Acoustic Radiation in Low Mach Number Flows

The research focuses on developing a boundary element approach to model acoustical radiation within non-uniform, low Mach number flows. The central problem is determining the acoustical field generated by a vibrating or reflecting body immersed in a mean flow. This is a simplified representation of more complex aeroacoustic scenarios, but incorporates key features such as the interaction between the sound field and the perturbed mean flow, interaction with vibrating and reflecting boundaries, and radiation into an unbounded domain. The model assumes a steady, isentropic, irrotational mean flow with a low Mach number. A critical aspect is the ratio of characteristic length scales for acoustic disturbances and the mean flow, influencing the validity of the proposed methods. The boundary (S) is considered to oscillate with a normal displacement, generating an acoustic disturbance propagating outwards. The 'radiation condition' assumes no reflected waves from infinity. The study aims to accurately predict the velocity potential due to this vibration, accounting for complex flow interactions.

A key innovation is the use of a transformation, valid at low Mach numbers for short-wavelength disturbances, that converts the complex flow problem into a simpler, analogous no-flow problem for the same geometry. This transformation accounts for non-uniform mean flow effects and is applicable to any irrotational mean flow where the velocity potential can be defined. This simplification allows for the application of the boundary element method (BEM), a computationally efficient numerical technique. The transformation significantly reduces the complexity of the governing acoustic equation, facilitating the use of established solution techniques typically used for no-flow problems. The accuracy and limitations of the low Mach number approximation are discussed, emphasizing the importance of considering the characteristic length scales of both the acoustic disturbance and the mean flow. The use of this transformation is central to the efficiency and applicability of the BEM to aeroacoustic problems.

The study employs two distinct boundary integral schemes within the BEM framework. The first is an overdetermined combined surface-interior formulation, while the second is a combined surface-surface derivative formulation. Both are used to compute the velocity potential stemming from the vibration of an arbitrarily shaped body within a uniform mean flow. The BEM's effectiveness is demonstrated with specific test cases involving pulsating and juddering spheres in low Mach number flows, comparing favorably against both analytical solutions and results obtained via alternative numerical schemes. The choice to use a BEM stems from its efficiency in handling the transformed no-flow problem, particularly for complex geometries where analytical solutions are unavailable. The method focuses on solving for the velocity potential on the surface of the vibrating body.

The study presents results for the test cases of pulsating and juddering spheres in low Mach number flows. These results are compared to analytical solutions and those obtained from an alternative numerical method (likely a Finite Element Method). The good agreement between the results obtained by the proposed boundary element formulations and those from the analytic solution and an alternative numerical scheme establishes the accuracy of the proposed methodology. This validation step is crucial for demonstrating the reliability and effectiveness of the new BEM formulation in solving aeroacoustic radiation problems in low Mach number flows. Specific details on the accuracy and divergence from analytical solutions at higher frequencies are also presented, highlighting the applicability limits.

The solution of the problem has significant implications for analyzing sound fields generated by aircraft. The abstract highlights that the proposed method is particularly relevant for predicting aircraft noise, including propeller-generated noise, noise from flow-fuselage interactions, and fan noise from the nacelle inlet of turbofan aircraft engines. These aeroacoustic problems typically involve radiating bodies with dimensions significantly larger than the acoustic wavelengths involved. The study emphasizes that the method provides a relatively simple and computationally efficient formulation for acoustic radiation predictions in non-uniform flows— a significant improvement over other existing methods which are computationally expensive or limited in scope. The methodology proposed addresses a long-standing challenge in aeroacoustics.

The research utilizes a low Mach number approximation to simplify the governing acoustic equation. This approximation involves discarding higher-order terms in the equation, resulting in a simpler, more readily solvable form. However, the text cautions that this simplification is not always strictly valid. The magnitudes of the various terms in the equation depend not only on the Mach number but also on the non-dimensional ratio of length scales associated with the mean flow and the acoustic disturbance. Therefore, careful consideration is necessary to ensure that discarded terms are indeed negligible compared to the retained terms. The validity of this approximation is directly tied to the relationship between these characteristic length scales and the frequency of the acoustic disturbances. The approximation's applicability is restricted to high-frequency limits and/or problems with a large geometric length scale.

A crucial aspect of the methodology is the transformation of the acoustic equation. This transformation converts the problem of acoustic propagation within a mean flow into an analogous no-flow problem. In the transformed space, the ordinary wave equation becomes applicable. This transformation is advantageous because there is a unique solution to the external radiation problem in the transformed domain. The existence of this unique solution is explicitly referenced, and a substantial number of established solution techniques become applicable to this simplified problem. The transformation includes the effects of non-uniform mean flow and can be applied to any irrotational mean flow for which the velocity potential can be defined. This transformation is key to making the boundary element method efficient for complex geometries.

