Numerical simulation of two-dimensional laminar unsteady flow past a right trapezoidal cylinder at low Reynolds number: study of sharpening angle, time step, grid independence and domain size

Numerical simulation of two-dimensional laminar unsteady flow past a right trapezoidal cylinder at low Reynolds number (Re = 100), zero of the flow approaching angle and sharpening angle of the right trapezoidal of 22.5 with a side ratio B/A = 1 are carried out to provide more applicable data for engineering design of barred tee in aspect of structural integrity. A finite volume method, non-uniform meshing with second-order implicit time discretization into eightnode quadratic quadrilateral finite elements is employed. An incompressible flow SIMPLEC code with constant fluid properties is used. The convective terms using a third-order QUICK scheme. The numerical simulation result is compared against the published results of flow past a square cylinder. The effect of sharpening angle on the response of the right trapezoidal cylinder is investigated. A special study of the effects of flow on significant factor for time step, grid independence, blockage ratio, domain size, upstream and downstream extents, size domain next to cylinder and size domain extent are performed systematically. The Strouhal number and RMS lift coefficients of fully saturated flow are calculated. The result shown that increasing of sharpening angle, the Strouhal number is negligible changed whilst the RMS lift coefficients significantly increased.


Introduction
In long piping or pipeline system normally used pigging for various internal activities such as separating fluids, cleaning and inspecting the pipeline. Pig is a spherically shaped device which can be propelled through a main pipeline by pressurized fluid flow from pig trap launcher through main line, branch line and together with all Fig. 1. Schematic of flow past barred tee associated piping, valves, pipe supports and instruments to pig trap receiver. The main line is major portion of a pipeline, between pig traps which compose many branch connection line. When piping or pipeline system is liquid filled into system to operate, the flow past main line and flow into branch line and past barred tee. In order to prevent the passing of pigs/spheres through branch pipe as show on Fig. 1. and Fig. 2. the barred tees (guided bars) is provided, filled-welded attached cross inside branch at connection point. Typically barred tees be installed on all branches larger than 50% or 25% of the main pipeline diameter where unwrapped foam pigs or spheres are to be used.
The barred tees have to integrity against to fluids flow induced vibration. In engineering design point of view, we concern to the synchronize between natural frequency of barred tee with vortex shedding frequency of fluid flow as past barred tee, the synchronize vibration of them cause barred tee experienced severe vibration then fatigue, detached from wall pipe into main pipe line which can be resulted pig stuck, valve face crash or scraper flow in to other inline equipment such as flow control valve and pump impeller. More additional cause  Fig. 2. the flow past a barred tee in which composed with several cross-section shape of a square, a rectangular and a right trapezoidal cylinder.
By numerical simulation, the enormous increase on the powerful processing of modern computers simulation has made it possible to identify and to study several mechanisms on flow past two dimensional cylindrical bluff body, many work used numerical simulation for flow past over cylindrical shape.
Sohankar et al. [1] carried out numerical simulations of two dimensional unsteady flow around a square cylinder at zero approaching angle α = 0 o , Re = 100 to study the influence of time step Δt, size domain next to cylinder ζ, grid size, extend of computation domain size Δ, upstream extends X u and downstream extends X d of the calculation domain, side wall height, respectively. The method for numerical simulation is an incompressible SIMPLEC finite volume code, nonstaggered grid arrangement and Neumann outlet boundary condition. The convective terms are a thirdorder QUICK scheme and a second-order Van Leer scheme. Implicit time discretization and second-order Crank-Nicolson scheme is employed. For accurate calculation [1] finally used of Δt = 0.025 and ζ = 0.004. The appropriate value of X u = 11.1 and X d ≥ 25 units.
Sohankar et al. [2] performed and provide a number of quantities same as [1]  Arnab et al. [5] performed numerical simulations of two-dimensional laminar flow past a triangular cylinder in which placed in free-stream at 10 ≤ Re ≤ 250 by method a finite volume, second-order accurate in space and time, employing non-staggered arrangement of the variables with momentum interpolation for the pressure-velocity coupling. Result shown that vortex shedding phenomena of a triangular cylinder is same as to the square cylinder. Effect of grid refinement, X u and X d of the calculation domain are investigated and also study on the RMS lift coefficient and Strouhal number.
Mittal et al. [6] investigated effect of the blockage ratio on numerical simulation for flow past circular cylinder at Re ≤ 150 with a stabilized space-time finite element formulation, γ = 1% and 5%. From literature as mention above and in the past, the most applicable engineering design data for flow past over circular, squre cylinder, rectangular, trangle and isosceles trapezoidal are emphasis intense investigated but more absence applicable engineering design data for right trapezoidal cylindrical cross section in which to used in barred tee engineering design.
In this study, numerical simulations are carried out to investigate Strouhal number, RMS lift coefficient, of a right trapezoidal cylinder in unsteady flow to provide further more applicable data for engineering design of barred tee in aspect of structural integrity. The effects of time step, grid independence, sharpening angle β, blockage ratio, domain size, upstream and downstream extents, size domain next to trapezoidal cylinder are studied systematically.  The confined boundary parameter which is blockage ratio is defined as γ = d/H, where velocities scaled with U ∞ . The origin of force coordinates is placed at the center of right trapezoidal cylinder geometry with drag and lift force positive in the D and L-direction, respectively.

