A Computational Fluid Dynamics Study on Characteristics of Flow Separation in Flow Rate Measurement Using Multi-Hole Plates ^†

Abstract

Flow rate measurement is a challenging task in the industry as there is no general-purpose measuring instrument for all appliances. However, orifice plates with multiple holes can be employed to measure the flow rate accurately. A computational fluid dynamics (CFD)-based numerical study was conducted to investigate the flow separation characteristics caused by the flow of water in multiple-hole orifice plates using ANSYS FLUENT R15.0 software. The study included single- and multiple-hole orifice plates, with orifices with a 36% area ratio, an equivalent diameter ratio (β-ratio) of 0.6, and hole number configurations of 1H, 4H, 9H, 16H, and 25H. The discharge coefficient for flow through multiple-hole orifices was obtained and compared for holes distributed in circular and square configurations. The significant parameters considered for the analysis were the hole number, distribution of holes, pressure drop, and reattachment points. A k-ε turbulence model was employed to study velocity fields, reattachment length, and discharge coefficient. We discuss the effects of hole numbers and their allocation on the reattachment length and discharge coefficient. Results are presented in the form of pressure variation comparisons, downstream recovery distance plots, recirculation zone plots, and percentage change in the coefficient of discharge. The study revealed that the number of holes in the plate significantly affects the pressure drop across the plate, the recirculation zone, and the orifice’s discharge coefficient.

1. Introduction

Fluid delivery systems in most industrial applications use a single-hole orifice as the main instrument for flow measurement, and the process involves measuring the pressure difference caused by the fluid moving through narrow restrictions in the pipe. Multi-hole plates or plates with perforations are plates consisting of several individual holes acting independently and parallelly and will have special flow characteristics compared to single-hole plates with the same area of flow [1]. An experimental investigation into multi-hole orifice throttling characteristics reported a higher loss coefficient for plates with fewer holes [2]. An investigation into perforated plates of different aspect ratios showed that their discharge coefficient is higher compared to that of standard orifices [3]. The loss coefficient is significantly affected by the distribution of holes in the case of perforated plates under non-cavitation conditions [4]. An experimental study on multiple-hole orifice plates under turbulent flow reported that the diameter of the hole and aspect ratio influence the discharge coefficient [5]. Experimental work on perforated plates with a ratio of diameters ranging from 0.11 to 0.6 and an aspect ratio ranging from 0.25 to 0.33 showed the loss coefficient and critical cavitation numbers vary with aspect ratio [6]. A CFX solver with k-ε and SST models of turbulence was employed for the simulation of single-hole plates with various geometries to determine their coefficient of discharge [7]. The computational fluid dynamics (CFD) method has proved to be cost-efficient in the prediction of mass flow rate [8]. Perforated plates of concave nature have been found to provide the best flow performance in straight pipes, with an improved index of uniformity and a reasonable pressure drop [9]. A numerical investigation found that multi-hole orifice flowmeters are less sensitive to upstream disturbances than standard ones [10]. STAR-CCM+ 11.0 software was used to simulate orifice discharge coefficient performance at low Reynolds numbers. Results showed a correlation between pressure distribution and flow structure through the orifice plate [11]. Simulation of turbulent flow through a circular orifice with square edges showed that increasing orifice plate thickness leads to an increase in the discharge coefficient with a fixed β ratio [12]. A streamline pattern confirmed the presence of separated flow areas in a concentric orifice plate [13]. The contraction coefficient affects disturbance resistance, but no significant differences were found between single and multi-hole orifice plates for higher β ratios [14]. The uncertainty in determining the discharge coefficient remains consistent between centric orifices and multi-hole orifices with the same reduction [15]. Numerical simulations using different turbulence models have been carried out to investigate the turbulent flow through an orifice plate [16]. Multi-hole orifices show high flow rate stability but need further optimization and analysis of hole number and location [17]. Large recirculation zones behind standard orifices cause significant energy losses [18]. Literature studies reveal that many efforts have been made to study the characteristics of flow separation in plates with a single hole. However, there are limited numerical studies on the flow separation characteristics of plates with multiple holes in the open literature. Also, with the little-known facts existing about numerical methods and their ability to predict the flow characteristics in multiple-hole orifice plates, further investigation in this direction is highly important. Previous investigations have revealed that the value of the discharge coefficient depends on the ratio of plate thickness to the diameter of the hole (t/d), also called the aspect ratio; the ratio of the diameter of the hole to the diameter of the pipe (d/D), also called the beta ratio; and flow nature defined by means of the hole/pipe Reynolds number (Re). The present study deals with the investigation of velocity fields, prediction of the discharge coefficient, and pressure loss across plates through a CFD-based numerical analysis, which is further validated with results from previous experiments. The present work also provides the details of the numerical model and the effects of hole number and its allocation on the reattachment length and discharge coefficient of a single-hole plate over multiple hole plates.

