Application of airfoil parameterization methods to broadband noise reduction in rotors

Abstract

This paper employs a Kriging-based surrogate model combined with a multi-objective genetic algorithm to optimize rotor blade airfoils with respect to thrust, power, and broadband noise. Three airfoil parameterization methods—ParFoil, PARSEC, and CST—are compared in generating airfoil geometries for the surrogate modeling and optimization process. The optimized airfoil shapes and their corresponding aerodynamic and acoustic performance metrics are presented and analyzed. Low-fidelity aerodynamic analyses are performed using XFOIL and blade element momentum theory, while acoustic predictions are obtained using Lee’s wall-pressure spectrum model and Amiet’s turbulent boundary-layer trailing-edge noise model implemented in UCD-QuietFly. The study focuses on two configurations: a small-scale ideally twisted rotor and a scaled XV-15 rotor blade. Optimization of the ideally twisted rotor across the three parameterization methods demonstrates an A-weighted overall sound pressure level reduction of approximately 4 dBA, primarily attributed to decreases in the chordwise pressure gradient and wall shear stress. Similarly, optimization of the XV-15 blade using the ParFoil method achieves a noise reduction of 3.47 dBA. Further analyses conducted under vertical climb conditions reveal that these hover-optimized blades maintain noise reductions of 3.0–4.0 dBA relative to the baseline configuration.

Introduction

Optimizing rotorcraft blades is essential for improving performance while effectively balancing competing objectives such as thrust, power consumption, and noise generation.^1,2 Traditional optimization strategies have primarily focused on maximizing thrust and minimizing power requirements. However, in the context of advanced air mobility (AAM) vehicles, mitigating mid-to high-frequency broadband noise has emerged as a critical design consideration. ³ This is especially important for ensuring public acceptance and regulatory compliance in noise-sensitive urban environments.

Modifications to rotor operating conditions and blade geometry offer effective means for reducing rotor broadband noise. Thurman et al. ⁴ utilized artificial neural networks in conjunction with low-fidelity models to identify which design parameters most significantly influence both tonal and broadband noise emissions. Their study employed the Brooks, Pope, and Marcolini (BPM) model ⁵ to predict broadband overall sound pressure level (OASPL), revealing that rotation rate and rotor radius are the most critical factors due to their impact on tip Mach number. Similarly, Ingraham et al. ⁶ performed a thrust-constrained optimization of the X-57 Maxwell propeller and observed analogous trends. They found that decreasing the rotation rate while increasing chord length and twist resulted in meaningful noise reductions without sacrificing thrust. Their optimization strategy led to a substantial increase in inboard chord length, effectively reshaping the radial thrust distribution, producing more thrust inboard and less outboard, which in turn lowered the radiated noise levels. However, these studies do not address the modification of broadband noise through airfoil design changes, which is an effective way to achieve noticeable noise reduction.

Since turbulent-boundary-layer trailing-edge (TBL-TE) noise is widely regarded as the dominant source of rotor broadband noise, ⁷ accurate prediction of this noise is essential for the development of effective noise-reduction strategies. Amiet’s theoretical model for TBL-TE noise ⁸ has served as a foundation for accurate noise prediction under idealized conditions. Building on this framework, Li and Lee^3,9 combined low-fidelity aerodynamic tools, Amiet’s theory, and a recent semi-empirical wall-pressure spectrum (WPS) model ¹⁰ to successfully predict broadband noise for small- and medium-scale rotors.

As this paper is part of a special issue honoring Stewart Glegg’s distinguished contributions to aeroacoustics, we highlight his influential work on TBL-TE noise. This encompasses foundational advances in noise prediction methodologies, ¹¹ noise reduction strategies, ¹² and applications across a diverse range of configurations, including cascades, ¹³ ducted fans, ¹⁴ and wind turbines. ¹⁵ However, these prior studies have largely focused on the acoustic characterization of fixed, pre-existing airfoil and blade geometries. The integration of such predictive capabilities into a design and optimization framework, where airfoil shape and rotor blade geometry can be systematically tuned for noise reduction, remains an important and underexplored avenue for advancing rotorcraft acoustic performance. In particular, the critical role of airfoil geometry in influencing TBL-TE noise has also been emphasized in recent studies, ¹⁶ underscoring the sensitivity of TBL-TE noise generation to subtle variations in airfoil shape.

The U.S. Army has conducted several studies on helicopter blade optimization aimed at enhancing rotor thrust and minimizing power consumption.^17,18 These efforts employed ParFoil, which is an airfoil parameterization tool that modifies baseline airfoil geometries by scaling specific attributes such as camber, camber crest position, thickness, thickness crest location, leading-edge radius, trailing-edge camber, trailing-edge camber crest location, and boat-tail angle. ¹⁷ These studies demonstrated that ParFoil can effectively produce airfoil geometries that significantly reduce power requirements in both hover and forward flight conditions. Building on this, Liu and Lee ¹⁹ employed ParFoil to investigate how variations in geometric parameters influence a turbulent boundary layer, which in turn affects broadband noise levels. Their analysis revealed that airfoil thickness, trailing-edge camber, and boat-tail angle are critical in modifying far-field sound pressure levels (SPL). In a separate study, Liu and Lee ²⁰ combined ParFoil with Kriging surrogate modeling and a Genetic Algorithm (GA) optimization strategy, applying Howe’s trailing-edge noise model ²¹ to identify airfoil designs with reduced noise emissions. However, that work was limited to two-dimensional airfoil sections and did not extend to full rotor blade geometries.

Karimian et al. ²² employed the PARSEC airfoil parameterization method ²³ to perform shape optimization aimed at minimizing propeller broadband noise. The PARSEC framework defines airfoil geometries using polynomial functions for the upper and lower surfaces, governed by physically intuitive parameters such as curvature, thickness distribution, and crest positions. In their optimization approach, Amiet’s TBL-TE noise model was utilized to predict far-field broadband noise. The results demonstrated that an optimized airfoil shape achieved a noise reduction of approximately 2.7 dB without compromising aerodynamic performance.

In addition to the previously discussed ParFoil and PARSEC techniques, the Class/Shape Function Transformation (CST) method is popular for airfoil aerodynamic optimization. This was demonstrated in a recent study using tantem neural networks with the CST method for wind turbine airfoil optimization. ²⁴ CST offers a flexible and robust parameterization framework by combining a class function, which determines the general airfoil type, with a shape function that defines the surface geometry using Bernstein polynomials. ²⁵ The method’s versatility stems from the adjustable polynomial degree, which allows for the construction of complex geometries through higher-order terms. Kou et al. ²⁶ applied the CST method to airfoil optimization targeting wind turbine noise reduction. Their results indicated that designs featuring thinner leading edges and thicker trailing-edge regions led to a reduction in A-weighted overall sound pressure level (OASPL(A)). Through two-dimensional aerodynamic and acoustic analysis, they reported an approximate 1.75 dBA decrease in noise relative to a baseline NACA 0012 airfoil across a range of angles of attack.

The current research focuses on reducing rotor TBL-TE noise through airfoil shape optimization. As discussed earlier, various airfoil parameterization methods exist, each offering distinct advantages and limitations. To this end, we systematically evaluate the performance of three widely used parameterization techniques, ParFoil, PARSEC, and CST, within a low-fidelity, multi-objective optimization framework that simultaneously considers thrust, power, and broadband noise. The objective is to identify which methods and resulting airfoil geometries yield more optimal thrust and power performance, while producing less noise. Additionally, comparing the resulting geometries from each method will reveal the airfoil features and boundary-layer characteristics responsible for TBL-TE noise reduction. The optimization is applied to two rotor configurations: a small-scale rotor and a medium-scale tiltrotor, both operating in hover. To assess the robustness of the optimized designs, the aerodynamic and acoustic performance of the tiltrotor that is optimized for hover is also evaluated under vertical climb flight conditions.

