Simulation of Gearbox Sloshing Effect in ANSYS Fluent

Gear Box Sloshing Effect Simulation:

1. Introduction

Gearboxes are mechanical devices used to transmit power from an input shaft to an output shaft. Lubrication at the meshing teeth is vital for increasing the life of gears. Gearboxes are usually oil-lubricated. A number of methods are used to move lubricant over the gears and bearings of a gearbox. The most popular methods of gearbox lubrication include splash lubrication, pressure lubrication, and mist lubrication.

The splash lubrication relies on the sloshing effect of lubricating fluid inside the gearbox housing generated due to the motion of the mating gears. Theoretically, sloshing means any motion of the free liquid surface inside its container. It's caused by any disturbance to a partially filled liquid container. Depending on the type of disturbance and container shape, the free liquid surface can experience different types of motions including simple planar, non-planar, rotational, irregular beating, symmetric, asymmetric, quasi-periodic, and chaotic.

Inside a gearbox relying upon splash lubrication the lubricant from the lower portion of the housing is carried away by the moving gears and effectiveness is determined based on the extent of fluid that is applied to the mating teeth and how well the rising and coming back of fluid to the sump happens establishing a cycle for continuous lubrication. In simulation studying the sloshing effect usually, several rotations of the gears are achieved and the lubricant motion to the parts of the gearbox is studied.

1.1 Selection of Lubricants:

The analysis for selection of lubricants includes elements such as speed factors, elastohydrodynamic (EHD) lubrication, bearing-life calculations, extreme-pressure lubrication, emergency lubrication, and various special application requirements.

For the speed aspect of the selection, a parameter - speed factor is defined and the Stribeck curve is used to find the dynamic viscosity of the lubricant required for the particular application.

The temperature aspect is dealt with by grading of oils based on the dependency of their viscosity on temperature. Society of Automotive Engineers (SAE) has devised a grading system based on two numbers; the first number ends with the letter 'W' which stands for winter. This measurement is related to how oil flows when its cold such as at engine startup. The second number is defined as how oil flows when at higher temperatures-like during normal operation of the engine. For example, 5W-30 oil will flow easier than a 10W-30 oil at startup and 10W-30 oil will flow easier than 10W-40 oil at normal engine operating temperatures.

1.2 The Study

The aim of this project was to simulate the sloshing effect in a gearbox with a provided fluid acting as a lubricant inside it. A total of four simulations were done with the following setups in Ansys Fluent:

1. 20% Immersion, Fluid - Engine oil.
2. 30% Immersion, Fluid- Engine oil.
3. 20% Immersion,Fluid-n-heptane (c7h16).
4. 30% Immersion,Fluid-n-heptane (c7h16)

The results of each simulation are obtained as an animation file and the achieved level of lubrication are compared for all the cases.

2. Geometry:

A CAD model of gearbox was made for the project and was imported to SpaceClaim for processing.

Fig 2.1: Cut section view of the gearbox model used in the project.

For further use, fluid volume was extracted from the solid model, and split command was used to split the model along the midplane. After that, for converting the problem to 2D, the planar section was copied and pasted to a newly created plane. This planar geometry was used for further activities.

Fig 2.2: Extracted 2D geometry used for simulation.

3. Meshing:

The triangular mesh was created in Mesher and a face sizing of 2 mm was used to get a working mesh. Triangular meshing elements were used due to their compatibility with the remeshing scheme of the dynamic meshing technique available in Ansys. Named selections were made as shown in the figure for assigning motions using user-defined function.

Fig3.1: Meshed model and named selection applied to gear teeth.

To capture the fluid flow properly proximity sizing function was chosen with the following settings:

Size Function	Proximity
Relevence Centre	Fine
Span angle centre	Fine
No. of cells across gap	1
Proximity Min. Size	0.004 mm
Max Face Size	0.2 mm

The meshing yielded a total of 493088 elements at a quality above 50% which is sufficient for this study.

Fig 3.2: mesh quality generated.

