Tonal noise prediction of propellers in ambient flow using steady-state CFD and Ffowcs Williams-Hawkings formulations

Abstract

As electric vertical take-off and landing (eVTOL) concepts and urban air mobility gain prominence, the mitigation of propeller noise has become essential. This study presents a comprehensive assessment of acoustic prediction methodologies, focusing on the application of both Formulation-1A and the more recent Formulation-1C of the Ffowcs Williams-Hawkings (FWH) equation. The work demonstrates the use of steady-state Reynolds-averaged Navier-Stokes (RANS) simulations with a single rotating frame (SRF) approach for efficient acoustic analysis, enabling the transformation of steady flow solutions into time-varying acoustic surface data. The methodology is implemented as a new class within the open-source libAcoustics library for OpenFOAM, leveraging its parallelization capabilities. Validation with experimental data and comparison against numerical results were performed for a range of propeller configurations. Comparative analysis with direct noise computation (DNC) highlights the improved accuracy of Formulation-1C, especially in scenarios with ambient flow, and demonstrates the limitations of Formulation-1A in such conditions. The results confirm the robustness of the proposed framework for propeller noise prediction across a range of operating regimes.

Keywords

aeroacoustics acoustic analogy single rotating frame (SRF)formulation-1A formulation-1C monopole dipole OpenFOAM libAcoustics UH-1H rotor SR-2 propeller

Introduction

The accurate prediction of noise for propellers and other aircraft components plays a key role in development of overall quieter air vehicles. The development of numerical tools and experimental techniques that can predict the sources and propagation of acoustic pressure waves efficiently and with higher accuracy has become essential and remains active field of research necessary for aircraft development and certification. Both aerodynamic and acoustic performance analysis can be combined in an optimization framework for designing low noise systems.¹ Most of the methodologies involving acoustic analysis utilize, as input, aerodynamic data which is generated either through experiments or through numerical analysis such as computational fluid dynamics (CFD) and so the accuracy of acoustic analysis depends on the fidelity of the preceding aerodynamic analysis.² There are two main approaches which numerical noise prediction frameworks can generally follow: direct noise computation (DNC) approach and surface integration-based hybrid approach. In the first approach i.e. DNC, high-fidelity transient CFD simulations are performed using large eddy simulations (LES) or detached eddy simulations (DES) turbulence models and pressure fluctuations are stored during simulation in pre-defined probes on which spectral analysis is performed later.^3–5 Although the DNC approach offers high accuracy, it is mostly restricted to near-field noise prediction due to requirement of detailed mesh and large computational resources. The second approach, involving volume and surface integral-based methodologies, utilizes Lighthill’s analogy^6,7 which is derived from Navier-Stokes equations. The two major extensions are Curle’s analogy⁸ which is restricted to static solid boundaries and Ffowcs Williams-Hawkings (FWH) methodology⁹ which is more general and caters for moving medium and arbitrary movement of both permeable and impermeable surfaces. These analogies, involving surface integrals, can predict noise propagation in the farfield with less computational time. FWH surface-integral methodology has been widely applied for far-field broadband jet noise prediction¹⁰ and have also been successfully used for propeller and rotor acoustic analyses, including recent low-Reynolds-number propeller studies.¹¹ Apart from the aforementioned time-domain approaches, low fidelity frequency-domain based acoustic theories for propeller noise prediction have also been formulated by Gutin,¹² Deming¹³ and Hanson.¹⁴

Comprehensive experimental and numerical research has been conducted in parallel over the years for characterization of propeller blade noise. Purcell¹⁵ conducted experiments on scaled down version of UH-1H rectangular planform rotor operating at hover condition and high tip speed with transonic flow. A geometrically similar rotor was used by Boxwell et al.¹⁶ in their experiments to measure impulsive noise for both hover and forward flight conditions. Farassat and Brentner¹⁷ used the experimental results by Purcell to perform numerical comparison between Formulation–1A of FWH equation and Kirchhoff’s Formulation, finding Formulation-1A to perform better even if the surface is permeable. Later, Morgans et al.¹⁸ extended the analysis for non-lifting forward flight. Extensive wind tunnel tests were also performed, both aerodynamic and acoustic, for designing a fuel-efficient group of NASA’s SR-series turboprop propellers.^19–23 Tan et al.²⁴ used transient CFD results, whereas, Gennaro et al.²⁵ used steady-state CFD results of SR-2 propeller to obtain noise propagation using Formulation–1A and its steady-state version respectively. However, it was noted by Gennaro et al.²⁵ that few corrections were required in order to properly compare the steady-state acoustic results with experimental data. Marinus et al.²⁶ developed a novel truncated methodology for dealing with the problem of sonic singularity and performed acoustic analysis utilizing steady-state CFD results of SR-3 propeller geometry. A detailed comparison study of different frequency-domain based acoustic methods was performed by Herniczek et al.²⁷ for nine published experiment cases including the case of SR-2 propeller. They concluded that Hanson’s model¹⁴ exhibited the maximum accuracy. Both experimental and numerical analysis were performed by Casalino et al.¹¹ with the purpose of setting a benchmark problem for low Reynolds number propeller acoustics. In the numerical part of their study, they utilized high fidelity as well as low-fidelity computational models. Recently, Baskaran et al.²⁸ performed detailed experiments to study the effect of number of blades on propeller broadband noise.