Methods

Aerodynamic and aeroacoustic predictions

Aerodynamic analysis is conducted using Blade Element Momentum Theory (BEMT) in conjunction with XFOIL. ²⁷ Each rotor blade is discretized into 100 uniformly spaced elements, where the local effective angle of attack and relative velocity are evaluated. Airfoil performance data, including lift and drag coefficients, are obtained from XFOIL and used to compute thrust and power distributions within the BEMT framework. For aeroacoustic predictions, the study employs UCD-QuietFly, a validated computational tool previously benchmarked against a range of rotor configurations.^3,9,28–30 UCD-QuietFly is based on Amiet’s TBL-TE theory, ⁸ with modifications introduced by Roger and Moreau. ³¹ In addition, the code also supports the Brooks, Pope, and Marcolini (BPM) model ⁵ for simulating additional airfoil self-noise mechanisms such as stall, laminar-boundary-layer vortex shedding, and bluntness vortex shedding noise. However, this study primarily focuses on TBL-TE noise under the assumption of fully turbulent flow, and does not consider airfoil designs exhibiting significant flow separation or other noise sources.

Amiet’s theory requires knowledge of turbulent wall-pressure fluctuations near the trailing edge to predict TBL-TE noise. Although such fluctuations can only be accurately captured using scale-resolving simulations, an alternative modeling approach is adopted in this study. Specifically, WPS near the trailing edge is estimated based on two-dimensional steady boundary-layer properties obtained from XFOIL. For the acoustic analysis, the rotor blade is divided into 21 uniformly spaced radial segments. At each segment, boundary-layer properties are extracted at the 99% chordwise location and used as input to the WPS model. In this study, Lee’s semi-empirical WPS model ¹⁰ is employed to estimate the near-trailing-edge pressure fluctuations required for TBL-TE noise prediction. This model has been shown to provide accurate WPS estimates under a wide range of pressure gradient conditions, including zero, adverse, and favorable gradients for boundary-layer flows without large separation or stall. The model requires several input parameters: boundary-layer displacement and momentum thicknesses, skin-friction coefficient, chordwise pressure gradient, and the edge velocity near the trailing edge. Notably, these parameters directly influence the amplitude of the WPS and, consequently, the predicted TBL-TE noise levels. As such, this approach enables an assessment of how specific airfoil design features alter boundary-layer characteristics and their overall impact on noise generation.

Airfoil parameterization methods

Three airfoil parameterization methods—ParFoil, PARSEC, and CST—are considered and compared for use in airfoil optimization aimed at broadband noise reduction. The details of each method are described in this subsection.

ParFoil method

ParFoil, as described in Refs. 17–19,32, is a morphing-based airfoil parameterization method in which the coordinates of a baseline airfoil are modified to scale specific geometric features, rather than through direct redistribution of coordinate points. The parameters employed by ParFoil are illustrated in Figure 1. First, key geometric characteristics of the baseline airfoil are identified, including the leading-edge radius $(k)$ , camber magnitude $(m)$ , location of the leading-edge camber crest $(x_0)$ , thickness $(t)$ , location of the thickness crest $(p_0)$ , trailing-edge camber magnitude $(n)$ , location of the trailing-edge camber crest $(q_0)$ , and the boat-tail angle $(b)$ . These baseline values are then adjusted according to user-defined inputs that scale or translate the corresponding features. A primary advantage of this method is that it offers intuitive control using physically meaningful attributes of the airfoil, enabling targeted investigations of how individual shape features influence aerodynamic and acoustic performance. In addition, ParFoil allows for independent modification of the leading- and trailing-edge regions, divided by $y_R$ , which is particularly useful in TBL-TE broadband noise optimization studies through the separate modification in the trailing-edge region.

Figure 1.

Diagram of ParFoil parameters (adapted from ¹⁹ ).

In the rotor airfoil optimization, seven parameters are selected for adjustment: the leading-edge radius scaling factor $(f_k)$ , camber scaling factor $(f_m)$ , thickness crest location $(p_0)$ , thickness scaling factor $(f_t)$ , trailing-edge camber offset $(\mathrm{\Delta }n)$ , trailing-edge camber crest position $(q_0)$ , and boat-tail angle offset $(\mathrm{\Delta }b)$ . In this context, the symbol $f$ denotes a scaling factor applied to the original geometric feature, while $\mathrm{\Delta }$ indicates an additive or subtractive modification relative to the baseline values shown in Figure 1. To simplify the parameterization, in this study, the leading-edge camber crest location $(x_0)$ is assumed to coincide with the thickness crest location $(p_0)$ . Additionally, the leading-edge/trailing-edge partition location $y_R$ and the measurement location for the boat-tail angle, $y_B$ , are fixed at $x/c=0.6$ and $x/c=0.8$ , respectively. The lower and upper bounds used for each parameter in the ParFoil optimization are summarized in Table 1.

Table 1.

ParFoil parameter range.

Parameter	Minimum bound	Maximum bound
Camber scaling factor $(f_m)$	0.00	2.00
LE radius scaling factor $(f_k)$	0.80	1.80
Thickness scaling factor $(f_t)$	0.50	1.50
Thickness crest position $(p_0)$	0.30	0.45
TE camber offset $(\mathrm{\Delta }n)$	$-0.015$	0.015
TE camber crest position $(q_0)$	0.60	0.90
Boat-tail angle offset $(\mathrm{\Delta }b)$	$-2.00$	2.00

PARSEC method

PARSEC is a widely used airfoil parameterization method that defines the airfoil geometry through analytic functions constructed using three control points per surface.^23,33,34 Unlike ParFoil, which modifies specific geometric features such as camber or thickness through scaling or shifting, PARSEC directly alters the airfoil’s coordinate distribution by fitting parametric curves to shape-defining points at the leading edge, thickness crest position, and trailing edge. The method offers a total of 11 geometric parameters, as illustrated in Figure 2. Like ParFoil, PARSEC allows control over the leading-edge radius $(r_{le})$ , thickness crest location $(x_s)$ , and maximum thickness $(z_{s,\text{max}})$ , where the subscript $s$ denotes the surface of interest (either upper or lower). However, rather than adjusting the camber line explicitly, PARSEC modifies the curvature of the upper or lower surfaces at the thickness crest location using the second derivative of $z_s$ with a value of $z_{\mathit{xxs}}$ . In addition, PARSEC provides control over trailing-edge characteristics, including the vertical position $(z_{te})$ , trailing-edge thickness $(\mathrm{\Delta }{z}_{te})$ , inclination angle $({\alpha }_{te})$ , and wedge angle $({\beta }_{te})$ , enabling detailed shaping near the trailing edge for aerodynamic or acoustic performance optimization.

Figure 2.

PARSEC parameterization: (a) diagram of PARSEC parameters and (b) PARSEC optimization design space.

The equation used to obtain the $z$ coordinates for each surface is

$z_s=\sum _{i=1}^{n=6}{a}_s\cdot x^{i-\frac{1}{2}}.$

(1)

Equation (1) defines a polynomial curve that passes through three prescribed points: the leading edge, the trailing edge, and the location of the maximum thickness (thickness crest). The coefficients $a_s$ are determined by solving the linear system $C_s{a}_s=b_s$ , where $C_s$ is the constraint matrix constructed so that the curve passes through the leading edge, trailing edge, and thickness crest location $(x_s)$ smoothly.

