© The Author(s), published by EDP Sciences, 2023

1 Introduction

Modern lithography machines work with nanometer precision to produce silicon chips at a breakneck pace. The power consumption of the machines can be more than a hundred kilowatts. However, just a few watts are used to project a pattern onto a silicon wafer. Much of the power consumed is dissipated as heat from various locations within the system, which has to be evacuated to avoid thermal expansion. Therefore, water is pumped through its cooling system. Within these cooling circuits, orifices are used as flow control devices. However, orifices are also a major source of pressure disturbances. While local hydrodynamic pressure fluctuations can be a nuisance, in the present application propagating acoustic waves are also of concern as they can reach sensitive components of these machines. The pressure drop across the orifices at given mass-flow should be tuned to obtain a desired flow distribution within the water-cooling network. One, therefore, aims at minimising the noise production for a given static pressure drop and fixed mass flow rate. In particular, low-frequency plane acoustic waves are of interest in this respect. Thus, the objective of this study is to estimate the local hydrodynamic forces and to model the far-field (acoustic) pressures due to orifices in water circuits and the influence of orifice geometry in the measured pressures.

The role of orifices as a source of sound in air has been extensively studied [1–16]. There are only few studies on water flow in the absence of cavitation [17, 18]. These studies concern thin orifices with acute edge. Most other studies consider sharp-edged orifices with 90° edges and an air flow. The authors are not aware of studies on the influence of a chamfer on the upstream edge on broadband noise production, which is the main focus of the present study. In an earlier paper a model has been developed for sharp-edged orifices to predict the broadband noise production [19]. This model makes use of estimated fluctuating drag force by means of an incompressible large-eddy simulation (LES) as an input for a plane-wave acoustic model. A simplified model has also been proposed which uses the measured or estimated drag coefficient as an input [19]. This corresponds to the model proposed by Nelson and Morfey [1]. The current paper uses a plane-wave model with input data obtained from LES for thick and thin orifices or the empirical source models proposed by Gibert [2] and Moussou [17] for thin orifices. Three orifices, within a pipe of diameter D = 9 mm, are the main focus of the study: one thick orifice with a diameter D_o = 4.0 mm and sharp square (90°) edges, both upstream and downstream, a second thin orifice of diameter D_o = 4.0 mm with sharp square edges and a third thick orifice with diameter of D_o = 3.0 mm and a 45° chamfer on the upstream edge. For these geometries LES simulations have been carried out and the measured steady-flow drag-coefficient $C_{\mathrm{drag}}$ compared to RANS and LES flow simulation results. For the other geometries, with both upstream and downstream 45° chamfers, experimental data on wall pressure fluctuations are shown. Particular attention is given to the D_o = 4.0 mm sharp-edge thick orifice and the D_o = 3.0 mm orifice with upstream chamfer, as the predicted drag coefficient (from both RANS and LES) deviates strongly from the experimental value. In an attempt to understand these deviations, for the thick sharp-edge orifice an additional LES simulation without sub-grid model was performed on a coarse mesh, which is referred as the Implicit Large Eddy Simulation (ILES).

In Section 2, the experimental setup is described both for wall-pressure fluctuations (Sect. 2.1) and for the measurement of the steady-flow drag coefficient of the orifice (Sect. 2.3). These measured drag coefficients are compared to results from RANS and LES predictions for three orifice geometries: a thick sharp-edge orifice and a thin sharp-edged orifice both of 4.0 mm diameter and a thick 3.0 mm diameter orifice with upstream chamfer. Details of the LES model is provided in Section 3. The acoustical model is described in Appendix A. Results are displayed and analysed in Section 4. The final part of Section 4 is dedicated to experiments on the influence of chamfers on the broadband wall-pressure fluctuations. Conclusions are summarised in Section 5.

2 Pressure measurements setup

2.1 Measurement of wall pressure fluctuations

Figure 1 shows the schematic of the experimental setup for measuring wall-pressure fluctuations. The measurement section (further also referred as the test-section) comprises a thick-walled straight stainless steel duct with a total length of 500 mm and an inner diameter D = 9.0 mm. The test section has a wall thickness of at least 2D. An orifice plate, with orifice diameter D_o and thickness δ_o is mounted with its upstream side at 350 mm from the inlet of the measurement section. The positive x-direction is in the direction of the main flow. The duct has six piezoelectric pressure transducers (PCB^® 105C02). The pressure transducers have a finite probe diameter D_probe = 2.5 mm. The transducers are mounted such that the center of the transducer surfaces are at a radial distance of 4.50 mm from the central axis of the test-section. The axial positions of the transducers are $\frac{x}{D}=-\left(2+\frac{{\delta }_{\mathrm{o}}}{D}\right),1, 2, 3, 6$ and 10, where the coordinate x is measured from the origin x = 0 at the downstream surface (exit) of the orifice plate.

Figure 1 Schematic of the test setup to measure wall pressure fluctuations. The flow of the water is from left to right through the test section. In first approximation the junctions between the steel test section and the long polyurethane (PU) tubes act as low impedance boundary conditions.

A volume flow rate $\left(\frac{\pi }{4}{D}^{2}U\right)$ of demineralized water, enters the test-section through a 30 m long polyurethane (PU) pipe with inner diameter D = 9 mm and wall thickness of 1.5 mm. The flow exits the test-section through a 15 m long PU pipe to reach a reservoir with capacity of 0.04 m³. The water in the reservoir has a free surface at atmospheric pressure. The polyurethane (PU) tube at exit also has the same cross-sectional dimensions as the PU tube at test-section inlet. The reservoir is an open-type, i.e., it is maintained at atmospheric pressure. The PU tubes terminations (inlet and outlet) are submerged deep within the reservoir and at the other end are connected to the test-section using 12 mm SERTO^® straight connectors. The upstream PU tube reduces the high frequency pump noise in the test section to negligible levels [19]. The long PU tubes at both ends of the steel test section behave as open-ends (low impedance conditions). Furthermore, it is assumed that due to absorption in these tubes, waves radiated from the test-section into the long PU tubes are not reflected back into the test section. A model for the acoustic response of the test section based on these assumptions is presented in Appendix A. The FLEXIM^® FLUXUS F601 ultrasonic flowmeter (UFM) was used to measure the volume flow rate $\left(\frac{\pi }{4}{D}^{2}U\right)$ with an accuracy of 1%. The pressure measurements were performed for a time-period of 500 s at sampling frequency of 30 kHz using the PICOScope^® 4000 series high resolution oscilloscope. This allows reliable measurements in the frequency range 1 Hz < f < 15 kHz.

Only single-hole orifice plates were used in the measurements. The orifice plates are designated as a combination of four normalised parameters denoted $\left(100\beta -\frac{{\delta }_{\mathrm{o}}}{D_{\mathrm{o}}}-\frac{{\delta }_{\mathrm{ch},\mathrm{up}}}{{\delta }_{\mathrm{o}}}-\frac{{\delta }_{\mathrm{ch},\mathrm{dn}}}{{\delta }_{\mathrm{o}}}\right)$ to recognise the individual plate properties. The first parameter $\beta ={\left(\frac{D_{\mathrm{o}}}{D}\right)}^2$ is the ratio of orifice area $\left(\frac{\pi }{4}{D}_{\mathrm{o}}^2\right)$ to pipe cross-sectional area $\left(\frac{\pi }{4}{D}^2\right)$. Here D_o is the orifice diameter and D = 9.00 mm is the pipe diameter. The second parameter $\frac{{\delta }_{\mathrm{o}}}{D_{\mathrm{o}}}$ is the ratio of orifice plate thickness (δ_o) to the orifice diameter (D_o). The third parameter $\frac{{\delta }_{\mathrm{ch},\mathrm{up}}}{{\delta }_{\mathrm{o}}}$ is the ratio of the width of the 45° chamfer on the upstream face of the orifice (δ_ch,up) to the thickness of the orifice plate (δ_o). The fourth parameter $\frac{{\delta }_{\mathrm{ch},\mathrm{dn}}}{{\delta }_{\mathrm{o}}}$ is the ratio of the width of the 45° chamfer on the downstream face of the orifice (δ_ch,dn) to the thickness of the orifice plate (δ_o). The four parameters mentioned give a description of the orifice geometry of each orifice used in the measurements. As an example the δ_o = 1.50 mm thick orifice plate with open area ratio of β = 11% having an upstream chamfer width of δ_ch,up = 0.50 mm and sharp downstream edge is referred to as the (11–0.5–0.33–0) orifice plate.

