorithm proceeds in three steps (see Fig. 12). First, an initial block structured mesh is created, without any treatment on the gearing region (as it would be done for an overlapping mesh strategy). For the
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Fig. 12 Example of mesh morphing strategy: first step (left); second step (centre); third step (right).
Fig. 13 Reduction of mesh nonorthogonality.
current time, the rotational speed is integrated to determine the rotation angle θ and the meshes corresponding to both gears are rigidly rotated. In the last step, given the position of both gears, an “interface” line is calculated, lying always between both gear wall boundaries without touching any of them. Both meshes are then ‘projected’ to this common interface. The quality of the resulting mesh partly depends on the quality of the “un-projected” mesh. With the purpose of reducing the nonorthogonality of the mesh particularly near the solid boundaries, Martı´nez (2018) also proposed an algorithm to generate the simple block corresponding to half a tooth, which is replicated to complete both full gear-meshes. As exemplified in Fig. 13, the proposed method moves the nodes to satisfy orthogonality at the boundaries, blending the required displacement for internal points.
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The next step is the determination of the “interface” between the two solid boundaries (see Fig. 14). Several methods with general applicability are presented in Martı´nez (2018), all based on geometric considerations, and involving the blending of numerically calculated curves (see Fig. 15). The last step consists of the numerical calculation of the required nodes displacements to restrict each of the meshes to the region limited by the interface (see Fig. 16). The final meshes both contain cell boundaries at the interface, but they are nonconformal. An interpolation algorithm (Arbitrary Mesh Interface) AMI is used to implicitly interpolate the fields between both meshes as solution proceeds. As the gears rotate, the boundary faces belonging to this interface change in time, which is the only topological change to be applied to the
Fig. 14 Overview of the interface between the two gears.
Fig. 15 Calculation of the interface.
Fig. 16 Mesh projection algorithm overview.
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mesh, reducing the computational cost with respect to mesh replacement or refinement/agglomeration techniques. The main advantage of this method is that it is not only applicable to 3D straight-cut gear pumps but can be easily extended to 3D helical gears. In the straight-cut case, the calculation only needs to be done in one 2D plane. The field of displacements can then be mapped for the rest of planes as shown in Fig. 17. When a helical mesh is considered the same strategy can be applied to the points in each of the 2D planes in the z direction. For each of them, the rotation angle will be different, but the strategy can be applied without any additional modifications, as shown in Fig. 18. As it happened for strategies such as remeshing, the distance between gears centres is here enlarged and a contact point does not really exist. However, the advantage of this methods, is that mesh shape in the vicinity of the contact point is oriented with the flow, increasing the stability of the contact point modelling method commented before, and the quality of the mesh in that region. As the distance between centres is reduced towards a realistic value, the gap reduces together with the need for any modelling. However, the aspect ratio of the cells increases and can limit the stability of the solution algorithms for the NS equations. The previously commented method can be easily parallelized, and such procedure is also described in Martı´nez (2018).
Fig. 17 Application of morphing strategy to a 3D spur-gear mesh.
Fig. 18 Application of morphing strategy to a 3D helical gear pump mesh.
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Fig. 19 Helical gear pump mesh as a series of spur-gear meshes.
The adequacy of this method for helical gear pumps can be easily understood when compared to other recent algorithms found in the literature. For instance, Heisler, Moskwa, and Fronczak (2009), proposed the use of series of spur gears rotated according to a helix angle, to approximate the simulation of a helical gear pump (see Fig. 19). This obviously eliminates the smoothness of the surface of the gears leading to a simpler but not very realistic method. Summing up, morphing algorithms are the main tools that currently allow us to properly simulate spur and helical gear pump meshes with the proper treatment of the gears contact point without incurring a too large computational cost.
Discussion The present chapter has described the different strategies found in the literature for the simulation of external gear pumps, starting from simplified 0D
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and 1D models and moving towards the full 3D CFD simulations of these pumps. Some of the different mesh handling strategies available have been described paying attention to the benefits and drawbacks of each of them, as well as their adequacy for simple 2D simulations, 3D simulations of spurgears or 3D helical gears. In conclusion, it can be noted that: • Theoretical and semiempirical models can be calibrated for a particular pump design, but they are difficult to extend to a new geometry and cannot provide a full description of the flow in its entire complexity. • While 0D and 1D models can be useful to study a given pump under different working condition, the inclusion of three-dimensional effects, helical gear pumps, cavitation, relief grooves and other phenomena cannot be easily included, limiting their applicability. • CFD is the only method that provides a complete description of the flow and working conditions of a gear pump. Meshing is probably the greatest challenge when considering the CFD simulation of gear pumps. Several strategies are available in the literature. Their adequacy depends on the particular application and the degree of detailed required for the simulation. • Remeshing methods incur in the highest computational cost among the strategies considered in this chapter. They are simple and can generate good quality meshes. A proper description of the contact point cannot be obtained without a dramatic increase in the resolution in the contact area, leading to a much higher cost. They can be convenient when leakage study is not important or for straight-cut gears, for which the algorithm can be simplified to the 2D case with an extrusion. • Dynamic refinement/agglomeration methods slightly reduce the cost with respect to the previous ones but generate triangular/tetrahedral meshes, which are known for the limited achievable accuracy. Then can allow for a proper treatment of the contact point and can also be simplified for spur gears. • Mesh superposition methods offer a fast alternative. Mesh quality is not limited but the problems arise in the mesh-to-mesh interpolation algorithms and the numerical diffusion introduced by these operations. They might not be useful if a very detailed description is required, in particular near the contact point, but they can benefit from stability and general applicability. • Mesh morphing is probably the most convenient alternative in terms of mesh quality, complexity and prediction capability. Mesh motion algorithms can be complex, expensive and difficult to generalize to any kind
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of gear pumps but offer significant benefits on mesh quality when compared to the previous methods and can be faster than mesh replacement and refinement/agglomeration methods.