$C_s=\left[\begin{array}{cccccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill x_s^{1/2}\hfill & \hfill x_s^{3/2}\hfill & \hfill x_s^{5/2}\hfill & \hfill x_s^{7/2}\hfill & \hfill x_s^{9/2}\hfill & \hfill x_s^{11/2}\hfill \\ \hfill 1/2\hfill & \hfill 3/2\hfill & \hfill 5/2\hfill & \hfill 7/2\hfill & \hfill 9/2\hfill & \hfill 11/2\hfill \\ \hfill \frac{1}{2}{x}_s^{-1/2}\hfill & \hfill \frac{3}{2}{x}_s^{1/2}\hfill & \hfill \frac{5}{2}{x}_s^{3/2}\hfill & \hfill \frac{7}{2}{x}_s^{5/2}\hfill & \hfill \frac{9}{2}{x}_s^{7/2}\hfill & \hfill \frac{11}{2}{x}_s^{9/2}\hfill \\ \hfill -\frac{1}{4}{x}_s^{-3/2}\hfill & \hfill \frac{3}{4}{x}_s^{-1/2}\hfill & \hfill \frac{15}{4}{x}_s^{1/2}\hfill & \hfill \frac{35}{4}{x}_s^{3/2}\hfill & \hfill \frac{63}{4}{x}_s^{5/2}\hfill & \hfill \frac{99}{4}{x}_s^{7/2}\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],$

(2)

The $b_s$ vector contains the desired values that define the surface.

$\begin{array}{cc}\hfill b_{\mathit{upper}}& ={\left[\begin{array}{cccccc}\hfill z_{te}+\frac{1}{2}\mathrm{\Delta }{z}_{te}\hfill & \hfill z_{u,\text{max}}\hfill & \hfill \tan \left(\frac{2{\alpha }_{te}-{\beta }_{te}}{2}\right)\hfill & \hfill 0\hfill & \hfill z_{\mathit{xxu}}\hfill & \hfill \sqrt{r_{le}}\hfill \end{array}\right]}^T,\hfill \\ \hfill b_{\mathit{lower}}& ={\left[\begin{array}{cccccc}\hfill z_{te}-\frac{1}{2}\mathrm{\Delta }{z}_{te}\hfill & \hfill z_{l,\text{max}}\hfill & \hfill \tan \left(\frac{2{\alpha }_{te}+{\beta }_{te}}{2}\right)\hfill & \hfill 0\hfill & \hfill z_{\mathit{xxl}}\hfill & \hfill \sqrt{r_{le}}\hfill \end{array}\right]}^T.\hfill \end{array}$

(3)

For the present optimization, all PARSEC parameters are employed except for the trailing-edge coordinate and thickness, resulting in a total of nine active parameters. The trailing-edge vertical position $(z_{te})$ and trailing-edge thickness $(\mathrm{\Delta }{z}_{te})$ are fixed at $z_{te}=0.0$ and $\mathrm{\Delta }{z}_{te}=0.00252$ , respectively, consistent with the geometry of the NACA 0012 airfoil. Unlike ParFoil, which requires the coordinates of a baseline airfoil to perform morphing transformations, PARSEC is capable of generating airfoil geometries directly from its parameter set. This feature allows for greater flexibility in producing a wide range of airfoil shapes without relying on a predefined geometric template. The upper and lower bounds for the PARSEC parameters used in the optimization are summarized in Table 2.

Table 2.

PARSEC parameter range.

Parameter	Minimum bound	Maximum bound
LE radius $(r_{le})$	0.01	0.03
Upper crest location $(x_u)$	0.25	0.40
Upper thickness $(z_{u,\text{max}})$	0.01	0.06
Upper curvature $(Z_{\mathit{xxu}})$	−0.30	−0.45
Lower crest location $(x_l)$	0.25	0.40
Lower thickness $(z_{l,\text{max}})$	−0.04	−0.08
Lower curvature $(Z_{\mathit{xxl}})$	0.01	0.06
TE direction angle $(\alpha )$	$-15.00$	−15.00
TE wedge angle $(\beta )$	5.00	30.00

Class shape transformation (CST) method

The CST method, introduced by Kulfan, ²⁵ provides a flexible and efficient means of airfoil parameterization. This method constructs the airfoil surface using a product of a class function and a shape function, with an additional term to account for finite trailing-edge thickness. Letting $\zeta =z/c$ and $\psi =x/c$ denote the non-dimensional airfoil coordinates normalized by the chord length, the general expression for the surface geometry is given by:

$\zeta (\psi )=C(\psi )S(\psi )+\psi {\zeta }_T,$

(4)

where ${\zeta }_T$ represents half of the trailing-edge thickness. To ensure consistency with the trailing-edge geometry of the NACA 0012 airfoil, ${\zeta }_T$ is held constant at 0.00126 for all CST-based designs, similar to the approach adopted for the PARSEC parameterization.

$C(\psi )$ is the class function represented by

$C(\psi )={(\psi )}^{N_1}{(1-\psi )}^{N_2}.$

(5)

The parameters $N_1$ and $N_2$ in the CST method govern the geometric behavior of the airfoil near the leading and trailing edges by shaping the class function. Specifically, these parameters control how the surface coordinate $\zeta$ near the edges varies with the normalized chordwise location $\psi$ . For example, setting $N_1=0.5$ and $N_2=1.0$ yields a class function that produces a rounded leading edge and a sharp trailing edge. Conversely, assigning $N_1=1.0$ and $N_2=0.5$ results in a sharp leading edge and a rounded trailing edge. By adjusting these exponents, the CST framework offers fine control over the edge characteristics of the airfoil profile.

The shape function is

$S(\psi )=\sum _{i=0}^n{a}_i{K}_{i,n}{\psi }^i{(1-\psi )}^{n-i},$

(6)

where $K_{i,n}$ is defined as

$K_{i,n}\equiv \frac{n!}{i!(n-i)!}.$

(7)

The coefficients $a_i$ control the amplitude of the corresponding Bernstein basis functions, defined as $K_{i,n}{\psi }^i{(1-\psi )}^{n-i}$ , where $K_{i,n}$ is the binomial coefficient. The parameter $n$ represents the order of the Bernstein polynomial, and an $n^{\text{th}}$ -order polynomial yields a total of $n+1$ basis functions, which collectively define the shape function in equation (6).

The shape function in the CST method is particularly powerful due to its ability to generate complex surface curvatures, making the parameterization both adaptable and robust for a wide range of airfoil geometries. The complexity of the resulting shape can be controlled by selecting the order of the Bernstein polynomial and tuning the corresponding amplitude coefficients $a_i$ . Conceptually, the CST approach constructs the airfoil surface as a summation of weighted shape functions, yielding a smooth and continuous curve for either the upper or lower surface. Figure 3(a) illustrates how the CST parameterization accurately reconstructs the upper surface of a NACA 0012 airfoil by appropriately choosing the $a_i$ coefficients and combining the basis functions defined by equations (4)–(6).

Figure 3.

CST parameterization: (a) six component curves using Berstein polynomials $(n=5)$ to approximate the NACA 0012 coordinates and (b) the CST optimization design space.

For the optimization using the CST method, a total of 12 parameters are employed, corresponding to a fifth-order polynomial $(n=5)$ , with six shape coefficients assigned to each of the upper and lower surfaces. The class function exponents are fixed at $N_1=0.5$ and $N_2=1.0$ , ensuring a rounded leading edge and a sharp trailing edge, consistent with conventional airfoil shapes. Additionally, the trailing-edge thickness is constrained to match that of the original NACA 0012 airfoil. A key advantage of the CST framework is that it employs continuous basis functions, allowing for a smooth and analytically defined design space. This feature enables a clear graphical representation of parametric bounds, as illustrated in Figure 3(b), where the envelope formed by the shaded regions defines the range of achievable airfoil shapes within the specified parameter limits.

Kriging surrogate model

The DACE MATLAB Kriging Toolbox developed by Lophaven et al. ³⁵ is employed to construct a surrogate (approximate) model of the design space. A surrogate model can predict the performance of designs that were not explicitly analyzed, thus reducing the number of costly acoustic evaluations required using UCD-QuietFly. This approach is consistent with the methodology previously applied by Lee et al. ³⁶ for the aerodynamic optimization of wing sails and Liu and Lee ²⁰ for two-dimensional airfoil acoustic optimization. Initially, a discrete set of design points is sampled and evaluated to quantify their acoustic performance. A Kriging surrogate model is then constructed by fitting a continuous response surface to this data, enabling accurate predictions of the performance for unevaluated designs. In this study, Kriging is chosen due to its capability to interpolate sparse data and quantify uncertainty, making it particularly well-suited for design space exploration and optimization under limited computational budgets.

