On the validity of the Reynolds equation for the simulation of wet grinding processes

Wet grinding is a process with a complex interaction of individual sub‐processes such as fluid mechanics, continuum mechanics and thermodynamics and branched coupling of corresponding properties. In order to be able to represent these properties in a model, this paper develops a description of the hydrodynamic properties in the grinding gap as a building block for a simple but accurate model to describe the process of wet grinding. To this end, it is shown that the Reynolds equation, which is well‐established in the lubrication theory, is also suitable for describing an exemplary grinding wheel with abrasive grains or pores. This is done by demonstrating the feasibility of a calculation and by comparing the results with Computational Fluid Dynamics (CFD) simulations.


INTRODUCTION
For the production of parts with high surface quality, as well as for the machining of hard surfaces, the manufacturing process of grinding is indispensable, especially for the production of large quantities [1].In this process with geometrically undefined cutting edges, a large part of the energy supplied to the system is converted into heat [1], which, if not dissipated from the contact zone, can lead to high machining temperatures that change the microstructure of the workpiece surface area.To prevent this, the use of cutting fluids is necessary, especially in large-scale industrial applications.In addition, the cutting fluid has the further task of removing the grinding chips.However, one disadvantage of using cutting fluid is that it also creates a hydrodynamic load-bearing effect in the grinding gap.As a result, the contact pressure of the grinding wheel is not transmitted directly to the contact zone of the cutting abrasive grains in engagement, but rather proportionally, over the entire fluid domain.Depending on the size of the hydrodynamic pressure build-up, the abrasive grains in engagement are thereby less pressed against the workpiece.The hydrodynamic load-bearing effect thus counteracts the actual process of material removal, so that the use of cutting fluid creates a conflict of objectives between optimum heat production and removal, and hydrodynamic floating of the grinding wheel.In order to optimally resolve this conflict of objectives, a deeper understanding of the process interrelationships of wet grinding is necessary [2].
The same applies to the development of sustainable cutting fluids.Since large proportions of the cutting fluids currently used are fossile-based and use additives that might have harmful properties [3], a substitution of these components with organic or water-based components is desirable in order to reduce efforts and costs, as well as negative impacts from an environmental point of view.However, for a successful substitution of the oil-based components, an understanding of the influences of the fluid properties on the grinding processes and their results is necessary in order to be able to carry out tailored substitutions.Since the process of wet grinding consists of different individual processes on a micro-and macroscopic level, mathematical modelling of the process is extremely complex.Due to the large number of geometrically indeterminate cutting edges, the pressure and temperature curves, as well as cavitation and abrasion considerations in connection with a mostly porous grinding wheel, a simulation of the processes is highly demanding.This makes simulations with conventional simulation tools only possible with the highest modelling effort and computing power, making optimisation routines unfeasible.For this reason, the grinding parameters are nowadays mostly set on the basis of experience or experimental trial-and-error procedures.Therefore, it is desirable to develop a model that is simple enough to model the processes and optimisation issues, but accurate enough to adequately represent them.Against this background, a basic model for the description of wet grinding is to be created on the basis of isogeometric analysis and the Reynolds equation from known considerations of hydrodynamic lubrication.In connection with the use of Non-uniform rational B-Splines (NURBS), the isogeometric analysis has already proven to be an extremely efficient analysis method for questions of solid contact kinematics [4,5] as well as in the consideration of hydrodynamic friction in ref. [6].