While Formulation–1A remains conventional and produces results within acceptable tolerances, its inherent limitation lies in its inability to account for ambient velocity effects. To address this, Najafi et al.²⁹ proposed a modified version, termed Formulation–1C, which explicitly incorporates ambient velocity considerations. Garrick Triangle (GT) Formulation³⁰ is the simplified version of Formulation-1C for stationary surfaces. Brès et al.³¹ applied the GT method to conduct acoustic analysis of a fan by constructing a stationary permeable surface around it. Building on this work, Zhang et al.³² developed an open-source FORTRAN code which integrates GT Formulation and leverages the OpenMP parallelization to increase computational speed of acoustic analysis. Although Formulation–1A has been used to perform acoustic analysis on steady-state CFD data, to the best of the authors’ knowledge the Formulation-1C has never been utilized in similar manner. Applying Formulation–1C to steady-state CFD data enables rapid and computationally efficient evaluation of propeller noise in the presence of ambient flow, overcoming the limitations of earlier methods and allowing for usage of both impermeable and permeable surface data for inclusion of nonlinear noise sources. This research brings a notable novelty to the existing literature, as it is the first to demonstrate and validate the use of Formulation–1C on steady-state numerical data for propeller aeroacoustics analysis.

The aims of this study are addressed through the following key objectives:

1. Summarize the foundational acoustic analogy forming the basis of the presented approach (Subsection: Acoustic analogy).

2. Establish the distinctions between Formulation-1A and Formulation-1C within the context of steady-state computational analysis (Subsection: Formulation – 1A and formulation – 1C).

3. Outline the range of verification and validation cases selected to demonstrate the effectiveness of the proposed methodology (Section: Verification and validation).

4. Highlight the comparative evaluation against both conventional Formulation-1A results and DNC from unsteady (transient) analysis (Section: Formulation comparison).

5. Integrate the developed methodology as a dedicated module within the libAcoustics suite for OpenFOAM, exploiting native parallel processing to enable efficient acoustic data handling.

Acoustic analysis methodology

In this section, the foundational FWH theory, which serves as the backbone of our methodology, is detailed. Both its convective and non-convective formulations are explained. Subsequently, the two principal integral solution Formulation-1A and Formulation-1C are discussed. The section concludes with a description of the steady-state methodology, integrating these developments into the overall computational approach.

Acoustic analogy

The FWH equation is derived from Lighthill’s acoustic analogy and is also its most widely used form. The differential form of FWH equation used for permeable surface is given as:

\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} [c^{2} (ρ - ρ_{0})] - \nabla^{2} [c^{2} (ρ - ρ_{0})] = \frac{\partial}{\partial t} [Q_{n} δ (f)] - \frac{\partial}{\partial x_{i}} [L_{i} δ (f)] + \frac{\partial}{\partial x_{i} x_{j}} [T_{i j} H (f)]

(1)

where:

$Q_{n} = Q_{i} {\hat{n}}_{i} = ρ_{0} v_{i} {\hat{n}}_{i} + ρ (u_{i} - v_{i}) {\hat{n}}_{i}$

$L_{i} = L_{i j} {\hat{n}}_{j} = P_{i j} {\hat{n}}_{j} + ρ u_{i} (u_{j} - v_{j}) {\hat{n}}_{j}$

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

$P_{i j} = (p - p_{0}) δ_{i j} - τ_{i j}$

The first term on right hand side of FWH equation is monopole source and contributes to loading noise, whereas the second term is dipole source which contributes to thickness noise. Dirac delta function ( $δ$ ) shows that both of these terms are utilized in surface integration. The last term is a quadrupole source, and it is associated with volume integral as evident by the presence of Heaviside function ( $H)$ . The surface is defined by $f = 0$ , whereas the unit normal vector field to the surface is $\hat{n} = \nabla f$ . The other symbols such as $ρ, ρ_{0}, p, p_{0}, u_{j}, v_{j}$ describe density field, ambient density, pressure field, ambient pressure, velocity field and surface velocity, respectively. The indices $i$ and $j$ denote the element number of vector or tensor. It should be noted that in stress tensor $P_{i j}$ , the contribution of viscous stress $τ_{i j}$ is assumed to be negligible. If the FWH surface coincides with an impermeable surface with no-slip condition, then $u_{i} = v_{i}$ , so loading noise is determined exclusively by the surface pressure distribution. The quadrupole source term is typically neglected in most applications due to its minimal contribution. However, in cases where its contribution becomes significant, integration can still be avoided by strategically constructing a permeable FWH surface that encloses all nonlinear flow effects. Equation (1) represents the non-convective form of FWH equation, however, for cases with ambient flow, Najafi et al.²⁹ derived the convective version, which for an arbitrary ambient velocity vector $U_{0}$ is as follows:

[\frac{\partial^{2}}{\partial t^{2}} - c_{0}^{2} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} + 2 U_{0, j} \frac{\partial^{2}}{\partial t \partial x_{j}} + U_{0, i} U_{0, j} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}}] [H (f) (ρ - ρ_{0})]

= (\frac{\partial}{\partial t} + U_{0, j} \frac{\partial}{\partial x_{j}}) [Q_{n} δ (f)] - \frac{\partial}{\partial x_{i}} [L_{i} δ (f)] + \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} [H (f) T_{i j}]

(2)

where:

$Q_{n} = ρ_{0} (v_{i} - U_{0, i}) {\hat{n}}_{i} + ρ (u_{i} + U_{0, i} - v_{i}) {\hat{n}}_{i}$

$L_{i} = P_{i j} {\hat{n}}_{j} + ρ u_{i} (u_{j} + U_{0, j} - v_{j}) {\hat{n}}_{j}$

The tensor terms ( $T_{i j}$ and $P_{i j}$ ) retain the same form as in the non-convective version of the FWH equation.

Formulation – 1A and formulation – 1C

Integral formulations of the FWH equation, both non-convective and convective forms, are essential for practical implementation. Formulation–1 A, introduced by Farassat and Succi³³ for both permeable and impermeable surfaces, is derived from the non-convective FWH equation. The pressure perturbation ( $p^{'}$ ) evaluated at observer location $(x)$ and time $(t)$ is expressed as:

p^{'} (x, t) = p_{L}^{'} (x, t) + p_{T}^{'} (x, t)

(3)

In the sum, $p_{L}^{'} (x, t)$ is loading noise and $p_{T}^{'} (x, t)$ is thickness noise component of pressure and both are computed for Formulation–1A as follows:

4 π p_{T}^{'} (x, t) = \int_{f = 0} {[\frac{\frac{\partial Q_{n}}{\partial τ}}{r {(1 - M_{r})}^{2}}]}_{ret} d S + \int_{f = 0} {[\frac{Q_{n} (r \frac{\partial M_{i}}{\partial τ} r_{i} + c_{0} (M_{r} - M^{2}))}{r^{2} {(1 - M_{r})}^{3}}]}_{ret} d S

(4)

4 π p_{L}^{'} (x, t) = \frac{1}{c_{0}} \int_{f = 0} {[\frac{\frac{\partial L_{i}}{\partial τ} r_{i}}{r {(1 - M_{r})}^{2}}]}_{ret} d S + \frac{1}{c_{0}} \int_{f = 0} {[\frac{L_{r} - L_{M}}{r^{2} {(1 - M_{r})}^{2}}]}_{ret} d S

+ \frac{1}{c_{0}} \int_{f = 0} {[\frac{L_{r} (r \frac{\partial M_{i}}{\partial τ} r_{i} + c_{0} (M_{r} - M^{2}))}{r^{2} {(1 - M_{r})}^{3}}]}_{ret} d S

(5)

For moving surfaces, the components of the Mach vector $M$ are defined as $M_{i} = v_{i} / c_{0}$ . The geometric distance between source and observer is denoted by $r$ , while the radiation direction unit vector is $\hat{r}$ . The retarded time $τ$ , defined as $τ = t - r / c_{0}$ , accounts for propagation delay. Additionally, the following variables are defined:

L_{r} = L_{i} r_{i}, M_{r} = M_{i} r_{i}, L_{M} = L_{i} M_{i}

Formulation – 1C is derived from convective form of FWH equation by Najafi et al.²⁹ and the respective thickness and loading noise terms are defined as:

4 π p_{T}^{'} = \int_{f = 0} {[\frac{\frac{\partial Q_{n}}{\partial τ}}{R^{*} {(1 - M_{R})}^{2}}]}_{ret} d S + \int_{f = 0} {[\frac{- \frac{\partial R^{*}}{\partial τ} Q_{n}}{R^{* 2} {(1 - M_{R})}^{2}}]}_{ret} d S + \int_{f = 0} {[\frac{Q_{n}}{R^{*} {(1 - M_{R})}^{3}} \frac{\partial M_{R}}{\partial τ}]}_{ret} d S - M_{0} \int_{f = 0} {[\frac{\frac{\partial R_{1} Q_{n}}{\partial τ}}{R^{*} {(1 - M_{R})}^{2}}]}_{ret} + M_{0} \int_{f = 0} {[\frac{\frac{\partial R^{*}}{\partial τ} Q_{n}}{R^{* 2} {(1 - M_{R})}^{2}}]}_{ret} d S - M_{0} \int_{f = 0} {[\frac{\frac{\partial M_{R}}{\partial τ} Q_{n}}{R^{*} {(1 - M_{R})}^{3}}]}_{ret} d S