The mathematical representation of the response surface is given by:

$\widehat{y}(x)=f(x){\beta }^{*}+r{(x)}^T{\gamma }^{*},$

(8)

where $x$ denotes the input vector of dimension $n$ , and $\widehat{y}(x)$ represents the predicted output vector of dimension $q$ from the Kriging response surface. This equation serves as a general form of the Kriging predictor, where $x$ may include both evaluated and unevaluated design points within the user-defined parameter bounds. The first term on the right-hand side is a linear combination of basis functions $f(x)$ and their associated coefficients ${\beta }^{*}$ , which collectively model the global trend of the underlying function and are often interpreted as the mean or expected value of the response. The second term captures the local deviation or error from the global trend and is estimated through statistical modeling of the spatial correlation between sampled data points. This error term enables the Kriging model to interpolate known data exactly while providing uncertainty quantification for untested designs. Equation (8) can be further expanded as follows:

$f(x)={\left[\begin{array}{cccc}\hfill {\mathtt{f}}_1(x),\hfill & \hfill {\mathtt{f}}_2(x),\hfill & \hfill \cdots ,\hfill & \hfill {\mathtt{f}}_p(x)\hfill \end{array}\right]}^T,$

(9)

${\beta }^{*}={(F^T{R}^{-1}F)}^{-1}{F}^T{R}^{-1}Y,$

(10)

${\gamma }^{*}=R^{-1}(Y-F{\beta }^{*}).$

(11)

Equation (9) defines the vector of basis functions $\mathtt{f}$ , which is determined by the order of the polynomial selected for the regression model. In this study, a zeroth-order (constant) polynomial (or $p$ = 1) is employed, corresponding to ${\mathtt{f}}_1(x)=1$ , which assumes a uniform mean across the design space. The design matrix $F$ , which is shown in equation (12), consists of the evaluated basis functions at each sampled design location $s_i$ . In this case, $F$ becomes a column vector of ones, simplifying the estimation of the regression coefficients ${\beta }^{*}$ while maintaining sufficient flexibility in the Kriging model through its spatial correlation structure.

$F=\left[\begin{array}{cccc}\hfill {\mathtt{f}}_1(s_1)\hfill & \hfill {\mathtt{f}}_2(s_1)\hfill & \hfill \cdots \hfill & \hfill {\mathtt{f}}_p(s_1)\hfill \\ \hfill {\mathtt{f}}_1(s_2)\hfill & \hfill {\mathtt{f}}_2(s_2)\hfill & \hfill \cdots \hfill & \hfill {\mathtt{f}}_p(s_2)\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill {\mathtt{f}}_1(s_m)\hfill & \hfill {\mathtt{f}}_2(s_m)\hfill & \hfill \cdots \hfill & \hfill {\mathtt{f}}_p(s_m)\hfill \end{array}\right]$

(12)

Note that $s_i=(S_{i,1},S_{i,2},\dots ,S_{i,j},\dots S_{i,n})$ with $n$ dimensions, where $i=1,\dots ,m$ , with $m$ denoting the total number of design samples and $n$ representing the input dimensionality—that is, the number of airfoil parameters associated with a given parameterization method. $S_{i,j}$ indicates $j$ -th parameter value, such as the airfoil maximum thickness or boat-tail angle, for the $i$ -th sample. Thus, each $s_i$ represents a unique airfoil geometry using $n$ design parameters for evaluation in the surrogate modeling process. Note that $s_i$ itself does not have a numeric value. Only $S_{i,j}$ has a numeric value.

$R$ in equations (10) and (11) is an $m\times m$ matrix of spatial correlations between evaluated locations.

$R=\left[\begin{array}{ccc}\hfill \mathtt{R}(\theta ,s_1,s_1)\hfill & \hfill \cdots \hfill & \hfill \mathtt{R}(\theta ,s_1,s_m)\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill \mathtt{R}(\theta ,s_m,s_1)\hfill & \hfill \cdots \hfill & \hfill \mathtt{R}(\theta ,s_m,s_m)\hfill \end{array}\right]$

(13)

The correlations are modeled using the Gauss correlation function

$\mathtt{R}(\theta ,s_i,s_k)=\prod _{j=1}^n\exp (-{\theta }_j{(S_{i,j}-S_{k,j})}^2,$

(14)

where $j$ refers to the index of the design parameter, such as camber, thickness, or other airfoil shape descriptors. The parameter ${\theta }_j$ appears in the correlation model and governs the rate at which correlation decays with distance along the $j^{\text{th}}$ input dimension. The values of ${\theta }_j$ are determined by minimizing the objective function $|R{|}^{1/m}{\sigma }^2$ , where $|R|$ is the determinant of the correlation matrix $R$ , $m$ is the number of evaluated design points, and ${\sigma }^2$ is the estimated variance of the model residuals, which is given as

${\sigma }^2=\frac{1}{m}{(Y-F{\beta }^{*})}^{\top }(Y-F{\beta }^{*}).$

(15)

$Y$ is an $m\times q$ matrix containing the observed outputs, corresponding to the performance metrics evaluated using BEMT and UCD-QuietFly for each of the $m$ design samples. As previously noted, $q$ denotes the number of response variables of interest. In this study, $q=3$ , as the optimization simultaneously considers thrust, power consumption, and A-weighted overall sound pressure level (OASPL(A)) as key performance indicators.

The vector $r(x)$ in equation (8) represents the correlation between the unevaluated design point $x$ and all previously evaluated design points $s_i$ , serving as a key component in the interpolation and uncertainty estimation of the Kriging predictor.

$r(x)={\left[\begin{array}{ccc}\hfill \mathtt{R}(\theta ,s_1,x)\hfill & \hfill \cdots \hfill & \hfill \mathtt{R}(\theta ,s_m,x)\hfill \end{array}\right]}^T.$

(16)

The initial set of evaluated design points may be insufficient to construct an accurate and reliable response surface. To address this, the mean-squared error (MSE), denoted by $\phi (x)$ , can be computed at any candidate point $x$ to quantify the local predictive uncertainty of the Kriging model. This MSE is given by

$\phi (x)={\sigma }^2(1+u^T{(F^T{R}^{-1}F)}^{-1}u-r{(x)}^T{R}^{-1}r(x))$

(17)

where $u=F^T{R}^{-1}r(x)-f(x)$ . Regions exhibiting high values of MSE are preferentially targeted for additional evaluations, as these locations are associated with greater uncertainty in the surrogate model’s predictions. By incorporating new data points in such regions, the accuracy and reliability of the response surface are systematically improved. Figure 4 illustrates this behavior, where the surrogate model’s prediction of the A-weighted OASPL (in dBA) evolves and becomes more refined as additional data points are incorporated into the training set.

Figure 4.

Surrogate response surface of OASPL(A) as a function of ParFoil’s thickness scaling factor $(f_t)$ and boat-tail angle $(\mathrm{\Delta }b)$ fitted with: (a) 100, (b) 1000, (c) 5000, and (d) 10,000 evaluations.