The orifices used in the measurements have open area ratios of β = 11% (D_o = 3.0 mm, D = 9.0 mm) and β = 20% (D_o = 4.0 mm, D = 9.0 mm). Three β = 11% orifice plates of thickness δ_o = 1.5 mm were considered. The first plate has square sharp edges. The second plate has a 45 mm chamfer with a width of δ_ch,up = 0.5 mm on the upstream edge of the orifice hole and a (square sharp edge) 90° edge on the downstream surface. The third plate has 45° chamfers on the upstream edge and downstream edge of the orifice hole with δ_ch,up = 0.5 mm and δ_ch,dn = 0.5 mm. The upstream edge chamfer of β = 11% orifices increases the vena-contracta factor $\alpha ={\left(\frac{D_{\mathrm{jet}}}{D_{\mathrm{o}}}\right)}^2$, where D_jet is the diameter of the jet formed by flow separation at the inlet of the orifice.

Two types of β = 20% orifice plates were considered, the first is a thick $\left(\frac{{\delta }_{\mathrm{o}}}{D_{\mathrm{o}}}=0.5\right)$ orifice. The second is a thin $\left(\frac{{\delta }_{\mathrm{o}}}{D_{\mathrm{o}}}=0.125\right)$ orifice. Both the plate types are tested with various edge configurations, which involves varying the edge-chamfer width of upstream or downstream orifice edges. The tip thickness δ_tip = δ_o − δ_ch,up − δ_ch,dn was varied systematically by changing the upstream and downstream chamfer width, δ_ch,up and δ_ch,dn, respectively. The configurations of all the orifice plates tested is listed in Table 1.

Table 1 provides an overview of the orifice geometries and their static drag coefficients (measured as described in Sect. 2.3). The diameters D_o were measured with an accuracy of 0.01 mm by the use of a microscope. The measured diameters D_o deviate less than $1\%$ from the nominal values D_o = 3.0 mm or 4.0 mm respectively. The nominal values were used in the numerical calculations described below.

2.2 Effect of air on the broadband noise

Residual air within the water circuit can have significant effect on the measured wall-pressure fluctuations. In order to investigate this effect, the setup is initially (at time t = 0 s) filled with water and the wall-pressure fluctuations are measured for 500 s at intervals of 1200 s during a total of 2 h measurement time. Then the setup is let to rest for a long pause with water within the test setup. This promotes dissolution of trapped air-bubbles. The measurement is then repeated for a 2 h period with time intervals of 1200 s. The procedure has been carried out with a pause of 20 h and repeated with a pause of 44 h. The results of these two tests are plotted in Figure 2.

Figure 2 Influence of air in the system on the PSD of the measured wall pressure fluctuations at $\frac{x}{D}=-2, 1$ and 10 for the thick β = 20% orifice plate with 0.1 width upstream and downstream chamfers (20–0.5–0.05–0.05). The PSD and the frequencies are normalised with respect to dynamic pressure in the orifice $\left(0.5\rho U_{\mathrm{o}}^2\right)$, the average flow velocity within the orifice Uo and the orifice diameter Do.

Figure 2 shows the normalised PSD of pressure fluctuations as a function of the Strouhal number St_o = fD_o/U_o based on the orifice diameter D_o and the cross-sectional averaged flow velocity in the orifice U_o. The PSD is normalised using the dynamic pressure $\frac{1}{2}\rho U_{\mathrm{o}}^2$ and Strouhal number St_o, where ρ is density of water. A definition of the PSD and discussion of different dimensionless forms of the PSD is provided by Kottapalli et al. [19]. One can see that the presence of air in the system has a significant effect on the PSD of the measured wall pressures for St_o > 0.1. The measured PSD converges and becomes fairly reproducible up to St_o = 3 after 2 h of continuous flushing. However, the measured PSD still shows some deviations at Strouhal numbers St_o ≥ 3. If a bubble is present at the test-section PU-hose junction it does not strongly affect the acoustic behaviour of the setup, because it is close to a pressure node p_ac ≃ 0. If a bubble is present close to the orifice it will strongly affect the longitudinal acoustic modes 1, 2, 5 and 6 of the test section, for which the orifice is close to a pressure anti-node. The effect of the bubble is observed clearly on the first two modes when measurements are carried out immediately after filling the setup with water t = 0 h. With increasing flushing time (t ≥ 2 h) or waiting a night t > 20 h, the lower acoustic modes (1 and 2) are less affected. However, the higher modes (5 and 6) are still affected. Data for St_o > 4 remains very sensitive to remaining small air bubbles, and is less reproducible.

To allow more measurements, as a compromise an initial flushing of the system for 2 h was carried out at the highest Reynolds number followed by a pause of at least 12 h and up to 24 h. Before measuring, the system is let to flush again for about 2 h. Then measurements are performed at various Reynolds numbers during 500 s at intervals of 1200 s.

2.3 Static pressure measurements

The static pressure drop measurements were carried out using the same setup shown in Figure 1. The static pressure drop was measured between two points at positions $\frac{x}{D}=-72.23$ and $\frac{x}{D}=17.7$ from the downstream surface of the orifice plate (x = 0). The static pressures were measured using the Wöhler^® DM2000 digital manometer. The manometer measures pressure difference within an accuracy of 0.01 mbar. Reference static pressure drop measurements ΔP_pipe were performed for the test-section without an orifice (for a straight pipe). These results were subtracted from the static drop (ΔP) measured across individual orifices for different volume flow rates. Since the pressures were measured sufficiently far from the orifice (both upstream and downstream) the measured drag coefficient of the orifice is:$$C_{\mathrm{drag}}=\frac{\left(\Delta P-\Delta P_{\mathrm{pipe}}\right)}{\left(\frac{1}{2}\rho U^2\right)}$$(1)

The vena-contracta factor (α) is estimated from the drag coefficient C_drag using the theoretical formula for the vena-contracta [2] limit at very high Reynolds numbers:$$\alpha =\frac{1}{1+\sqrt{C_{\mathrm{drag}}}}{\left(\frac{D}{D_{\mathrm{o}}}\right)}^2$$(2)

The values for α and C_drag are normalized using the reference values of vena-contracta factor (α_ref) and the high-Reynolds-number limit of drag coefficient ${\left(C_{\mathrm{drag}}\right)}_{\mathrm{ref}}$ for thin sharp-edged orifices estimated by means of the empirical formula by Idelchik [20, 21]:$${\alpha }_{\mathrm{ref}}=\frac{1}{1+\sqrt{\left(\frac{1}{2}\left(1-{\left(\frac{D_{\mathrm{o}}}{D}\right)}^2\right)\right)}}$$(3) $${\left(C_{\mathrm{drag}}\right)}_{\mathrm{ref}}={\left(\frac{1}{{\alpha }_{\mathrm{ref}}}{\left(\frac{D}{D_{\mathrm{o}}}\right)}^2-1\right)}^2$$(4)

3 Large-eddy simulations (LES) and Reynolds averaged Navier–Stokes simulations (RANS)

3.1 Governing equations for LES

Large eddy simulations (LES) calculate flow properties by solving the incompressible Navier–Stokes (N–S) equations, for the larger scales. By introducing resolved and modelled flow variables, the N–S equations are written as:$$\frac{\partial {\tilde{u}}_i}{\partial x_i}=0,$$(5) $$\frac{\partial {\tilde{u}}_i}{\partial t}+{\tilde{u}}_j\frac{\partial {\tilde{u}}_i}{\partial x_j}=-\frac{1}{\rho }\frac{\partial \tilde{p}}{\partial x_i}+\nu \frac{{\partial }^2{\tilde{u}}_i}{\partial x_j^2}+\frac{\partial {\tau }_{\mathrm{ij}}^r}{\partial x_j},$$(6)where, ${\tilde{u}}_i$ and $\tilde{p}$ are the so-called resolved velocity and pressure components, x_i is the spatial coordinate, ρ is the density and ν is the kinematic viscosity. The term ${\tau }_{\mathrm{ij}}^r$ is the residual stress tensor. The resolved quantities $\widetilde{u_i}$ and $\tilde{p}$ correspond to the large and inertial scale eddies of the flow, which are solved numerically. The residual stress tensor corresponds to the eddies of the smaller unresolved scales down to the dissipative scale (Kolmogorov scale). These are solved using a sub-grid scale (SGS) model. The equations were solved using the commercial code StarCCM+^®.

