1

Optimization of under-relaxation factors

and Courant numbers for the simulation of

sloshing in the oil pan of an automobile

Swathi Satish*, Mani Prithiviraj† and Sridhar Hari⁰

*National Institute of Technology, Surathkal, Karnataka, 575025.

† CD-adapco, Bangalore, Karnataka, 560066.

⁰ CD-adapco, Bangalore, Karnataka, 560066.

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INDEX

I. List of Figures . . . . . . . . 3

II. List of Tables . . . . . . . . 4

Abstract . . . . . . . . . 5

1. Background . . . . . . . . 5

2. Introduction . . . . . . . . 6

3. Geometry modelling and simulation . . . . 7

4. Results and Discussion . . . . . . . 9

5. Conclusion . . . . . . . . 11

6. References . . . . . . . . 12

7. Figures . . . . . . . . . 13

8. Tables . . . . . . . . . 19

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I. List of Figures

1. Illustration of grids that are unsuitable (left) and suitable (right) for two-phase flows using

the VOF model.

2. 3D model of the oil pan.

3. Section plane showing the polyhedral mesh generated.

4. Scalar scene after initialization showing the 6 point probes.

5. Scalar scene after the attainment of steady state.

6. (a) Trimmer mesh with base size 8 mm, (b) Trimmer mesh with base size 4 mm.

7. Plots of URF values versus simulation time at the 6 probe points.

8. Plot of (a) Volume fraction of Oil at the 6 probe points versus the simulation time, (b)

Pressure at the pressure outlet versus the simulation time for a Velocity URF 0.9.

9. Plot of (a) Volume fraction of Oil at the 6 probe points versus the simulation time, (b)

Pressure at the pressure outlet versus the simulation time for a Pressure URF 0.5.

10. Plot of (a) Volume fraction of Oil at the 6 probe points versus the simulation time, (b)

Pressure at the pressure outlet versus the simulation time for a Segregated VOF URF 0.8.

11. Plot of (a) Volume fraction of Oil at the 6 probe points versus the simulation time, (b)

Pressure at the pressure outlet versus the simulation time for a K-ɛ turbulence URF 0.8.

12. Plot of Courant number versus simulation time for a polyhedral mesh.

13. Plot of Courant number versus simulation time for a trimmer mesh of base size 8 mm.

14. Plot of Courant number versus simulation time for a trimmer mesh of base size 4 mm.

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II. List of Tables

1. Meshing parameters for the polyhedral mesh.

2. Physics models.

3. Material properties in the multiphase mixture.

4. Meshing parameters for the two trimmer meshes.

5 (a). Convergence time for different values of Velocity URF at the 6 probe points.

(b). Convergence time for different values of Pressure URF at the 6 probe points.

(c). Convergence time for different values of Segregated VOF URF at the 6 probe points.

(d). Convergence time for different values of K-ε turbulence URF at the 6 probe points.

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Abstract

The phenomenon of sloshing in the oil pan of an automobile moving with a constant

acceleration of 3 m/s^2 has been simulated in STAR-CCM+ v7.02.008, a multi-disciplinary

engineering simulation tool. The Volume Of Fluid multiphase model has been used for the

same. This study consists of two parts, the first one being, optimization of the under-

relaxation factors (URFs) involved in the flow simulation. The optimum values for the

velocity URF, pressure URF, segregated VOF URF, K-Epsilon turbulence URF and K-

Epsilon turbulent viscosity URF have been obtained. The maximum, mean and minimum

Courant numbers for the optimum case have been found out. The second part of the study is

the analysis of the variation of the maximum, mean and minimum Courant numbers with a

change in mesh type from polyhedral to trimmer mesh. This has been performed on two cases

of the trimmer mesh, with base mesh sizes of 8 mm and 4 mm respectively. The results

obtained show that the Courant numbers decrease when the mesh type is changed from

polyhedral to trimmer. Also, the Courant numbers are lower for the trimmer mesh with 4 mm

base mesh size as compared to the trimmer mesh with 8 mm base mesh size. Therefore,

according to the requirements of the user, a balance can be achieved between mesh type,

mesh size and Courant numbers.

1. Background

Oil pans are major engine cooling system parts. They are usually constructed of thin

steel and shaped into a deeper section to fully perform its function. It is also where the oil

pump is placed. When an engine is not running or at rest, oil pans collect the oil as it flows

down from the sides of the crankcase. In other words, oil pans that are mounted at the bottom

of the crankcase serves as an oil reservoir. They also act as a source of structural strength for

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the crank case. Engine oil is used for the lubrication, cooling, and cleaning of internal

combustion engines [1-2]

.

At the bottom of the pan is the oil drain plug that can be usually removed to allow old

oil to flow out of the car during an oil exchange. After the used oil drains out, the plug is

screwed back into the drain hole. Drain plugs are often made with a magnet in it, collecting

metal fragments from the oil. Some contains a replaceable washer to avoid leakage caused by

corrosion or worn threads in the drain hole.

