1. Computational Hydrodynamics
Red Sea University Faculty of Engineering Department of Mechanical Engineering
Computational Hydrodynamics
Moataz Abdelgadir Ali Abdelgadir

2. Intro
The differential equations which arose from the theories of Reynolds rapidly exceeded the capacity of analytical solution.
Numerical solutions to hydrodynamic lubrication problems can now satisfy most engineering requirements for prediction of bearing characteristics and improvements in the quality of prediction continue to be found.

3. Intro
A popular numerical technique, the 'finite difference method', is introduced and its application to the analysis of hydrodynamic lubrication is demonstrated.
The steps necessary to obtain solutions for different bearing geometries and operating conditions are discussed.

4. Dimensionless Reynolds equation
Non-dimensionalization is the substitution of all real variables in an equation, e.g., p, h, etc., by dimensionless fractions of two or more real parameters.
A basic disadvantage of a numerical solution is that data are only provided for specific values of controlling variables, e.g., one value of friction force for a particular combination of sliding speed, lubricant viscosity, film thickness and bearing dimensions.

5. Dimensionless Reynolds equation
Analytical expressions, on the other hand, are not limited to any specific values and are suited for providing data for general use, for example, they can be incorporated in an optimization process to determine the optimum lubricant viscosity.
The benefit of non dimensionalization is that the number of controlling parameters is reduced and a relatively limited data set provides the required information on any bearing.

6. Dimensionless Reynolds equation
The Reynolds equation is expressed in terms of film thickness 'h', pressure 'p', entraining velocity 'U' and dynamic viscosity '' as follow
It’s non-dimensional forms of the equation's variables are:
h* = h/c
x* = x/R
y* = y/L
p* = pc2/ 6UR 1

7. Dimensionless Reynolds equation
The Reynolds equation in its non-dimensional form is:
All terms in equation (2) are non-dimensional apart from 'R' and 'L' which are only present as a non-dimensional ratio.
Although any other scheme of non-dimensionalization can be used, this particular scheme is the most popular and convenient.
For planar pads, 'R' is substituted by the pad width 'B' in the direction of sliding. 2

8. The Vogelpohl parameter Mv
It was developed to improve the accuracy of numerical solutions of the Reynolds equation and was introduced by Vogelpohl in the 1930's.
The Vogelpohl parameter 'Mv' is defined as follows:
Substitution into the non-dimensional form of Reynolds equation (2) yields the 'Vogelpohl equation': 3 4

9. The Vogelpohl parameter Mv
where parameters 'F' and 'G' for journal bearings are as follows:
The Vogelpohl parameter facilitates computing by simplifying the differential operators of the Reynolds equation, and furthermore it does not show high values of higher derivatives in the final solution, i.e., dnMv/dx*n where n > 2, unlike the dimensionless pressure 'p*'. 5

10. Finite difference method
Journal and pad bearing problems are usually solved by a 'finite difference' method,
Although a 'finite element' method has also been employed
The finite difference method is based on approximating a differential quantity by the difference between function values at two or more adjacent nodes.
For example, the finite difference approximation to Mv/x* and for the second differential 2Mv/x*2 are given by

11. Finite difference method
where the subscripts i-1 and i+1 denote positions immediately behind and in front of the central position 'i' and 'x*' is the step length between nodes. 6

12. Finite difference method
The finite difference equivalent of equ. (4) is found by considering the nodal variation of 'Mv' in two axes, i.e., the 'x' and 'y' axes.
A second nodal position variable is introduced along the 'y' axis, the 'j' parameter.
The expressions for Mv/y* and 2Mv/y*2 are exactly the same as the expressions for the 'x' axis but with 'i' substituted by 'j'.

13. Finite difference method
The equation (4) can be written in it’s final form as:
Where

14. Definition of Solution Domain and Boundary Conditions
After establishing the controlling equation, the next step in numerical analysis is to define the boundary conditions and range of values to be computed.
For the journal or pad bearing, the boundary conditions require that 'p*' or 'Mv' is zero at the edges of the bearing and also that cavitation can occur to prevent negative pressures occurring within the bearing.

15. Definition of Solution Domain and Boundary Conditions
The range of 'x*' is between 0- 2 (360 angle) for a complete bearing or some smaller angle for a partial arc bearing. T
he range of 'y*' is from-0.5 to +0.5 if the mid-line of the bearing is selected as a datum.
A domain of the journal bearing where symmetry can be exploited to cover either half of the bearing area, i.e., from y* = 0 to y* = 0.5, or the whole bearing area

16. Definition of Solution Domain and Boundary Conditions
Nodes on the edges of the bearing remain at a pre-determined zero value while all other nodes require solution by the finite difference method.
When symmetry is exploited to solve for only a half domain, it should be noted that nodes on the mid-line of the bearing are also variable and the finite difference operator requires an extra column of nodes outside the solution domain, as zero values along the edge of the solution domain cannot be assumed.

17. Definition of Solution Domain and Boundary Conditions
This extra column is generated by adopting node values from the column one step from the mid-line on the opposite side. In analytical terms this is achieved by setting:

18. Definition of Solution Domain and Boundary Conditions

19. Definition of Solution Domain and Boundary Conditions
After establishing the controlling equation, the next step in numerical analysis is to define the boundary conditions and range of values to be computed.
For the journal or pad bearing, the boundary conditions require that 'p*' or 'Mv' is zero at the edges of the bearing and also that cavitation can occur to prevent negative pressures occurring within the bearing.
The range of 'x*' is between 0- 2 (360 angle) for a complete bearing or some smaller angle for a partial arc bearing.
The range of 'y*' is from-0.5 to +0.5 if the mid-line of the bearing is selected as a datum.
A domain of the journal bearing where symmetry can be exploited to cover either half of the bearing area, i.e., from y* = 0 to y* = 0.5, or the whole bearing area

20. Solution Algorithm

21. Solution Algorithm - 2