3.2 Computational domain

The computational domain comprises a cylinder of diameter D = 9.00 mm and length 297 mm (33D), excluding the thickness of the orifice. The orifice is placed 20D from the inlet. The flow direction is from left to right. The 20D long upstream duct length ensures a fully developed turbulent flow at the orifice. The length of the downstream duct is 13D, which corresponds to the steel pipe segment downstream of the orifice. LES computations were performed for three orifices. The first is the D_o = 3.0 mm upstream chamfered orifice (11–0.5–0.33–0). The second and third are the square sharp edges D_o = 4.0 mm orifices, thick (20–0.5–0–0) and thin (20–0.125–0–0) orifice. The geometries are shown in Figures 3 and 4. For the (11–0.5–0.33–0) orifice a calculation was also performed without a sub-grid-scale model, which is referred to as Implicit LES (ILES).

Figure 3 Dimensions of the computational domain of LES with the β = 11% (Do = 3.0 mm) orifice with δo = 1.5 mm and a 45° upstream chamfer of width δch,up = 0.5 mm also designated as the (11–0.5–0.33–0) orifice.

Figure 4 Dimensions of the computational domain of LES for β = 20% (Do = 4.0 mm) orifices with sharp edges. The orifice thickness δo = 0.125 Do =0.5 mm for the thin (20–0.125–0–0) orifice and δo = 0.5 Do = 2.0 mm for the thick (20–0.5–0–0) orifice.

3.3 Meshing, discretization and boundary conditions

The mesh parameters of the computational domain were estimated through a steady-state RANS simulation of the orifice flows with same domain and boundary conditions. The procedure for the estimation of the mesh parameters is explained in Kottapalli et al. [19]. The meshing is performed using the automatic polyhedral meshing algorithm of StarCCM+^®. The polyhedral meshing algorithm aims to form uniform mesh cells such that (imaginary) lines connecting the neighboring mesh cell-centers are normal to the adjoining surfaces. The meshing algorithm requires a reference mesh size (s) as an input which is obtained from the calculated Taylor microscales. In the straight pipe region, the mesh size is increased radially from s = 0.02D close to the wall to s = 0.04D towards the pipe center. The domain walls were lined with 5 layers of prismatic hexahedral cells. The cell spacing of each successive layer was increased in the radial direction. In the region between positions $\frac{x}{D}=-1$ to 3.5, of the mesh was refined to s = 0.009D. The width of the cells in the first prismatic layer in radial direction were such that (Δy⁺)_min < 1 for the straight pipe section and (Δy⁺)_min ≈ 2 in the refined grid region $\left(-1<\frac{x}{D}\le 3.5\right)$. The coordinate Δy⁺ is orientated normal to the pipe wall, towards the interior of the flow domain, and is defined as:$$(\Delta y^{+}{)}_{\min }=\frac{u^{\mathrm{*}}(\Delta y{)}_{\min }}{\nu },$$(7)where, $u^{\mathrm{*}}=\sqrt{\frac{{\tau }_{\mathrm{wall}}}{\rho }}$ is the friction velocity with τ_wall being the wall shear stress and ρ the fluid density. The coarse LES mesh of β = 11% orifice has approximately 16 million mesh cells, while β = 20% had 26 million mesh cells.

The LES stores the instantaneous pressure at the chosen probe locations at every time step. The instantaneous velocity field u can be used to calculate the mean flow properties. The pressure field is also used to calculate the drag force over the orifice (F_drag) and the acoustic source pressure Δp_source. In order to allow comparison with experiments, the hydrodynamic wall pressures from LES are averaged over a finite circular probe surface with diameter D_probe = 2.5 mm corresponding to that of the pressure transducers.

The convection terms of the N–S equations were discretized using the bounded-central-differencing (BCD) scheme. The BCD scheme is a composite normalized-variable-diagram (NVD) scheme [22]. It comprises a pure central-differencing scheme and, a blended scheme of central-differencing and second-order upwind schemes. However, when the convection-boundedness-criterion (CBC) is violated BCD uses first-order scheme. An implicit 2nd order time-stepping method was used to discretize the temporal terms. The time-steps for the LES computations were adjusted such that mean average Courant number (Co) in the region between $-1\le \frac{x}{D}<3.5$ is Co ≈ 0.5 [23]. The Courant number is defined as follows:$$\mathrm{Co}=\frac{u_x\Delta t}{\Delta x},$$(8)where u_x is the local fluid x-velocity in a given mesh cell, Δt is the time-step, and Δx is the length of the mesh cell in x-direction. The Courant number is calculated for each cell locally to calculate the average Courant number over a given region.

The inlet boundary of the fluid domain was set to a static pressure of 25.0 kPa for (20–05–0–0) orifice and 31 kPa for (20–0.125–0–0) orifice which corresponds to a volume flow rate of 80.5 cm³ s⁻¹. Similarly, a static pressure of 48.0 kPa was applied at the inlet for the (11–05–033–0) orifice corresponding to a volume flow rate of 66 cm³ s⁻¹. The outlet boundaries in both cases were set to a pressure of 0 Pa. A no-slip boundary condition was applied at the walls. Note that a limited study of grid-size convergence was carried out for the β = 20% orifice case by comparing with the results of a four times coarser grid. This has been reported by Kottapalli et al. [19].

3.4 LES mesh convergence

The convergence of the LES was verified by performing three simulation of upstream chamfered (11–0.5–0.33–0) orifice. Two of three simulations were with a coarse mesh and one with a fine mesh with a 2 times the mesh density of the coarse mesh. Of the two coarse mesh simulations, one was simulation was LES and other was Implicit-LES (ILES). Implicit-LES does not use a SGS model and relies on numerical dissipation to perform a similar function as an SGS model [24, 25]. Figure 5 shows the comparison between PSD plot of the fluctuating drag force on the orifice surface calculated from LES (fine and coarse mesh) and the ILES (no sub-grid-scale model, using coarse mesh) for the chamfered β = 11% orifice.

Figure 5 Predicted PSD of acoustic sound source due to fluctuating drag force for the (11–0.5–0.33–0) orifice obtained using LES with the fine mesh and the coarse mesh. The third simulation is an Implicit-LES (ILES) with the coarse mesh.

The amplitude of the drag force for high frequencies ${\mathrm{St}}_{\mathrm{jet}}=\frac{f(D-D_{\mathrm{jet}})}{U_{\mathrm{jet}}}>3$ are comparable for the two different meshes. At frequencies between 0.2 ≤ St_jet ≤ 3 the fine mesh results show lower fluctuations in drag force within a factor 2. The LES results show a hump around St_jet = 3, which is not observed the result from ILES. The hump has been observed for the LES of (20–0.5–0–0) orifice. A similar hump is reported for thick sharp square edged orifices in experiments [6, 26, 27] and LES results [10]. Furthermore sound producing hydrodynamic modes with similar Strouhal numbers have been observed in Linearized Navier–Stokes Equation (LNES) results [27, 28]. An ideal grid convergence is difficult to assess for an LES calculation because the sub-grid turbulence model is affected by grid size. Globally the results are reasonably independent of the mesh grid with significant differences in certain parts of frequency spectrum (2 ≤ St_jet ≤ 6). At frequencies (St_jet < 0.1) the results are strongly dependent on the total simulation time. For the current set of simulations, the computational time was already in the range of 3–6 months and further calculation was not possible.

3.5 RANS simulations

In addition to LES, RANS simulations are also performed to estimate the steady-flow drag coefficients C_drag at different Reynolds numbers for the (20–0.5–0–0), (20–0.125–0–0) and (11–0.5–0.33–0) orifices. The RANS simulations were performed using the same computational domain as the LES with coarse-mesh. The details of the computational domain and the mesh is provided in Section 3.3. Most RANS simulations were performed using the Realizable two-layer k-ϵ model [29]. The Realizable k-ϵ model is also compared to the Low Reynolds (Low-Re) k-ϵ model [30] for the (11–0.5–0.33–0) orifice.

The inlet boundary of the computational domain was set to a time-averaged uniform velocity boundary condition in the normal direction. The RANS computations were performed for orifice Reynolds number ranging from 5 × 10³ ≤ Re_o ≤ 3 × 10⁴ for all the orifices. The turbulent kinetic energy intensity of the flow is set to 5% with a turbulent length scale of 4.5 mm at the inlet boundary. The outlet boundary is given a time-averaged uniform pressure of 0 Pa. At the wall of the domain, the low-y⁺ treatment model was used, where the friction velocity is estimated as:$$u^{\mathrm{*}}=\sqrt{\frac{\nu u_t}{\Delta y}},$$(9)where, u_t is the velocity of the fluid tangential to the wall normal and Δy is the distance of the centroid of the near-wall cell to the wall.