ISOGEOMETRIC ANALYSIS USING NURBS
The isogeometric analysis first introduced by Hughes et al. [7] is a procedure that uses the same functions both for the description of the geometry and for the representation of the solution.Intended as a concept for simplifying FEM modelling by combining geometry description and calculation, in most cases the representation of the geometry via a parametrisation using NURBS forms the basis [8].However, the use of other basis functions such as B-Splines or T-Splines is also possible, as are other groups of functions, if the isogeometric analysis is defined only by the use of the same functions for geometry description and solution approximation [8].The advantage of isogeometric analysis using NURBS or B-/T-Splines is that it is possible to describe the geometry exactly using only a few discrete control points.By adopting this description in the Galerkin approach within the isogeometric analysis using the same NURBS basis functions as ansatz functions, the calculation can thus also be evaluated at only a few discrete control points.Due to the continuous character of NURBS without direct specification of a curve shape in fixed polynomial order, even this small number of control points can be sufficient for an adequate representation of the solution.
The principle of local support [9] applies to the NURBS basis functions, which means that only the according to the knot vector on the respective parametrisation section active basis functions have an influence on the course of the parameterisation curve.A suitable option can be the reformulation of this summation via a vector notation, since especially with regards to the local support property, only a limited number of the basis functions and control points have an influence on the currently considered parameterisation section.Thus, the desired parametrisation can be defined via with  ∈ ℝ +1×1 being the active entries of the NURBS basis functions and  ∈ ℝ +1×1 being the corresponding control points where  would have to be redetermined with each evaluation point and  element wise.Another advantage of this notation is that this same vector with entries of the control points can be used for an evaluation of the derivatives [9].The vector notation can additionally be used for the representation of the state variables in the isogeometric analysis.This results in the said requirement for the NURBS basis functions to be able to represent not only the geometry but also the solution.For this reason, the number of control points is usually increased via a refinement so that the solution to be calculated can also be described.Due to the property of NURBS to be continuous functions, steep gradients or abrupt strong changes of the gradient of the solution usually require a higher number of NURBS control points in order to adequately represent the solution.

MODELLING OF EXEMPLARY GRINDING WHEELS
Modelling exemplary grinding wheels in the microscopic range with protruding abrasive grains or recessing pores like shown in Figures 1 and 3 is a challenge in the field of NURBS-based geometry representation, since representing sharp Microscopic view of a grinding wheel with 45 mm diameter and 125 μm grain size.

F I G U R E 2
Exemplary grinding wheel with model geometry 1, which models the pores.Mind the different scales of the axes.
edges contradicts the very character of NURBS to represent continuous geometries, especially with higher order basis functions.An option to do this anyway is the representation with the help of multiplicities within the knot vector.The thereby reduced continuity at the corresponding control points opens up the possibility of representing said sharp edges within the framework of the isogeometric analysis.This allows the hydrodynamic properties to be investigated using the exemplary geometries 1 and 2 as shown in Figures 2 and 4, which are intended to represent the geometries of the measured grinding wheels in an idealised way.
To create them with NURBS, the basic geometry of the grinding wheel is first modelled as a section of a circle.This geometry is then refined in such a way that at least all the points necessary for the representation of the exemplary grinding wheel are present.If the ratio between grain/pore-width and their separating distance is to be other than one, the discretisation must be correspondingly finer.In the following step, unneeded control points of the refinement are discarded and the target knot vector with multiplicities at the transition from original wheel shape to grain/pore is created accordingly.Finally, the control points located in the centre of the grain/pore are shifted in their position according to the desired deflection along the normal direction.The resulting geometry can then be adjusted in height to fit the requirement of the minimum gap height and refined with regards to the requirements of the solution for the distribution of control points.This procedure is shown as an example in Figures 5-8.The corresponding knot vector for the geometry with five artificial abrasive grains is accordingly  = [0, 0, 0, 1, 1, 2, 2, … , 10,10,11,11,11] with said multiplicities within the knot vector.Note that the axes are not equally scaled due to depictability.As the basis of the creation is the 22.5 mm radius of the grinding wheel with the center lying in the coordinate origin, the y-axis describing the gap height shows negative values.This does not effect the procedure as the final geometry is shifted to the desired minimum gap height.