Numerical details
The following numerical parameter were used in this study, unless otherwise specific, Re =100, α = 0 o , β = 22.5 o , B/A = 1, γ = 0.05, X u = 10 and X d = 15, ζ = 0.004 and extends all side of 5 units from cylinder, the mesh distribution was made non-uniform with a vary mesh size of 0.004-0.5 and outside a region from the cylinder which extends 5 units, the mesh distribution was made uniform with a constant mesh size Δ = 0.5, the number of nodes on each side of cylinder N b = 20 nodes as shown in Fig. 4.
A finite volume method, non-uniform meshing with second-order implicit time discretization into eightnode quadratic quadrilateral elements is employed. An incompressible flow SIMPLEC code with constant fluid properties such as kinematic viscosity and density are used and the convective terms using a third-order QUICK scheme same as [1][2][3]. At the inlet which is located X u units upstream from the cylinder, a uniform flow velocity was definded in streamwise direction U = 1, in tranverse streamwise direction V = 0. At the outlet which is located X d units downstream from the cylinder,

Result and discussion
To avoid inaccuracy from numerical simulation, the study effect of modeling parameter are performed. By using the Strouhal number and RMS lift coefficients as indicator the strong sensitivity [2][3] represent to another sensitivity numerical parameters from fully saturated flow calculation.

Influence of time step
In order to capture the shedding phase and amplitude accuracy, the influence of the time step is investigated as presented in Table 1. These results indicated that with the decrease of time step the Strouhal number and RMS lift coefficient are increased. Decreasing it from 0.2 to 0.1, the Strouhal number and RMS lift coefficient are increased of 3.03% and 2.29% respectively. Decreasing it further to 0.05, the Strouhal number and RMS lift coefficient are increased of 0.76%, 0.87%, respectively. It negligible changed and next decreasing to 0.025, the Strouhal number and RMS lift coefficient are negligible changed. The time step of 0.05 and Courant number of 12.5 is chosen to used for grid independence study further from reason as mention above and minimize the Courant number. This value less than [1][2][3] which them used 0.025.

Grid independence study
To avoid rapidly changing in high gradient area [1] the grid distribution is studied, the result is shown in Table  2. Decreasing the size domain next to cylinder from 0.004 to 0.002, consequently grid density changed 150x138 to 151x139. From the result both the Strouhal number and RMS lift coefficient are decreased of 0.76% and 0.24% respectively. More additional study, fixed the size domain next to cylinder of 0.002, increase the nodes on each side of cylinder to 40 nodes per cylinder side, the Strouhal number and RMS lift coefficient are again increased of 0.76% and 0.12% respectively. Finally it is decided to used the size domain next to cylinder by 0.004, grid density 150x138 and the nodes on each side of cylinder to 20 nodes for further investigate.   It is indicated that for β = 0, which is square cylinder, symmetry. Where sharpening angle is increased, RMS lift coefficient evidently increase but the Strouhal number constant that conformable to square cylinder at some approaching angle [4]. From Table 4. It can see that flow past the right trapezoidal cylinder is quite similar to the square cylinder on phase of transverse oscillation although in RMS lift coefficient is more effect than square cylinder at zero flow approaching but tend to similar to the square cylinder with flow approaching rather than zero.

Conclusions
Numerical simulation of two-dimensional laminar flow past a right trapezoidal cylinder in unsteady flow at Re = 100, α = 0 o and β = 22.5 o with B/A = 1 have been carried out. The flow past the right trapezoidal cylinder is slightly difference to the square cylinder on phase of transverse oscillation at zero flow approaching, The RMS lift coefficient is increased follow sharpening angle increasing and to seem to similar square cylinder at flow approaching rather than zero. The increasing of sharpening angle, the Strouhal number is negligible changed whilst the RMS lift coefficients significantly increased.
The Strouhal number of 0.1527 and RMS lift coefficient of 0.7641 in this study are provided for more applicable data for engineering design of barred tee in aspect of structural integrity further.