The validity of both the low Mach number approximation and the transformation is highly dependent on the characteristic length scales of the problem. The geometric length scale is typically intuitive, but defining a characteristic length scale for the acoustic disturbance is less straightforward, especially at low frequencies (long wavelengths). The paper discusses the concept of an 'acoustically compact' reflecting body, where the reflected wavelength is large compared to the body's dimensions. This raises the question of whether the characteristic length scale should be the wavelength itself or some other geometric measure. This detailed consideration of length scales highlights the importance of selecting appropriate approximation techniques for the specific acoustic problem being studied. The interplay between the high frequency (short wavelength) approximation and the low Mach number regime dictates the conditions under which this approach is valid.

The research utilizes the Combined Helmholtz Integral Equation Formulation (CHIEF) method, a combined surface integral and exterior integral scheme. This method employs an iterative overdetermination procedure to solve for the acoustic potential. While the CHIEF method offers potentially reliable solutions, it is noted that it's not without disadvantages. The accuracy of the CHIEF method is dependent on the careful placement of interior points. To fully validate the CHIEF method, it must be implemented as part of a less efficient residual least-squares procedure. The method's performance, particularly its accuracy relative to the Burton-Miller formulation, is discussed based on comparisons to analytical solutions and alternative numerical methods. The CHIEF method appears to produce more accurate results for the chosen interior points in this study, but this advantage comes at the cost of higher computational expense.

The study also employs the Burton-Miller Formulation (BMF), a method that linearly combines the surface integral equation and its normal derivative with respect to a field point. This approach addresses the issue of non-uniqueness at critical wavenumbers, where the two implicit methods used have only one solution in common. The BMF method, further modified by Meyer et al., is shown to consistently deliver good results for the considered wavenumbers. A key parameter in the BMF is the coupling constant (α). The study explores the impact of different values of α (α = i and α = 1), revealing a trade-off between accuracy and computational cost. The BMF (α = i) and BMF (α = 1) methods are compared with each other and to the CHIEF method, demonstrating their individual strengths and weaknesses in terms of accuracy and computational efficiency. The choice of the coupling constant α impacts the weighting of different terms in the equation, especially at higher wavenumbers. The BMF method, while potentially less accurate than CHIEF in this specific application, offers a significant computational advantage.

The paper compares and contrasts the CHIEF and Burton-Miller Formulation (BMF) methods. While the CHIEF method demonstrates higher accuracy in specific test cases, particularly for carefully chosen interior points, it involves a more computationally intensive residual least-squares procedure for validation. In contrast, the BMF method, while potentially less accurate than CHIEF in some instances, is computationally more efficient and does not require the same level of careful positioning of interior points. The selection of the optimal method depends on the desired balance between accuracy and computational resources. The study concludes that both methods offer reliable solutions for the considered problem, but the Burton-Miller formulation with α = i shows consistently good results and greater computational efficiency for the studied wavenumbers. The performance of each method is assessed based on comparing results to the analytical solutions and findings of other researchers.

The numerical implementation of the boundary element method (BEM) involves discretizing the boundary of the three-dimensional body into triangular elements. This approach allows for a close approximation of arbitrarily shaped surfaces. The choice of element type (constant, linear, or higher-order) impacts the accuracy and computational cost. The paper notes that while higher-order elements are generally preferred, computational difficulties can arise when using linear elements, particularly regarding the precise definition of normals at nodal points located on vertices or edges. The use of triangular elements is a common and efficient technique for representing complex geometries in BEM simulations. The accuracy of the solution depends on both the size and shape of the elements used in the discretization of the boundary.

Accurate numerical integration is crucial for the BEM, especially near singularities where the integrands become unbounded. The study explores various numerical integration schemes developed by Silvester, Irons, and Cowper. Silvester's quadrature formulas are simple and symmetric but relatively inefficient compared to Gaussian formulas. Irons' conical product formulas, while highly efficient, lack the symmetry of Silvester's approach. Cowper's formulas, which are of the Gaussian type and fully symmetric, are chosen for this work because they offer a good balance between accuracy and efficiency. The paper compares the performance of Cowper's 12-point 6th-degree rule to Irons' 16-point 7th-degree rule, showing that they produce comparable results. The selection of the integration scheme significantly impacts the accuracy and computational time of the BEM simulation, especially in the presence of singularities in the integrands.