2. Materials and Methods

2.1. Geometry of the Fluid Flow Domain

Figure 1 shows the geometry of the computational fluid flow domain adopted for numerical analysis. To achieve a fully developed flow, the pipe diameter (D) was set to 50 mm at the inlet, and the fluid flow domain was 5 and 10 times larger than the pipe diameter upstream and downstream of the orifice plate, respectively. The thickness of the orifice plate was maintained at a constant value of 1.5mm across all configurations. The aspect ratio varied from 0.05 to 0.25. ANSYS Design Modeler was used to create the geometry of the computational domain.

Table 1. Geometric configurations of the orifice plates [5].

Number of Holes	Orifice Plate Thickness, t (mm)	Diameter of Hole, d (mm)	Aspect Ratio, t/d	Beta Ratio β = d/D
1	1.5	30	0.05	0.6
4	1.5	15	0.1	0.6
9	1.5	10	0.15	0.6
16	1.5	7.5	0.2	0.6
25	1.5	6	0.25	0.6

shows the geometric configurations of the orifice plates considered in the analysis, with values corresponding to those presented in the experiments in [5]. An illustration of the geometric configurations of all the orifice plates is presented in Figure 2.

2.2. Turbulence Model and Analysis Method

The ANSYS FLUENT pressure-based solver was used to run numerical simulations. A turbulent flow with a Reynolds number greater than 1 × 10⁴ was simulated using the standard k-ε turbulence model. The interpolation utilized the least-square cell-based spatial discretization scheme, while the SIMPLE scheme represented the pressure velocity coupling. A second-order upwind scheme was adopted for momentum to enhance the model’s accuracy. The simulations were conducted until the convergence criterion was met, which was set at less than 10⁻⁶ for mass, momentum, and turbulence.

2.3. Boundary Condition

For this study, it is assumed that the working fluid was incompressible. The type of inlet boundary condition used was a velocity inlet, with three different velocities applied to each orifice configuration. On the other hand, the outlet boundary condition was a pressure outlet, with the outlet pressure set to zero pascals. The walls were subject to a no-slip boundary condition. A summary of the boundary conditions can be found in

Table 2. Summary of the boundary conditions.

Parameter	Type	Value/Condition
Working Fluid	Water	Incompressible
Inlet	Velocity inlet	0.5–1.5 m/s
Outlet	Pressure outlet	0 pascals
Surrounding	Wall	No slip

.

2.4. Grid Independency Study

The computational domain is represented by a meshed model consisting of hexahedral elements. The meshed model of the flow domain for the nine-hole configuration plate can be seen in Figure 3. To ensure accurate results, a meshed model was put through a grid independency test by varying the number of elements between 450,000 and 1,600,000. The test showed that a mesh with 750,000 elements, each 1.3 mm in size, was sufficient for precise computation. This is illustrated in Figure 4, where it can be seen there was no significant change in the outcome beyond the chosen number of elements. The average aspect ratio and orthogonal quality of the mesh were 1.220 and 0.988, respectively, indicating that the mesh adopted was of good quality.