- U_{0} \int_{f = 0} {[\frac{R_{1} Q_{n}}{R^{* 2} (1 - M_{R})}]}_{ret} d S

(6)

4 π p_{L}^{'} = \frac{1}{c_{0}} \int_{f = 0} {[\frac{\frac{\partial L_{R}}{\partial τ}}{R^{*} {(1 - M_{R})}^{2}}]}_{ret} d S - \frac{1}{c_{0}} \int_{f = 0} {[\frac{\frac{\partial R^{*}}{\partial τ} L_{R}}{R^{* 2} {(1 - M_{R})}^{2}}]}_{ret} d S

+ \frac{1}{c_{0}} \int_{f = 0} {[\frac{\frac{\partial M_{R}}{\partial τ} L_{R}}{R^{*} {(1 - M_{R})}^{3}}]}_{ret} d S + \int_{f = 0} {[\frac{L_{i} R_{i}^{*}}{R^{* 2} (1 - M_{R})}]}_{ret} d S

(7)

In this formulation, the geometric distance $r$ is replaced by the effective acoustic distance $R$ between source and observer. It should be noted that the direction of ambient velocity $U_{0}$ is assumed to be in $x_{1}$ direction. The definitions of $R$ and auxiliary quantities, including $β^{*}$ , $R^{*}$ and unit radiation vector components $R_{i}$ and $R_{i}^{*}$ are given as:

R = \frac{- M_{0} (x_{1} - y_{1}) + R^{*}}{β^{2}},

R^{*} = \sqrt{{(x_{1} - y_{1})}^{2} + β^{2} [{(x_{2} - y_{2})}^{2} + {(x_{3} - y_{3})}^{2}]},

β = \sqrt{1 - M_{0}^{2}},

R_{1} = \frac{1}{β^{2}} (- M_{0} + R_{1}^{*}), R_{2} = \frac{x_{2} - y_{2}}{R^{*}}, R_{3} = \frac{x_{3} - y_{3}}{R^{*}},

R_{1}^{*} = \frac{x_{1} - y_{1}}{R^{*}}, R_{2}^{*} = β^{2} \frac{x_{2} - y_{2}}{R^{*}}, R_{3}^{*} = β^{2} \frac{x_{3} - y_{3}}{R^{*}} .

Steady-state acoustic analysis methodology

Acoustic analysis, whether using DNC or surface integral methodologies like FWH, requires transient pressure and velocity field data on the surface as input. Generating this data through CFD is computationally expensive, particularly with high-fidelity models like LES or direct numerical simulation (DNS). For propellers, LES is essential to capture broadband noise by resolving turbulent fluctuations, while unsteady RANS (URANS) suffices for low-frequency tonal noise.³⁴ However, URANS simulations remain time consuming. When propeller loading (i.e., pressure and velocity distributions) remains stable, a steady-state RANS model, with a single rotating frame (SRF) approach can be employed.³⁵ The converged flow fields can then be sampled on the FWH surface and rotated along with it to generate time-varying data for surface integration. The primary limitation of this methodology is its inability to accurately capture complex effects during noise-generating such as blade vortex interaction (BVI),^36,37 cavitation,³⁸ installation effects,^39,40 interference between counter-rotating propellers,⁴¹ and ground effects.⁴² These events involve highly unsteady and transient flow characteristics, which are not effectively modeled using a steady-state CFD. Sampled scalar fields (e.g., density and pressure) require no rotation, while vector fields, including fluid velocity $(\hat{u})$ and cell center $(\hat{y})$ , must be rotated. Given a unit rotation axis $\hat{k} = [k_{1}, k_{2}, k_{3}]$ and rotation angle $(θ)$ , the rotation matrix $(R_{rot})$ is computed by Rodrigues formula⁴³:

R_{r o t} = I + K \sin θ + K^{2} (1 - \cos θ)

(8)

where

I

is identity matrix and

K

is:

K = [\begin{array}{l} 0 & - k_{3} & k_{2} \\ k_{3} & 0 & - k_{1} \\ - k_{2} & k_{1} & 0 \end{array}]

The rotated vector fields can be obtained by multiplying the rotation matrix with the vector field e.g., ${\hat{u}}_{rot} = R_{rot} \hat{u}$ , whereas the surface velocity is obtained by taking the cross product of rotation rate vector with cell center field as follows:

{\hat{v}}_{r o t} = ω \times \hat{y}

(9)

Field variables on the permeable FWH surface were sampled using OpenFOAM’s surfaceSampling utility, which applies cell-centered inverse-distance weighted linear interpolation. The sampled surface data was obtained directly from the steady SRF solution.