4 Results

4.1 Static pressure measurements

Figure 6 shows LES simulations of the (20–0.5–0–0) orifice. The LES predicts a reattachment of the flow within the restriction for the (20–0.5–0–0) orifice. A separation bubble is formed with back flow, as seen from the velocity profile at $\frac{x}{D}=-0.25$. Before reaching the orifice exit $\left(\frac{x}{D}=0\right)$ the flow reattaches to the orifice walls. The plot of the pressure along the pipe axis shows a local minimum of pressure within the orifice, which is is typical for a separation bubble within the orifice.

Figure 6 (a) Static pressure drop (ΔP) across the (20–0.5–0–0) orifice. (b) Mean axial velocity within the orifice $\frac{x}{D_{\mathrm{o}}}=-0.5-0.25, 0$ and slightly downstream of the orifice at $\frac{x}{D_{\mathrm{o}}}=0.25$ from LES. The horizontal dotted (▪▪▪) line represents $\frac{1}{\alpha }$ calculated from RANS using equation (2).

RANS and LES computations of the flow in the upstream chamfered (11–0.5–0.33–0) orifice do not show flow separation within the orifice (see Fig. 7b). The chamfer at the upstream edge drastically affects the predicted flow profile within the orifice leading to a higher vena-contracta factor. One finds α = 0.90 for the Realizable k-ϵ model, α = 0.88 for the Low-Re k-ϵ RANS model and and α = 0.89 for the LES (fine-mesh) model. The C_drag factor predicted by Low-Re k-ϵ RANS model and LES is lower than results from the incompressible-potential-flow theory for (11–0.5–0.33–0) orifice of C_drag = 0.5 by 10% [31]. From the pressure increase in the recovery region downstream of the orifice one sees that the flow reattaches to the pipe wall after a distance in the order of two pipe diameters for both orifices (see Figs. 6 and 7).

Figure 7 (a) Comparison between static pressure drops across the (11–0.5–0.33–0) orifice computed using steady-state k-ϵ RANS model and Low-Re k-ϵ RANS model. (b) and (c) Mean axial velocity within (11–0.5–0.33–0) orifice at $\frac{x}{D_{\mathrm{o}}}=-0.5, -0.25, 0$ and slightly downstream of the orifice at $\frac{x}{D_{\mathrm{o}}}=0.25$ from k-ϵ RANS model and Low-Re k-ϵ RANS model. The horizontal dotted (▪▪▪) line represents $\frac{1}{\alpha }$ calculated from RANS using equation (2).

Figure 8 shows the comparison between the measured and computed normalised drag coefficients C_drag for the (11–0.5–0.33–0), (20–0.125–0–0),and (20–0.5–0–0) orifices. The drag coefficient is normalised using the high-Reynolds-number limit of the drag coefficient (equations (3) and (4)). This normalized drag $\frac{C_{\mathrm{drag}}}{(C_{\mathrm{drag}}{)}_{\mathrm{ref}}}$ is displayed as a function of the Reynolds number Re_o = U_oD_o/ν based on the orifice diameter D_o and the cross-sectional average flow velocity in the orifice. The kinematic viscosity of water is assumed to be ν = 10⁻⁶ m²/s. As the Reynolds number increases an increase in the normalised C_drag is observed which then reaches an asymptotic value. For the (20–0.125–0–0) orifice the measured values of C_drag and α and estimated values from RANS and LES approach the predicted values of $(C_{\mathrm{drag}}{)}_{\mathrm{ref}}$ and α_ref by Idelchik [20]. This indicates that flow does not reattach within the thin orifice. The results of LES and RANS agree globally with the measurements (within 10% in the drag coefficient) of the (20–0.125–0–0) orifice. Large deviations (20%) are observed between measured and computed values for the (11–0.5–0.33–0) orifice. The predicted value approaches the incompressible-potential-flow limit for a 2-D planar chamfered slit flow (convergent channel with a 90° angle) [31]. In this potential-flow model separation is only assumed at the end of the chamfer. A vena-contracta factor α = 0.82 is obtained and used to calculate C_drag using equation (4). This corresponds to a 2-D planar model for the (11–0.5–0.33–0) orifice.

Figure 8 Comparison between measured and computed normalised drag coefficient $\frac{C_{\mathrm{drag}}}{(C_{\mathrm{drag}}{)}_{\mathrm{ref}}}$ as a function of orifice Reynolds number Reo for the thick (20–0.5–0–0) orifice, the thin (20–0.125–0–0) orifice, and the chamfered (11–0.5–0.33–0) orifice. The dotted black line is the theoretical drag coefficient (potential flow theory) for a 90° converging nozzle (corresponding to the (11–0.5–0.33–0) orifice) [31]. The $(C_{\mathrm{drag}}{)}_{\mathrm{ref}}$ are calculated for thin orifices using the empirical equation (4) (proposed by Idelchik [20]).

For the (20–0.5–0–0) orifice, differences up to 30% were observed between measurement series with three different orifice samples with the same nominal geometry. The largest deviations occur at low Reynolds numbers. For sample 1, the deviation of drag measurements from RANS computations are within 10%, however, for the samples 2 and 3 the deviations are significantly higher at low Reynolds numbers. Observing the orifices under a microscope for fabrication tolerances did not reveal any significant deviation from the nominal values. These large deviations are expected to be related to the transition between “short-orifice behaviour” without reattachment and a “thick-orifice behaviour” for which the flow reattaches within the orifice. The reattachment of the jet reduces the drag coefficient of the orifice. This transition occurs for 0.5 < δ_o/D_o < 1. In this range the flow is very sensitive to apparently minor variation in the geometry [32]. Furthermore, the RANS models assume a uniformly turbulent flow, which tends to exaggerate reattachment and therefore underestimate the drag coefficient.

For the drag coefficient, the numerical results are fairly independent of the RANS model used (within 10%) and the LES results are independent of the mesh refinement (Tab. 2). The C_drag values from other RANS models showed similar results (see Tab. 1). The predicted values of C_drag from RANS and LES for the (11–0.5–0.33–0) orifice also shows high deviations between RANS or LES models and experiment. An inspection of the (11–0.5–0.33–0) orifice under microscope does show some imperfections (few hundreds of a mm) in the upstream chamfer. These small imperfections are not expected to cause the large deviations observed in C_drag values. The RANS model is not appropriate for moderately high Reynolds-number flows that display local re-laminarization. For this reason, a low Reynolds number (Low-Re) k-ϵ model was used as an alternative. The Low-Re RANS model slightly improves the results for the (11–0.5–0.33–0) orifice, but is not sufficient to explain the experimental results. For the (20–0.5–0–0) orifice no significant changes in C_drag were observed. An Implicit-LES was also performed using the coarse-mesh. The results obtained for this coarse-mesh ILES did not deviate significantly from that of the other models (RANS and LES). The over-simplified potential-flow model assuming separation at the end of the chamfer and ignoring reattachment provides a better prediction of the drag coefficient than the RANS or LES flow simulations.

4.2 Acoustics response of the setup

Using PSD of the axial component of the drag force calculated by means of the incompressible large-eddy simulations (LES). One can predict the PSD of the acoustic pressure fluctuations at any point in the test section by means of a one-dimensional acoustical model. The sound source is assumed to be a fluctuating pressure discontinuity $\Delta p_{\mathrm{source}}=-\frac{4F_{\mathrm{drag}}}{\left(\pi D^2\right)}$ placed just downstream of the orifice. The fluctuating drag (F_drag) is calculated as:$$F_{\mathrm{drag}}={\int }_{S_{\mathrm{o}}}\mathrm{}\left[p\right(-{\delta }_{\mathrm{o}},t)-p(0,t\left)\right] n_x\mathrm{d}{S}_{\mathrm{o}}.$$(10)

On the wall surface S_o of the orifice. Here, n_x is the x-component (in the axial direction) of the outer unit normal vector to the surface, with ${\int }_{S_{\mathrm{o}}}\mathrm{}{n}_x\mathrm{d}{S}_{\mathrm{o}}=\frac{\pi }{4}(D^2-D_{\mathrm{o}}^2)$. The difference in retarded time between the front and back of the orifice has been neglected because $\frac{f{D}_{\mathrm{o}}}{c_{\mathrm{w}}}\ll 1$, where c_w = 1.46 × 10³ ms⁻¹ is the speed of sound in the (water-filled) test section. The approximation used corresponds to the theory used by Tao et al. [12] for frequencies below the first cut-on frequency of transverse pipe modes.

