The role of compressibility in computing noise generated at a cavitating orifice

Abstract

The computational fluid dynamics calculation can be accomplished by solving either compressible or incompressible Navier–Stokes equations to determine the flow-field variables of the noise source. The proper assumption depends on both the physical situation and the Mach number. Although in cavitating devices usually we are dealing with low Mach number flow, cavitation is an acoustic phenomenon that can be affected by compressibility. Cavitation behaves acoustically as a monopole and it is mentioned by some researchers that incompressible solution is sufficient to study the dipole sources. However, in order to study the monopole (and quadrupole) sources a compressible solution may be required. In this study, the role of compressibility in computing noise generated at a cavitating single-hole orifice was investigated using large eddy simulation and Ffowcs Williams–Hawkings formulation. The fluid zone downstream of the orifice where the cavitation occurs was evaluated as the acoustic source which generates sound. Time-accurate solutions of the flow-field variables on source surfaces were obtained from both compressible and incompressible flow simulations. Three cases of cavitation were studied and the sound pressure signals far downstream of the orifice were computed by the Ffowcs Williams–Hawkings formulation. For a developed cavitation regime at low frequencies, there is a big discrepancy between the computed values of sound pressure level from compressible and incompressible simulations, and at higher frequencies greater than 6 kHz, both simulation methods provide almost the same values for sound pressure levels. For a super cavitation regime, both compressible and incompressible simulations provide similar values for sound pressure levels at frequencies greater than 2 kHz. The results of this work demonstrate that the compressibility has a significant role in terms of computing noise generated at a cavitating orifice and cannot be ignored, especially when the noise generated by developed cavitation regimes at low frequencies is investigated.

Keywords

Orifice large eddy simulation Ffowcs Williams–Hawkings compressible simulation noise developed cavitation super cavitation

Introduction

Computational methods for flow-generated sound can be divided into two broad categories: direct computation and hybrid computation. In the first case, the noise is defined together with the fluid dynamic field by solving the flow equations directly and the numerical scheme is dictated by the required assessment of the different flow scales. In the direct approach, the sound together with its fluid dynamic source field is computed by solving the compressible flow equations. Theoretically, a direct numerical simulation (DNS) which resolves all flow scales would be the best choice, but the present computational resources do not allow deploying such a simulation for every case. Large eddy simulation (LES), which resolves only the dynamically important flow scales while modelling the effects of smaller scales, can be applied. It is also possible to use unsteady Reynolds-averaged Navier–Stokes (RANS) methods to compute the noise generated by the largest flow features.^1–3 The development and the application range of these alternative methods such as LES and RANS still represent a research issue.² Applying the direct approach in terms of sound computation requires the solution of the compressible flow equations; sound is determined as a pressure fluctuation propagating in a medium at a finite speed and as it is under the isentropic flow hypothesis, and sound propagation speed c₀ is specified by $c_{0} = d p / d ρ$ . As a result, the incompressibility assumption ( $d ρ = 0$ ) prevents any propagation phenomena (c₀ becomes infinite) and is not physically compatible with any acoustic analysis.²

In the hybrid approach, the computation of sound is decoupled from the computation of flow. The far-field sound is calculated by integral or numerical solutions of acoustic analogy equations by applying computed source field data.^1,4 The one-way coupling of flow and sound is a principal assumption of acoustic analogy-based prediction, i.e. the unsteady flow induces sound and modifies its propagation, but the sound waves do not affect the flow in any significant way.²

The noise prediction can be simplified by applying the hybrid approach. The computational fluid dynamics (CFD) calculation can be accomplished by solving either compressible or incompressible Navier–Stokes equations. The proper assumption depends on both the physical situation and the Mach number. Specially, an incompressible simulation can be adequate to provide the solver with all the requested input data if no compressibility affects the acoustic phenomena. For example in air, the incidence of a shock wave related to a transonic/supersonic range of the blade speed is the only compressibility effect that can affect the noise generated by a propeller. Therefore, at a low rotational Mach number, the prediction of the fundamental loading noise component is successfully performed by using the unsteady air load distributions provided by an incompressible aerodynamic code.^2,5 At low Mach numbers, incompressible flow solutions can be suitable for computing acoustic source terms.¹ Layton and Novotny⁶ have mentioned that the direct approach is often costly, unstable, inefficient and unreliable for flows in the low Mach number regimes (below 0.3 according to Wilcox⁷). At low Mach numbers, using incompressible models is an efficient way to improve the simulation. Wang et al.¹ derived that incompressible flow solutions are sometimes adequate at low Mach numbers but there is no agreed Mach number threshold for CFD solutions in terms of identifying aero-acoustic sources.

Wang and Moin⁸ used incompressible LES in conjunction with Lighthill’s theory to simulate the trailing-edge aero-acoustic experiment which involves flow over a flat strut with an asymmetric trailing edge and this has been performed by Blake⁹ and Blake and Gershfeld.¹⁰ The simulation results show reasonable agreement with experimental data for velocity statistics, frequency spectra of surface pressure fluctuations and the far-field sound spectra.

Langthjem and Olhoff^11,12 determined the acoustic pressure field in a centrifugal pump. They developed a computationally simple and fast method which is capable of giving a useful estimate of the noise-generating ‘background flow’ and obtained flow-field variables from an incompressible flow analysis. They expressed the solution to the inhomogeneous wave equation in the frequency domain by using a Fourier transformation. Their numerical results were well supported with available experimental results. Also, they showed that the frequency-domain solution is very useful method to minimize the flow noise by design optimization.