References Bruce, D., Wilson, M., & Generalis, S. (1997). Flow field analysis of both the Trilobal element and mixing disc zones within a closely intermeshing co-rotating twin-screw extruder. International Polymer Processing, 12(4), 323–330. https://doi.org/ 10.3139/217.970323. Casoli, P., Vacca, A., & Franzoni, G. (2005). In A numerical model for the simulation of external gear pumps 6th JFPS Int. symposium of fluid power. Tsukuba, Japan. https://doi.org/ 10.5739/isfp.2005.705. Castilla, R., Gamez-Montero, P. J., del Campo, D., Raush, G., & Garcia-Vilchez, M. (2015). Three-dimensional numerical simulation of an external gear pump with decompression slot and meshing contact point. Journal of Fluids Engineering, 137(4), 041105. https://doi.org/10.1115/1.4029223. Castilla, R., Gamez-Montero, P. J., Ert€ urk, N., Vernet, A., Coussirat, M., & Codina, E. (2010). Numerical simulation of the turbulent flow in the suction chamber of a gear pump using deforming mesh and mesh replacement. International Journal of Mechanical Sciences, 52(10), 1334–1342. https://doi.org/10.1016/j.ijmecsci.2010.06.009. Castilla, R., Wojciechowsky, J., Gamez-Montero, P. J., Vernet, A., & Codina, E. (2008). Analysis of the turbulence in the suction chamber of an external gear pump using time resolved particle image velocimetry. Flow Measurement and Instrumentation, 19(6), 337–384. https://doi.org/10.1016/j.flowmeasinst.2008.06.005. Concli, F., & Gorla, C. (2012a). Analysis of the oil squeezing power losses of a spur gear pair by mean of CFD simulations. ASME 2012 11th Biennial conference on engineering systems design and analysis. ESDA, 2, 177–184. https://doi.org/10.1115/ESDA2012-82591. Concli, F., & Gorla, C. (2012b). Oil squeezing power losses in gears: A CFD analysis. WIT Transactions on Engineering Sciences, 73, 37–48. https://doi.org/10.2495/AFM120041. Concli, F., & Gorla, C. (2016). Numerical modeling of the power losses in geared transmissions: Windage, churning and cavitation simulations with a new integrated approach that drastically reduces the computational effort. Tribology International, 103, 58–68. https:// doi.org/10.1016/j.triboint.2016.06.046. Concli, F., Gorla, C., Della Torre, A., & Montenegro, G. (2016). A new integrated approach for the prediction of the load independent power losses of gears: Development of a meshhandling algorithm to reduce the CFD simulation time. Advances in Tribology, 2016. https://doi.org/10.1155/2016/2957151. del Campo, D. (2012). Analysis of the suction chamber of external gear pumps and their influence on cavitation and volumetric efficiency. Terrassa: Escola Te`cnica superior D’Esginyeries industrial i Aerona`utica de Terrasas. del Campo, D., Castilla, R., Raush, G., Gamez-Monter, P., & Codina, E. (2012). Numerical analysis of external gear pumps including cavitation. Journal of Fluids Engineering, 134(8), 081105. https://doi.org/10.1115/1.4007106. Donea, J., Huerta, A., Ponthot, J. P., & Rodrı´guez-Ferra´n, A. (2004). Arbitrary LagrangianEulerian methods. [Encyclopedia of computational mechanics]. Eaton, M., Keogh, P. S., & Edge, K. A. (2006). The modelling, prediction and experimental evaluation of gear pump meshing pressures with particular reference to aero-engine fuel pumps. Journal of Systems and Control Engineering, 220(5), 33–40. https://doi.org/ 10.1243/09596518JSCE183.
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Edge, K. A., & Johnston, D. N. (1990a). The ’secondary source’ method for the measurement of pump pressure ripple characteristics part 1: Description of the method. Institution of mechanicala Engineers Part A: Journal of Power and Energy, 204. https://doi.org/10.1243/ pime_proc_1990_204_006_02. Edge, K. A., & Johnston, D. N. (1990b). The ’Secondary Source’ method for the measurement of pump pressure ripple characteristics Part1: Description of method. Proceedings of the Institution of Mechanical Engineers Part A: Journal of Power and Energy. 204. https://doi. org/10.1243/PIME_PROC_1990_204_006_02. Ferziger, J. H., & Peric, M. (2002). Computational methods for fluid dynamics. Springer. Gamez-Montero, P. J., Castilla, R., del Campo, D., Ert€ urk, N., Raush, G., & Codina, E. (2012). Influence of Interteeth clearances on the flow ripple in a Gerotor pump for engine lubrication. Proceedings of the Institution of Mechanical Engineers, Part D, 226, 930–942. https://doi.org/10.1177/0954407011431545. Ghazanfarian, J., & Ghanbari, D. (2014). Computational fluid dynamics investigation of turbulent flow inside a rotary double external gear pump. Journal of Fluids Engineering, 132 (2), 021101. https://doi.org/10.1115/1.4028186. Gorla, C., Concli, F., Stahl, K. H., Michaelis, K., Schuliteiss, H., & Stemplinger, J. -P. (2013). Hydraulic losses of gearbox: CFD analysis and experiments. Tribology International, 66, 337–344. https://doi.org/10.1016/j.triboint.2013.06.005. Gutes, M., Gamez-Montero, R., Castilla, R., & Codina, E. (2000). Journal bearing performance in gear pumps. In 1st Int. FPNI-PhD Symposium. Germany. Heisler, A. S., Moskwa, J. J., & Fronczak, F. J. (2009). Simulated Helical Gear Pump Analysis Using a New CFD Approach. In ASME fluids engineering division summer meeting, volume 1: Symposia, parts a, B, and C (pp. 445–455). Houzeaux, G., & Codina, R. (2007). A finite element method for the solution of rotary pumps. Computers & Fluids, 36(4), 667–679. https://doi.org/10.1016/j. compfluid.2006.02.005. Ivantysyn, J., & Ivantysynova, M. (2003). Hydrostatic pumps and motors. New Delhi, India: Tech Books Int. Khalaf, A. H. (1989). The design and performance of gear pumps with particular reference to marginal suction conditions. Cranfield Institute of Technology. Khurram, R., & Masud, A. (2006). A multiscale/stabilized