4. Solver settings used in fluent

The meshed model was imported into Fluent and the following settings were used:

• Pressure based solver was used as the velocities involved were very low.
• The transient solver was used as we needed to capture the entire solution process and not only the end results.
• Viscous model: K- epsilon 2 equation model with enhanced wall treatment was used to capture the viscous effects properly.
• Multiphase model: Volume of Fluid (VOF) with two eulerian phases was used. The parameter formulation scheme was set to implicit as the explicit formulation is susceptible to divergence due to the Courant number getting too high.
• Gravity was provided in the negative Y direction according to the CAD model used.
• Material and phases: Air and another liquid acting as a lubricant were used.

4.1 Dynamic Meshing

In cases of simulations involving motions of cell zones, it is necessary to provide dynamic meshing along with a user-defined function containing details of the intended motion. For example, dynamic meshing can be used to simulate the floating of a body inside a fluid, simulation of compression and expansion inside a cylinder and simulation of flow inside a centrifugal pump, etc.

4.1.1 User Defined Function:

Its a program written in C which can be dynamically loaded with the fluent solver to enhance the standard features of the solver. The UDF file is either interpreted or compiled in fluent.

The interpreted option is used for small straightforward functions. The compiled form is used when dealing with complex functions which have significant CPU requirement. For example, a property UDF that is called per cell basis every iteration needs to be loaded with compiled option available under the User-defined tab of Fluent.

4.1.2: UDF used for the motion of gears:

1. #include "udf.h"

2. DEFINE_CG_MOTION(right_motion, dt, vel, omega, time, dtime)
3. {
4. vel[0] = 0.0;
5. vel[1] = 0.0;
6. vel[2] = 0.0;

7. omega[0] = 0.0;
8. omega[1] = 0.0;
9. omega[2] = 2.0e2; /* [rad/s]*/
10. }
11. DEFINE_CG_MOTION(left_motion, dt, vel, omega, time, dtime)
12. {
13. vel[0] = 0.0;
14. vel[1] = 0.0;
15. vel[2] = 0.0;

16. omega[0] = 0.0;
17. omega[1] = 0.0;
18. omega[2] = -2.0e2; /* [rad/s]*/
19. }

Explanation:

• Line 1: It is the header file meant for calling the "udf.h" library .
• Line 2. Define_CG_MOTION is a predefined function in the udf.h library. It will enable the motion of assigned zones with reference to C.G. of the body. It takes six arguments which are as follows: (1.) right_motion: It's the name of the UDF.

(2.) dt: Its a pointer to a structure that stores dynamic mesh data. ( Note: It can be either done by code or the Fluent does it by default.)

(3.) vel: Linear velocity that is provided to the dynamic mesh zone. The direction of motion is specified by the number inside the bracket as [0] stands for along X direction, [1] stands for motion along the Y direction, and [2] for motion along the Z direction.

(4) omega: It stores the value of rotational speed. The direction is assigned in a manner similar to linear velocity.

• As can be figured out from the body of code, linear velocity was zero and angular velocity was assigned 200 rad/s to left and right gears in clockwise and counter-clockwise directions respectively along the Z direction.

4.1.3 Dynamic Meshing Techniques:

Dynamic Meshing is used when a solid body in the flow domain is moved with a prescribed motion or under the hydrodynamic forces. There are three techniques available in Fluent for dynamic mesh generation. They are as follows:

1. Layering: It is used for linear motion and the mesh type has to be quad for 2D and hexahedral for 3D geometry. In this technique, the geometry must be partitioned and an interface must be defined if the moving object is within a fluid domain. In this technique, the interface's contact cells collapse layer by layer during the execution of the assigned motion.
2. Smoothing: It is used during a boundary motion when the fluid zone's general shape does not change and the number of nodes remains the same. It is usually used along with remeshing.
3. Remeshing: Its a branch of smoothening and is used when the displacement is large. Unlike the smoothing, it does not fail in such cases. It enables Fluent to remesh zones with low quality locally. It can be used with tetrahedral elements for 3D and triangular elements for 2D geometry.