Olivares¹³ studied and simulated the acoustic wave propagation on district heating pipes by applying 2D incompressible LES and Ffowcs Williams–Hawkings (FW–H) formulation and 3D compressible LES and FW–H formulation. He mentioned that 2D incompressible simulations cannot predict the real amplitudes of the noise level. It can only show at which frequency range the noise is strongest. The 3D compressible case showed that noise also exists in the low-frequency region and that the noise is clearly affected by temperature and pressure. Also, the power spectral densities resulting from 3D compressible LES and FW–H formulations provide valuable information about the locations of the largest peaks.

Liu⁵ used the unsteady RANS and the acoustic finite element method to simulate the whistle noise caused by vortex shedding. The flow-field variables were obtained from CFD with an aim to determine the aero-acoustic sources from Lighthill’s analogy. He applied the results of both compressible and incompressible CFD simulation to find flow-field variables. He compared the acoustic results of both analytical solutions and experimental data to develop a better understanding of the effects of fluid compressibility on the acoustic frequency response. The results verified that the incompressible flow assumption can be sufficient for some flow situations with a very low flow speed. However, he recommended assuming compressible flow instead of incompressible flow where the Mach number is bigger than 1.10. In such cases, the aero-acoustic source terms are very sensitive to density changes. Using incompressible flow will potentially result in solutions that do not capture the vortex shedding phenomenon.

It is notable that in all of the above cases where the incompressible Navier–Stokes equations were used for the source flow simulations, the Mach numbers are low. Also, since water practically behaves as an incompressible fluid, it is worth pointing out that for a hydrodynamic simulation of marine and/or maritime problems always incompressibility assumption is applied and the assumption $d ρ = 0$ results in a notable simplification of both the governing equations and the corresponding numerical schemes.² In cavitating devices such as a cavitating orifice usually we are dealing with a low Mach number flow but cavitation is an acoustic phenomenon and can be affected by compressibility.^14,15 An example of this effect is the collapse of vapour bubbles in a cavitating flow. In addition, cavitation behaves acoustically as a monopole and it is mentioned by some researchers that an incompressible solution is sufficient to study the dipole sources. However, in order to study the monopole (and quadrupole) sources a compressible solution may be required.^2,13,16 A monopole source is categorized as the simplest acoustic source and radiates sound equally in all directions. In aero-acoustics, monopoles are normally generated by a pulsating flow.⁵ Therefore, an incompressibility assumption in terms of source flow simulation for hybrid computation of generated sound by a cavitating flow is doubtful. Although an incompressibility assumption results in a notable simplification of both the governing equations and the corresponding numerical schemes, the effect of this assumption on the accuracy of the predicted noise by the hybrid computation method for cavitating flow cases should be investigated further. Several researchers applied compressible CFD simulations to study cavitating flow or noise generated by cavitating flow with a hybrid aero-acoustic computation.

Seo et al.¹⁷ presented a DNS procedure for the cavitating flow noise. They wrote the compressible Navier–Stokes equations for the two-phase fluid and employed a density-based homogeneous equilibrium model with a linearly combined equation of state. They computed cavitating flow noise for a 2D circular cylinder flow at a Reynolds number based on its diameter, 200 mm and cavitation numbers, σ = 0.7–2. Their results showed that at cavitation numbers σ = 0.7 and σ = 1, the cavitating flow and its noise characteristics were significantly affected by the shock waves. They derived an acoustic analogy based on a classical theory of Fitzpatrik and Strasberg to verify the direct simulation and further analysed the sources of cavitation noise. Then, the far-field noise calculated by direct simulation was compared with that of an acoustic analogy. Their results also confirmed $f^{- 2}$ decaying rate in the sound pressure level spectrum, like the model of Fitzpatrik and Strasberg which the Rayleigh–Plesset equation suggests.

Lacombe et al.¹⁸ characterized the whistling ability of an orifice by using a numerical approach. They used CFD to identify the aero-acoustic scattering matrix of an orifice in a duct. First, they performed a LES of a turbulent compressible flow, with superimposed broadband acoustic excitations. Then, a time series of acoustic data was extracted by using a specific filter and finally system identification techniques were applied. Comparisons between experimental and numerical results at two different Mach numbers showed a good agreement between the experiment and the CFD simulation. The simulation predicted the potential whistling frequency range in terms of frequency and amplitude.

Hickel¹⁹ performed a numerical modelling and simulations of compressible two-phase flows that involve phase transition between the liquid and gaseous state of the fluid. He stated that compressibility of both the liquid phase and the gaseous phase plays a decisive role and that the numerical simulations of cavitating flows should solve complex multi-physics and multiscale problems in a consistent way. He applied a thermodynamic equilibrium model for LES to solve the coarse-grained Navier–Stokes equations for a homogenized, cell-averaged fluid. He presented the results for the cavitating flow in a throttle valve. He showed that this thermodynamic equilibrium model can contribute to a better understanding of the mutual interaction of turbulence and cavitation.

In this study, the role of compressibility in computing of generated noise at a cavitating single-hole orifice has been investigated by using CFD and FW–H formulation. The fluid zone downstream of the orifice where the cavitation occurs is evaluated as the acoustic source which generates sound. Time-accurate solutions of the flow-field variables, such as pressure and velocity components on source surfaces, are obtained from both compressible and incompressible LES. Three cases of cavitation are investigated and the sound pressure signals far downstream of the orifice are computed by applying the FW–H formulation. Then the sound pressure levels far downstream of the orifice are presented. For each case, the acoustic results predicted by FW–H formulation based on both compressible and incompressible simulations for source flow are compared to investigate the effect of compressibility in computing the generated noise.