It is furthermore assumed that the upstream and downstream polyurethane tubes have anechoic terminations so that the acoustic boundary condition is assumed to be determined by the reflection coefficient of acoustic waves at the steel/polyurethane transitions delimiting the test section (R_s/pu). Convective effects are neglected because the Mach number is very low $M=\frac{U}{c_{\mathrm{w}}}\le 10^{-3}$. Damping of the acoustic waves is assumed to be due to viscous Stokes layers thinner than the viscous sub-layer of the turbulent flow. This is reasonable at frequencies of the order of the first longitudinal acoustic resonance of the setup or higher. The acoustic flow through the orifice is assumed to be locally incompressible, because ${\left(\frac{2\pi f{D}_{\mathrm{o}}}{c_{\mathrm{w}}}\right)}^2\ll 1$. The inertia of the acoustic flow through the orifice is taken into account by means of an effective length: the orifice plate thickness δ_o plus the inertial end correction 0.8D_o. The model is described in more detail in Appendix A, which also describes the expression used for the reflection coefficient at the steel/polyurethane transition. It is based on experimental data of Rodrigues [33] for f < 0.5 kHz and of Moonen et al. [34] for f > 0.5 kHz.

4.3 Comparison between wall pressure fluctuations of two square-edged orifice samples

Figures 9 and 10 show the comparison between the PSD of wall pressure fluctuations measured for the (20–0.5–0–0) orifice samples 1 and 3. The PSD of both samples show a very similar frequency response despite the significant deviations in the measured drag coefficients (see Fig. 8).

Figure 9 Comparison between the normalised PSD of wall pressure fluctuations measured for sample 1 and sample 3 of (20–0.5–0–0) orifice at transducer positions transducer positions $\frac{x}{D}=1, 2$ and 3 with orifice Reynolds number, Reo = 22,000 as a function of Strouhal number $\left(\frac{f{D}_{\mathrm{o}}}{U_{\mathrm{o}}}\right)$. As shown in Figure 8, sample 1 and 3 have drastically different steady flow drag coefficients but the PSD of the wall pressure fluctuations are quite similar.

Figure 10 Same as Figure 9 with measurements at transducer positions transducer positions $\frac{x}{D}=-2, 6$ and 10.

For sample 1 of the thick sharp-edged β = 20% orifice (20–0.5–0–0) a peak is observed at St_o = 0.18 for $\frac{x}{D}=1$ and 2 (Fig. 9). As this peak is not observed at other transducer positions, it must be local and hydrodynamic in nature. It is also observed for other thick sharp-edged orifices with open-area ratios β = 11% and 31% [19]. The peak appears to be more pronounced as the Reynolds number decreases. This peak is probably related to transversal oscillations of the jet within the orifice. These transversal oscillations do not radiate sound because the oscillation frequency is lower than the transversal pipe resonances and associated non-planar wave propagation. This is, therefore, a purely hydrodynamic instability, similar to an edge-tone oscillation [35, 36]. No such peaks were observed for samples 2 and 3.

The hydrodynamic instability is observed particularly at transducer position $\frac{x}{D}=1$. The PSD plots at other transducer positions do not show any deviations between the two samples. Hence overall the estimation of source pressure Δp_source from LES is valid for both the tested samples, which is shown later in Figures 13 and 14.

4.4 Reynolds dependency on PSD of fluctuating pressures.

The Reynolds number dependency of the wall pressure fluctuations has been extensive discussed in Kottapalli et al. [19]. In addition to results discussed in the aforementioned study, the current study also focuses on the effect of a chamfered orifice edge on the PSD of pressure fluctuations.

Similar to the previous study [19], the normalized PSD plots of wall pressure fluctuations at near-field positions $\frac{x}{D}=1, 2$ and 3 are not a strong function of Reynolds number for the β = 20% orifice at St_o < 0.7. However, the chamfered (11–0.5–0.33–0) orifice a significant Reynolds number dependency is observed at $\frac{x}{D}=1$ (see Fig. 11). The normalized PSD for St_o < 0.7 roughly proportional to the Reynolds number. For St_o > 0.7 acoustic pressures become dominant. Acoustic resonance peaks are observed around fixed frequencies which imply a decrease in the corresponding Strouhal number ${\mathrm{St}}_{\mathrm{o}}=\frac{f{D}_{\mathrm{o}}}{U_{\mathrm{o}}}$ with increasing velocity U_o (and Reynolds number). Hence, the apparent Reynolds dependency should in principle, for St > 0.7, be studied in a presentation of the data as a function of the Helmholtz number $\frac{\mathrm{fL}}{c}$ rather than the Strouhal number.

Figure 11 Reynolds number dependency of wall pressure fluctuations PSD for the chamfered β = 11% (11–0.5–0.33–0) orifice. The PSD are normalised using average flow velocity through the orifice Uo and the orifice diameter Do as a function of the orifice Strouhal number $\frac{f{D}_{\mathrm{o}}}{U_{\mathrm{o}}}$.

Similarly, for positions $\frac{x}{D}=-2, 6$ and 10 the acoustic pressure fluctuations are dominant. Additionally, some dependency on the Reynolds number is observed for both orifices. The upstream chamfer of (11–0.5–0.33–0) orifice suppresses the modulations observed in (20–0.5–0–0) orifice, which is unexpected.

Around St_o = 0.18 the dimensionless PSD of wall-pressure fluctuations at $\frac{x}{D}=1$ and 2 is significantly reduced by the upstream chamfer. A similar phenomenon is observed in wide gas transport systems where chamfering of flow conditioners affected the self-sustained acoustic oscillation due to lock-in to transversal acoustic resonances downstream of perforated plates [37]. These whistling flow straighteners had perforations of typical thickness to diameter ratio of order unity $\frac{{\delta }_{\mathrm{o}}}{D_{\mathrm{o}}}=O\left(1\right)$. Longer perforations did not whistle as the flow reattaches to the orifice wall, reducing the hydrodynamic instabilities.

The wall pressure pressure PSD for (11–0.5–0.33–0) orifice displays a peak around St_o = 2.3 × 10⁻². It is observed at all the transducers positions but is more clear at positions $\frac{x}{D}=-2$ and 1. The Strouhal number based on the pipe diameter D and pipe flow velocity U is of order unity $\frac{\mathrm{fD}}{U}={\mathrm{St}}_{\mathrm{o}}{\left(\frac{D}{D_{\mathrm{o}}}\right)}^3=0.6$. Hence, this could correspond to a global hydrodynamic oscillation of the upstream and/or downstream-flow separation region around the orifice.

Sharp peaks (or spectral lines) are also observed in the pressure PSD for the (11–0.5–0.33–0) orifice for St_o ≥ 3 at a critical Reynolds number (Fig. 11). These peaks are indicative of whistling. It might here be related to the modulation of an onset of cavitation [5, 18]. The non-linear saturation necessary to reach a limit cycle amplitude results in a line spectrum with harmonics of a fundamental oscillation frequency. One does indeed observe a second sharp peak (line) at the second harmonic of this oscillation frequency, next to the first sharp peak (line) at the whistling frequency.

4.5 Dipole sound source model

The scaling used by Nelson and Morfey [1] is used here to compare the dipole sound source of the orifices. Figure 12 shows the normalized PSD of the fluctuating (dipole) pressure source (Δp_source) predicted using LES of the thick (20–0.5–0–0) orifice and thin (20–0.125–0–0) orifice sharp-edged β = 20% (D_o = 4 mm) at Re_o = 22.5 × 10³ and the upstream chamfered (11–0.5–0.33–0) β = 11% (D_o = 3 mm) orifice at Re_o = 30 × 10³. The Δp_source has been normalized using drag pressure (Δp_drag) defined as:$$\Delta p_{\mathrm{drag}}=\frac{1}{2}\rho U_{\mathrm{o}}^2{\left(\frac{1}{\alpha }-{\left(\frac{D_{\mathrm{o}}}{D}\right)}^2\right)}^2,$$(11)where, α is the vena-contracta factor of the orifices obtained from the static pressure measurements. The frequency is normalized using the jet Strouhal number ${\mathrm{St}}_{\mathrm{jet}}=\frac{f(D-D_{\mathrm{jet}})}{U_{\mathrm{jet}}}$ as proposed by Gibert [2]. Here, $D_{\mathrm{jet}}=D_{\mathrm{o}}\sqrt{\alpha }$ is the jet diameter and U_jet = αU_o is the jet velocity.

Figure 12 Comparison between normalized dipole sound source from LES for the β = 20% sharp-edged (20–0.5–0–0) orifice at Reo = 2.25 × 104, the β = 11% upstream-chamfered (11–0.5–0.33–0) orifice at Reo = 3 × 104 and the β = 20% thin orifice (20–0.125–0–0) at Reo = 2.25 × 104. The LES source data is compared to measured source data for thin orifices with β = 9%, 16%, 25% by Gibert [2] and Moussou [17].