Multi-objective genetic algorithm (GA)

The study by Jones et al. ³⁷ represents an early example of employing a Genetic Algorithm (GA) to minimize airfoil noise. In contrast, the objective of the present study is to perform a multi-objective optimization that simultaneously maximizes rotor thrust and minimizes both power consumption and OASPL(A). After constructing the surrogate response surface defined by equation (8), this surface is subsequently used to compute the objective functions for the GA implemented in MATLAB’s Design and Optimization Toolbox. GAs belong to the broader class of evolutionary optimization (EO) techniques, which operate by evolving a population of candidate solutions over successive generations. Each generation applies selection, crossover, and mutation to improve the fitness of the population, thereby progressing toward optimal solutions. A key advantage of EO methods is that they do not require gradient information, making them well-suited for complex, nonlinear, or non-differentiable objective functions. Additionally, EO algorithms are known for their ability to identify global optima, a feature well-documented in the literature.^36,38,22 In the context of multi-objective optimization, GAs provide a diverse set of Pareto-optimal solutions, enabling trade-off analysis between competing objectives. This is particularly important in multi-disciplinary design problems, where improving one performance metric may lead to compromises in others.

Individuals within the population are encoded as genes, each representing a parameter that defines the airfoil geometry. For this study, each generation comprises 100 individuals. The initial population is generated randomly, and the performance of each individual is evaluated using the Kriging surrogate model described in equation (8). Individuals whose performance exceeds the population average are selected for genetic operations, including crossover, mutation, and elite preservation. Crossover is performed by selecting high-performing individuals (parents) and exchanging segments of their gene sequences to produce new offspring, thereby promoting the inheritance and mixing of advantageous traits. Mutation introduces random changes to certain genes in the offspring, enhancing the exploration capability of the algorithm by allowing it to search previously unexplored regions of the design space. Elite preservation ensures that the top-performing individuals are retained in the next generation without alteration, thereby safeguarding against the loss of high-quality solutions. The genetic algorithm proceeds iteratively until either the Pareto front converges with no further improvement or a predefined limit of 1000 generations is reached. A design is considered Pareto optimal if it is non-dominated by any other design in the population. Specifically, an individual $x_1$ dominates $x_2$ if (1) $x_1$ is no worse than $x_2$ in all objectives, and (2) $x_1$ is strictly better than $x_2$ in at least one objective. For a detailed discussion of multi-objective EO and GA methodologies, the reader is referred to Deb, ³⁸ which offers a comprehensive treatment of these techniques and their applications.

Rotor geometries and microphone locations

Two rotors are considered for the hover optimization study: the ideally twisted rotor (ITR) and the XV-15 tiltrotor. The ITR configuration is used as a platform to investigate the differences and similarities among three airfoil parameterization methods: ParFoil, PARSEC, and CST. In contrast, for the XV-15 blade optimization, only the ParFoil method is employed.

Pettingill et al. ³⁹ conducted experiments on a small-scale, four-bladed ITR, characterized by a constant-chord NACA 0012 airfoil and a radius of 0.1588 m. The blade’s radial twist distribution is shown in Figure 5(a), and the microphone arrangement used for broadband noise measurements is presented in Figure 5(b). Among the microphone positions, location 5 is used as the reference for evaluating optimized designs, as prior studies have demonstrated strong agreement between the UCD-QuietFly simulations and experimental data at this location. ²⁹ The baseline operating condition corresponds to a rotation rate of 5500 RPM, yielding a tip Reynolds number of approximately $0.20\times 10^6$ and a Mach number of 0.27. For the ITR, the optimization targets only the airfoil shape in the outer radial range $r/R=0.7\text{–}1.0$ , since the high-speed tip region is the dominant source of TBL-TE noise. The inboard region $(r/R=0.0\text{–}0.7)$ retains the unmodified NACA 0012 profile.

Figure 5.

Ideally twisted rotor experiments (adapted from ³⁹ ): (a) radial twist distribution and (b) microphone locations.

The original XV-15 blade spans approximately 3.81 m and rotates at 582 RPM, resulting in a tip Mach number of 0.69 and a Reynolds number of approximately $5.0\times 10^6$ . Since this tip Mach number is significantly higher than that of typical AAM rotors, we consider a resized XV-15 blade with a reduced radius of 1.9 m, yielding a tip Mach number of approximately 0.34 and a Reynolds number of $1.3\times 10^6$ . This shortened blade preserves the original blade solidity and linear twist of $-40.25°$ . The blade geometry is defined by the chord and airfoil distribution summarized in Table 3, where the first 25% of the span tapers linearly and the remaining 75% maintains a constant chord length. It is worth noting that the actual XV-15 rotor employs a thicker airfoil than the NACA 64-118 in its inboard region. However, due to convergence issues encountered when using thicker airfoils in XFOIL, the inboard section is modeled using the NACA 64-118. Since this region contributes minimally to both aerodynamic performance and acoustic emissions, this approximation is deemed acceptable. No interpolation is applied between the different airfoil profiles listed in Table 3; i.e., each segment transitions directly to the next.

Table 3.

Modeled XV-15 chord and airfoil distributions.

$r/R$	Chord (m)	Airfoil
$0.00\text{–}0.25$	$0.5021-0.5859\times r/R$	NACA 64-118
$0.25\text{–}0.70$	0.3556	NACA 64-118
$0.70\text{–}0.90$	0.3556	NACA 64-(1.5)12
$0.90\text{–}1.00$	0.3556	NACA 64-208

No acoustic measurement data are available for the shortened XV-15 blade. Nevertheless, the far-field observer is positioned at $270°$ relative to the rotor plane (i.e., directly beneath the rotor) and located at a distance of 40 rotor radii. Given the critical role of the blade’s outboard sections in determining both aerodynamic efficiency and acoustic characteristics, the optimization is confined to the radial region $r/R=0.7$ –1.0. The airfoil geometry optimized over the $r/R=0.7$ –0.9 span is designated as Airfoil 1, while Airfoil 2 corresponds to the $r/R=0.9$ –1.0 region. Although these airfoils are optimized specifically for hover conditions, their performance in vertical climb flight will also be assessed to determine whether additional optimization is warranted for that operational regime.

Results

Ideally twisted rotor

The ITR experiments conducted by Pettingill et al. ³⁹ are utilized to validate prediction tools and to support the airfoil optimization study. While noise reduction remains the primary objective, the Figure of Merit (FM) is also a critical performance metric for hover optimizations. As emphasized in Ref. 40, even a small decrease in FM of just 0.005 can lead to a reduction of one passenger in vehicle payload capacity. The Figure of Merit is defined as

$FM=\frac{C_T^{3/2}}{\sqrt{2}{C}_P},$

(18)

where $C_T$ and $C_P$ are the thrust and power coefficients.

The optimization results obtained using all three airfoil parameterization methods are presented in Figure 6(a), where OASPL(A) is plotted against FM. The baseline ITR design is denoted by a red hexagram and corresponds to an FM of 0.6320 and an OASPL(A) of 57.36 dBA. Many of the designs on the Pareto front lie to the southeast of the baseline point, indicating overall improvement characterized by simultaneous increases in FM and reductions in OASPL(A). However, several designs appear to be dominated in either FM or OASPL(A). This is because the Pareto front is constructed not solely based on FM and OASPL(A), but also includes nominal thrust and power coefficients as objectives, which influence the observed trade-offs.

Figure 6.

Optimized designs for the ideally twisted rotor: (a) all methods, (b) ParFoil, (c) PARSEC, and (d) CST method. Letters labeled within each subfigure correspond with the airfoil geometries provided in Figure 7 and performance metrics listed in Table 4.

Figure 6(a) reveals each method converges to a minimum OASPL(A) of around 53–54 dBA. Notably, the Pareto front generated by the CST-based optimization displays more aerodynamically efficient designs, characterized by higher FMs for the same OASPL(A). This may be a result of using an $n=5$ Bernstein polynomial, where the CST optimization utilizes more parameters when compared to the ParFoil and PARSEC methods, thus a larger design space. However, the ParFoil and PARSEC methods produce lower OASPL(A) designs, likely because both provide explicit control over trailing-edge features such as boat-tail angle and direction.