Theory

FW–H formulation

The FW–H formulation extracts the most general form of Lighthill’s acoustic analogy and is able to predict sound generated by equivalent acoustic sources like monopoles, dipoles and quadrupoles.⁴ The FW–H equation that can be derived by manipulating the continuity equation and the Navier–Stokes equations is an inhomogeneous wave equation. The FW–H equation can be written in the following format^4,20

\frac{1}{c_{0}^{2}} \frac{\partial^{2} p^{′}}{\partial t^{2}} - \nabla^{2} p^{′} = \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} {T_{i j} H (f)} - \frac{\partial}{\partial x_{i}} {[P_{i j} n_{j} + ρ u_{i} (u_{n} - v_{n})] δ (f)} + \frac{\partial}{\partial t} {[ρ_{0} v_{n} + ρ (u_{n} - v_{n})] δ (f)}

(1)

where p^{′}

represents the sound pressure at the far field (

p^{′} = p - p_{0}

). The surface (

f = 0

) corresponds to the source (emission) surface which can be a body (impermeable) surface or a permeable surface off the body surface. T_ij is the Lighthill stress tensor and determined as

T_{i j} = ρ u_{i} u_{j} + P_{i j} - c_{0}^{2} (ρ - ρ_{0}) δ_{i j}

(2)

$P_{i j}$ is the compressive stress tensor and for a Stokesian fluid can be defined given by

P_{i j} = p δ_{i j} - μ [\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}} - \frac{2}{3} \frac{\partial u_{k}}{\partial x_{k}} δ_{i j}]

(3)

By assuming a free-space flow and the absence of obstacles between the sound sources and the receivers, equation (1) can be integrated analytically and the complete solution includes both surface integrals and volume integrals. While the surface integrals indicate contributions from monopole, dipole and partially quadrupole acoustic sources, the volume integrals represent quadrupole sources from the region outside the source surface.

Cavitation number

A free jet surrounded by a dead water pressure region of uniform pressure is usually generated by an orifice in single-phase flow. In the jet region, the static pressure has its minimum value and large eddies are produced in the shear layer separating the jet in the dead water region. When the lowest static pressure in the fluid goes below the vapour pressure, two-phase flow transition occurs.²¹

The cavitation number is the parameter usually applied to show the level of cavitation. There are different definitions for the cavitation number especially for hydrodynamic cavitation. For the cavitation around a body such as a hydrofoil or generated by a slit, the cavitation number is generally specified as a function of the upstream conditions^21,22

σ = \frac{P_{0} - P_{V}}{(1 / 2) ρ_{L} U_{0}^{2}}

(4)

For the cavitation formed in a jet (mixing cavitation such as cavitation in pumps, valves, orifices), a similar cavitation number, as cavitation around a body, can be applied^21,22

σ = \frac{P_{r e f} - P_{V}}{(1 / 2) ρ_{L} U_{0}^{2}}

(5)

In this study common practice in industry has been followed and the cavitation number based on the pressure drop across the singularity generating the jet is used²³

σ = \frac{P_{2} - P_{V}}{Δ P}

(6)

If the static pressure at the jet has a low value, intermittent tiny cavitation bubbles grow in the centre of the turbulent eddies along the shear layer of the jet. This transient regime of the flow is called cavitation inception. Cavitation inception arises at a cavitation number around 1.^23–27 When the jet pressure decreases, more bubbles with larger diameters form and create a white cloud. The flow regime experiences more pressure fluctuations and a characteristic bang may be heard. Continued decreasing of the jet pressure causes a large vapour pocket formation downstream of the orifice, surrounding the liquid jet. The regime happening after this transition is called super cavitation and it causes the largest noise and vibration levels. In the super cavitation regime, the noise is mostly generated in a shock transition between the cavitation zone and the full liquid flow of the pipe, at some distance downstream of the orifice.^28,29 The occurrence of cavitation regimes can be predicted by using cavitation indicators. In this study, the same cavitation indicators applied by Testud et al.²⁸ to experimentally investigate noise generated by cavitating orifices in a water pipe have been deployed.

Model geometry and water characteristics

In this study a single-hole orifice that was previously investigated experimentally by Testud et al.²⁸ in terms of noise generated by cavitating flow has been modelled. The flow characteristics of the transient regime downstream of the orifice as the sound sources were computed by using CFD and far-field noise prediction based on the FW–H formulation. The geometry characteristics of the orifice are as follows:

It is a single-hole, circular, centred orifice, with right angles and sharp edges. Its thickness (t) is 0.014 m and its diameter (d) is 0.022 m. The orifice is located inside of a hydraulically smooth steel pipe of 0.074 m inner diameter (D) and 0.008 m wall thickness (t_p). Since $\frac{t}{d} \leq 2$ , the orifice is defined as a ‘thin’ orifice.

The length of the pipe is included in the CFD model to find time-accurate solutions of the flow-field variables on source surfaces (fluid zone downstream of the orifices). The pipe length is 5D upstream of the orifice and 15D downstream of the orifice. The water characteristics applied in this study are shown in Table 1.

Table 1.

Water characteristics.

