Numerical investigation of flow rate measure using vortex counting at low Reynolds number

You The trapezoidal bluff body is a typical configuration of vortex shedding bodies. The aim of this work is to study flow behaviour over a trapezoidal cylinder at low Reynolds number. The geometry was constructed from a prototype device for measuring the volumetric flow-rate by counting vortices. Simulations were run for this geometry under steady and unsteady flow conditions using finite volume discretization. Laminar flow was investigated in this model with rigid walls and homogeneous incompressible Newtonian fluid. Calculations were performed for Reynolds number range 5 Re 200 and several flow parameters were documented. The present computations are in good agreement with the experimental observations and the numerical calculations by several investigators. Keywords—bluff body, confined flow, numerical calculations, steady and unsteady flow, vortex shedding flow meter


Introduction
Flow around an obstacle has been subjected to numerous theoretical, experimental and computational investigations during several decades.The basis of flow around circular cylinders have been reported in the standard reference by Zdravkovich [1] where Reynolds numbers range were given for different regimes of flow around smooth circular cylinder in steady flow.Tian and Wu [2] generalized a relationship between Strouhal and Reynolds numbers by an extensive analytical and computational investigation of two-dimensional flow around regular polygons at low Reynolds number.Lee [3]studied numerically the symmetrical wake flow developments around a tapered trapezoidal cylinder.Kahawita and Wang [4] discussed briefly the influence of the trapezoidal shape on the vortex shedding and on the critical Reynolds number value.Dhiman and Hasan [5] studied the flow and heat transfer across a trapezoidal cylinder.Chung and Kang [6] reported that Strouhal number variation depends on Reynolds number and the height ratio.Pankanin [7] analyzed the various methods of investigating phenomena of the vortex flow meter and observed that the bluff body most suitable for the design of this type of device is a sharp edged trapezoidal cylinder.
The flow across this kind of obstacles is complicated and still not well documented.The present work is concerned with laminar flow at low Reynolds around a trapezoidal bluff body in both steady and unsteady regimes.

Computational geometry
The details of geometry are reported in a paper flow measurement by vortex shedding flow meter [8].It consists of a square-section pipe in which a two-dimensional trapezoidal cylinder is placed Fig. 1-a.Mesh generation was performed using different blocking strategies.Four blocks were used to generate an o-grid type mesh around a trapezoidal cylinder.A fifth block is attached to the o-grid mesh to extend the domain in the downstream region of the flow as shown in Fig. 1-b.In order to achieve mesh independent results, a mesh dependency study is done for the current case.Convergence of the grid is assessed by refining the mesh in terms of applying successively smaller elements.The grid is considered to be sufficiently fine when parameters do not differ by more than 3-5% between two succeeding grids.

Governing equation
The flow was described by the dimensionless Navier-Stokes equations for unsteady laminar incompressible flow.Equations may be written as follows: The continuity equation: The momentum equation: Where, represents the velocity component, is the pressure, is the fluid density and is the dynamic viscosity.

Boundary conditions
Boundary conditions used are summarized in the table 1: The inlet condition is a uniform velocity at the inlet and a zero normal pressure gradient.The trapezoidal cylinder, upper and lower wall are considered a rigid and impermeable wall and hence the no-slip velocity and a zero normal pressure gradient are appropriate.At the domain outlet a zero normal velocity gradient and an absolute pressure of zero are prescribed.

Numerical solution
Solution was performed using the pisoFOAM solver [9].It is a transient solver for incompressible laminar or turbulent flow.It is based on the Implicit with Splitting of Operators (PISO) algorithm for coupling pressure-velocity.In this algorithm: momentum equations are solved for intermediate pressure field in the predictor step without respect of continuity condition, then velocity and pressure fields are corrected in a way to fulfill both momentum and continuity equations in the two predictor steps.A standard second order finite volume discretization of Gaussian integration scheme is applied for the gradient terms using a linear interpolation scheme from cell centers to face centers.An explicit nonorthogonal correction scheme is used to compute surface normal gradients which were evaluated at cell faces.The linear corrected Gauss scheme is applied to for the Laplacian and divergence terms.The implicit first order Euler scheme is applied for the time derivative.Convergence is assessed when the value of any parameters should not differ by more than 1-3% between two subsequent time step, between two subsequent element size or between to subsequent grids.To ensure numerical stability in the solution, the Courant-Friedrichs-Lewy condition was fulfilled:

Conclusions
In the present study, the two dimensional laminar flow around a trapezoidal bluff body was investigated at low Reynolds.For our configuration study for a confined flow, there are two flow regimes for low Reynolds numbers, stable with two different topologies of flows adhesive or unstuck and unstable with the appearance of the first instability which is the Von Karman.For the stable flow in the adhesive regime the current lines stay adhere to the obstacle while for unstuck regime two symmetric vortices stay just adhere behind the obstacle.Regarding the unstable flow, there is instability of Von Karman which appears.The detachment frequency of vortices must be determined having regard to the link with a flow measurement.The obtained results are in perfect agreement with numerical and experimental previous studies.The simulations are underway to determine the exact value of the critical Reynolds number characterizing the transition between the two flow regimes.

v
: .The kinematic viscosity En écoulement confiné le débit de fluide est donné par : qv = v.S S: The cross-section of the test tunnel.
q v : Volume flow rate.
can only be measured from a laminar flow with Von Karman instability.The characteristic of the flow meter is given by the frequency of detachment of the vortices as a function of the flow rate (Tab 2).

Fig. 2 .
Fig. 2. Streamline for flow around the trapezoidal cylinder projected on the Q-criterion results.As it is shown in the Fig.3, flow separates at trailing edge and forms a recirculation region which consists of two symmetric vortices.Size of recirculation zone increases with an increase of Reynolds number and by reaching the critical Reynolds number, Von Kármán Vortex Street with periodic vortex shedding happens as a result of high positive pressure gradients.Vortex shedding causes fluctuation of pressure distribution on the body surface.

Fig. 3 .
Fig. 3. Temporal evolution of the Velocity field around the trapezoidal cylinder.