The Layering and Remeshig schemes were used with the values as shown in the figure 4.1.3.1:

Fig 4.1.3.1: Remeshing settings used in the simulation.

Note: Such a working set of values are obtained with the iterative approach based on the time step size and mesh element sizes.

Assigning Zone motion:

The zone motion were assigned using the create option under the dynamic mesh zone window. The UDF names were assigned to named selections as follows:

The assigned motion was checked using the display the zone motion option available in the dynamic mesh zone window.

4.1.4 Other Solver settings:

Marking cells for patching:

Cells were marked for patching the desired percentage of the total gearbox area with the fluid selected as lubricant. Simple mathematical approach was used to calculate the Coordinates of bounding rectangle's vertices.

Fig:4.1.4.1: Sketch used for obtaining the height of line (x) from the base.

The obtained equation was:

`y/100*R^2*[pi+4] = [((2*pi*R^2)/360)*cos^-1(1-x/R)]-[sqrt(2*R*x-x^2)*(R-x)]+2*R*x`

Where:

1. R: radius of outer circle
2. x: height of the line AB
3. y: percentage of immersion needed.

The above equation was solved for R= 67.5 and y= 20 for 20% immersion and y=30 for 30% immersion yielding the values of x as 30.36 and 43.127 respectively.

Based on these values the coordinates of points C and B were obtained and their values were used for marking the area for patching.

Fig 4.1.4.1: Marked region for patching.

Solution Method Settings:

1. Pressure-velocity coupling

Scheme: SIMPLE

2. Spatial Discretization

Gradient: Least Squares Cell Based

Pressure: PRESTO!

Momentum: Second order upwind

Turbulent K.E. Dissipation rate: First Order Upwind

3. Transient Formulation: First Order Implicit

Hybrid initialization was done and the patch option was used to provide a volume fraction of 1 for the lubricant in the marked region.

Finally, a time step size of 9e-05 was used with 1000 iterations to obtain the results.

5. Results and discussion:

To visualize the result of sloshing a contour was plotted for the volume fraction of lubricant and a solution animation was created prior to starting the simulation.

Fig 5.1: The initial level of lubricating fluid in the gearbox sump.

After that, a total of 1000 iterations were run with a time step size of 9e-05.

Fig 5.2: Scaled residuals for the transient simulation.

The residuals show a steady pattern with periodic changes which hints that the solution has converged.

5.1.1 Results for 20% Immersion of gears in engine oil:

Properties of the engine oil selected for the study:

Castrol Optigear 1710 range oil was used and its properties were obtained from the official website of Castrol.

Fig 5.1.1.1 Properties of engine oil used.

A flow time of 0.09 seconds was obtained which translates to almost 3 rotations at a speed of 200 rad/s or 1909 rpm.

Fig 5.1.1.2 Final contour for the 20% immersion in engine oil.

Video 5.1.1.1: animation for the 20% immersion in engine oil.

5.1.2 Results for 30% immersion in Engine oil

Fig 5.1.2.1 Final contour for the 30% immersion in engine oil.

Video 5.1.2.1: animation for the 30% immersion in engine oil.

5.1.3: Results for 20% immersion in n-heptane (C7H16)

Fig 5.1.3.1: properties of the n-heptane used from the Fluent database.

Fig 5.1.3.2: Final contour for 20% immersion in n-heptane.

Video 5.1.3.1: animation for 20% immersion in n-heptane (C7H16)

5.1.4: Results for 30% immersion in n-heptane (C7H16)

Fig 5.1.4.1: Final contour for 30% immersion in n-heptane.