An oil pan is more prone to leaking compared to any other car part. It is because it

holds oil which is being thrown around due to sloshing. Sloshing refers to the movement

of liquid inside another object (which is, typically, also undergoing motion). Here, sloshing

of oil occurs in the oil pan of an automobile that is under motion. The dynamics of the oil

interact with the pan to alter the system dynamics significantly. Other examples

include propellant slosh in spacecraft tanks and rockets (especially upper stages), and cargo

slosh in ships and trucks transporting liquids (for example oil and gasoline).

2. Introduction

The phenomenon of sloshing in the oil pan of an automobile is a multiphase flow

problem. It consists of two phases, air and oil. Multiphase flow refers to the flow and

interaction of several phases within the same system where distinct interfaces exist between

the phases. Multiphase flows can be modelled using the Lagrangian approach or the Eulerian

approach.

For simulating sloshing in an oil pan, Volume Of Fluid (VOF) approach is used [3]

.

The VOF model is used for immiscible fluids and makes use of Eulerian phases. It is well

suited for problems where each phase constitutes a large structure, with relatively small

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contact area between the phases (Figure 1). It acts as an efficient tool for tracking and

locating the position and shape of the fluid-fluid interface.

Under-relaxation factors are significant parameters affecting the convergence of a

numerical scheme. They represent the fraction of the solution being carried forward from one

iteration to the next for the various equations being solved during the simulation. The

Courant number expresses the ratio of the distance travelled by a disturbance in one time step

to the length of a computational distance step. It influences the accuracy and stability of the

solution.

The phenomenon of sloshing in the oil pan of an automobile has been a topic of

intense research, as can be seen evinced from even a cursory glance at literature [4-8]

. The

current exercise has been undertaken with an objective to optimize the values of the various

under-relaxation factors as well as the Courant numbers involved in the simulating sloshing

in an oil pan in order to minimize the simulation time and hence the cost of the simulation.

3. Geometry modelling and simulation

The geometry of the oil pan was created using the STAR-CCM+ 3D-CAD modeller.

The dimensions of the model are shown in Figure 2. The oil pan has a length 59 cm, width 28

cm and height 19 cm. The base of the pan has three steps at depths of 9 cm, 13 cm and 19 cm.

A pressure outlet is present on the top face of the oil pan in order to simulate the atmospheric

pressure acting on the oil surface. The reference pressure was taken as 101325.0 Pa and all

the pressure values were specified with respect to this value.

The mesh generation process was performed with a base size of 8 mm using

polyhedral cells with a prism layer mesh at the boundaries (Figure 3). The surface remesher

was used to improve the quality of the mesh. The meshing parameters used for the process

are shown in Table 1. The physics models were specified to the region as shown in Table 2.

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A 3-dimensional, implicit unsteady model was chosen the VOF method was selected as the

multiphase flow model. The material for the region was specified as a multiphase mixture

consisting of Air and Oil, whose material properties are shown in Table 3. The RANS k-

epsilon turbulence model was used for the simulation along with the additional gravity

model.

Initial and boundary conditions were assigned as follows. The initial level of oil in the

pan was specified as 7 cm from the top surface. The oil pan was given an initial acceleration

of 3 m/s in the forward direction as shown in Figure 2. The pressure at the pressure outlet was

taken as 0 Pa with respect to the reference pressure. Six point probes were created as shown

in Figure 4 to monitor the volume fraction of oil at the points. Pressure at the pressure outlet

was also monitored. Plots were created for the above two quantities versus simulation time.

The solver parameters included an implicit unsteady time-step of 0.03 s and

maximum inner iterations of 20. The tolerance value for determining steady-state was taken

as 0.001 for 100 iterations, that is, the change in the value of the physical quantity should be

less than 0.001 for 100 iterations. The flow was simulated until steady state was obtained

(Figure 5).

The process of optimization of the URFs includes varying the value of one URF,

while keeping the others constant and selecting the value for which the simulation time is the

least. This was carried out for the velocity URF, pressure URF, segregated VOF URF, K-

Epsilon turbulence URF and K-Epsilon turbulent viscosity URF. The maximum, mean and

minimum Courant numbers were then determined for the polyhedral mesh.

The mesh type was changed from polyhedral to trimmer and the maximum, mean and

minimum Courant numbers were found out for two trimmer meshes, one with a base size 8

mm and the other with a base size 4 mm (Figure 6). The meshing parameters for the two

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trimmer meshes are shown in Table 4. The flow was simulated with the default values of

URFs provided by STAR-CCM+.