formulation of the incompressible Navier-stokes equations for moving boundary flows and fluid-structure interaction. Computational Mechanics, 38(4), 403–416. https://doi.org/10.1007/s00466-006-0059-4. Kim, H., Marie, H., & Patil, S. (2007). Two-dimensional CFD analysis of a hydraulic gear pump. American Society for Engineering Education. Liping, C., Yan, Z., Fanili, Z., Jianjun, Z., & Xianzhao, T. (2011). In Modeling and simulation of gear pumps based on Modelica/Mworks 8th Modelica Conference. Dresden, Germany. Manring, N. D., & Kasaragadda, S. B. (2003). The theoretical flow ripple of an external gear pump. Journal of Dynamic Systems Measurement and Control transactions of the ASME, 125(3), 396–404. https://doi.org/10.1115/1.1592193. Martı´nez, J. (2018). Multidimensional simulations of external gear pumps. [Politecnico di Milano]. Mucchi, E. A. (2012). Simulation of the running in process in external gear pumps and experimental verification. Meccanica, 47(3), 621–637. https://doi.org/10.1007/s11012-0119470-9. Myllykyl€a, J. (1999). Semi-empirical model for the suction capability of an external gear pump. Tampere University of Technology. Paltrinieri, F., Borghi, F., & Milani, M. (2004). Studying the flow field inside lateral clearances of external gear pumps. In 3rd FPNI-PhD Symposium on Fluid Power. Spain. Qi, F., Dhar, S., Nichani, V., Srinivasan, C., Wang, D., Yang, L., et al. (2016). A CFD study of an electronic hydraulic power steering helical external gear pump: Model
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development, validation and application. SAE International, 2016. https://doi.org/ 10.4271/2016-01-1376. Rundo, M. (2017). Models for flow rate simulation in gear pumps: A review. Energies, 10, 1261. https://doi.org/10.3390/en10091261. Rundo, M., & Corvaglia, A. (2016). Lumped parameters model of a crescent pump. Energies, 9(11), 876. Strasser, W. (2006). Investigation of gear pump mixing using deforming/agglomerating mesh. Journal of Fluids Engineering, 129(4), 476–484. https://doi.org/ 10.1115/1.2436577. Vande Voorde, J., Vierendeels, J., & Dick, E. (2004). Development of a Laplacian-based mesh generator for ALE calculation in rotary volumetric pumps and compressors. Computational methods in Applied Mechanics and Engineering, 193(39–41). https://doi.org/ 10.1016/j.cma.2003.12.063. Wang, S., Sakura, H., & Kasarekar, A. (2011). Numerical modelling and analysis of external gear pumps applying generalized control volumes. Mathematical and Computer Modelling of Dynamical Systems, 17(5), 501–513. https://doi.org/10.1080/13873954.2011.577556. Wilson, W. E. (1948). Performance criteria for positive displacement pumps and fluid motors. In ASME semi-annual meeting.
Mesh handling for the CFD simulation of external gear pumps Javier Martínez Politecnico di Milano, Dipartimento di Energia, Milano, Italy
Contents Introduction EGP working principle Numerical simulation of external gear pumps Theoretical and semiempirical methods 0D and 1D models Computational fluid dynamics Remeshing Mesh superposition Mesh morphing Discussion References
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Introduction External gear pumps (EGP) play a major role in the framework of positive displacement turbomachinery (Ivantysyn & Ivantysynova, 2003). The low cost derived from a simple manufacturing, their robustness, high volumetric efficiency and wide pressure operation range (from low-pressure lubricating pumps to medium-high pressure open and close loop hydraulic circuits), make EGP a popular and reliable choice for a wide variety of applications. In fact, they are commonly found in the petrochemical industry for the pumping of high viscosity fluids, possible thanks to their rigid design, but their tight internal clearances make them also suitable for applications where precise flow control is required, i.e., metering applications of polymers, fuels or chemical additives. Despite their relatively simple manufacturing process, the operation of gear pumps involves complex phenomena both from the mechanical and Positive Displacement Machines https://doi.org/10.1016/B978-0-12-816998-8.00012-1
© 2019 Elsevier Inc. All rights reserved.
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from the fluid motion points of view. As a result, EGP have captured the attention of many researchers both in the experimental and numerical modelling fields. Available investigations focus on several aspects of these pumps, from inter-teeth pressure and forces distribution (Eaton, Keogh, & Edge, 2006) to the fluid leakage (Paltrinieri, Borghi, & Milani, 2004), noise production (Gutes, Gamez-Montero, Castilla, & Codina, 2000), cavitation (Ghazanfarian & Ghanbari, 2014) or the effect of the teeth geometry on the flow ripple (Manring & Kasaragadda, 2003). This chapter will focus on the numerical approach to the problem. While early models were based on semiempirical methods and one-dimensional models, with the advent of low-cost large-scale computing, the use of Computational Fluid Dynamics (CFD) has become a promising solution. In fact, CFD allows for a full three-dimensional description of the fluid region, extremely useful to understand the flow characteristics through the pump and to use such information to improve the efficiency and reduce noise and vibration produced by the flow and pressure pulsations in EGPs. As a general overview, this chapter will first give some basic details of the flow motion through EGPs to describe then the different simulation approaches available in literature, starting with some of the classical simplified 0D or 1D models, to move then into the difficulties that come up in the CFD approach, mainly due to the manipulation of the computational mesh.