The dipole sources from LES of the thick $\left(100\beta -\frac{{\delta }_{\mathrm{o}}}{D_{\mathrm{o}}}-\frac{{\delta }_{\mathrm{ch},\mathrm{up}}}{{\delta }_{\mathrm{o}}}-\frac{{\delta }_{\mathrm{ch},\mathrm{dn}}}{{\delta }_{\mathrm{o}}}\right)=$ (20–0.5–0–0) orifice, the thin (20–0.125–0–0) orifice and the thick (11–0.5–0.33–0) orifice with upstream chamfer are compared to the measured source data for thin orifices with β = 9%, 16% and 25% by Gibert [2] for low Strouhal numbers (0.01 < St_o < 0.1) and to the correlation proposed by Moussou [17] for high Strouhal numbers (St_o > 0.1). The LES of the (20–0.125–0–0) orifice is computed to a simulation time of 3.5 × 10⁻² s, which gives a reliable Δp_source data from f > 200 Hz (>15 oscillations). The PSD of Δp_source of the thin orifice source is in the same order of magnitude as thick orifices for St < 0.1 [2]. At St_o > 0.5 the PSD of Δp_source for thin orifices is two orders of magnitude lower than that of thick orifices [17]. The PSD of Δp_source from LES of (20–0.125–0–0) orifice agrees with the data by Moussou [17] at high Strouhal numbers and approaches the data by Gibert [2] at low Strouhal numbers. The results are consistent with the low noise observed for the thin β = 20% orifice in Figure 12.

4.6 Prediction of acoustic pressure based on a dipole sound-source model

The plane wave model predicts the measured PSD of pressure fluctuations within about a factor 3 when acoustic pressures are dominant. The acoustic pressure fluctuations dominate the wall pressure fluctuations upstream and in the far field $\left(\frac{x}{D}=-2, 6 \mathrm{and} 10\right)$ for St_o ≥ 0.1 (Fig. 13) and in the near-field $\left(\frac{x}{D}=1\right)$ for St_o > 1 (Fig. 14). The acoustic predictions with Δp_source from the coarse ILES provides the best estimate of acoustic pressures between 1 < St_o < 5. The LES estimates lower pressures between St_o < 0.5 by factor 3.

Figure 13 Comparison between PSD of measured pressures and estimated acoustic pressures for the β = 20% sharp-edged thick (20–0.5–0–0) orifice (left column) at Reo = 2.25 × 104, the β = 11% upstream-chamfered thick (11–0.5–0.33–0) orifice at Reo = 3 × 104 (center column) and the β = 20% sharp-edged thin (20–0.125–0–0) orifice at Reo = 2.25 × 104 (right column) at positions $\frac{x}{D}=-2, 6$ and 10. The acoustic pressures are estimated from LES and data from Gibert [2] and Moussou [17].

Figure 14 Same caption as Figure 13 for positions $\frac{x}{D}=1, 2$ and 3.

At $\frac{x}{D}=-2$ and 10 between 0.01 ≤ St_o ≤ 0.1 the acoustic pressures estimated by the source data from Gibert [2] agrees with the measured data for the thin (20–0.125–0–0) orifice within 30%. The acoustic pressures estimated with the source data from Moussou [17] drastically overestimates the acoustic pressures for St_o < 1, however, for higher Strouhal numbers St_o > 1 the estimated acoustic pressures agree with the measured pressures within factor 3.

At the near-field position $\frac{x}{D}=1, 2$ and 3 for low Strouhal numbers the acoustic pressures predicted by the model are almost two orders of magnitude lower than the measured pressure fluctuations (see Fig. 14). At these positions the hydrodynamic pressure fluctuations dominate for Strouhal numbers St_o < 0.3. For St_o > 0.5 the acoustic pressure fluctuations dominate even at the near field microphones x/D = 1, 2 and 3. However, the integrated total r.m.s. values of the pressure fluctuations are dominated by the hydrodynamic fluctuations, as reported earlier by Kottapalli et al. [19].

Figures 13 and 14 also show the PSD of the electronic noise-floor of the microphone. The noise-floor PSD is normalized with the dynamic pressure $\Delta p_{\mathrm{ref}}=0.5\rho U_{\mathrm{o}}^2$. Hence, depending on the orifice and the Reynolds number the normalized noise-floor varies because of U_o. For the thick upstream chamfered β = 11% orifice the PSD of the measured and estimated acoustic-pressures fluctuations is above the noise-floor for the relevant frequency range. For the thin (20–0.125–0–0) orifice the PSD of the measured pressures reach the noise-floor at high frequencies St_o ≃ 1 depending on the microphone position.

Whistling involves a feedback from the acoustic field on the flow which determines the sound production. Sharp peaks in the PSD of wall pressure fluctuations related to whistling (Fig. 13) are not predicted by the theory, because the incompressible flow model with a constant pressure as boundary condition ignores the influence of the acoustical feedback on the flow.

4.7 Effect of orifice-edge chamfers on PSD of the wall pressures fluctuations

Figure 15 shows the effect of upstream and downstream chamfer in the measured wall pressures for the $\left(100\beta -\frac{{\delta }_{\mathrm{o}}}{D_{\mathrm{o}}}-\frac{{\delta }_{\mathrm{ch},\mathrm{up}}}{{\delta }_{\mathrm{o}}}-\frac{{\delta }_{\mathrm{ch},\mathrm{dn}}}{{\delta }_{\mathrm{o}}}\right)$ = (11–0.5–0–0), (11–0.5–0.33–0) and (11–0.5–0.33–0.33) orifices. Chamfering of the upstream edge does not drastically change the low frequency behaviour of the PSD of measured wall pressures at the upstream $\left(\frac{x}{D}=-2\right)$, near-field $\left(\frac{x}{D}=1, 2, 3\right)$ and far-field $\left(\frac{x}{D}=6, 10\right)$ positions. The chamfering shows an increase in pressure amplitude for St_o < 0.1 compared to sharp edged (11–0.5–0–0) orifice. For St_o > 0.5 the acoustic pressures are not affected by the chamfering of the upstream edge of the (11–0.5–0.33–0) orifice compared to the sharp-edged (11–0.5–0–0) orifice.

Figure 15 Comparison between PSD of measured pressures pressures for the Do = 3 mm sharp-edged (11–0.5–0–0) orifice, the Do = 3 mm upstream-chamfered (11–0.5–0.33–0) orifice and the Do = 3 mm upstream-downstream-chamfered (11–0.5–0.33–0.33) orifice at Reo = 3 × 104 at positions $\frac{x}{D}=-2, 1, 2, 3, 6$ and 10.

The downstream-chamfered edge in the (11–0.5–0.33–0.33) orifice shows a 10² times reduction in the order of the pressure PSD over the complete frequency range at positions $\frac{x}{D}=-2, 6$and 10. A similar drop in acoustic pressures is also observed for the near-field positions at $\frac{x}{D}=1, 2, 3$ for St_o > 0.3.

Figure 16 shows the effect of orifice thickness (and downstream-chamfer) on the measured wall pressures for $\left(100\beta -\frac{{\delta }_{\mathrm{o}}}{D_{\mathrm{o}}}-\frac{{\delta }_{\mathrm{ch},\mathrm{up}}}{{\delta }_{\mathrm{o}}}-\frac{{\delta }_{\mathrm{ch},\mathrm{dn}}}{{\delta }_{\mathrm{o}}}\right)$ = (20–0.5–0–0), (20–0.5–0.05–0.7), (20–0.125–0–0) and (20–0.125–0–0.4) orifices. The thin and strongly chamfered orifices show similar acoustic behaviour. The thin (20–0.125–0–0) and (20–0.125–0–0.4) orifices produce drastically lower (acoustic) noise compared to the thick sharp-edged (20–0.5–0–0) orifice. The downstream-chamfering of the thick orifice (20–0.5–0.05–0.7) reduces the effective thickness δ_tip = (δ_o − δ_ch,up − δ_ch,dn) of the orifice, thus making it behave like a thin orifice. The data of Figure 16 confirm the observation that the dipole sound source (Fig. 12) for the thin orifices is lower than that of thick orifices.

Figure 16 Comparison between PSD of measured pressures pressures for the Do = 4 sharp-edged thick (20–0.5–0–0) orifice and thick-chamfered (20–0.5–0.05–0.7) orifice, thin sharp-edged (20–0.125–0–0) orifice and thin-chamfered (20–0.125–0–0.4) orifice at Reo = 2.2 × 104 at positions $\frac{x}{D}=-2, 1, 2, 3, 6$ and 10.