HYDRODYNAMIC PROPERTIES IN WET GRINDING
One property of the narrower contact zone in the grinding gap is that the dimension in the gap height direction is significantly smaller than in the longitudinal direction of the gap.Particularly from a macroscopic point of view, the system thus shows similarities to the hydrodynamic issues of journal bearings.Since in these systems the pressure can be considered as constant over the gap height [10], and the Reynolds number for the considered systems is clearly below the critical Reynolds number, the pressure build-up in the lubrication gap can be described by the Reynolds equation in connection with further assumptions.The main advantage over a description of the pressure build-up via the Navier-Stokes equations is that, due to the assumption of constant pressure over the height, it is sufficient to discretise only the dimensions of the plane orthogonal to the gap height direction.A two-dimensional domain can thus be described by the one-dimensional Reynolds equation.Especially in connection with the isogeometric analysis, this results in a clear runtime advantage compared to calculations with the full Navier-Stokes equations.The stationary Reynolds equation states when separated into its components from the continuity equation and the Navier-Stokes equation.With   being the vector containing the mean fluid velocities of the couette flow induced by the movement of the surrounding geometries, in the following considerations the fluid velocity is induced by the motion of the lower plate giving the ability to neglect the squeeze flow.
In the derivation of the Reynolds equation from the Navier-Stokes equations, several assumptions are made, the fulfilment of which is a prerequisite for a meaningful application of the Reynolds equation to wet grinding.While the assumptions that the inertia of the fluid is negligible and that any compression or expansion occurs frictionless do not differ in the case of wet grinding from journal bearing considerations or adhesive flows to which the Reynolds equation has been successfully applied [11], the assumption of a laminar flow field requires deeper consideration.Due to the geometry of the exemplary grinding wheels, vortices occur in the flow field, but only on the macroscopic scale.Due to the high viscosity of the fluid, the formation of vortices is clearly inhibited at the microscopic level.As the Reynolds number also suggests, a laminar flow can therefore be assumed even when considering the exemplary grinding wheels, despite the obvious vortices.
Other assumptions must again be examined in greater depth for the case of wet grinding.These include the assumptions that the pressure is assumed to be constant over the gap height and that there are no significant components of the flow velocity in the height direction.Both assumptions are mutually dependent to a certain extent, but the assumption of constant pressure over the gap height can only be fully verified with the help of CFD simulations.For the fluid velocity, it applies that in the area of the main pressure variation in the vicinity of the minimum gap height, the surface gradient is very flat, which means that there are no significant flow components in the gap height direction induced by the geometry.This applies both to geometries of the grinding wheel with mounted abrasive grains, but also to the case with modelled pores.Likewise, these restrictions apply in relation, so that a strong pressure increase along the lubrication gap deprives any pressure gradients in the gap height direction of their significance.This is the case, among other things, with very small gap heights.
For these reasons, the Reynolds equation can also be used for the description of the hydrodynamic properties in the grinding gap using the explicit modelling of an exemplary grinding wheel.Even though these assumptions of the Reynolds equation are limiting, their use opens up the possibility of fast calculations that are numerically stable, since only a linear system of equations has to be solved as the result of an FEM formulation or other approaches.
The disadvantage of this consideration is that the Reynolds equation in its original form calculates backflows induced by negative pressures to ensure mass conservation in the region of strongly diverging gaps, resulting in nonphysical, strongly negative pressures.In real systems, however, cavitation occurs in these areas.In this case, when the pressure in the fluid drops below a certain cavitation pressure, gas bubbles form which contain gases dissolved in the fluid as for a cavitation mechanism in the steady state case [12].For the implementation of these cavitation phenomena, there are a number of considerations that more or less violate the validity of the Reynolds equation or conservation of mass.To describe the cavitation in the grinding gap, a mass-conserving cavitation model with Jakobsson-Floberg-Olsson (JFO) cavitation conditions according to ref. [6] is used in the following.This approach uses the void fraction  as the fraction of the non-filling with fluid to describe the cavitation with the correlation for the density  =  0 (1 − ) based on the basic density  0 and substitutes this into the Reynolds equation.The same procedure can also be applied for the viscosity , whereby this description is shortened by the structure of the Reynolds equation in the observation.Thus the Reynolds equation for the one-dimensional case is and therefore obtains an additional term to describe the cavitation.