EGP working principle External gear pumps belong to the so-called positive displacement pumps. As many other fluid power systems, their function is to supply hydraulic energy to the fluid, obtained from the mechanical energy introduced typically by means of a shaft. The working principle of EGP pumps is not different from any other volumetric displacement machine. An expanding cavity in the suction side brings the fluid towards the pump and drives it to the discharge side, where a collapsing cavity impulses the fluid out of the pump. As a result of the narrow clearances, the volumetric efficiency of these pumps is very high, leading to fluid flow through the pump almost independent of the discharge pressure. For external gear pumps, the reduction of the volume in the suction cavity is obtained by means of two generally identical gearwheels that mesh and rotate towards the suction chamber, as schematized in Fig. 1. On the discharge or pressure side, the fluid is squeezed out as the two gears rotate against each other. The power is externally introduced to the pump through a drive shaft, which is connected to the so-called “driving gear”. While the
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Fig. 1 Fluid motion phases in an EGP.
other is dragged by meshing with the former and is referred to as “driven gear”. Considering the small number of parts involved in the system, this simplicity leads to an increase in the reliability of this kind of pumps, as compared to other more complex designs. As a result, gear pumps can operate at high speeds and up to moderate pressure heads. Following the fluid motion shown in Fig. 1, the energy transformation in the pump occurs in three phases: • Suction: The motion of both gears generates an opening volume that is continuously filled by the fluid. Inlet pressure, typically atmospheric, pushes the liquid towards the vacuum created by the opening volume. • Displacement: Volumes of fluids are trapped in the region between gears and the casing and are transported towards the discharge side. • Impulsion: The volume containing the liquid is here reduced, forcing the fluid to move towards the outlet pump. As the fluid is pushed into the chambers and squeezed out in the pressure side, the pressure field continuously varies. The motion it experiments leads to an oscillatory behaviour generating the so-called flow and pressure ripple. The pressure distribution is of major importance. In fact, the minimum value determines the possible presence of cavitation. Furthermore, the pressure distribution on the teeth can be used to determine the chances of mechanical failure of the gears and pump casing, typically related to a fatigue process. In order to reduce the stress suffered by gears and casing, indentation of relief grooves is a common choice. In fact, decompression slots allow improving the fatigue life of the different elements of the pump. However, they
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also increase the liquid leakage, reducing the suction capability of the pump by decreasing its volumetric efficiency. Nonetheless, leakage is just one of the factors that can influence the volumetric efficiency of the pump (del Campo, 2012). • Leakage: As shown in Fig. 2, the fluid can find its way from the pressure side to the suction side through several paths, that lead to the definition of: ▪ Relief groove leakage: Fluid from the meshing inter-teeth volume that leaks through the volume trapped between contact points, due to the effect of the relief grooves. ▪ Axial leakage: Fluid passage between the gears and the side plates. ▪ Radial leakage: Movement of fluid through the teeth and the casing towards adjacent inter-teeth volumes. • Compressibility: In a real pump, the volumetric efficiency also depends on the fluid working conditions. • Cavitation: When cavitation occurs, volumetric efficiency can be drastically reduced.
Fig. 2 Fluid leakage: relief groove leakage (grey); axial leakage (dark grey); radial leakage (black).
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As it will be detailed in the next sections, different simulation approaches are more or less suitable for the correct prediction of volumetric efficiency, flow and pressure ripple, teeth pressure distribution. A complete description of all the phenomena involved is only possible through a 3D CFD simulation of the flow and therefore will be analyzed in detailed.
Numerical simulation of external gear pumps This section describes the state of the art for the available methods in the numerical study of EGP.
Theoretical and semiempirical methods Different degrees of accuracy can be obtained with simplified theoretical and semiempirical methods. To cite some examples, Manring and Kasaragadda (2003) studied the theoretical mass flow through external gear pumps neglecting leakages or cavitation effects. Simple techniques were however proposed to study cavitation phenomena too, including for instance the semiempirical methods presented by Myllykyl€a (1999), which focused on the suction capabilities of these pumps before cavitation occurs. Marginal suction conditions were also studied by Khalaf (1989), who paid attention to the effect on pump performance. However, most of the existing models try to replicate the pressure and flow pulsation occurring in these pumps, given the importance of this phenomena on the deterioration of the gearwheels, affecting the lifetime of the pump. Eaton et al. (2006) used an analytical model to predict the pressure ripple based on an equivalent hydraulic circuit, while Edge and Johnston (1990a, 1990b) worked on the estimation of the flow ripple from pressure ripple measurements. While theoretical and semiempirical methods can provide accurate prediction in some cases, their application is typically restricted to very particular designs and difficult to generalize without the appropriate calibration.