Figure 17 compares the pressure PSD obtained at Re_p = 10⁴ for the orifices with small relative tip thickness $\frac{{\delta }_{\mathrm{tip}}}{D_{\mathrm{o}}}=0.167, 0.167, 0.125$ and 0.075 for the $\left(100\beta -\frac{{\delta }_{\mathrm{o}}}{D_{\mathrm{o}}}-\frac{{\delta }_{\mathrm{ch},\mathrm{up}}}{{\delta }_{\mathrm{o}}}-\frac{{\delta }_{\mathrm{ch},\mathrm{dn}}}{{\delta }_{\mathrm{o}}}\right)$ = (11–0.5–0.33–0.33), (20–0.5–0.05–0.7), (20–0.125–0–0) and (20–0.125–0–0.4) orifices respectively. The acoustic pressures generated by the $\beta =11\%$ and β = 20% orifices are globally similar, as long as the relative effective thickness $\left(\frac{{\delta }_{\mathrm{tip}}}{D_{\mathrm{o}}}\right)$ remains small. However, the thick chamfered orifice (11–0.5–0.33–0.33) is more silent than the thin orifices at low Strouhal numbers St_o < 1. At higher Strouhal number the β = 20% orifices are more silent than the β = 11%. It is more difficult to decide which of the β = 20% orifices (in Fig. 17) is more silent, because the PSD approaches the electronic noise level.

Figure 17 Comparison between PSD of measured wall pressures fluctuations for δtip/Do ≪ 1 obtained with the chamfered thick (11–0.5–0.33–0.33) orifice and thick-chamfered (20–0.5–0.05–0.7) orifice, thin sharp-edged (20–0.125–0–0) orifice and thin-chamfered (20–0.125–0–0.4) orifice at Reo = 2.2 × 104 at positions $\frac{x}{D}=-2, 3$ and 10.

5 Conclusion

In the measurements of fluctuating pressure carried out here, a strong effect of trapped air bubbles can be observed, even after flushing for several hours. Still after this drastic procedure, the PSD observed for high Strouhal numbers St_o ≥ 4 is not fully reproducible.

Drag measurements of the thick (20–0.5–0–0) orifice with three different samples showed significant variations. The deviations could possibly be due to variation in flow reattachment within the orifice between the samples. The RANS and LES predictions tend to overestimate the reattachment within the orifice because they assume a turbulent flow everywhere. This leads to underestimation of the drag coefficient at low Reynolds numbers. The observed large deviation in drag coefficients between the different samples of the (20–0.5–0–0) orifice did not significantly affect the PSD of the far field wall-pressure fluctuations (dominated by the acoustic field).

Drag measurements for the upstream-chamfered orifice (11–0.5–0.33–0) also indicate that RANS and LES overestimate reattachment within the orifice. A potential-flow model [31] assuming flow separation at the end of the chamfer provides a better prediction of the drag coefficient than RANS and LES models.

The dipole sound source obtained by means of LES can be used to predict the order of magnitude of acoustic pressures due to a low Mach number flow through an orifice. The accuracy of the prediction is reduced by limitations of the LES model made apparent by the mesh dependency of the results.

Chamfering the orifice inlet reduces lateral jet oscillations within thick orifices δ_o/D_o = 0.5 observed around St_o = 0.18 for sharp-edged thick orifices (31–0.5–0–0), (20–0.5–0–0) and (11–0.5–0–0). These transversal oscillations do not radiate sound because the oscillation frequency is lower than the first transversal pipe resonance and associated non-planar wave propagation. The LES model does not predict those transversal oscillations. Upstream chamfering reduces near-field wall-pressure fluctuations around orifice Strouhal number St_o = 0.18. Additional samples of orifice (20–0.5–0–0) had different drag coefficient than the original sample and did not display the near field peak in PSD at St_o = 0.18. This indicates that the phenomenon is sensitive to small changes in geometry.

The narrowest orifices (100β − δ_o/D_o − δ_ch,up/δ_o − δ_ch,dn/δ_o) = (11–0.5–0–0) and (11–0.5–0.33–0), display sharp peaks at high Strouhal numbers, which are probably due to whistling combined with cavitation. This effect is suppressed by the downstream chamfer. This whistling is not predicted by the incompressible flow LES model, which excludes acoustical feedback from the acoustic field to the source. Such a feedback drives whistling.

Thin sharp-edged orifices are much less noisy than the sharp-edged thick orifices. The use of a thin orifice does also suppress whistling observed for sharp edged narrow thick orifices. The sound source data of Gibert [2] for thin orifices at low Strouhal numbers (St_o < 0.1) is quite accurate. The thin-orifice sound-source data correlation of Moussou [17] provides a fair prediction for thin orifices at high Strouhal numbers (St_o > 1). At lower Strouhal numbers it overestimates the sound production.

Large chamfers, leading to a significant reduction of the effective orifice thickness δ_tip/D_o = (δ_o − δ_ch,up − δ_ch,dn)/D_o ≪ 1, do make the broadband sound source approach that of thin orifices δ_o/D_o ≪ 1. The strongly chamfered narrow orifice (11–0.5–0.33–0.33) appears to be quieter than wider thin orifices at equal jet dynamic pressure $0.5\rho U_{\mathrm{o}}^2$. The present results can be used for a further study to find optimised orifice shapes.

Conflict of interest

Author declared no conflict of interests.

Data availability statement

Data are available on request from the authors.

Acknowledgments

The research was funded by ASML, Philips and TNO under the Flow Induced Vibrations Consortium Agreement – 20150911. The authors would like to thank ASML for providing us access to their high-performance cluster and research labs. The authors would also like to thank the lab technicians Henri Vliegen and Gerrit Fermius for their excellent support.

References

Appendix

A. Acoustic response of the test section

Consider the schematic of the test-section in Figure A.1. As the cut-off frequency for non-planar waves in the test-section is f_c = 95 kHz, only propagation of plane waves with amplitudes $p_m^{\pm } (m=1, 2)$ is considered. The index m = 1 refers to the segment of length L₁ upstream from the orifice and m = 2 to the downstream segment with length L₂. In the proposed one dimensional acoustical model, the sound source due to the flow is presented by a fluctuating pressure discontinuity Δp_source. The drag force fluctuation is related to this sound source by $F_{\mathrm{drag}}=\Delta p_{\mathrm{source}}\frac{\pi }{4}{D}^2$. Neglecting the wave propagation in the orifice, by conservation of momentum across the orifice, one gets:$$\left(p_1^{+}+p_1^{-}\right)-\left(p_2^{+}+p_2^{-}\right)=-\Delta p_{\mathrm{source}}+\mathrm{i}\omega \rho u_{\mathrm{o}}{L}_{\mathrm{o}}.$$(A.1)

Figure A.1 Schematic of the test-section.

By flux continuity:$$p_1^{+}-p_1^{-}=p_2^{+}-p_2^{-}=\rho c_{\mathrm{w}}{u}_0{\left(\frac{D_{\mathrm{o}}}{D}\right)}^2,$$(A.2)where, ρ is the density of water, c_w is the speed of sound in water, L_o = δ_o + 0.8D_o is the orifice thickness plus inertial end corrections and u_o is the acoustic velocity fluctuation in the orifice. At the boundaries of the test-section (at positions x = −L₁ and x = L₂) using a known reflection coefficient (assuming an infinitely long polyurethane tube on both ends) R_s/pu the ratios $\frac{p_1^{-}}{p_1^{+}}$ and $\frac{p_2^{-}}{p_2^{+}}$ are:$$\frac{p_1^{-}}{p_1^{+}}=\frac{\exp \left(2\mathrm{i}k{L}_1\right)}{R_{\mathrm{s}/\mathrm{pu}}},$$(A.3) $$\frac{p_2^{-}}{p_2^{+}}=\frac{R_{\mathrm{s}/\mathrm{pu}}}{\exp \left(2\mathrm{i}k{L}_2\right)},$$(A.4)where, k is the wave number considering viscous dissipation and the tube material, given by:$$k=k_0\left(1+(1+i)\frac{{\delta }_{\nu }}{D}\right).$$(A.5)

Here, $k_0=\frac{\omega }{c_{\mathrm{w}}}$ and ${\delta }_{\nu }=\sqrt{\frac{2\nu }{\omega }}$ is the characteristic length of the Stokes layer. In this approximation, one ignores the interaction of the acoustic boundary layers with the flow. In other words it is assumed that the Stokes layer is thin compared to the viscous sub-layer. As the first longitudinal resonance frequency of the setup is f₀ = c_steel/(2L) ≃ 1.4 kHz, one has for the relevant frequencies: δ_ν < 2 × 10⁻⁵. The PSD of the drag force fluctuations calculated by means of the LES simulation as input for the equations (A.1)–(A.4), one can calculate the PSD of the acoustic pressures at transducer positions upstream (p₁(x)) and downstream (p₂(x)) of the orifice.