The design points obtained using ParFoil are presented in Figure 6(b). The quietest design, labeled as (a), achieves an OASPL(A) of 53.36 dBA and an FM of 0.6206. This corresponds to a notable noise reduction of 4.00 dBA and a modest decrease in FM of 0.0114 compared to the baseline NACA 0012 configuration. In contrast, the design labeled (d) yields an OASPL(A) of 58.49 dBA and an FM of 0.6637, representing a noise increase of 1.13 dBA but a substantial FM improvement of 0.0317. Several designs in Figure 6(b) also demonstrate balanced trade-offs between noise and aerodynamic performance. For example, one such design shows an OASPL(A) of 53.95 dBA and an FM of 0.6408, achieving a 3.41 dBA noise reduction and a slight FM gain of 0.0088 relative to the baseline. While this study emphasizes the extremes in the performance or broadband noise space, the optimization methodology is capable of yielding designs that simultaneously improve both metrics.

Figure 6(c) presents the optimization results obtained using the PARSEC airfoil parameterization method. The Pareto front generated with PARSEC closely resembles that of the ParFoil method, although it appears to be slightly less optimal overall. The quietest PARSEC-derived design, labeled as (b), achieves an OASPL(A) of 53.41 dBA and an FM of 0.6140, corresponding to a 3.95 dBA reduction in noise and a 0.0180 decrease in FM compared to the baseline. The design with the highest FM among the PARSEC results, labeled as (e), shows a modest performance trade-off: a noise increase of 0.92 dBA and an FM improvement of 0.0241. The underlying causes for the observed differences between the PARSEC and ParFoil results are ascribed to the parameter bounds used in this optimization and will be discussed later in this section.

Finally, Figure 6(d) shows the optimization results obtained using the CST airfoil parameterization method. The distribution of CST-derived designs deviates noticeably from the patterns observed in the ParFoil and PARSEC results, with a greater number of configurations exhibiting improved FM values while maintaining or reducing noise levels. Moreover, the quietest configuration (c) achieves a 3.42 dBA reduction in OASPL(A) and is accompanied by a FM increase of 0.0144. The highest FM design (f) yields a noticeable increase of 0.0409 in the FM with a noise increase of 1.62 dBA.

For each parameterization method, the airfoils corresponding to the quietest and highest-FM designs are depicted in Figure 7, following the same labeling scheme used throughout this section (a–f). In addition, Table 4 presents detailed performance metrics for these designs, including thrust, power, FM, and OASPL(A). This comprehensive presentation facilitates a direct comparison of how different airfoil geometries influence both aerodynamic efficiency and acoustic performance, thereby highlighting the capability of each parameterization method to achieve optimized outcomes across multiple design objectives.

Figure 7.

Optimized airfoils for quietest design (top) and highest FM design (bottom) for: ParFoil (a and d), PARSEC (b and e), and the CST method (c and f).

Table 4.

Performance of various designs for the optimized ideally twisted rotor.

Design	Thrust $(C_T\times 10^2)$	Power $(C_P\times 10^3)$	FM	OASPL (dBA)
Baseline	1.382	1.818	0.6320	57.36
ParFoil (a)	1.382 $(+0.00\%)$	1.851 $(+1.82\%)$	0.6206 $(-1.80\%)$	53.36 $(-4.00)$
ParFoil (d)	1.622 $(+17.37\%)$	2.201 $(+21.07\%)$	0.6637 $(+5.02\%)$	58.49 $(+1.13)$
PARSEC (b)	1.376 $(-0.43\%)$	1.859 $(+2.26\%)$	0.6140 $(-2.85\%)$	53.41 $(-3.95)$
PARSEC (e)	1.927 $(+39.44\%)$	2.883 $(+58.58\%)$	0.6561 $(+3.81\%)$	58.28 $(+0.92)$
CST (c)	1.457 $(+5.43\%)$	1.924 $(+5.83\%)$	0.6464 $(+2.27\%)$	53.94 $(-3.42)$
CST (f)	1.649 $(+19.32\%)$	2.225 $(+22.38\%)$	0.6729 $(+6.48\%)$	58.98 $(+1.62)$

Figure 8 compares the quietest airfoils obtained from the ParFoil, PARSEC, and CST parameterization methods. The optimized airfoils exhibit distinct geometric features, but they also share common features in trailing-edge profile from $x/c=0.95$ , which largely overlap. Even though airfoil shapes are different for each method, they all result in an approximate 4 dBA reduction in OASPL(A). Notably, the quietest CST design is thinner and more cambered than the other airfoils, likely contributing to the favorable FM displayed in Table 4. In contrast, the PARSEC airfoil is thicker and shows noticeably less curvature on the upper surface, contributing to its lower FM. The poor FM performance of the PARSEC (b) design may be attributed to the parameter bounds imposed during optimization. For instance, small localized shape changes may require substantial adjustments in the parametric vector defined in equations (1)–(3). Expanding the parameter bounds to improve convergence was not pursued, as it risked generating invalid airfoil geometries—such as surface crossings between the upper and lower profiles. Because the surrogate model is constructed from sampled data, including invalid parameter combinations could introduce regions of high uncertainty into the response surface, ultimately reducing the model’s predictive accuracy and robustness.

Figure 8.

Optimized airfoils that yield the lowest OASPL(A) for each parameterization method.

Figure 9 presents the airfoils with the highest FM identified from each parameterization method. The fact that each method converges to distinctly different geometries also suggests that the parameter bounds imposed during optimization may have been overly restrictive, potentially constraining the exploration of the full design space. This is particularly true for the ParFoil (c) and PARSEC (e) designs, as they only achieve an increase of 0.0241 and 0.0291 in their FM, respectively, while the CST (f) airfoil increases substantially by 0.0409. Nevertheless, these distinctions highlight how different parameterization frameworks navigate the trade-off space and underscore the influence of parameter structure and bounding constraints on optimization outcomes.

Figure 9.

Optimized airfoils that yield the highest FM for each parameterization method.

Figure 10 presents the broadband noise spectra for the ITR. It shows good agreement between baseline predictions and experimental measurements, except for the SPL hump associated with trailing-edge bluntness vortex shedding noise between 10 and 20 kHz.^29,41 In addition, the noise generated by the baseline blade is evaluated against that of the quietest designs (a–c) obtained from each airfoil parameterization method. Each OASPL(A)-optimized design demonstrates a similar TBL-TE noise reduction, but the CST (c) design is slightly louder above 5 kHz. Despite this, we do note that, because of its greater FM and thrust coefficient when compared to baseline, the rotation rate or collective pitch can be reduced. These changes may yield further reductions in noise as shown in previous studies.^6,4 Therefore, a future multi-objective optimization study that simultaneously considers rotation rate, collective pitch, and airfoil geometry may yield further improvements.

Figure 10.

Ideally twisted rotor TBL-TE sound spectra for the baseline and optimized blades compared with experimental data. ³⁹

The radial distribution of OASPL for the quietest designs obtained using each parameterization method is illustrated in Figure 11. A noticeable discontinuity at $r/R=0.7$ arises from the use of piecewise-constant airfoils without interpolation between the inboard and outboard sections. As expected, the ParFoil (a) and PARSEC (b) designs demonstrate comparable acoustic performance, while the the CST-optimized airfoil is louder at the tip. Notably, the outboard region $(r/R=0.7-1.0)$ generates lower OASPL levels than the adjacent $r/R=0.6-0.7$ segment, despite being subject to higher velocities. This observation suggests that further noise reductions may be possible if the inner portion is included in the optimization process.

Figure 11.

Radial distribution of OASPL of each blade section for the baseline and quietest designs from each parameterization method.