0D and 1D models The next degree of generalization consists in the use of more sophisticated 0D and 1D models. Generally speaking, 1D codes model gear pumps by considering the fluid region to be divided into control volumes, each of which represents the volume of the fluid trapped between two teeth and
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the casing, or between the teeth of both gears in the gearing region. Properties within the chamber are uniform and the evolution of these properties depend on the interaction with the adjoining chambers by variable orifices (Rundo, 2017). In 0D models, also known as lumped parameter models, each control volume is simulated by a capacitive element, which calculates the pressure as a function of the net ingoing flow rate. Volumes are connected by resistive components calculating the flow rate as a function of the pressure drop. 0D models can be further divided (see Fig. 3). In single-chamber models (Wang, Sakura, & Kasarekar, 2011), a control volume is associated to each volume chamber, whereas in multichamber models (Rundo & Corvaglia, 2016), all chambers are connected to the same volume and are lumped together, in order to obtain two main volumes: inlet and outlet. In 1D models, also known as distributed parameter models, the discretization continues further than in 0D models. Suction and delivery ducts for instance, which are simulated with a single capacitive component in 0D models, are here split into many volumes connected in series along the pump axis allowing, for instance, the simulation of pressure waves in the pipes. The simplification, despite the sometimes-extreme assumptions, has been successfully applied to the description of simple straight-cut gears (Casoli, Vacca, & Franzoni, 2005; Liping, Yan, Fanili, Jianjun, & Xianzhao, 2011; Mucchi, 2012). The main drawbacks of such methods are, among others: the difficult description of the geometrical volumes and throat areas as a function of the shaft angular position for a general realistic profile, and the lack of a detail description of the pressure distribution in the teeth. Besides, the consideration of three-dimensional effects and the extension to helical gears becomes critical.
(A)
(B)
Fig. 3 0D models: single chamber (left and centre); multichamber (right).
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Computational fluid dynamics CFD is probably the only approach that allows us to treat the problem in its full complexity. A detailed CFD simulation can take into account not only the 3-dimensionality of the problem but also the fluid compressibility, turbulence, cavitation and the fluid–structure interaction (FSI), where even a two-way coupling with the gear-wheels vibration can be considered. In CFD the fluid region (and the solid too for the cases of FSI) is divided into discrete cells (computational mesh) where the governing equations are solved (Ferziger & Peric, 2002). Depending on the approach to solve these equations and the discretization methods, several options are common nowadays, including among others: spectral-element method (SEM), finite element method (FEM), finite volume method (FVM) and finite-difference method (FDM). Finite volume methods are the preferred choice for most commercial codes dealing with the solution of the fluid motion conservation equations (Navier– Stokes NS equations). FVM is robust, simple and a natural choice when the partial differential equations to be solved are conservation laws. FVM is based on the application of the integral form of the NS equations on each of the small control volumes defined by the computational mesh. The critical aspect when CFD is used to simulate EGP lies in the meshing process. As the gears rotate the fluid domain changes and the computational mesh should adapt to the gear motion. In order to deal with a mesh that is continuously changing, most methods rely on the so-called Arbitrary Lagrangian–Eulerian (ALE) formulation (Donea, Huerta, Ponthot, & Rodrı´guez-Ferra´n, 2004), of extensive applicability in the literature (Castilla, Wojciechowsky, Gamez-Montero, Vernet, & Codina, 2008; Houzeaux & Codina, 2007; Strasser, 2006; Wilson, 1948). In the usual Eulerian method, equations, including convective terms, are solved in a steady mesh. With pure Lagrangian methods, the mesh moves with the fluid particles and the convective term vanishes. ALE approach tries to adopt an intermediate solution, the mesh can be independently moved, leading to additional terms when the NS equations are solved. Nonetheless, the generation and motion of the computational mesh are not straight forward. The accuracy and convergence of the solution depend primarily on the quality of the computational mesh. Several parameters are typically used to determine the quality of a mesh to be used in an FV CFD code, including for instance: nonorthogonality, skewness, aspect ratio and growth ratio. Nonorthogonality: For every face of a computational cell (see Fig. 4), refers to the angle γ between the surface normal Sf and the line that joins the
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P o Sf
Q
no g
Fig. 4 Definition of nonorthogonality.
two centroids sharing the face PQ. Depending on the method used for the solution of NS equations, the computation of surface gradients at the cell boundaries might be required. For instance, in the calculation of diffusion terms in a collocated grid for FV methods, the required surface gradients are typically decomposed in an orthogonal component o, taken implicitly, and an explicit correction for the nonorthogonal component no, that is sometimes limited since it can lead to instabilities. Ideally, γ should be kept as close to 0 as possible. Convergence speed and stability close depend on how low γ is kept in the mesh. Skewness: for a given cell, the skewness expresses the vicinity of the cell to the optimal shape for the same volume. Several definitions can be found in literature depending on the type of cell. For instance, for a 2D triangular mesh (see Fig. 5), the skewness can be obtained as the relative difference d and an ideal triangular cell with in area between the current cell ABC 0 0 0 d the same circumradius A B C , following equation: s¼
Ad Ad 0 0 0 ABC
ABC
Ad 0 0 0
(1)
ABC
Alternatively, it can also be obtained following equation: γ γ e γ e γ min , s ¼ max max 180 γ e γe
(2)
where γ max and γ min are respectively the maximum and minimum angles in the current cell, and γ e is the angle for the equilateral optimal cell (i.e., 60 degrees for a triangular cell or 90 degrees for a quadrilateral cell).
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Fig. 5 Definition of skewness.
Aspect ratio: for a given cell, is a measurement of the stretching of the cell. Low aspect ratio should be kept in the bulk regions, while high aspect ratio cells might be useful for highly anisotropic regions, where it will allow a reduction in the number of cells (consider for instance boundary layers). Again, several definitions can be found in the literature. A common one is the ratio between the largest and the smallest edges, as defined in Eq. (3) and Fig. 6. AR ¼
dmax dmin
(3)
Cell size growth: The influence of this parameter depends a lot on the method used to solve NS equations. However, the change in the cell or element size should be gradual as sudden changes in the cell/element sizes can typically lead to low accuracy and stability issues. In order to obtain a satisfactory mesh for fast convergence of the equations, several mesh motion strategies have been applied in literature for the simulation of EGP and will be commented next. For the different strategies, mesh quality is not the only issue to consider, but the simplicity of the method, the speed of the process, possible requirement of additional modelling effort, and the parallelizability are also major factors to determine the most appropriate method for the intended simulation.