Parameter	Value
Temperature (K)	310
Dynamic viscosity of water (μ_w (310 K))(kg m⁻¹ s⁻¹)	8.86 × 10⁻⁴
Density of water (ρ_w (310 K))(kg m⁻³)	994
Speed of sound in pure water (m s⁻¹)	1,420
Water vapour pressure (P_w (310 K))(Pa)	5.65 × 10³

Numerical procedures

Mathematical models, mesh and unsteady simulation parameters

Three-dimensional Navier–Stokes equations are solved for compressible and incompressible flow. The LES approach is applied to simulate the unsteady large-scale structures and separation zones. In LES, large eddies are solved directly, and small eddies are simulated with a subgrid-scale model. A main function of the small eddies is to dissipate the turbulent energy which is transferred from the larger scales to the smaller scales by the energy cascade.²⁰ In the LES method, low-pass spatial filtering of the governing equations is deployed to achieve the separation between the resolved and unresolved scales. The subgrid-scale stresses which result from the filtering operation are uncertain and need to be modelled.³⁰ LES is the most appropriate approach for acoustic applications. The approach is a compromise solution between DNS and RANS. All scales are numerically solved in DNS, while in RANS, all scales are modelled. In LES, the transport equations are filtered, only larger eddies are resolved and the smaller eddies modelled. Therefore, LES is an efficient method of achieving good results in turbulent flows.³¹ As LES requires that only larger eddies resolve, coarser mesh and a larger time step can be applied compared to DNS, but still a much finer mesh is needed compared to other turbulent models. LES has to run for a long flow time to obtain statistics so the flow can be modelled and achieve good results. As a result, computational costs in terms of RAM and CPU are higher than RANS models and high-performance computing is required.¹³

In this case, wall adapting local eddy-viscosity (WALE) model is used. 0.5 is the published value for the WALE constant (C_w) but validation during a European Union research project has presented consistently superior results in ANSYS Fluent with C_w = 0.325, and therefore this value is used as the default setting. The WALE model is designed to provide the correct wall asymptotic (y³) behaviour for wall-bounded flows. In addition, the WALE model provides a zero turbulent viscosity for laminar shear flows and allows the correct treatment of laminar zones in the domain.²⁰

Schnerr and Sauer^32,33 model is utilized to include cavitation effects in two-phase flows while the mixture model is applied. Water-liquid and water vapour are used as primary phase material and secondary phase material, respectively. In compressible simulation, the ideal gas equation of state is applied for the vapour domain and Tait’s equation of state is deployed for the liquid domain.¹⁹ Schnerr and Sauer model is applied following general form for the vapour volume fraction

\frac{\partial}{\partial t} (α ρ_{v}) + \nabla . (α ρ_{v} \vec{V}) = \frac{ρ_{v} ρ_{l}}{ρ} \frac{D α}{D t}

(7)

Where the net mass source term is as follows

R = \frac{ρ_{v} ρ_{l}}{ρ} \frac{d α}{d t}

(8)

Schnerr and Sauer use the following expression to connect the vapour volume fraction to the number of bubbles per volume of liquid

α = \frac{n_{b} \frac{4}{3} π ℜ_{B}^{3}}{1 + n_{b} \frac{4}{3} π ℜ_{B}^{3}}

(9)

then they derived the following equations

R = \frac{ρ_{v} ρ_{l}}{ρ} α (1 - α) \frac{3}{ℜ_{B}} \sqrt{\frac{2}{3} \frac{(P_{v} - P)}{ρ_{l}}}

(10)

ℜ_{B} = {(\frac{α}{1 - α} \frac{3}{4 π} \frac{1}{n_{b}})}^{\frac{1}{3}}

(11)

where

$R =$ mass transfer rate

$ℜ_{B} =$ bubble radius

Equation (10) is also used to model the condensation process. The final form of the model is as follows

when $P_{v} \geq P$

R_{e} = \frac{ρ_{v} ρ_{l}}{ρ} α (1 - α) \frac{3}{ℜ_{B}} \sqrt{\frac{2}{3} \frac{(P_{v} - P)}{ρ_{l}}}

(12)

when

P_{v} \leq P

R_{c} = \frac{ρ_{v} ρ_{l}}{ρ} α (1 - α) \frac{3}{ℜ_{B}} \sqrt{\frac{2}{3} \frac{(P - P_{v})}{ρ_{l}}}

(13)

The frequency domain approach of FW–H formulation which is applicable to the computed domain is used as the governing equation for the noise prediction far downstream of the orifices.

The ANSYS-Fluent software with parallel processing, version 16.1, is utilized to solve the filtered governing equations that are discretized by the finite volume method. The ‘SIMPLE’ scheme is used for pressure–velocity coupling. For the momentum spatial discretization, ‘Bounded Central Differencing’ scheme is applied and ‘Least Squares Cell Based’ scheme, ‘PRESTO!’ scheme, ‘QUICK’ scheme and ‘Second-Order Upwind’ scheme are used for gradient spatial discretization, pressure spatial discretization, volume fraction and density spatial discretization, respectively. The ‘Bounded Second-Order Implicit’ formulation is deployed for the temporal discretization.²⁰

When meshing the computational domain, the multi-block structured hexahedral mesh created by ANSYS Meshing tools is used. The mesh is refined at the walls to capture the boundary layer. The grid is concentrated near the walls and the measured y+ is around 1. As the grid moves further away from the walls, it is enlarged by an expansion ratio of 1.2:1. The initial instability of the shear layer controls the aero-acoustic interaction, therefore the resolution close to the edge can be a crucial parameter and the mesh is refined around the orifice.³⁴ In the radial direction, the grid size is refined as well at the orifice wall and in the area of the jet downstream of the orifice. Table 2 shows the features of the computational grids. This choice of the grid size guaranteed grid-independent simulations.