Verification and validation

In the present work, four cases were used to verify and validate the proposed methodology. The verification case involves baseline monopole and dipole acoustic sources, and the remaining three cases focus on propeller studies, with the results from the Formulation-1C method validated against existing experimental and numerical data. For the validation cases, steady-state CFD simulations were performed using a single reference rotating frame (SRF) approach in StarCCM+. The $k - ω$ SST turbulence model was employed, and spatial discretization was handled using a third-order MUSCL scheme. The viscous region was resolved using the wall function and the solution tolerance was set below $1 \times 10^{- 4}$ .

The computational domain was cylindrical, with a radius of 10 and a length of 20 propeller radii, centered on the propeller. For two-bladed geometries (UH-1H and Casalino Propeller), a half-domain with periodic boundaries was used, while a quarter-domain with periodic boundaries was implemented for the four-bladed SR-2 propeller. Surface meshing on the blades maintained a baseline cell size of $0.005$ propeller radii, with localized refinement to $0.001$ radii at the tip for the UH-1H case to resolve shock formation. Total cell counts reached approximately $1.3 \times 10^{6}$ for the UH-1H configuration, while remaining close to $1 \times 10^{6}$ for the other two cases. Domain and mesh schematics are presented in Figures 1 and 2, respectively. Although the code is implemented as an OpenFOAM library, it is designed to be versatile and can process surface data from various sources, including custom formats for baseline cases, as well as outputs from StarCCM+ and Fluent, making it a generic and adaptable tool.

Figure 1.

(a) Semi-cylindrical computational domain used for the UH-1H and TUDelft Casalino propeller cases; (b) Quarter-cylindrical domain employed for the SR-2 propeller case.

Figure 2.

Tip mesh for propeller cases: (a) UH-1H tip mesh showing localized refinement for shockwave resolution; (b) SR-2 propeller; (c) TUDelft Casalino propeller.

Baseline monopole and dipole

Verifying the methodology and code functionality for transient analysis using baseline cases, such as monopole and dipole acoustic sources, is essential. These sources were selected for their simplicity and availability of their exact analytical solutions which can be derived without computationally intensive simulations. Furthermore, the superposition of monopole and dipole sources effectively models most real-world complex acoustic phenomena, making them ideal for verification. Formulation-1C’s unique capability to predict noise propagation in moving media, requires validation through analytical solutions for monopoles and dipoles in such environments. The potential function solutions for both sources are given below, as detailed in the work of Najafi et al.²⁹

Monopole

ϕ (x, t) = \frac{A}{4 π R^{*}} \exp [i ω (t - \frac{R}{c_{0}})]

(10)

Dipole

ϕ (x, t) = \frac{\partial}{\partial x_{2}} {\frac{A}{4 π R^{*}} \exp [i ω (t - \frac{R}{c_{0}})]}

(11)

The pressure and velocity perturbation can be calculated using the following expressions:

p^{'} (x, t) = - ρ_{0} [\frac{\partial}{\partial t} + M_{0} c_{0} \frac{\partial}{\partial x_{1}}] ϕ

(12)

u (x, t) = \nabla ϕ (x, t)

(13)

The FWH surface was defined as a sphere with a 1 m radius, with surface data computed using the equations (10)–(13). Verification involved 20 receivers arranged in a 30 $m$ radius circular array, where noise propagation from the FWH surface to these receivers, calculated using Formulation-1C, was compared against direct analytical solution. The observers were placed along fixed azimuth lines, where the azimuth angle remained constant and only the elevation varied, ensuring reproducible directivity plots. The amplitude $(A)$ was set to $1 m^{2} s^{- 1}$ , the speed of sound $(c_{0})$ to $347 m s^{- 1}$ and angular frequency $(ω)$ to 31.419 $r a d / s$ . Two scenarios were examined: one with no ambient flow and another with an ambient Mach number $(M_{\infty})$ of 0.8. The polar plots of $(p_{\max})$ directivity are presented in Figures 3 and 4 for both stationary and moving flow cases, respectively.

Figure 3.

Directivity pattern $(p_{\max}$ ) for (a) monopole and (b) dipole sources at $M_{\infty} = 0$ and $R = 30$ m observer distance. (): Formulation-1C results; (): Analytical reference.

Figure 4.

Directivity pattern $(p_{\max}$ ) for (a) monopole and (b) dipole sources at $M_{\infty} = 0.8$ and $R = 30$ m observer distance. (): Formulation-1C results; (): Analytical reference.