Remeshing A mesh replacement strategy is probably the most common approach (Castilla, Gamez-Montero, del Campo, Raush, & Garcia-Vilchez, 2015; Castilla, Gamez-Montero, Vernet, Coussirat, & Codina, 2010; Kim,
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dmin
dmax
Fig. 6 Definition of aspect ratio.
Marie, & Patil, 2007). Typically, an unstructured mesh is created for the initial position of the gears. As the gears rotate, mesh points are displaced following the gears motion, and the quality of the resulting mesh is controlled. When the quality lies below a given threshold, a new mesh is generated for that particular position of the gears. Variables are then interpolated from one mesh into the other and the simulation proceeds with the new mesh repeating the process. The approach is frequently used mainly due to its simplicity. However, it is clear that the mesh generation process can consume a significant amount of computational power since it needs to be performed every small angle of rotation (the frequency depends obviously on the degree of refinement of the mesh and the quality thresholds). Besides, interpolation of variables between meshes generates numerical errors which are difficult to control. Furthermore, a good mesh quality in the regions near the contact points or the circumferential clearance gaps is hard to achieve. This strategy is generally used with unstructured hexahedral dominant meshes (see Fig. 7). For these, the quality is high in internal regions where a perfectly orthogonal uniform mesh can be considered and slightly reduced when increasing the refinement level (nonorthogonality is increased), and near the mesh boundaries, in the connection between the boundary layer and internal cells. It is a common method, and available in a number of open-source CFD codes, and it has been developed for several years, increasing its robustness and resulting mesh quality. However, the computational cost to generate each mesh is not negligible in the simulation solution process. When considering the parallelizability of the method, while these algorithms work properly in parallel, redistribution of generated cells between processors is generally required as the mesh generation process continues every time a new mesh is needed. This obviously increases the computational cost. Regarding the mesh-to-mesh interpolation errors, the effect
Mesh handling for the CFD simulation of external gear pumps
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(B)
Fig. 7 Example of mesh-replacement strategy with a hex-dominant mesh.
will not be very significant far from the gearwheels’ boundaries, where the mesh is almost unaffected, but it will near the gearing region, which might be the most important when considering the computation of the volumetric efficiency. In fact, with such a method, the contact region is hardly resolved properly without significantly increase the computational cost, and additional modelling is generally required. Since the actual solution of the contact point as a real contact (separation of the chambers connected by this point in two regions) increases the complexity of the method, the distance between gears centers is artificially increased and contact does not exist anymore (upper and lower gear wall boundaries are not in contact so that the fluid could freely move through the contact point), and a model is applied to mimic the behaviour of physical contact. This modelling is typically based on an artificial increase of the fluid viscosity in the surroundings of the contact point (Castilla et al., 2015; del Campo, Castilla, Raush, GamezMonter, & Codina, 2012; Gamez-Montero et al., 2012), or by introducing a porous media, which acts as an additional force to impede the flux between the regions connected by the contact point. A sudden increase in the viscosity or a too large porous explicit source term can typically lead to instabilities in the solution. Therefore, both methods are blended so that the additional terms are smoothly added to the equations (both in time and space). Depending on the artificially increased distance between gears centers, and how sudden is the addition of the extra terms, the solution might be affected, particularly when there is interest in the determination of the volumetric efficiency since it directly affects the computation of the leakage.
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It must be mentioned that the mesh-replacement method becomes even more interesting when applied to spur gears (straight-cut) where the efficiency of this strategy can be maximized (Concli, Gorla, Della Torre, & Montenegro, 2016; Concli & Gorla, 2016). This is not only applicable to the mesh-replacement strategy but also to the other methods that will be described next. In such cases, the generation of the 3D mesh can be simplified to the extrusion of a 2D mesh. The generation of a 2D mesh and its extrusion is much faster, reducing one of the drawbacks of this method, the computational cost. This will however not apply to the case of helical gears, which cannot be generated through direct extrusion. Dynamic mesh refinement-agglomeration The mesh substitution does not need to be performed in the entire mesh, or even in the entire gearing region. In fact, the mesh can be locally refined or coarsened according to the requirements given by the gears motion. It is the case of the method used in (Strasser, 2006; Gorla et al., 2013; Concli & Gorla, 2012a, 2012b). Fig. 8 shows an example of such a method, of common use in commercial codes. A 2D triangular mesh is slowly deformed following the rotation. Due to the continuous deformation, distorted elements appear and must be substituted. This is taken care of by proceeding with a local adjustment of the triangle sizes in those regions, which can be quickly performed using the available algorithms that generate a tetrahedral grid patch. New elements are however not regularly distributed compared to the initial mesh size. To reduce these issues, and maintained lower aspect ratio and mesh skewness, triangular/tetrahedral cells are created/agglomerated using spring-based smoothing to maintain grid quality. New cells are created on the trailing tooth edge, while cells are destroyed in the leading tooth edge. In the gearing region, this creation/destruction process must
(A)
(B)
Fig. 8 Example of the dynamic mesh refinement strategy in a 2D case: mesh overview (left); zoom in the gear meshing region (right).
