Numerical Study of Viscous Flow in the Hydraulic System of Electro Optical Tracking System

Abstract

In this chapter, we present a study of numerical simulation of centerline velocity, velocity contour and wall shear stress for a two dimensional viscous and incompressible fluid flow in rectangular pipe. The numerical results have been corroborated through a scaling law and asymptotic analysis. It deals with simulation of viscous flow in a typical hydraulic control system can be used in Electro Optical Tracking System (EOTS). Due to geometric constraint, the typical piping can be used in hydraulic circuit of EOTS is of rectangular (with aspect ratio p factor = 1) cross section. The two dimensional governing equation of laminar flow of highly viscous fluid is solved in the present work by using finite difference method. Through extensive simulation, the grid independence of centerline velocity and wall shear stress has been established in the present study. In addition a scale analysis approach and asymptotic analysis of the problem have been carried out. The axial velocity profile in 3D space and corresponding contour has been computed here. It has been demonstrated that the velocity contour is parabolic in nature. The present work also establishes the fact that the velocity profile remains parabolic for rectangular pipe with varying cross sectional aspect ratio (p factor). At different p factor, the centerline velocity and wall shear stress have also been presented in this chapter.

Appendices

Appendix: A

Tri - diagonal Matrix Algorithm.

The partial differential equation after discretization using finite difference method has been casted into the following form.

$$ b_{i} u_{i} = a_{i} u_{i - 1} + c_{i} u_{i + 1} + d_{i} $$

(1)

The arrangement of the grids are as shown in the following figure

Taking boundary nodes into considerations we can easily conclude that $a_{1} = 0,\,c_{N} = 0$

Now for i = 1

$$ b_{1} u_{1} = c_{1} u_{2} + d_{1} $$

$$ u_{1} = \frac{{c_{1} }}{{b_{1} }}u_{2} + \frac{{d_{1} }}{{b_{1} }} $$

Similarly for i = 2:

$$ \begin{aligned} b_{2} u_{2} &= a_{2} u_{1} + c_{2} u_{3} + d_{2} \\ &= \left( {\frac{{c_{1} }}{{b_{1} }}} \right)a_{2} u_{2} + \left( {\frac{{d_{1} }}{{b_{1} }}} \right) + c \\ \end{aligned} $$

It observed that u _i depends on u _i+1, so following above we can write

$$ u_{i} = P_{i} u_{i + 1} + Q_{i} $$

(2)

$$ u_{i - 1} = P_{i - 1} u_{i} + Q_{i - 1} $$

(3)

$$ b_{i} u_{i} = a_{i} u_{i} P_{i - 1} + a_{i} Q_{i - 1} + c_{i} u_{i + 1} + d_{i} $$

$$ \left( {b_{i} - a_{i} P_{i - 1} } \right)u_{i} = c_{i} u_{i + 1} + d_{i} + a_{i} Q_{i - 1} $$

$$ {u}_{i} = \frac{{c}_{i} }{{b}_{i} - {a}_{i} {p}_{{i} - 1}} {u}_{i + 1} + \frac{{d}_{i} + {a}_{i}{Q}_{i} - 1} {{b}_{i} - {a}_{i}{p}_{i} - 1} $$

(4)

Now comparing Eqs. (2) and (4) we can write,

$$ {p}_{i} = \frac{{c}_{i} }{{{b}_{i}} - {a}_{i} {p}_{{i} - 1}} $$

$$ {Q}_{i} = \frac{{d}_{i} +{a}_{i} {Q}_{{i} - 1}} {{b}_{i} - {a}_{i}{p}_{{i} - 1}} $$

$$ P_{N} = 0; {Q}_{N} = \frac{{d}_{N} + {a}_{N} {Q}_{N} - 1} {{b}_{N} } $$

Now the algorithm can be summarized as,

Algorithm:

• 1. Calculate P₁, Q₁; $P_{1} = \frac{{c_{1}}}{{b_{1}}},Q_{1} = \frac{{d_{i}}}{{b_{i}}}$

• 2. Compute P_i, Q_i for i = 1,2,3 … N

• 3. u_N ← Q

• 4. u _i  = P _i u _i+1 + Q _i i = N−1, 1, −1.

Appendix: B

The TDMA described in appendix can give direct solution of the discrete system obtained by discretizing on dimensional equation in FDM. However the matrix obtained for two dimensional systems is penta-diagonal. Although direct solution can be possible using Gauss-elimination method (GEM), however the computation cost of GEM is of order N ³. So to take the advantage of TDMA whose computation cost is of order N only, line-by-line TDMA may be used for such system of equation. This uses the TDMA in conjunction with Gauss-Sedial iteration and the detail is explained in the following figure:

$$ \begin{gathered} a_{p} u_{p} = a_{E} u_{E}^{ * } + a_{w} u_{w}^{ * } + a_{s} u_{s} + a_{N} u_{N} + b \hfill \\ a_{i} = a_{s} ;b_{i} = a_{p} ;c_{i} = a_{N} ;d_{i} = b + a_{E} u_{E}^{*} + a_{w} u_{w}^{*} \hfill \\ \end{gathered} $$