Table 2.

Features of the computational grids.

Number of cells	2.3 × 10⁶
Smallest cell size (m)	1.48 × 10⁻⁴
Largest cell size (m)	2.8 × 10⁻³

High grid resolution with enough small time steps is applied to resolve the relevant fluctuations. The unsteady calculations are performed with a time-step size of 5 × 10⁻⁵ s. This time step is sufficient to keep the Courant number less than 1 for the majority of computational cells and accurately resolve pressure spectra within human hearing range. Unsteady simulation parameters are listed in Table 3. The simulations are run by using 30 cores (Intel® Xeon® Processor E5-2640). At each case, the transient solution is run for 1000 time steps before enabling the acoustics model to obtain a statistically steady-state solution. After the unsteady flow field under consideration has become fully developed, the FW–H acoustics model is enabled and acoustic source data are exported for 0.05 s flow time (1000 time steps).

Table 3.

Unsteady simulation parameters.

Parameter	Value
Total simulated flow time (s)	0.1
Simulated flow time before enabling the acoustics model (s)	0.05
Time step (s)	5 × 10⁻⁵
Number of total time steps	2000
Number of time steps before enabling the acoustics model	1000
Max. number of iterations per time step for incompressible simulation	50
Max. number of iterations per time step for compressible simulation	70
Average hardware operating time per case for incompressible simulation (s)	518,400
Average hardware operating time per case for compressible simulation (s)	777,600

Boundary and initial conditions

The mass flow inlet boundary condition with pressure is employed at the inlet of the pipe and a pressure outlet boundary condition is applied at the domain outlet. To simulate different cavitation cases, values of the inlet mass flow rate and outlet pressure at different cases were calculated based on the values presented by Testud et al.²⁸ resulting from their experiments (Table 4). The converged steady-state results are applied as initial conditions for unsteady simulation.

Table 4.

Flow conditions in the single-hole orifice.²⁸

	Developed cavitation			Super cavitation
U (m s⁻¹)a	1.91	2.38	2.90	3.75	4.08	4.42
St. deviation (m s⁻¹)	0.06	0.04	0.04	0.03	0.03	0.05
P₁ (10⁵ Pa)b	6.3	9.2	13.3	21.4	25.0	29.5
St. deviation (10⁵ Pa)	0.3	0.3	0.6	1.4	0.7	0.8
P₂ (10⁵ Pa)b	2.7	2.7	2.7	2.8	2.8	2.8
St. deviation (10⁵ Pa)	0	0	0	2.0	0.2	1.5
$σ$	0.74	0.41	0.25	0.15	0.12	0.10

^aThe Reynolds number is based on the pipe diameter and the water viscosity that varies from 2 × 10⁵ to 5 × 10⁵.

^bThe static pressures P₁ and P₂ are measured 11D upstream and 40D downstream of the orifice, respectively.

Results and discussion

Table 4 shows the flow conditions for the developed cavitation and the super cavitation in the single-hole orifice experiments conducted by Testud et al.²⁸ in which $σ_{i} \geq 0.93$ , $σ_{ch} = 0.25$ and all experiments based on the provided flow conditions for a single-hole orifice are cavitating ( $σ < 0.74$ ). The last three experiments experience a super cavitation regime as well ( $σ < 0.25$ ).

In this section the simulated results for the developed cavitation conditions (σ = 0.74 and σ = 0.41) and the super cavitation condition (σ = 0.15) for both compressible and incompressible simulations are presented and analysed.

Figures 1 and 2 show visualization of the developed and super cavitation regimes, respectively, for a single-hole orifice from other experiments at EDF R&D.³⁵ For the developed cavitation regime, white bubble clouds are observed around the jet which formed at the orifice. In the super cavitation condition, vapour pockets are formed in the jet region and expand downstream of the orifice.

Figure 1.

Visualization of the developed cavitation regime for a single-hole orifice (d/D = 0.30, t/d = 0.10, D = 2.66 × 10⁻¹ m) with σ = 0.49 and U = 1.5 m s⁻¹ (ΔP = 3.1 × 10⁵ Pa, P₁=4.6 × 10⁵ Pa) from other experiments at EDF R&D.³⁵

Figure 2.

Visualization of the super cavitation regime for a single-hole orifice from other experiments at EDF R&D.³⁵

Figure 3 shows the vapour volume fraction at different flow times downstream of the single-hole orifice for the developed cavitation condition (σ = 0.41) from both compressible and incompressible simulations. The results of both methods show that similar to the experimental visualization (Figure 1), the bubbles are formed around the jet at the orifice. However, the incompressible simulation shows lots of vapour pockets downstream of the orifice while the compressible simulation shows that just a few bubbles form downstream of the orifice. Compared to the visualization of the developed cavitation regime from the experiment (Figure 1), it appears that the incompressible simulation overestimates the quantity of bubbles which form downstream of the orifice and the compressible simulation provides more similarity to the visualizations.

Figure 3.

Vapour volume fraction at different flow times downstream of the single-hole orifice for developed cavitation condition (σ = 0.41): (a) 0.0525 s, incompressible simulation; (b) 0.08 s, incompressible simulation; (c) 0.0525 s, compressible simulation and (d) 0.08 s, compressible simulation. (A) Close up view showing vapour volume fraction at 0.08 s flow time downstream of the single-hole orifice for developed cavitation condition (σ = 0.41) resulting from compressible simulation.