Since the data on the FWH surface and the pressure data at the receivers were both computed using the same analytical equations, the resulting error is almost negligible. This comparison confirms that the methodology performs correctly for both monopole and dipole sources. It can also be observed that, in moving flow cases, $p_{\max}$ values increase opposite to the direction of the ambient velocity and decrease in the direction of the flow. This is evident in Figure 4, where the otherwise circular directivity pattern of the monopole extends further opposite to the flow direction, similarly both lobes of the dipole pattern bend away from it. Since the FWH surface in this analysis was stationary, Formulation-1C effectively reduces to the Garrick triangle formulation.

UH-1H rotor

The UH-1H rotor has been extensively used in experimental studies for thorough categorization of both thickness and loading noise components. As a result, a substantial amount of experimental data is available for comparison with numerical studies. The rotor features a simple rectangular planform with a NACA 0012 airfoil section, a radius of 1.045 m, and an aspect ratio of 13.7, as described in Purcell.¹⁵ The test model was a $1 / 7^{t h}$ scale replica of the original rotor and had no twist. This propeller has served as a baseline case in numerous numerical studies of propeller noise. Consequently, it also serves the purpose here to evaluate current method’s ability to predict noise at transonic tip speeds. The rotor CAD model, along with the permeable surfaces utilized in the current study, is shown in Figure 5. The comparison was conducted under hover operating condition for two tip Mach numbers $M_{H}$ , 0.85 and 0.88, at which the flow becomes supercritical. Figure 6 illustrates the relative Mach and pressure contours.

Figure 5.

CAD geometry of the UH-1H rotor with the permeable surfaces.

Figure 6.

Relative Mach number and pressure contours for the UH-1H rotor: (a) relative Mach contour at $M_{H} = 0.85$ ; (b) relative Mach contour at $M_{H} = 0.88$ ; (c) pressure contour at $M_{H} = 0.85$ ; (d) pressure contour at $M_{H} = 0.88$ .

Analysis of the pressure drop profile at an in-plane receiver positioned $3.09 R$ from the source (Figure 7) reveals a 150 Pa increase in magnitude as the tip Mach number rises from 0.85 to 0.88. This sharp rise coincides with the onset of supercritical flow, where shock formation near the blade tip induces a significant pressure jump. The FWH surface sensitivity analysis reveals that selecting the propeller wall as the FWH surface leads to under-prediction of the pressure drop in both Morgans study¹⁸ and the current work. However, extending the permeable surface into the flow region reduces this error significantly. Beyond a certain extent, further error reduction becomes negligible, suggesting that nonlinear flow effects outside the domain have been adequately captured. The numerical results for permeable surface exhibit strong agreement with those reported by Morgans.¹⁸

Figure 7.

Pressure time histories for UH-1H rotor in hover at observer location $3.09 R$ : (a) $M_{H} = 0.85$ ; (b) $M_{H} = 0.88$ . (): Wing (Current); (): Permeable surface 1 (Current); (): Permeable surface 2 (Current); (): Permeable surface 3 (Current); (): Wing (Morgans¹⁸); (): Permeable surface (Morgans¹⁸); (): Experiment (Purcell¹⁵).

SR-2 propeller

The validation case involving the SR-2 turboprop propeller has been selected to allow the feasibility of the formulation to be assessed on complex propeller geometries, as well as the presence of upstream velocity in the corresponding experimental data. The geometry utilized in this study is a 4-bladed propeller with diameter of 0.591 m. The CAD geometry and distribution of geometric parameters with respect to radial ratio $(r / R)$ are shown in Figure 8 whereas, experimental setup is shown in Figure 9, respectively. Further details about the experiments are described by Soderman and Horne.²² The aerodynamic validation of CFD with experiment was considered for two test points with different $β_{r e f}$ values and the results are presented in Table 1.

Figure 8.

(a) SR-2 CAD geometry; (b) SR-2 geometric parameters distribution.²²

Figure 9.

Experimental setup for SR-2 propeller noise measurements by Soderman and Horne²²: (a) front view; (b) top view.

Table 1.

Comparison of experimental and numerical results for SR-2 propeller.

$U_{\infty}$ $(m / \sec)$	$N$ (RPM)	$β_{r e f}$	$J$	$M_{t i p}$	$C_{T}$ exp.	$C_{T}$ num.	$C_{T}$ error %	$C_{P}$ exp.	$C_{P}$ num.	$C_{P}$ error %
62	6000	30	1.06	0.545	0.16	0.14	12.5	0.23	0.19	17.4
62	8200	21	0.77	0.745	0.08	0.068	15	0.09	0.07	22.2