The presence of singularities in the integrands presents a significant challenge in numerical integration. The paper notes that a better approximation can be obtained by increasing the number of integration points around the singularity. The use of two separate integration schemes is proposed— one for regular integrals and a higher-order scheme for elements containing singularities. While a specialized 42-point integration scheme for handling singularities is mentioned, the results suggest that it's unnecessary for the specific formulations used in this research. The choice to use different integration schemes based on the presence of singularities demonstrates a strategy of optimizing accuracy and computational time by tailoring the integration scheme to the characteristics of the integration region. The study highlights the practical considerations in the selection of an integration scheme.

The accuracy of the proposed boundary element method is evaluated using two canonical test cases: pulsating and juddering spheres in low Mach number flows. These simple geometries allow for comparison against analytical solutions, providing a benchmark for assessing the method's accuracy. The results obtained using the boundary element formulations demonstrate good agreement with the analytical solutions and an alternative numerical method, indicating the reliability of the proposed technique. The choice of these test cases allows for a thorough assessment of the method's accuracy without the complexities of more intricate geometries. The agreement extends to a range of frequencies within the low Mach number regime, thus bolstering confidence in the BEM's ability to accurately predict acoustic radiation in more complex situations.

Computed results from the boundary element method (BEM) are rigorously compared against both analytical solutions and results from an alternative numerical scheme, likely a finite element method (FEM). This comparative analysis serves as a critical validation step. The comparison reveals a strong correlation between the BEM results and both the analytical and FEM solutions for the test cases. For example, the CHIEF method closely matches the finite element wave envelope solution for a pulsating sphere. The level of agreement is assessed quantitatively. This validation confirms the accuracy and efficacy of the proposed BEM formulations for solving aeroacoustic radiation problems within the specified flow conditions and frequency ranges. Any discrepancies observed are analyzed to identify limitations of the approach or potential sources of error in the simulations.

While the BEM shows good agreement with analytical solutions and finite element methods, the study acknowledges limitations. For the pulsating and juddering sphere test cases, the values of M(ka) (a non-dimensional frequency parameter) are given as 0.93 and 1.35, respectively. These values are beyond the small parameter limit required for the validity of the analytical solution. The study notes significant divergence from the analytic solution at higher M(ka) values, highlighting the limits of the low Mach number approximation. Nevertheless, the comparison to the alternative numerical scheme (FEM) strongly indicates the BEM's continued accuracy in the higher frequency range. This demonstrates that the BEM’s validity extends beyond the limitations of the analytical solution, offering a useful tool for aeroacoustic predictions in a broader range of conditions.

The study concludes that a transformed boundary element scheme offers a simple and computationally inexpensive formulation for predicting acoustical radiation in non-uniform, low Mach number flows. The analysis and results demonstrate the validity and accuracy of this approach. Comparisons with analytical solutions (where available) and an alternative numerical (finite element) scheme confirm the reliability of the proposed method. The two boundary integral formulations considered – the CHIEF and Burton-Miller methods – both prove effective, with the CHIEF method showing slightly higher accuracy for the specific test cases and interior point selection used. However, the Burton-Miller formulation (with a coupling constant of i) offers a good balance between accuracy and computational efficiency, making it a practical choice for many applications. The research successfully validates a novel approach to a computationally challenging problem in aeroacoustics.

The CHIEF method, while potentially more accurate for the chosen interior points, necessitates a less efficient residual least-squares procedure for validation. In contrast, the Burton-Miller formulation provides consistently good results without relying on the precise positioning of interior points. The computational expense of the methods is discussed, with the Burton-Miller formulation highlighted as being significantly less expensive than the residual least-squares procedure needed to support the CHIEF method. The choice between these methods represents a trade-off between accuracy and computational efficiency, and the research provides guidance on when each method might be preferred depending on the priorities of a particular simulation. The study thoroughly examines the strengths and weaknesses of both methods.

While the study demonstrates the success of the transformed boundary element scheme for low Mach number flows, future research could explore its applicability to higher Mach numbers and more complex flow geometries. Extending the methodology to more realistic scenarios, such as those involving complex shapes and higher flow velocities, would broaden the practical impact of the findings. This suggests that this work provides a strong foundation for more advanced modeling techniques within computational aeroacoustics. Further investigation into the robustness of the methods under varying conditions and with different boundary element configurations would be valuable for continued development and application of the methodology. The authors implicitly recommend further research to push the boundaries of the technique.