4. Results and discussion

a. Optimization of the URFs:

(i) Velocity URF: The values of simulation time obtained by varying the value of velocity

URF and keeping the other URFs constant (default values) are shown in Table 5.a. The

number of iterations taken to attain a steady state is shown is found to be the least for a

Velocity URF of 0.9 (Figure 7.a). The same behaviour is observed in the plots obtained for

volume fraction of oil at the six probe points and outlet pressure versus simulation time

(Figure 8).

(ii) Pressure URF: The values of simulation time obtained by varying the value of pressure

URF, keeping the value of velocity URF at 0.9 and the other URFs constant (default values)

are shown in Table 5.b. The number of iterations taken to attain a steady state is shown is

found to be the least for a Pressure URF of 0.5 (Figure 7.b). The same behaviour is observed

in the plots obtained for volume fraction of oil at the six probe points and outlet pressure

versus simulation time (Figure 9).

(iii) Segregated VOF URF: The values of simulation time obtained by varying the value of

segregated VOF URF, keeping the value of velocity URF at 0.9, pressure URF at 0.5 and the

other URFs constant (default values) are shown in Table 5.c. The number of iterations taken

to attain a steady state is shown is found to be the least for a Segregated VOF URF of 0.8

(Figure 7.c). The same behaviour is observed in the plots obtained for volume fraction of oil

at the six probe points and outlet pressure versus simulation time (Figure 10).

(iv) K-ɛ turbulence URF: The values of simulation time obtained by varying the value of K-ɛ

turbulence URF, keeping the value of velocity URF at 0.9, pressure URF at 0.5, segregated

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VOF URF at 0.8 and the K-ɛ turbulent viscosity URF at 1.0 (default value) are shown in

Table 5.c. The number of iterations taken to attain a steady state is shown is found to be the

least for a K-ɛ turbulence URF of 0.9 (Figure 7.d). The same behaviour is observed in the

plots obtained for volume fraction of oil at the six probe points and outlet pressure versus

simulation time (Figure 11).

b. Variation of Courant numbers with the type and size of mesh:

For the polyhedral mesh, using the optimized values of the under-relaxation factors, at

the end of 1000 iterations we obtain the following values for the maximum, mean and

minimum Courant numbers (Figure 12):

Maximum Courant Number = 0.0611.

Minimum Courant Number = 0.0092.

Mean Courant Number = 0.0511.

Total CPU time = 8567 s.

For the trimmer mesh with base size 8 mm (Figure 6.a), using the default values of

URFs, at the end of 1000 iterations we obtain the following values of maximum, mean and

minimum Courant numbers (Figure 13):

Maximum Courant Number = 7.0575.

Minimum Courant Number = 4.5014E-4.

Mean Courant Number = 0.01556.

For the trimmer mesh with base size 4 mm (Figure 6.b), using the default values of

URFs, at the end of 2000 iterations we obtain the following values of maximum, mean and

minimum Courant numbers (Figure 14):

Maximum Courant Number = 5.8065.

Minimum Courant Number = 1.2924E-4.

Mean Courant Number = 0.0080.

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For the two trimmer meshes used, it was observed that the number of iterations

required to attain a steady state is more for the trimmer mesh with base size 4 mm. Due to its

smaller mesh size, the amount of computation involved is more. The maximum, mean and

minimum Courant numbers obtained for this mesh are significantly lower compared to the

trimmer mesh with base size 8 mm.

5. Conclusion

Optimization of the values of under-relaxation factors was performed for the case of

sloshing in the oil pan of an automobile moving with a constant acceleration of 3 m/s2 was

performed and the URF values after optimization were found to be as follows:

Velocity URF : 0.9

Pressure URF : 0.5

Segregated VOF URF : 0.8

K-ɛ turbulence URF : 0.9

K-ɛ turbulent viscosity URF : 1.0 (default)

The maximum, mean and minimum Courant numbers for a polyhedral mesh were

found to be lower than those for a trimmer mesh of the same base mesh size (8 mm). With a

decrease in the base size of the trimmer mesh to 4 mm, there was a significant decrease in the

Courant numbers. The simulation time, however, was higher. Therefore, according to the

requirements of the developer such as mesh type and mesh size, optimum values of Courant

numbers are obtained.

In terms of future scope, optimization of URFs can be performed for various other

conditions such as sloshing in the oil pan during braking, varying accelerations, etc.

Optimization is a necessity for almost all simulations and hence the same methodology can

be utilized to reduce the simulation time for a variety of problems.

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6. References

[1] http://en.wikipedia.org/wiki/Crankcase

[2] http://www.carpartswholesale.com/cpw/oil_pan.html

[3] STAR-CCM+ Code User Manual Version 7.02.008 (2012), CD-adapco, NY 11747, USA.

[4] E Kopec, W Oberknapp, “Slosh baffle for oil pan of internal combustion engine.” US

Patent 4,449,493, 1984.