The reflection coefficient of the steel-polyurethane tube junction is calculated, for f < 500 Hz, from tests by Rodriguez [33]. A third order extrapolation of the reflection coefficient data by Rodriguez was performed for frequencies up to 1 × 10⁴ Hz such that a reasonable reproduction of the standing wave ratio observed in experiments by Moonen et al. [34]. Here, the standing wave ratio is defined as ratio of maximum and minimum acoustic pressure amplitude observed between a successive peak and trough $\left(\mathrm{SWR}=\frac{p_{\max }}{p_{\min }}\right)$. The reflection coefficient is calculated using a third order polynomial function of the logarithm of the frequency, given by:$$\left|R_{\mathrm{s}/\mathrm{pu}}\right|=A_3{\left({\log }_{10}\left(\frac{f}{f_{\mathrm{ref}}}\right)\right)}^3+A_2{\left({\log }_{10}\left(\frac{f}{f_{\mathrm{ref}}}\right)\right)}^2+A_1\left({\log }_{10}\left(\frac{f}{f_{\mathrm{ref}}}\right)\right)+A_0,$$(A.6)where A₃ = −0.0044, A₂ = 0.023, A₁ = −0.075, A₀ = 0.893 and f_ref = 1 Hz.

All Tables

All Figures

	Figure 1 Schematic of the test setup to measure wall pressure fluctuations. The flow of the water is from left to right through the test section. In first approximation the junctions between the steel test section and the long polyurethane (PU) tubes act as low impedance boundary conditions.
In the text

Figure 2 Influence of air in the system on the PSD of the measured wall pressure fluctuations at $\frac{x}{D}=-2, 1$ and 10 for the thick β = 20% orifice plate with 0.1 width upstream and downstream chamfers (20–0.5–0.05–0.05). The PSD and the frequencies are normalised with respect to dynamic pressure in the orifice $\left(0.5\rho U_{\mathrm{o}}^2\right)$, the average flow velocity within the orifice Uo and the orifice diameter Do.

In the text

	Figure 3 Dimensions of the computational domain of LES with the β = 11% (Do = 3.0 mm) orifice with δo = 1.5 mm and a 45° upstream chamfer of width δch,up = 0.5 mm also designated as the (11–0.5–0.33–0) orifice.
In the text

	Figure 4 Dimensions of the computational domain of LES for β = 20% (Do = 4.0 mm) orifices with sharp edges. The orifice thickness δo = 0.125 Do =0.5 mm for the thin (20–0.125–0–0) orifice and δo = 0.5 Do = 2.0 mm for the thick (20–0.5–0–0) orifice.
In the text

	Figure 5 Predicted PSD of acoustic sound source due to fluctuating drag force for the (11–0.5–0.33–0) orifice obtained using LES with the fine mesh and the coarse mesh. The third simulation is an Implicit-LES (ILES) with the coarse mesh.
In the text

	Figure 6 (a) Static pressure drop (ΔP) across the (20–0.5–0–0) orifice. (b) Mean axial velocity within the orifice $\frac{x}{D_{\mathrm{o}}}=-0.5-0.25, 0$ and slightly downstream of the orifice at $\frac{x}{D_{\mathrm{o}}}=0.25$ from LES. The horizontal dotted (▪▪▪) line represents $\frac{1}{\alpha }$ calculated from RANS using equation (2).
In the text

Figure 7 (a) Comparison between static pressure drops across the (11–0.5–0.33–0) orifice computed using steady-state k-ϵ RANS model and Low-Re k-ϵ RANS model. (b) and (c) Mean axial velocity within (11–0.5–0.33–0) orifice at $\frac{x}{D_{\mathrm{o}}}=-0.5, -0.25, 0$ and slightly downstream of the orifice at $\frac{x}{D_{\mathrm{o}}}=0.25$ from k-ϵ RANS model and Low-Re k-ϵ RANS model. The horizontal dotted (▪▪▪) line represents $\frac{1}{\alpha }$ calculated from RANS using equation (2).

In the text

Figure 8 Comparison between measured and computed normalised drag coefficient $\frac{C_{\mathrm{drag}}}{(C_{\mathrm{drag}}{)}_{\mathrm{ref}}}$ as a function of orifice Reynolds number Reo for the thick (20–0.5–0–0) orifice, the thin (20–0.125–0–0) orifice, and the chamfered (11–0.5–0.33–0) orifice. The dotted black line is the theoretical drag coefficient (potential flow theory) for a 90° converging nozzle (corresponding to the (11–0.5–0.33–0) orifice) [31]. The $(C_{\mathrm{drag}}{)}_{\mathrm{ref}}$ are calculated for thin orifices using the empirical equation (4) (proposed by Idelchik [20]).

In the text

Figure 9 Comparison between the normalised PSD of wall pressure fluctuations measured for sample 1 and sample 3 of (20–0.5–0–0) orifice at transducer positions transducer positions $\frac{x}{D}=1, 2$ and 3 with orifice Reynolds number, Reo = 22,000 as a function of Strouhal number $\left(\frac{f{D}_{\mathrm{o}}}{U_{\mathrm{o}}}\right)$. As shown in Figure 8, sample 1 and 3 have drastically different steady flow drag coefficients but the PSD of the wall pressure fluctuations are quite similar.

In the text

	Figure 10 Same as Figure 9 with measurements at transducer positions transducer positions $\frac{x}{D}=-2, 6$ and 10.
In the text

	Figure 11 Reynolds number dependency of wall pressure fluctuations PSD for the chamfered β = 11% (11–0.5–0.33–0) orifice. The PSD are normalised using average flow velocity through the orifice Uo and the orifice diameter Do as a function of the orifice Strouhal number $\frac{f{D}_{\mathrm{o}}}{U_{\mathrm{o}}}$.
In the text

Figure 12 Comparison between normalized dipole sound source from LES for the β = 20% sharp-edged (20–0.5–0–0) orifice at Reo = 2.25 × 104, the β = 11% upstream-chamfered (11–0.5–0.33–0) orifice at Reo = 3 × 104 and the β = 20% thin orifice (20–0.125–0–0) at Reo = 2.25 × 104. The LES source data is compared to measured source data for thin orifices with β = 9%, 16%, 25% by Gibert [2] and Moussou [17].

In the text

Figure 13 Comparison between PSD of measured pressures and estimated acoustic pressures for the β = 20% sharp-edged thick (20–0.5–0–0) orifice (left column) at Reo = 2.25 × 104, the β = 11% upstream-chamfered thick (11–0.5–0.33–0) orifice at Reo = 3 × 104 (center column) and the β = 20% sharp-edged thin (20–0.125–0–0) orifice at Reo = 2.25 × 104 (right column) at positions $\frac{x}{D}=-2, 6$ and 10. The acoustic pressures are estimated from LES and data from Gibert [2] and Moussou [17].

In the text

	Figure 14 Same caption as Figure 13 for positions $\frac{x}{D}=1, 2$ and 3.
In the text

	Figure 15 Comparison between PSD of measured pressures pressures for the Do = 3 mm sharp-edged (11–0.5–0–0) orifice, the Do = 3 mm upstream-chamfered (11–0.5–0.33–0) orifice and the Do = 3 mm upstream-downstream-chamfered (11–0.5–0.33–0.33) orifice at Reo = 3 × 104 at positions $\frac{x}{D}=-2, 1, 2, 3, 6$ and 10.
In the text

	Figure 16 Comparison between PSD of measured pressures pressures for the Do = 4 sharp-edged thick (20–0.5–0–0) orifice and thick-chamfered (20–0.5–0.05–0.7) orifice, thin sharp-edged (20–0.125–0–0) orifice and thin-chamfered (20–0.125–0–0.4) orifice at Reo = 2.2 × 104 at positions $\frac{x}{D}=-2, 1, 2, 3, 6$ and 10.
In the text

	Figure 17 Comparison between PSD of measured wall pressures fluctuations for δtip/Do ≪ 1 obtained with the chamfered thick (11–0.5–0.33–0.33) orifice and thick-chamfered (20–0.5–0.05–0.7) orifice, thin sharp-edged (20–0.125–0–0) orifice and thin-chamfered (20–0.125–0–0.4) orifice at Reo = 2.2 × 104 at positions $\frac{x}{D}=-2, 3$ and 10.
In the text

	Figure A.1 Schematic of the test-section.
In the text