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proceed aggressively but smoothly. Mesh skewness is the main concern in this method. In order to reduce this, other ideas have been proposed, such as reducing the motion of smaller cells (Khurram & Masud, 2006), that adjust relatively less than larger cells from time step to time step. The main drawback of this method is that the mesh type is restricted to the triangular cells in two-dimensions or tetrahedral cells in 3D. These meshes are quick to generate and are very useful when very complex geometries are intended. However, the average nonorthogonality and skewness are much higher than the ones that can be obtained with a structured grid. As a result, convergence issues can appear, and more dissipative schemes are required, affecting the quality of the results. Besides, the restriction of using triangular-tetrahedral cells impedes the resolution of the boundary layer. Cell-shape is therefore not oriented with the flow near solid boundaries, necessary if we intend to resolve the large gradients near the wall in a wall-resolved simulation. Regarding the parallelizability and computational cost of this method, even if the replacement and agglomeration occur only in some regions of the mesh, the number of cells and faces continuously changes. The change in the topology forces the recalculation of the mesh connectivity, which increases the computational time, and difficulties the proper balancing of the processor cells distributions. One of the benefits that should be mentioned for his method is that, given the continuous mesh modification that involves the addition and subtraction of cells and faces, the consideration of internal baffles and wall boundary conditions for the contact point does not produce a significant increment in the computational cost. Therefore, a realistic approximation of the contact point, completely impeding the flow, can be easily considered for these algorithms (Castilla et al., 2010).
Mesh superposition Most of the problems presented in the previous methods are linked to the fact that the continuous motion of the gears forces the deformation/replacement of the computational mesh. An alternative method to avoid these problems is to consider overlapping meshes, also known as overset meshing (Bruce, Wilson, & Generalis, 1997). If the volumes of stationary and rotational components are constructed and gridded separately, so that grid cells are free to overlap, mesh quality is preserved in each of the separate domains (see Fig. 9). In the solution procedure, for a given time step, some of the
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(A)
(B)
Fig. 9 Example of overlapping grids.
(A)
(B)
Fig. 10 Active cells in the overlapping grid method.
computational cells are temporarily or permanently de-activated (passive cells). The active cells along the interface between overlapping meshes refer to donor cells at another grid, instead of to the passive neighbours on the same grid. Grids are therefore implicitly coupled, and the solution is computed on all grids simultaneously. As an example, Fig. 10 shows only the active cells in an overset mesh calculation performed with commercial software. The problem arises when considering how to interpolate the information accordingly. As the gears rotate, the fluid area covered by the different mesh regions changes. If interpolation is performed between a very refined area in one of the meshes and a very coarse one in other mesh, the diffusive effect of this interpolation is clear. As a result, the quality of the solution is significantly affected. Furthermore, the required resolution in the vicinity of the contact region is much higher than the necessary far from it. When overlapping grids are used, the grids are not deformed, therefore the resolution is fixed, and might not be sufficient if the contact point needs to be properly studied. Despite the complexity in the handling of several meshes, the absence of the need of any mesh topological modification (the total number of cells and faces remains constant) typically increases the speed with respect to the
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remeshing and mesh refinement/agglomeration algorithms. However, the detection of active and passive cells must still be done every time step, as well as the interpolation scheme allocation between grids.
Mesh morphing The last method worth mentioning in this chapter is the so-called mesh morphing. The described issues related to the changes in the topology of the mesh in the remeshing and mesh refinement/agglomeration methods (which increases the computational cost), and the problems of the meshto-mesh interpolation in the overlapping algorithms, would be both eliminated if we were able to handle a deforming mesh that adapts to the motion of the gears, but still maintains a constant number of cells and faces. Furthermore, this is the only method that allows studying the leakage gaps (which can be considered in the order of the micron) accurately without an extreme increase in the computational cost. This is the main reason behind the extensive use of mesh deformation algorithms in the recent years (Martı´nez, 2018; Qi et al., 2016; Vande Voorde, Vierendeels, & Dick, 2004). In the mesh morphing method, the complex motion of the gears in the gearing region is solved by continuously deforming a block-structured grid. While the use of structured grids generally satisfies very high-quality grids, particularly in the vicinity of the leakage gaps, and a very simple datastructured (the grid can be easily defined just with the position of the nodes). the problem lies now on the determination of the field of displacement of the grids to simulate the gears motion. While the method is used by several commercial codes, some algorithms performing this kind of mesh motion strategy are also available in the open literature. In particular, two of the existing methods will be exposed here. Vande Voorde, Vierendeels & Dick, (2004) recently proposed an innovative method. In order to determine the field of displacements of a structured grid, they decided to use an additional unstructured grid that is moved according to the mesh refinement/agglomeration method described earlier but is only sued to solve for a Laplace equation (see Fig. 11). The constrains on the unstructured mesh quality are therefore relaxed (NS equations will not be solved in this mesh), and the complex motion algorithms of the structured grid points is simplified since it is just directly exported from the solution of the Laplace equation in the unstructured grid. The method however also presents some clear disadvantages. First of all, it forces the use of two simultaneous meshes, doubling the problems linked to
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(A)
(B) Fig. 11 Unstructured grid for the solution of the Laplace equation (left); Structured morphing mesh (right).
the motion algorithm. While the mesh in the unstructured grid continuously changes its topology (the number of cells and faces changes as the mesh adapts to the gears motion), the deformation of the structured grid also implies possible changes in the processor decomposition of the mesh, since the processor-to-processor boundaries might need to be modified. Secondly, the structured grid nodes displacement must be obtained from interpolation from the unstructured grid, which might still incur in a nonnegligible computational cost. Recently, an example of an algorithm to directly determine the field of displacements of a block-structured grid was proposed in Martı´nez (2018). The objective of that work was to minimize the computational cost of the mesh motion algorithm, while still maintaining a good quality mesh, using a general method that could be applied to a given gear profile both in spur-cut and helical gears. The process will be now described for a two-dimensional case, while the extension to the 3D simulations will be commented later. The motion algorithm proceeds in three steps (see Fig. 12). First, an initial block structured mesh is created, without any treatment on the gearing region (as it would be done for an overlapping mesh strategy). For the
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Fig. 12 Example of mesh morphing strategy: first step (left); second step (centre); third step (right).