Video 5.1.4.1: animation for 30% immersion in n-heptane (C7H16)

5.2 Discussion and interpretation of the results

• The engine oil used was heavier than the n-heptane in terms of density and viscosity. This led to the quick ascend of lubricant in the n-heptane simulation as compared to the Engine oil simulation.
• In the n-heptane simulation, the portion containing the mating teeth can be seen sometimes running dry. It can be attributed to the fact that being a lighter fluid it flows better and hence a higher level of fluid in the sump is needed for better lubrication.
• After around one rotation the lubricant can be seen flowing upward to the casing edge; it can be explained from the pressure contour as follows:

Fig 5.2.1: Pressure contour on the geometry.

• As clear from fig 5.2.1 there exists a high-pressure region near mating teeth and surrounding regions on the upper side have significantly low pressure hence an upward flow can be observed.
• Similarly, the upward rise of the fluid from the lower side of the mating region up to the pressure point can be observed due to the existence of negative pressure in the region.
• The level of fluid in the sump has a more significant role while using a lighter lubricant as can be seen from the animations that for n-heptane the 20% immersion case has almost a dry-sump at a point during the second rotation which is not desirable.

6. Validation

Theoretically, the pressure variation is expected as follows:

During the initial engagement, the pressure must rise as the gap between teeth goes on decreasing leading to compression of fluid inside it. And after the engagement is over the gap between teeth begins to expand leading to expansion and subsequent low pressure. This trend can be visualized from the fig 5.2.1. so, the simulation is correct up to a good extent.

7. Difficulties encountered during the simulation:

Dynamic mesh failed error:

Negative cell volume detected: Negative cell volume usually means that the mesh moved too far in a single time step so that the cells have collapsed. In other words, the remeshing done by the Fluent in the dynamic zone is too coarse and the distance moved per time step may be greater than the edge or cell size in the dynamic region. Reducing the size of the timestep usually solves this problem. In this simulation, this error was solved by reducing the time step size to 9e-05 and setting the minimum length scale to 0.17 mm and maximum length scale to 1.9 mm. This solved the problem as both are below the mesh size of 2 mm for the triangular elements.

Floating-point exception: It simply means that the solver has encountered an arithmetic operation that it can not compute. For example division by zero or a mathematical expression for which the obtained solution is out of defined range. Out of many reasons behind this error, one of the basic causes is zero in the initial conditions like zero turbulent K.E., Zero temperature, zero pressure in a compressible flow problem, etc. Usually to solve it one of the following remedies work:

Limit the time step size, Reduction of the under relaxation factor- Its a technique that enables the replacement of values during the solution to be done based on the weighted average between the old and the new values. An under-relaxation factor of 1 means that the old values were totally replaced with the new set of values obtained at the end of the current iteration. Whereas an under-relaxation of 0 means completely ignoring the new results and carrying forward with the old values. Generally, values between 0 and 1 are used to stabilize the convergence process. The suggested values are 0.7 for velocity and 0.3 for pressure and other scalar fields.

Selection of correct solver settings: Implicit Vs explicit scheme in VOF: In VOF methods, explicit is simply an interface tracking scheme. It sees the interface at time t, obtains the momentum at that point, then extrapolates to get a continuous interface at time t+1. Whereas the implicit scheme actually calculates the interface location by making use of the volume fraction equation. The implicit scheme is significantly more stable and hence bigger time step sizes can be used.

Failure to load the UDF: Sometimes the Fluent fails to load the UDF from the file. This can be solved by launching the workbench from the command prompt of Microsoft Visual C++ for windows.

8. Future scope:

• The energy equation was not used in the simulation to simplify the problem and to reduce the computational requirements. This resulted in complete ignorance of the temperature dependency of the properties of lubricants used. A more realistic approach for the simulation will be the consideration of this aspect.
• With the available resources for computation, the flow time was merely for three rotations which are not sufficient for doing a detailed study of the problem. So, a study with more number of iterations will be needed to get a better picture of the flow patterns inside the gearbox.

9. References:

• (20-03-2020) Slosh dynamics, Available at: https://en.wikipedia.org/wiki/Slosh_dynamics
• Optigear Synthetic 1710/320, Available at: https://thelubricantoracle.castrol.com/product/optigear-synthetic-1710-320/industrial/en-JP