The maximum error for $C_{T}$ is $15 %$ , while for $C_{P}$ it is $22.8 %$ . These discrepancies may be attributed to differences between the CAD model and the experimental setup. Nevertheless, the overall agreement with the experimental results is satisfactory. For the acoustic analysis, four microphones (numbered 7, 8, 9, and 12), which are also circled in red in Figure 9, were selected. Looking at Figure 10, we can see that the results obtained using Formulation-1C are in good agreement with the frequency method results of Gutin and Hanson for all probe locations in the case of the first harmonic. For the second harmonic, a deviation is observed at probe 9 $(θ = 15^{o})$ , while the agreement remains strong for all other probes. The frequency method results for the SR-2 propeller were extracted from the work of Herniczek et al.²⁷ The significant differences between the frequency method and experimental results are attributed to errors in the experimental data, as demonstrated by Herniczek. Given that probes 7 and 12 are separated by nearly 3.1 m, their measured noise levels should not be nearly identical, as seen in the experimental data, which suggests the presence of measurement inaccuracies. A similar explanation accounts for the deviations observed between the experimental results and those obtained with Formulation-1C. It is also observed that the utilization of either a permeable or impermeable surface results in negligible differences on the SPL scale for this specific case. The slight discrepancy beyond 90° may be related to microphone calibration uncertainty present in the Soderman and Horne²² measurements. In the current study, the BPF harmonics were extracted using FFT-based processing, consistent with typical tonal analysis practice.

Figure 10.

Sound pressure level (SPL) as a function of $θ$ for the SR-2 propeller at 8200 RPM: (a) first blade passing frequency (BPF) SPL values; (b) second BPF SPL values. (): Formulation-1C (Impermeable surface); (): Formulation-1C (Permeable surface); (): Gutin (Frequency method)²⁷; (): Hanson (Frequency method)²⁷; (): Experimental results.²².

Casalino et al. study (TUDelft)

Casalino et al.¹¹ conducted a series of experiments in TUDelft A-Tunnel on small-scale propellers designed typically for eVTOL applications. Their primary objective was to provide a comprehensive comparison of acoustic analysis results obtained from experimental measurements, high-fidelity large eddy simulations (LES), and low-fidelity blade element momentum theory (BEMT) calculations. The inclusion of this validation is essential to test the proposed methodology’s accuracy for propellers operating at low subsonic speeds. The experimental setup depicted in Figure 11 illustrates the placement of a vertical receiver array located at a distance equal to four times the propeller diameter. This array consists of approximately 13 uniformly spaced receivers. Notably, this configuration differs from the conventional arc arrangement typically used in such experiments. The two-bladed propeller has a radius of 0.15 m, with its geometric details present in Casalino et al.¹¹ The operating conditions used for both the experiments and numerical validation are presented in Table 2. It should be noted that the propeller RPM was held constant at 5000.

Figure 11.

Experimental setup used by Casalino et al.¹¹ for propeller noise measurements.

Table 2.

Operating conditions.

$U_{\infty} (m / \sec)$	$J$
0	0
6	0.24
10	0.4
15	0.6
20	0.8

For performance validation, two key parameters were analyzed: the coefficient of thrust $(C_{T})$ and the coefficient of torque $(C_{Q})$ . As shown in Figure 12, the variation of these coefficients with respect to the advance ratio (J) indicates that the steady-RANS results from the current study align more closely with the experimental data than the LES results. This finding is consistent with transient results reported by Abuhanieh.⁴⁴ However, the observed deviations remain within acceptable error thresholds. Data from only two microphones was utilized to compare accuracy of noise assessment: microphone 7, positioned in the propeller plane, and microphone 1, located downstream. The resulting power spectral density (PSD) plots are presented in Figure 13 for two advance ratio (J) values, 0.24 and 0.6. Only the results from the permeable surface are included in these plots. Examining Figure 13 reveals that the PSD values at the blade passing frequency (BPF) increases as the observer moves closer to the propeller plane. Additionally, the overall PSD values decrease with increasing advance ratio for both the in-plane and downstream locations. The current methodology over-predicted PSD values for both the first and second harmonics of the blade passing frequency (BPF), while the LES approach consistently underpredicted the noise relative to experimental measurements. These acoustic discrepancies correspond to the under-prediction of aerodynamic coefficients by LES, indicating that underestimation of the aerodynamic results leads to a similar underestimation in the acoustic predictions.

Figure 12.

Comparison of aerodynamic coefficients with advance ratio for validation against Casalino’s experimental and numerical results: (a) thrust coefficient $C_{T}$ ; (b) torque coefficient $C_{Q}$ . (): CFD (Current); (): LBM-LES¹¹; (): Experimental.¹¹

Figure 13.