[5] TM Bishop, “Oil pan with vertical baffles for oil flow control.” US Patent 6,845,743,

2005.

[6] RK Shier, “Engine having oil pan with deflection vanes.” US Patent 3,425,514, 1969.

[7] M Beer, “Oil container and a process for the production thereof.” US Patent 7,077,285,

2006.

[8] JR Lang, “Oil level sensing apparatus.” US Patent 4,091,895, 1978.

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7. Figures

Figure 1. Illustration of grids that are unsuitable (left) and suitable (right) for two-phase

flows using the VOF model.

Figure 2. 3D model of the oil pan

Figure 3. Section plane showing the polyhedral mesh generated

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Figure 4. Scalar scene after initialization showing the 6 point probes

Figure 5. Scalar scene after the attainment of steady state

Figure 6. (a) Trimmer mesh with base size 8 mm, (b) Trimmer mesh with base size 4 mm

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Figure 7. Plots of URF values versus simulation time at the 6 probe points

Figure 8. Plot of (a) Volume fraction of Oil at the 6 probe points versus the simulation time,

(b) Pressure at the pressure outlet versus the simulation time for a Velocity URF 0.9

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Figure 9. Plot of (a) Volume fraction of Oil at the 6 probe points versus the simulation time,

(b) Pressure at the pressure outlet versus the simulation time for a Pressure URF 0.5

Figure 10. Plot of (a) Volume fraction of Oil at the 6 probe points versus the simulation time,

(b) Pressure at the pressure outlet versus the simulation time for a Segregated VOF URF 0.8

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Figure 11. Plot of (a) Volume fraction of Oil at the 6 probe points versus the simulation time,

(b) Pressure at the pressure outlet versus the simulation time for a K-ɛ turbulence URF 0.8

Figure 12. Plot of Courant number versus simulation time for a polyhedral mesh

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Figure 13. Plot of Courant number versus simulation time for a trimmer mesh of base size 8

mm

Figure 14. Plot of Courant number versus simulation time for a trimmer mesh of base size 4

mm

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8. Tables

Mesh Parameter Value

Base size 8 mm

Number of prism layers 2

Prism layer stretching 1.2

Prism layer thickness 4 mm

Surface curvature 18 pts/circle

Surface growth rate 1.3

Table 1. Meshing parameters for the polyhedral mesh

Table 2. Physics models

Material Density (kg/ ) Viscosity (E-5 Pa.s)

Air 1.18415 1.85508

Oil 890 8.55

Table 3. Material properties in the multiphase mixture

Mesh Parameters Trimmer mesh

with base size 8

mm

Trimmer mesh

with base size 4

mm

Base size 8 mm 4 mm

Maximum cell size 10000% of base 100% of base

Number of prism layers 2 2

Prism layer stretching 1.2 1.2

Surface curvature 18 pts/circle 36 pts/circle

Surface minimum size 2 mm 1 mm

Surface target size 8 mm 4 mm

Table 4. Meshing parameters for the two trimmer meshes

Physics parameter Type of model

Space Three dimensional

Time Implicit unsteady

Material Multiphase mixture

Multiphase flow model Volume Of Fluid model

Viscous Regime Turbulent

Turbulence model RANS K-epsilon turbulence

Optional model Gravity

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Velocity URF Number of iterations for convergence

Top1 Top2 Mid1 Mid2 Bottom1 Bottom2

0.5 1063 1127 1250 1007 1591 1871

0.6 1143 798 849 928 1353 1907

0.7 846 768 862 727 1101 1090

0.8 813 660 600 661 1086 1148

0.9 612 725 560 636 883 983

Table 5. (a) Convergence time for different values of Velocity URF at the 6 probe points

Pressure URF Number of iterations for convergence

Top1 Top2 Mid1 Mid2 Bottom1 Bottom2

0.1 1300 1301 802 1289 1364 1842

0.2 980 1043 681 910 1328 1713

0.3 668 796 600 681 1162 1429

0.4 612 725 560 636 883 983

0.5 588 571 538 593 926 723

Table 5. (b) Convergence time for different values of Pressure URF at the 6 probe points

Segregated VOF

URF

Number of iterations for convergence

Top1 Top2 Mid1 Mid2 Bottom1 Bottom2

0.9 732 703 570 743 769 1211

0.8 588 571 538 593 926 723

0.7 568 798 549 781 995 834

0.6 606 650 579 780 756 786

Table 5. (c) Convergence time for different values of Segregated VOF URF at the 6 probe

points

k-ε turbulence

URF

Number of iterations for convergence

Top1 Top2 Mid1 Mid2 Bottom1 Bottom2

0.9 584 572 545 599 696 759

0.8 588 571 538 593 926 723

0.7 627 693 538 606 786 1086

0.6 607 607 545 597 1182 1122

Table 5. (d) Convergence time for different values of K-ε turbulence URF at the 6 probe

points