Numerical Simulation of flow in pipe with cross jet effects

A numerical method is developed to obtain two-dimensional velocity and pressure distribution through a cylindrical pipe with cross jet flows. The method is based on solving partial differential equations for the conservation of mass and momentum by finite difference method to convert them into algebraic equations. This well-known problem is used to introduce the basic concepts of CFD including: the finite-difference mesh, the discrete nature of the numerical solution, and the dependence of the result on the mesh refinement. Staggered grid implementation of the numerical model is used. The set of algebraic equations is solved simultaneously by “SIMPLE” algorithm to obtain velocity and pressure distribution within a pipe. In order to verify the validity for present code, the flow behavior predicted by this code is compared with these of another studies and there is a good agreement is obtained


‫اﻟﻤﺘﻘﺎﻃﻊ‬ ‫اﻟﺠﺮﻳﺎن‬ ‫ﺧﻼل‬ ‫ﺟﺎﻧﺒﻲ‬ ‫ﻧﻔﺚ‬ ‫ﺑﻴﺘﺄﺛﻴﺮ‬ ‫اﻧﺒﻮب‬ ‫ﻓﻲ‬ ‫ﻟﻠﺠﺮﻳﺎن‬ ‫اﻟﻌﺪدي‬ ‫اﻟﺘﻤﺜﻴﻞ‬
Presented numerical study to predict hydrodynamic and thermal characteristics in a pipe with sinusoidal wavy surface for steady laminar flow . The integral forms of governing equations are discretized using control volume based Finite Volume method with collocated variable arrangement. SIMPLE algorithm is used and TDMA solver is applied for solution of system of equations. (Shimomukai and Kanda 2008). Studies the computation of pipe flow in the entrance region. The pressure distribution and flow characteristics, particularly the effect of vorticity in the vicinity of the wall, are analyzed for Reynolds numbers ranging from 500 to 10000. (Smith et al., 2008). Described the flow through circular orifice by using computational fluid dynamics (CFD) with various turbulence modeling. Effects of orifice diameter ratios (d/D = 0.5, 0.6, and 0.8) on flow field characteristics is extensively investigated. To study the influence of turbulence model on the predicted results, the standard k-ε turbulence model was employed to compare with the Reynolds Stress Model.

(Voronova and Nikitin 2006). Present
complete Navier-Stokes equations for the turbulent flow in a pipe of elliptical crosssection with semiaxis ratio b/a = 0.5 is directly calculated for the Reynolds number Re = 6000 (determined from the mean-flow velocity and the hydraulic diameter). The distribution of the average and pulsatory flow characteristics over the pipe cross-section are obtained. In particular, the secondary flow in the crosssection plane, typical of turbulent flows in noncircular pipes, is calculated. In the present study a two-dimensional numerical method is presented to obtain velocity and pressure distribution through a cylindrical pipe. The method is based on solving partial differential equations for the conservation of mass and momentum by finite difference method.

The governing equations
The equations for conservation of mass and momentum for incompressible, steady and laminar flow in cylindrical coordinate system of the r and z directions are given below (Graebel, 2007) (i) Mass Conservation .

Finite Difference Formulation of the Equations
The basic of the numerical method is the conversion of the differential equations (1, 2 and 3) into algebraic equations relating the value of the dependent variables at the considered grid point to the its values at the neighboring grid points. This was done by finite difference method.
After treating the governing equations by (FDM), the general form of the resultant equation can be termed as ( Patankar 1980) : Where the subscript ( P ) denotes the corresponding grid point. The small letters subscript (e, w, n and s ) denote the value of the variable at the faces of the control volume. Fig (1) shows the corresponding grid point and its neighbor grid points, and the u velocity component.
Table (1) illustrates the value of (B) for Equation (4) and the value of (B 1 ) for Equation (5e) for the governing equations.