Fig. 13 Reduction of mesh nonorthogonality.
current time, the rotational speed is integrated to determine the rotation angle θ and the meshes corresponding to both gears are rigidly rotated. In the last step, given the position of both gears, an “interface” line is calculated, lying always between both gear wall boundaries without touching any of them. Both meshes are then ‘projected’ to this common interface. The quality of the resulting mesh partly depends on the quality of the “un-projected” mesh. With the purpose of reducing the nonorthogonality of the mesh particularly near the solid boundaries, Martı´nez (2018) also proposed an algorithm to generate the simple block corresponding to half a tooth, which is replicated to complete both full gear-meshes. As exemplified in Fig. 13, the proposed method moves the nodes to satisfy orthogonality at the boundaries, blending the required displacement for internal points.
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The next step is the determination of the “interface” between the two solid boundaries (see Fig. 14). Several methods with general applicability are presented in Martı´nez (2018), all based on geometric considerations, and involving the blending of numerically calculated curves (see Fig. 15). The last step consists of the numerical calculation of the required nodes displacements to restrict each of the meshes to the region limited by the interface (see Fig. 16). The final meshes both contain cell boundaries at the interface, but they are nonconformal. An interpolation algorithm (Arbitrary Mesh Interface) AMI is used to implicitly interpolate the fields between both meshes as solution proceeds. As the gears rotate, the boundary faces belonging to this interface change in time, which is the only topological change to be applied to the
Fig. 14 Overview of the interface between the two gears.
Fig. 15 Calculation of the interface.
Fig. 16 Mesh projection algorithm overview.
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mesh, reducing the computational cost with respect to mesh replacement or refinement/agglomeration techniques. The main advantage of this method is that it is not only applicable to 3D straight-cut gear pumps but can be easily extended to 3D helical gears. In the straight-cut case, the calculation only needs to be done in one 2D plane. The field of displacements can then be mapped for the rest of planes as shown in Fig. 17. When a helical mesh is considered the same strategy can be applied to the points in each of the 2D planes in the z direction. For each of them, the rotation angle will be different, but the strategy can be applied without any additional modifications, as shown in Fig. 18. As it happened for strategies such as remeshing, the distance between gears centres is here enlarged and a contact point does not really exist. However, the advantage of this methods, is that mesh shape in the vicinity of the contact point is oriented with the flow, increasing the stability of the contact point modelling method commented before, and the quality of the mesh in that region. As the distance between centres is reduced towards a realistic value, the gap reduces together with the need for any modelling. However, the aspect ratio of the cells increases and can limit the stability of the solution algorithms for the NS equations. The previously commented method can be easily parallelized, and such procedure is also described in Martı´nez (2018).
Fig. 17 Application of morphing strategy to a 3D spur-gear mesh.
Fig. 18 Application of morphing strategy to a 3D helical gear pump mesh.
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Fig. 19 Helical gear pump mesh as a series of spur-gear meshes.
The adequacy of this method for helical gear pumps can be easily understood when compared to other recent algorithms found in the literature. For instance, Heisler, Moskwa, and Fronczak (2009), proposed the use of series of spur gears rotated according to a helix angle, to approximate the simulation of a helical gear pump (see Fig. 19). This obviously eliminates the smoothness of the surface of the gears leading to a simpler but not very realistic method. Summing up, morphing algorithms are the main tools that currently allow us to properly simulate spur and helical gear pump meshes with the proper treatment of the gears contact point without incurring a too large computational cost.
Discussion The present chapter has described the different strategies found in the literature for the simulation of external gear pumps, starting from simplified 0D
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and 1D models and moving towards the full 3D CFD simulations of these pumps. Some of the different mesh handling strategies available have been described paying attention to the benefits and drawbacks of each of them, as well as their adequacy for simple 2D simulations, 3D simulations of spurgears or 3D helical gears. In conclusion, it can be noted that: • Theoretical and semiempirical models can be calibrated for a particular pump design, but they are difficult to extend to a new geometry and cannot provide a full description of the flow in its entire complexity. • While 0D and 1D models can be useful to study a given pump under different working condition, the inclusion of three-dimensional effects, helical gear pumps, cavitation, relief grooves and other phenomena cannot be easily included, limiting their applicability. • CFD is the only method that provides a complete description of the flow and working conditions of a gear pump. Meshing is probably the greatest challenge when considering the CFD simulation of gear pumps. Several strategies are available in the literature. Their adequacy depends on the particular application and the degree of detailed required for the simulation. • Remeshing methods incur in the highest computational cost among the strategies considered in this chapter. They are simple and can generate good quality meshes. A proper description of the contact point cannot be obtained without a dramatic increase in the resolution in the contact area, leading to a much higher cost. They can be convenient when leakage study is not important or for straight-cut gears, for which the algorithm can be simplified to the 2D case with an extrusion. • Dynamic refinement/agglomeration methods slightly reduce the cost with respect to the previous ones but generate triangular/tetrahedral meshes, which are known for the limited achievable accuracy. Then can allow for a proper treatment of the contact point and can also be simplified for spur gears. • Mesh superposition methods offer a fast alternative. Mesh quality is not limited but the problems arise in the mesh-to-mesh interpolation algorithms and the numerical diffusion introduced by these operations. They might not be useful if a very detailed description is required, in particular near the contact point, but they can benefit from stability and general applicability. • Mesh morphing is probably the most convenient alternative in terms of mesh quality, complexity and prediction capability. Mesh motion algorithms can be complex, expensive and difficult to generalize to any kind
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of gear pumps but offer significant benefits on mesh quality when compared to the previous methods and can be faster than mesh replacement and refinement/agglomeration methods.
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