Figure 4 shows the vapour volume fraction at different flow times downstream of the single-hole orifice for the super cavitation condition (σ = 0.15) from both compressible and incompressible simulations. The results of both methods show that the vapour pockets mainly form in the jet region and expand downstream of the orifice. For the incompressible simulation, the vapour volume fraction is around 1 in the majority of areas downstream of the orifice while the compressible simulation predicts fewer bubbles downstream of the orifice. However, comparison between Figures 2 and 4 clarifies that the predicted results by the compressible simulation are more similar to the visualization of the super cavitation regime captured in the experiment.

Figure 4.

Vapour volume fraction at different flow times downstream of the single-hole orifice for the super cavitation condition (σ = 0.15): (a) 0.055 s, incompressible simulation; (b) 0.0855 s, incompressible simulation; (c) 0.055 s, compressible simulation and (d) 0.085 s, compressible simulation.

In Figure 5, the sound pressure levels at 28D and 280D downstream of the single-hole orifice from both compressible and incompressible simulations are presented for the developed cavitation regime with σ = 0.74. Repeatability of the provided spectrums has been verified by repeating of the noise simulation and sound pressure level computing. The graphs of the sound pressure levels generated from different simulation methods have a totally different trend for both distances downstream of the orifice. The values of the sound pressure level computed by a compressible simulation generally are increased in low frequencies less than 800 Hz while there are few peaks and troughs at this frequency range (less than 800 Hz). Then the average value of the sound pressure level is fairly steady at a frequency range between 800 and 6000 Hz. At frequencies greater than 6 kHz the computed values for sound pressure level from compressible simulation start to decrease.

Figure 5.

Sound pressure levels downstream of the single-hole orifice for both compressible and incompressible simulations for a developed cavitation regime with σ = 0.74: (a) 28D downstream of the orifice and (b) 280D downstream of the orifice.

The results computed by the incompressible simulation show that the values of the sound pressure level increase from 100 to 2020 Hz to meet its maximum values around 2020 Hz. The average values of the sound pressure level are steady at a frequency range between 2020 and 4000 Hz and then it decreases at frequencies greater than 4 kHz. The incompressible simulation has a maximum sound pressure level at 2020 Hz. The compressible simulation has a maximum sound pressure level at 505 Hz.

At low frequencies less than 1000 Hz there is a large discrepancy between computed values of sound pressure level between compressible and incompressible simulations. The discrepancy between the calculated sound pressure levels from both simulation methods is around 35 dB at 500 Hz. This discrepancy is decreased at frequency ranges greater than 1 kHz and both compressible and incompressible simulations provide almost the same values of sound pressure level at frequency ranges greater than 5 kHz. For both compressible and incompressible simulations, the sound pressure level is broad in the range of 1–5 kHz. This broadband sound induces from the explosions of small-size bubbles. It is notable that in the range which human hearing is more sensitive to sound, both simulation methods provide more similar results. The human hearing range is generally stated as 20 Hz to 20 kHz, but it is far more sensitive to sounds between 1 and 4 kHz.³⁶

Figure 6 shows the sound pressure levels at 28D and 280D downstream of the single-hole orifice for both compressible and incompressible simulations for the developed cavitation regime with σ = 0.41. Similar to Figure 5, the graphs of the sound pressure levels resulted from different simulation methods have a totally different trend for both distances downstream of the orifice. The average values of the sound pressure level computed by compressible simulation are fairly steady at a frequency range between 400 and 2500 Hz and start to decrease at frequencies greater than 2500 Hz. The compressible simulation shows that at 384 Hz the sound pressure level has its maximum value. The computed results by the incompressible simulation show that the values of the sound pressure level increase from a low frequency to meet its maximum value at a frequency range between 2000 and 2150 Hz and then the average value of the sound pressure level almost is fairly constant.

Figure 6.

Sound pressure levels downstream of the single-hole orifice for both compressible and incompressible simulations for a developed cavitation regime with σ = 0.41: (a) 28D downstream of the orifice and (b) 280D downstream of the orifice.

Similar to Figure 5, at low frequencies there is a large discrepancy between the computed values of sound pressure level from compressible and incompressible simulations. Although the computed values of both simulation methods are not close for all frequencies less than 7 kHz, there is a big difference between the values from both methods at frequencies less than 2 kHz. This discrepancy is around 55 dB at 400 Hz. Both compressible and incompressible simulations predict that the sound pressure level is broad in the range of 2–5 kHz which induces from the explosions of small-size bubbles. At a frequency range greater than 7 kHz, both simulation methods provide almost the same values for sound pressure levels. It is notable that unlike the cavitation regime with σ = 0.74, for this case the results of the different simulation methods are different in the frequency range which human hearing is more sensitive to sound.

Figure 7 shows the sound pressure levels at 28D and 280D downstream of the single-hole orifice for both compressible and incompressible simulations for the super cavitation regime with σ = 0.15. The graphs from both compressible and incompressible simulations show the same trend at both distances downstream of the orifice. In low frequencies (below 700 Hz) there are few peaks and troughs and generally the sound pressure levels increase very slowly. The values of sound pressure level start to increase rapidly around 700 Hz and reach their maximum peak around 2500 Hz. More peaks and troughs can be observed at this range (1–2.5 kHz). After meeting their maximum values around 2.5 kHz, the sound pressure levels decrease. However, for both distances downstream of the orifice the compressible simulation shows the maximum sound pressure level at 2470 Hz while the maximum sound pressure level is at 2360 Hz for the incompressible simulation.