Power spectral density (PSD) versus frequency for acoustic validation with Casalino’s experimental and numerical data: (a) Mic. 1 at $J = 0.24$ ; (b) Mic. 1 at $J = 0.60$ ; (c) Mic. 7 at $J = 0.24$ ; (d) Mic. 7 at $J = 0.60$ . (): Formulation-1C (Steady); (): LBM-LES¹¹; (): Experimental.¹¹

Formulation comparison

In this section, the propeller geometry from the study by Casalino et al.¹¹ is used to compare results from DNC with those from the steady-state method. This comparison aims to clarify the differences between Formulation-1A and Formulation-1C, as well as to highlight the limitations of Formulation-1A. For the transient analysis, the HiSA solver in OpenFOAM, using a RANS turbulence model, was employed. A timestep of $Δ t = 5 \times 10^{- 5} s$ was used for the transient reference simulation, giving approximately 240 steps per revolution at 5000 RPM. This falls within accepted ranges for tonal propeller simulations.⁴⁵ A short timestep sensitivity check confirmed that BPF SPL varied by less than 1 dB, indicating converged tonal behaviour. Second-order discretization was applied for both spatial and temporal terms, along with the Van-Leer limiter. The domain size and mesh settings were kept consistent with the steady-state cases; however, a full-domain geometry was implemented to eliminate inconsistencies in probe data. The transient analysis was performed over 10 complete cycles of propeller rotation, with surface data from the final timestep used to obtain acoustic results from the steady-state formulation. The propeller RPM was kept to 5000 as during the case of validation however during this time the advance ratio (J) was kept to 1.2 to demonstrate the effects of ambient velocity on pressure calculation. The receiver or probe locations are shown in Figure 14 and were placed as such to study the effect on in-plane and off-plane noise prediction.

Figure 14.

Receiver locations relative to the propeller used for the comparison study.

Analysis of Figure 15 reveals that Formulation-1C achieves minimal error when compared with direct noise results, while Formulation-1A shows significant discrepancies. The figure displays pressure time histories for two in-plane receivers (2 and 3) and one out-of-plane receiver (8). For receiver 2 (Figure 15(a)), both direct results and Formulation-1C exhibit pressure variations of approximately 10 Pa, while Formulation-1A shows fluctuations of 50 Pa. As distance increases (Figure 15(b)), Formulation-1A’s pressure range decreases substantially, though residual errors persist. For the out-of-plane receiver (Figure 15(c)), both formulations demonstrate minimal error and yield accurate predictions. These results demonstrate that pressure prediction errors for Formulation-1A peak within the propeller plane. This trend is clearly visible in Figure 16(a) (maximum SPL vs $x$ -location in the propeller plane) and Figure 16(b) (maximum SPL vs $z$ -location). Figure 16(a) shows Formulation-1A’s error decreasing from 20 dB to 5 dB with increasing distance from the propeller, while Formulation-1C maintains negligible error throughout. Figure 16(b) further confirms significantly higher error at in-plane locations $(z = 0)$ compared to out-of-plane receivers, substantiating our conclusion that Formulation-1A exhibits maximum error within the propeller plane.

Figure 15.

Pressure versus time at $J = 1.2$ and 5000 RPM for (a) receiver 2, (b) receiver 3, and (c) receiver 8. (): Formulation-1C; (): Formulation-1A; (): Direct results.

Figure 16.

Variation of maximum sound pressure level (Max SPL) with respect to (a) x-location and (b) z-location. (): Formulation-1C; (): Formulation-1A; (): Direct results.

Conclusion

This study establishes a framework for tonal noise prediction of propeller by FWH formulations (both 1A and 1C) with steady-state CFD. Through verification against analytical and validation with numerical, and experimental benchmarks, the approach demonstrates reasonable accuracy for acoustic sources and complex propeller configurations. The implementation within the libAcoustics library for OpenFOAM ensures computational efficiency and parallelization, making the methodology accessible for both academic and industrial applications.

The comparative analysis underscores the superiority of Formulation-1C over Formulation-1A, especially in scenarios involving significant ambient flow, where Formulation-1A’s neglect of convective effects leads to notable discrepancies. By leveraging a steady-state RANS model with a single rotating frame (SRF), the methodology offers substantial computational savings while reliably capturing tonal noise components. However, it is important to note that this approach is inherently limited in its ability to resolve highly transient phenomena such as blade-vortex interaction and cavitation noise, which require high-fidelity unsteady simulations for accurate prediction.

In general, the developed framework provides a practical and efficient tool for propeller noise analysis and optimization. Its successful validation across a range of propeller types and operating conditions demonstrates its potential to support the design and certification of next-generation low-noise propulsion systems. Future work may focus on extending the methodology to address broadband and transient noise mechanisms and its integration with the optimization framework, further enhancing its applicability to the evolving demands of aircraft noise prediction and mitigation. Although the present work focuses on tonal noise, approximate broadband estimates may be obtained using semi-empirical WPS models, such as Schlinker et al.⁴⁶ and Goody⁴⁷ which can complement steady-state SRF predictions.

Footnotes

Author’s note

First author’s current affiliation: Adelaide University, Adelaide, SA 5005, Australia. The research was conducted while the author was with Turkish Aerospace Industries, Ankara, Turkey. Updated email address: abdulhaseeb.lodhi@adelaide.edu.au.

ORCID iDs

Abdul Haseeb Lodhi

Saleh Abuhanieh

Sahin Yigit

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

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