Boundary layer parameters, which are critical inputs for the WPS model, exhibit significant variation along the blade span. These parameters are extracted at $x/c=0.99$ for each radial location. Figure 12 presents the displacement thickness distributions on the suction and pressure sides (i.e., the upper and lower surfaces, respectively) across the blade radius. It is observed that the optimized airfoils from the PARSEC and ParFoil methods exhibit larger displacement thicknesses compared to the baseline configuration. Interestingly, the ordering of displacement thickness magnitudes among the optimized designs does not correspond with their respective OASPL trends shown in Figure 11. Based on classical expectations, ⁵ one would anticipate that increasing displacement thickness would be associated with higher OASPL values. A similar lack of correlation is noted for boundary layer thickness and momentum thickness, although these values are not presented here. These findings suggest that displacement thickness, momentum thickness, and general boundary layer growth may not be the dominant factors driving the observed noise reductions in the optimized airfoil designs. It also suggests that semi-empirical models that rely primarily on the boundary layer displacement thickness, such as the BPM model, ⁵ may yield inaccurate results.

Figure 12.

Radial distribution of boundary layer displacement thickness near the trailing edge: (a) suction side and (b) pressure side.

Consequently, the reduction in OASPL can be primarily attributed to other boundary layer characteristics—most notably the chordwise pressure gradient and the friction coefficient. Figure 13 illustrates the magnitude of the pressure gradient along the blade span, presented in terms of its absolute value, as this quantity serves as a key input in Lee’s WPS model. ¹⁰ The spanwise distribution of the skin-friction coefficient is also shown in Figure 14. Although the ParFoil (a) and PARSEC (b) designs exhibit increases in boundary layer, displacement, and momentum thicknesses, they also show significantly lower pressure gradients and friction coefficients near the trailing edge, both of which contribute substantially to the observed reductions in OASPL. Furthermore, the CST (f) design exhibits comparable reductions in pressure gradient; however, it does not show a substantial change in friction coefficient, which accounts for the comparatively smaller improvement displayed in Figure 11.

Figure 13.

Radial distribution of chordwise pressure gradient near the trailing edge: (a) suction side and (b) pressure side.

Figure 14.

Radial distribution of friction coefficient near the trailing edge: (a) suction side and (b) pressure side.

In Lee’s WPS model, ¹⁰ the wall shear stress, represented by the friction coefficient, plays a critical role in determining two key terms: the amplitude correction and the non-dimensional pressure gradient, also known as the Clauser parameter. ⁴² As wall shear stress decreases, the amplitude correction term diminishes, while the Clauser parameter increases. Except in cases where the Clauser parameter becomes exceptionally large, the overall WPS tends to decrease as wall shear stress is reduced. For a given pressure gradient, the amplitude correction effect typically dominates over the influence of the Clauser parameter. Physically, higher wall shear stress reflects intensified energy and momentum exchange in the near-wall turbulence, which generally corresponds to higher aerodynamic noise levels. The pressure gradient is another important contributor to the Clauser parameter. As the pressure gradient decreases, the WPS typically declines in parallel. While traditional airfoil studies (e.g., those involving NACA sections at varying angles of attack) often show that boundary layer thickness and pressure gradient evolve in parallel, these parameters can have a different trend. In such cases, it is generally the pressure gradient or wall shear stress that exerts the dominant influence on WPS levels, overriding the effects of boundary layer thickness. This is because pressure gradients directly affect flow unsteadiness and susceptibility to separation, both of which are crucial factors in TBL-TE noise generation.

XV-15 rotor in hover

Figure 15(a) presents a comparison of FM as a function of thrust coefficient $(C_T)$ in hover between experimental data and model predictions for both the original full-span XV-15 blade and its scaled (shortened) counterpart. The blade is shortened to reduce the tip Mach number by approximately 50% relative to the original configuration. Experimental results indicate a maximum FM of approximately 0.78 at $C_T=0.0105$ . Predictions for the full blade yield a maximum FM of about 0.80 at $C_T=0.011$ , while the shortened blade achieves a maximum FM of 0.76 at the same thrust coefficient. At higher $C_T$ values, FM decreases due to aerodynamic stall. At lower $C_T$ values, the FM predictions for the full blade align well with the experimental data. In contrast, the shortened blade tends to underpredict FM in this range, likely due to differences in tip Mach and Reynolds numbers between the scaled configuration and the experimental setup. Nonetheless, the overall agreement between predictions and measurements is reasonable.

Figure 15.

XV-15 blade results in hover: (a) FM versus thrust coefficient and (b) optimized design for a shortened blade.

Optimization is performed on the shortened blade at two outboard radial sections ( $r/R=0.7$ –0.9 and 0.9–1.0), at the operating condition corresponding to its maximum FM, which occurs at a collective pitch angle of $14.0°$ , with $C_T=0.0114$ and FM = 0.7585. The optimization employs the ParFoil method using the same parameterization settings as in the ITR study. The baseline blade predicts an OASPL(A) of 52.08 dBA. The quietest optimized design achieves an OASPL(A) of 48.61 dBA, corresponding to a 3.47 dBA noise reduction, but with a slight FM decrease of 0.0047, yielding an FM of 0.7538.

The airfoils shown in Figure 16 illustrate the baseline and optimized geometries for the $r/R=0.7$ –0.9 and 0.9–1.0 blade sections. The corresponding ParFoil parameter values are provided in Table 5, including camber scaling factor $(f_m)$ , camber crest location $(p_0)$ , thickness scaling factor $(f_t)$ , trailing-edge camber offset $(\mathrm{\Delta }n)$ , trailing-edge camber crest location $(q_0)$ , boat-tail angle offset $(\mathrm{\Delta }b)$ , and leading-edge radius scaling factor $(f_k)$ . For the definitions and formulations of these parameters, readers are referred to Liu and Lee. ¹⁹ A notable feature of the optimized airfoils is the introduction of negative camber near the trailing edge for the $r/R=0.7$ –0.9 section. This design choice aligns with findings by Liu and Lee, ¹⁹ who showed that integrating Lee’s WPS model with Howe’s TBL-TE model ²¹ predicts noise reduction with negative trailing-edge camber applied to a NACA 0012 airfoil. This approach was further validated by Kang and Lee ⁴³ using large-eddy simulations, which confirmed reductions in TBL-TE due to the introduction of negative camber. At the $r/R=0.7$ –0.9 section, the optimized airfoil becomes more symmetric, the location of maximum thickness shifts forward toward the leading edge, and the trailing-edge camber becomes negative. At $r/R=0.9$ –1.0, the optimized airfoil becomes thinner overall and features a greater magnitude of trailing-edge camber.

Figure 16.

Comparison of baseline XV-15 airfoils to optimized airfoils: (a) $r/R=0.7-0.9$ and (b) $0.9-1.0$ .

Table 5.

ParFoil parameters for quietest XV-15 optimized design with respect to corresponding baseline airfoils.

Baseline airfoil	$f_m$	$p_0$	$f_t$	$\mathrm{\Delta }n$	$q_0$	$\mathrm{\Delta }b$	$f_k$
NACA64-(1.5)12	0.6769	0.3316	1.062	−0.0025	0.6567	1.428	0.8305
NACA64-208	1.3383	0.3883	0.895	0.0042	0.8668	1.566	1.3287

A comparison of the one-third octave band sound spectra for TBL-TE is presented in Figure 17. As experimental data are not available for the XV-15 configuration, the analysis is limited to a comparison between the baseline-shortened blade and its optimized counterpart. Notably, the optimized design exhibits a substantial reduction in sound pressure level across all frequency bands. The primary mechanism driving this noise reduction is consistent with that observed for ITR—namely, the reduction in the chordwise pressure gradient. Figure 18 illustrates the spanwise distribution of the magnitude of the pressure gradient on the suction side. The optimized airfoils, particularly in the tip region, display significantly lower pressure gradients compared to the baseline airfoils, thereby contributing to the overall reduction in TBL-TE.

Figure 17.

One-third octave band sound spectra comparing the baseline rotor to the optimized rotor.

Figure 18.

Radial distribution of chordwise pressure gradient at the trailing edge on the suction side for the baseline rotor and the optimized XV-15 blade.