Figure 7.

Sound pressure levels downstream of the single-hole orifice for both compressible and incompressible simulations for a super cavitation regime with σ = 0.15: (a) 28D downstream of the orifice and (b) 280D downstream of the orifice.

It can be concluded from the graphs that in low frequencies less than 1000 Hz, the discrepancy between calculated sound pressure levels, by using compressible and incompressible simulations is almost 1.5 times compared to the discrepancy between calculated values at frequencies greater than 1000 Hz. While for the low frequencies, the discrepancy between resulted values from compressible and incompressible simulations is around 15 dB, this discrepancy is decreased to almost 10 dB in higher frequencies. This may be due to a lower ability of the incompressible simulation to predict the bubbles which create and collapse at lower frequencies. It is notable that in the range which human hearing is more sensitive to sounds, the calculated sound pressure levels by using compressible and incompressible simulations have less of a discrepancy and both methods provide similar results.

As mentioned in the ‘Introduction’ section, cavitation noise generates by the volume variation and cavitation behaves acoustically as a monopole. Some researchers suggest that a compressible solution may be required to study the monopole sources.^2,16 In addition, comparison between Figure 3 with Figures 1 and 4 with Figure 2 shows that the compressible simulation can predict the hydrodynamics conditions more accurately, especially for a developed cavitation regime. The accuracy of the computed noise by FW–H method directly depended on the accuracy of the computed source field data.^1,4 Therefore, it can be concluded that the compressible simulation can provide more accurate acoustic results. Comparison between Figures 6 and 7 clearly shows that for a developed cavitation regime, applying the incompressible simulation to find time-accurate solutions of the flow-field variables can result in underestimated values of sound pressure levels induced by a cavitating orifice. While for a super cavitation regime, both compressible and incompressible simulations provide similar values for sound pressure levels at frequencies greater than 2 kHz (including the frequency range which human hearing is more sensitive to sounds), there is a big discrepancy between computed sound pressure values from compressible and incompressible simulations for a developed cavitation regime in wide frequency ranges.

Although the FW–H formulation adopts the most general form of Lighthill’s acoustic analogy, there are some limitations with the FW–H method.^13,20 This method does not account for the walls and reflection in the pipe. The pipe geometry affects traveling waves which are influenced by dispersions and other factors. Also, this method focuses on the sound sources rather than the environment, in which reflection and scattering dispersion are not included and heterogeneity and the flow effect are ignored.¹³

Conclusions

In this work, the role of compressibility in computing noise generated at a cavitating single-hole orifice was investigated. The fluid zone downstream of the orifice where the cavitation occurs was evaluated as the acoustic source which generates sound. Time-accurate solutions of the flow-field variables, such as pressure and velocity components on source surfaces, were obtained from both compressible and incompressible simulations. Three cases of cavitation were investigated and the sound pressure signals far downstream of the orifice were computed by applying FW–H formulation. For each case, the acoustics results predicted by FW–H formulation based on both compressible and incompressible simulations for source flow were compared to investigate the effect of compressibility in computing noise generated at a cavitating orifice.

The results confirm that the computed values of sound pressure levels generated by a cavitation orifice totally have been influenced by the methods applied in terms of finding time-accurate solutions of the flow-field variables of the noise source. For a developed cavitation regime, at low frequencies there is a large discrepancy between computed values of sound pressure level from compressible and incompressible simulations and at higher frequencies greater than 6 kHz, both simulation methods provide similar values for sound pressure levels. For a super cavitation regime, both compressible and incompressible simulations provide similar values for sound pressure levels at frequency greater than 2 kHz and compared to a developed cavitation regime, at low frequencies there is less discrepancy between computed values of sound pressure levels generated from both simulation methods. This discrepancy is around 55 dB at 400 Hz for a developed cavitation regime while for a super cavitation regime, the discrepancy is about 25 dB at 350 Hz. Unlike to the super cavitation regime with σ = 0.15, for the developed cavitation regime with σ = 0.41, the results of the compressible and incompressible simulation methods are not close in the frequency range which human hearing is more sensitive to sound. The results of this work demonstrate that the compressibility has a significant role in terms of computing noise generated at a cavitating orifice and cannot be ignored, especially when the noise generated by developed cavitation regimes at low frequencies is investigated. In this work, the sound pressure signals far downstream of the orifice are computed by applying the FW–H formulation. There are some limitations with the FW–H method. This method focuses on the sound sources rather than the environment, in which reflection and scattering dispersion are not included and heterogeneity and the flow effect are ignored. Further research should be performed to apply other acoustic analogies to include reflection and scattering dispersion.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Appendix

References

Wang

Freund

Lele

SK.

Computational prediction of flow-generated sound. Annu Rev Fluid Mech 2006; 38: 483–512.

Ianniello

Muscari

Di Mascio

Ship underwater noise assessment by the acoustic analogy. Part I: nonlinear analysis of a marine propeller in a uniform flow. J Mar Sci Technol 2013; 18: 547–570.

Said

Philip

JM.

Acoustic streaming: from Rayleigh to today. Int J Aeroacoust 2003; 2: 255–292.

Ffowcs Williams

Hawkings

DL.