COMPARISON TO CFD RESULTS
The correct implementation of the isogeometric analysis and the observation modes can be verified by comparing the results with an equivalent CFD simulation in Ansys Fluent.An additional advantage is that the simulation in Ansys Fluent is based on the solution of the full Navier-Stokes equations and thus it is also checked whether the application of the Reynolds equation is possible in the considered case.The comparison for the considered cases of grains and pores is made separately.On the one hand, the system with applied abrasive grains (geometry 2), and on the other hand, the system with pores embedded in the grinding wheel (geometry 1) are considered.To ensure that both the Ansys Fluent modelling and the MATLAB implementation refer to the same geometry, the geometry of the grinding wheel created via NURBS was saved from the MATLAB implementation as an Initial Graphics Exchange Specification (IGES) geometry and used as the initial geometry of the CFD simulation.The used parameters are simplified oil properties for the cutting fluid with a density  0 = 1000 kg∕m 3 and a dynamic viscosity  0 = 1 Pas or  0 = 0.01 Pas as a variation in Figure 11.The surface velocity of the grinding wheel is set to 35 m∕s, therefore the input velocity for the Reynolds equation is   = 17.5 m∕s.
In all cases a 45 mm diameter grinding wheel forms the basis, with either a fine surface topology with 125 μm grains size or a coarse surface with 400 μm grain size.
The CFD simulations show on the one hand that the assumed constant pressure distribution over height as a prerequisite for the Reynolds equation is sufficiently given as can be seen in Figure 9 and even in Figure 10, and on the other hand that the results of the IGA-based Reynolds equation and the CFD-results are in good agreement.Especially the qualitatively comparison reveals the good abilities of the Reynolds equation to describe the pressure drops within cavities or behind the grains as shown in Figures 11 and 12.In theses comparisons for the spiked grinding wheel (geometry 2) the agreement for both the fine case and the coarse case can be seen, even though pressure differences slightly increased in the case of the coarse grinding wheel or a lower viscosity as the pressure build-up is little more dictated by the single grains than the overall curvature.The same can be stated for the grinding wheel with pores (geometry 1) as can be exemplarily seen in Figure 13.
For the consideration with cavitation, which ultimately also represents the status of the system that occurs in real operation, the comparison is made for the geometry 1, whereby geometry 2 follows the same behaviour.Again the comparison shows good quantitative and very good qualitative agreement of IGA-based results and CFD results as shown in Figure 14 even more regarding the massive run time advantages with the IGA cavitation algorithm only taking 0.24 s on a desktop PC compared to minutes with the pure CFD simulation.
Although the Reynolds equation gives good results in the cases considered, caution should be taken when attempting to apply the Reynolds equation to systems that are more prone to turbulence effects, such as ones with fluids having much lower viscosity or systems in which the pressure builds up mainly due to flow in cavities or around obstacles.

CONCLUSION AND OUTLOOK
In this paper, it was shown that the Reynolds equation can represent complex geometries with discontinuous shapes.The utilisation of the multiplicities within the knot vector offers the possibility to generate geometries with locally reduced continuity requirements.For an exemplary grinding wheel generated in this way, both with exemplary pores and with exemplary abrasive grains, it was shown that the results from the Reynolds equation are in very good agreement with the results of a CFD simulation.Therefore, the Reynolds equation can be used not only to model the macroscopic cylindrical grinding wheel, but also to study the microscopic behaviour, provided that the conditions for the application of the Reynolds equation are fulfilled.Since not only the pressure but also the void fraction is correctly predicted, the cavitation algorithm can be used as a basis for evaluating the temperature properties in the fluid in future works.

A C K N O W L E D G M E N T S
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -498570013.Furthermore, we would like to express our gratitude for the financial support of the Publication Fund of the TU Braunschweig.
Open access funding enabled and organized by Projekt DEAL.

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I G U R E Measurement of the line roughness of a grinding wheel with 45 mm diameter and 125 μm grain size.F I G U R E Exemplary grinding wheel with model geometry 2, which models the grains.Mind the different scales of the axes.F I G U R E Step 1: Creation of the basic part of the grinding wheel.F I G U R E Step 2: Insertion of additional control points according to desired end geometry.F I G U R E Step 3: Selection of needed control points for width ratios.F I G U R E Step 4: Correction of control point coordinates according to final geometry.

F I G U R E 9
Pressure result in Pascal of Ansys Fluent for the fine exemplary grinding wheel.F I G U R E 1 0 Pressure result in Pascal of Ansys Fluent for the coarse exemplary grinding wheel.

F I G U R E 1 1
Pressure comparison between Reynolds equation and CFD-results for spiked grinding wheel.CFD, Computational Fluid Dynamics.F I G U R E 1 2Pressure comparison between Reynolds equation and CFD-results for spiked grinding wheel.CFD, Computational Fluid Dynamics.

F I G U R E 1 3
Pressure comparison between Reynolds equation and CFD-results for porous grinding wheel.F I G U R E 1 4Pressure comparison between Reynolds equation and CFD-results for porous grinding wheel considering cavitation phenomena.CFD, Computational Fluid Dynamics.