XV-15 rotor in vertical climb

The assessment of the XV-15 hover-optimized blades is extended to additional flight conditions—specifically vertical take-off or climb—instead of performing a separate optimization tailored to this flight. This approach is adopted because, although a tiltrotor designed for AAM is expected to spend the majority of its operational time in propeller mode (i.e., cruise), the broadband noise generated during hover and vertical climb is likely to have greater ground impact. This is due to both the directivity of TBL-TE and the proximity of the rotor to ground-based observers during these phases. In contrast, during cruise flight, the strongest broadband noise is directed normal to the rotor plane, resulting in less noise being radiated downward toward the ground. Climb freestream velocities considered in this study include 0.0 m/s (hover), 5.0 m/s, 10.0 m/s, and 20.0 m/s. In all cases, the observer location is fixed at a distance of $40R$ from the rotor center, and the rotation rate is held constant at 582 RPM. To maintain consistent thrust across different flight conditions, the collective pitch angle is adjusted. This is necessary because the effective sectional angle of attack decreases with increasing climb velocity, which in turn reduces the thrust generated. Matching thrust through collective pitch rather than modifying the rotation rate is essential, as the latter would significantly alter the Mach and Reynolds numbers, thereby affecting the noise characteristics.

Figure 19 illustrates the directivity of OASPL(A) for the baseline shortened XV-15 blade. In this representation, $90°$ corresponds to a location normal to the rotor plane (i.e., directly above the rotor during vertical climb), while $270°$ denotes a location directly below. The OASPL(A) trends reveal a modest increase in noise levels downstream of the rotor with increasing climb velocity, accompanied by a slight reduction in noise levels upstream. For instance, at the $270°$ observer location, the OASPL(A) for the $V_c=20.0$ m/s case is 1.36 dBA higher than that of the hover condition. This modest rise in radiated noise can be attributed to the increased freestream velocity encountered by the blade sections during climb. However, because the blade’s tangential velocity, driven by the rotation rate, remains significantly greater than the axial freestream velocity, the overall effect of climb velocity on noise generation is relatively limited. These results indicate that although climb velocity contributes to variations in radiated noise, the dominant source remains the blade’s rotational motion. This observation is consistent with findings by Brown et al., ⁴⁴ who predicted TBL-TE noise from a propeller–wing configuration and similarly concluded that tangential motion of propeller blades plays the primary role in broadband noise generation under comparable operating conditions.

Figure 19.

Directivity pattern of OASPL(A) for the baseline rotor at various climb velocities $(V_c)$ .

The directivity of OASPL(A) for the optimized XV-15 design compared to the baseline is shown in Figure 20 for climb velocities of 10.0 and 20.0 m/s. Across both flight conditions, the hover-optimized rotor achieves noise reductions ranging from 3 to 5 dBA, depending on the observer location. Table 6 provides detailed noise reductions at the $270°$ observer location, with values ranging from approximately 3.47 to 4.27 dBA relative to the baseline. This consistent level of noise reduction across different climb speeds can be attributed to the similar Reynolds and Mach numbers experienced by each blade section, as well as the matched thrust coefficients. These trends are corroborated by the effective angle of attack distributions shown in Figure 21, which indicate that the tip sections maintain comparable effective angles of attack across all evaluated flight conditions. As a result, the boundary layer characteristics remain relatively unchanged. Although small variations in freestream velocity may induce slight differences in edge velocity, these effects appear minimal in comparison to the dominant contributions from rotational motion. Therefore, airfoils optimized for hover conditions demonstrate robust performance in vertical climb scenarios in terms of TBL-TE noise.

Figure 20.

Noise directivity for XV-15 rotor in climb flight with $V_c=10.0$ m/s and $V_c=20$ m/s.

Table 6.

Noise reduction of the optimized XV-15 compared to the baseline rotor at the 270 $°$ observer location.

Climb velocity (m/s)	$\mathrm{\Delta }$ OASPL(A) (dB)
0.0	$-3.47$
5.0	$-3.95$
10.0	$-4.27$
20.0	$-3.85$

Figure 21.

Radial distribution of effective angle of attack for various flight conditions for the optimized blades.

Conclusions

Rotor blade airfoil optimization was conducted using a multi-objective GA in conjunction with a Kriging surrogate model to improve thrust, power, and TBL-TE noise for an ITR blade and a scaled XV-15 rotor blade. Using BEMT and UCD-QuietFly, the optimization focused on the tip sections of the blades, which are critical for both aerodynamic performance and noise generation.

Three airfoil parameterization methods—ParFoil, PARSEC, and CST method—were compared to evaluate their effectiveness in simultaneously optimizing aerodynamic and acoustic objectives. All three parameterization methods produced comparable results in terms of Pareto-optimal performance, with maximum FM increases of approximately 0.0409 and OASPL(A) reductions of up to 4.00 dBA. However, despite these similarities in overall performance trends, the individual airfoil geometries corresponding to optimal designs varied significantly. In particular, the quietest CST-optimized design exhibited an OASPL(A) approximately 0.6 dBA higher than those obtained via PARSEC and ParFoil. A detailed analysis of the airfoil shapes revealed that ParFoil and PARSEC produced geometries with trailing edge features, while CST-generated airfoils were markedly thinner and more highly cambered. These discrepancies suggest that differences in the definition of parametric bounds across methods may have restricted the exploration of a consistent design space, thereby affecting the optimization outcomes. To ensure a fair comparison and maximize the utility of each method, future studies should prioritize the careful calibration of parametric bounds. This approach would help to avoid the generation of non-physical geometries, thereby improving the robustness and validity of the optimization process.

Despite the differences in optimized airfoil geometries among the parameterization methods, all approaches effectively reduced TBL-TE noise by modifying boundary layer characteristics in a consistent manner. Specifically, the quietest designs achieved significant reductions in OASPL(A) by minimizing the chordwise pressure gradient and wall shear stress near the trailing edge. Notably, these reductions were accomplished even if the boundary layer, displacement thickness, and momentum thickness increased. This outcome underscores the dominant influence of pressure gradient and friction coefficient, rather than boundary layer thickness alone, on TBL-TE noise generation.

Furthermore, the effectiveness of the hover optimization was demonstrated in the XV-15 hover tests using the ParFoil method, where a significant reduction in the OASPL(A) by 3.47 dBA was observed. This reduction was primarily attributed to the decrease in the chordwise pressure gradient in the optimized airfoils compared to the baseline geometries. Evaluations of these optimized airfoils were conducted under climb flight conditions, demonstrating the versatility and robustness of the hover-optimized designs. In vertical climb flight, the introduction of a climb velocity component naturally reduces the effective angle of attack. To compensate for this reduction and maintain performance, an increase in collective pitch was implemented instead of adjusting the rotation rate. This strategic adjustment allowed the hover-optimized tip section airfoils to operate efficiently at similar Reynolds and Mach numbers across different climb velocities while sustaining the same thrust levels as in the hover case. The performance of these optimized airfoils under varied climb velocities consistently showed a decrease in OASPL(A) ranging from approximately 3.47–4.27 dBA when compared to the baseline XV-15 blade. This success indicates that optimizations aimed at reducing specific aerodynamic parameters, such as the chordwise pressure gradient during hover, can yield beneficial outcomes that extend beyond the initial conditions.

While this study has made significant strides in utilizing low-fidelity tools for the acoustic evaluation of optimized airfoils, further research is imperative to fully ascertain and enhance the effectiveness of these methodologies. Although the theoretical underpinnings, such as the impact of reducing the pressure gradient on minimizing WPS, are well understood and supported by prior studies, the current aerodynamic and acoustic models employed are still categorized as low-fidelity. To bridge the gap between theoretical predictions and practical applications, future investigations should incorporate high-fidelity simulations, such as large eddy simulations, or experimental approaches to directly measure WPS and associated noise. These advanced methodologies would provide a more detailed and accurate validation of the optimization techniques developed, confirming their utility and effectiveness in real-world scenarios.