3. Results

Numerical simulations were carried out for plates with different hole configurations (1H, 4H, 9H, and 25H) using both circular and square arrangements to analyze the influence of the number of holes in the plate on the discharge coefficient and pressure drop and reattachment length.

Table 3. Pressure drop for plates with circular and square arrangements.

Number of Holes	Pressure Drop (Pa)
	Circular Arrangement	Square Arrangement
1	8602.70	8602.70
4	7516.67	7651.14
9	7227.20	6440.14
16	6870.48	7186.89
25	6642.44	6467.27

shows the total pressure drop for plates with circular and square arrangements.

Table 4. Reattachment length for plates with circular and square arrangements.

Number of Holes	Reattachment Length (mm)
	Circular Arrangement	Square Arrangement
1	87.5	87.5
4	58.5	59.5
9	56.5	40.5
16	40.5	48.5
25	41.5	39.5

displays the reattachment length for plates with circular and square arrangements.

4. Discussion

Based on the data presented in Table 3, it is clear that there was a decrease in the pressure drop of about 12.5%, 16%, 20%, and 23% in plates with hole configurations of 4H, 9H, 16H, and 25H, respectively. Additionally, the multi-hole plate experienced lower pressure drops as compared to the single-hole plate. Figure 5 illustrates the relationship between pressure drop and the number of holes in circular and square arrangements. The data show that an increased number of holes in the circular configuration resulted in a decrease in pressure drop. On the other hand, the square configuration showed a decreasing trend in pressure drop up to 9 holes, followed by an increasing trend up to 16 holes, but then decreased again with an increasing number of holes.

In Figure 6, we can see how the Percentage Shift in the discharge coefficient was affected by the number of holes in both circular and square arrangements. From the data, it is evident that an increased number of holes in the circular arrangement leads to an increase in the Percentage Shift of the discharge coefficient. However, the square arrangement showed an increasing trend in discharge coefficient up to 9 holes, followed by a decreasing trend up to 16 holes, before increasing again with more holes. The data presented in Figure 7 demonstrate that the reattachment length decreased when the number of holes increased. This decrease helped to minimize the loss caused by the interaction between the incoming and recirculating flows. In the circular configuration, the reattachment length consistently decreased as the number of holes increased. However, in the square configuration, the reattachment length decreased up to 9H but remained relatively constant from 9H to 16H. After 16H, the reattachment length started to decrease again. From Figure 8, it can be seen that pressure recovery happened much closer to the orifice on the downstream side of the multi-hole plate compared to the single-hole plate. Specifically, for the single-hole plate, pressure recovery occurred at a distance of 2.5 times the pipe diameter, while for the multi-hole plate, it occurred at a smaller distance of just 1.9 times the pipe diameter. In Figure 9, the contours of velocity indicate the region where the flow reattached. This region was larger downstream of the single-hole plate than the multi-hole plate, and it became smaller with an increasing number of holes.

Validation

The numerical simulation results were compared to the experimental results and standard values from ISO-5167 [19]. Figure 10 displays a difference of approximately 10% to 12% in the pressure drop when comparing the experimental and numerical results to those from the Stolz and Reader-Harris equations [20]. However, the numerical and experimental results align well. The maximum difference in pressure drop between computed and experimental values was only 3%.

5. Conclusions

This research involved conducting numerical simulations on plates featuring different hole configurations (1H, 4H, 9H, and 25H) arranged in circular and square shapes. The purpose of the study was to examine how the distribution of holes in multiple-hole orifice plates affects flow characteristics such as the discharge coefficient, pressure drop, and reattachment length. The number of holes had a substantial impact on the pressure drop and discharge coefficient of the orifice, regardless of the beta ratio. In plates with hole configurations of 4H, 9H, 16H, and 25H, there was a decrease in the pressure drop of about 12.5%, 16%, 20%, and 23%, respectively, when compared to the single-hole plate. Multi-hole plates also experienced lower pressure drops compared to single-hole plates. Moreover, the number of holes significantly affected the reattachment regions and flow separation downstream of the orifice plate.