Method of Solution
The first step in the solution is dividing the flow field into grid points, then the partial differential equations would be transformed into an algebraic form by finite-difference method as illustrated in the previous section. The discretized procedure of the equation is based on the power law scheme ( Patankar 1980) and the discretized equations are solved by (TDMA) ( Try Diagonal Matrix Algorithm) with underrelaxation factor 0.75 for pressure and 0.45 for velocity. The pressure and velocity are linked by the SIMPLE algorithm ( Patankar 1980).

Computer Program Descriptions
A computer program in FORTRAN-90 is written to solve a set of the partial differential equations that govern the flow field. The field is divided into grid points, which are distributed in r and z coordinates. There are four sets of grid without clustering are tested, where (30*15) is chosen because this set ensure good results in addition is a time saving as shown in Figure (2

Staggered grid
Co-located storage of the pressure and velocity variables at the cell centers leads to the problem of checker boarding. This is because the cell centre values of pressure and velocity get cancelled out on expanding the face gradient terms. To overcome this problem staggered grid has been used for discretization of the momentum equations. The staggered grid for the u momentum equation is shown in Figure (1) along with the neighboring velocity vectors for calculation of velocity gradients. Staggered grid in vertical direction is used for v momentum equation. Pressure is stored on the original grid and the pressure difference terms are evaluated as a difference of cell centre pressure values.

Under-relaxation
The velocity corrections are approximated by dropping the velocity part of the corrected momentum equations which places the entire burden of the velocity correction on pressure correction. Large pressure corrections might lead to poor pressure iterates so the pressure correction is under-relaxed to correct p*.It is necessary to under-relax the momentum equations due to the nonlinear nature of the equations.

SIMPLE Solver Algorithm
Semi-Implicit Method for Pressure-Linked Equations was first proposed by Patankar and Spalding (1972). Here we start with the discrete continuity equation and substitute into this the discrete u and v momentum equations containing the pressure terms resulting in a equation for discrete pressures. SIMPLE actually solves for a relative quantity called pressure correction. We guess an initial flow field and pressure distribution in the domain. The set of momentum and continuity equations are coupled and are nonlinear so we solve the equations iteratively. The pressure field is assumed to be known from the previous iteration. Using this u and v momentum equations to solve for the velocities. At this stage the newly obtained velocities don't satisfy continuity since the pressure field assumed is only a guess. Corrections to velocities and pressure are proposed to satisfy the discrete continuity equation.
Where u* v* and p* are the guess values and u′ , v′ and p′ are the corrections.

Results
In order to validate the present code, the velocity profile is compared with that of (Muppidi, and Mahesh 2006) in figures (4) and (5) . These figures shown good agreement for velocity contours especially at the cross flow region and down stream up to the pipe end. This code was them used to simulate a laminar flow in pipe, where figure (6) shows the velocity vector and axial velocity contour along the pipe from inlet to exit. The velocity is reducing towards the pipe wall and becomes zero a the wall, while the maximum value of the velocity is at the center of the pipe. Figure  (7) shows the pressure contour along the pipe. From this figure the pressure is reducing with the pipe length as a result of the friction losses inside the pipe thus the velocity increasing towards the exit of the pipe . Figure (8) show the another case of cross flow which is found in a several engineering applications, where the velocity vector for this application for two cases are predicted , a) Cross flow from the upper wall of the pipe and b) Cross flow from both sides of the pipe. The effect of cross flow for both cases is seen by the separation and formation of vortex in the down stream regions. Therefore, cross flow can be used to eliminate the effect of boundary layer as seen in figure (6) of developing pipe flow which is unwanted in many engineering applications.

Conclusions
Two-dimensional numerical study of laminar flow in a pipe with cross flow is presented. Finite difference is used to solve two-dimensional Naviar Stokes equations; Staggered grid is used to avoid the difficulties that result from calculating the velocity and pressure at the grid point. The resultant algebraic equations are solved by "SIMPE algorithm" to define velocity and pressure distribution. The interaction of cross jet with the main pipe flow was predicted accurately .the vortices that formed at downstream region was seen to destroy the boundary layer originally generated in developing pipe flow.