Sound generated by turbulence and surfaces in arbitrary motion. Phil Trans R Soc Lond Se A Math Phys Sci 1969; 264: 321–342.

Liu

Simulation of whistle noise using computational fluid dynamics and acoustic finite element simulation. Lexington: University of Kentucky, 2012.

Layton

Novotny

On Lighthill’s acoustic analogy for low Mach number flows. In: Fursikov AV, Galdi GP and Pukhnachev VV (eds) New directions in mathematical fluid mechanics. Basel: Birkhäuser Verlag, 2010, pp.247–279.

Wilcox

DC.

Basic fluid mechanics. San Diego, CA: Birmingham Press Inc., 2007.

Wang

Moin

Computation of trailing-edge flow and noise using large-eddy simulation. AIAA J 2000; 38: 2201–2209.

Blake

A statistical description of pressure and velocity fields at the trailing edge of a flat strut. Bethesda, MD: David Taylor Naval Ship Res. Dev. Cent, 1975.

10.

Blake

Gershfeld

JL.

The aeroacoustics of trailing edges. Frontiers in experimental fluid mechanics. Berlin: Springer-Verlag, 1989, pp.457–532.

11.

Langthjem

Olhoff

A numerical study of flow-induced noise in a two-dimensional centrifugal pump. Part I. Hydrodynamics. J Fluids Struct 2004; 19: 349–368.

12.

Langthjem

Olhoff

A numerical study of flow-induced noise in a two-dimensional centrifugal pump. Part II. Hydroacoustics. J Fluids Struct 2004; 19: 369–386.

13.

Olivares

PAV.

Acoustic wave propagation and modeling turbulent water flows with acoustics for district heating pipes. Uppsala, Sweden: Uppsala University, 2009.

14.

Geoffrey

ML.

The source of aerodynamic noise. Int J Aeroacoust 2003; 2: 241–253.

15.

Sandro

Enrico De

Farassat’s formulations in marine propeller hydroacoustics. Int J Aeroacoust 2015; 14: 87–103.

16.

Bistafa

Lauchle

Reethof

Noise generated by cavitation in orifice plates. J Fluids Eng 1989; 111: 278–289.

17.

Seo

Moon

Shin

BR.

Prediction of cavitating flow noise by direct numerical simulation. J Comput Phys 2008; 227: 6511–6531.

18.

Lacombe

Föller

Jasor

et al . Identification of aero-acoustic scattering matrices from large eddy simulation: application to whistling orifices in duct. J Sound Vib 2013; 332: 5059–5067.

19.

Hickel

DNS and LES of two-phase flows with cavitation. Direct and large-eddy simulation IX. Switzerland: Springer International Publishing, 2015, pp.595–605.

20.

ANSYS, Inc. ANSYS Fluent User's Guide. Canonsburg, PA: ANSYS, Inc., 2017.

21.

Brennen

CE.

Cavitation and bubble dynamics. Oxford: Oxford University Press, 1995.

22.

Young

FR.

Cavitation. London: Imperial College Press, 1999.

23.

Tullis

JP.

Hydraulics of pipelines: pumps, valves, cavitation, transients. New York: Wiley-Interscience Ed.x, 1989.

24.

Yan

Thorpe

RB.

Flow regime transitions due to cavitation in the flow through an orifice. Int J Multiphase Flow 1990; 16: 1023–1045.

25.

Kugou

Matsuda

Izuchi

et al . Cavitation characteristics of restriction orifices (experiment on characteristics of piping vibration and noise). In: ASME, Fluids Engineering Division Summer Meeting, 1996.

26.

Sato

Saito

Unstable cavitation behavior in a circular–cylindrical orifice flow. JSME Int Journal Series B 2002; 45: 638--645.

27.

Pan

Xiang

Investigating to cavitation behavior of orifice tunnel. In: CAV2001, fourth symposium on cavitation. Pasadena: California Institute of Technology, 2001, pp. 1--8

28.

Testud

Moussou

Hirschberg

et al . Noise generated by cavitating single-hole and multi-hole orifices in a water pipe. J Fluids Struct 2007; 23: 163–189.

29.

Wijngaarden

One-dimensional flow of liquids containing small gas bubbles. Annu Rev Fluid Mech 1972; 16: 369–396.

30.

Meneveau

Katz

Scale-invariance and turbulence models for large-eddy simulation. Annu Rev Fluid Mech 2000; 32: 1–32.

31.

Chung

TJ.

Computational fluid dynamics. Huntsville; Cambridge: University of Alabama in Huntsville; Cambridge University Press, 2002.

32.

Sauer

Schnerr

GH.

Development of a new cavitation model based on bubble dynamics. Z Angew Math Mech 2001; 81: 561–562.

33.

Schnerr

Sauer

Physical and numerical modeling of unsteady cavitation dynamics. In: Fourth international conference on multiphase flow, New Orleans, USA, 2001.

34.

Hofmans

GCJ

Ranucci

Ajello

et al . Aero-acoustic response of a slit-shaped diaphragm in a pipe at low Helmholtz number, 2: unsteady results. J Sound Vib 2001; 244: 57–77.

35.

Archer

Boyer

Nimanbeg

et al.

Résultats des essais hydrauliques et hydroacoustiques du troncon PTR 900 sur la boucle EPOCA (in French). Paris, France: EDF R&D, 2002.

36.

Smith

SW.

The scientist and engineer’s guide to digital signal processing. 1st ed. San Diego: California Technical Publishing, 1997.