Conventional time-domain algorithms that solve the augmented Burgers equation describing nonlinear propagation in a relaxing fluid have a high computational cost associated with discretizing thin shocks. The Burgers equation can be expressed using intrinsic coordinates [Hammerton and Crighton, J. Fluid Mech. 252, 585–599 (1993)], in which waveforms remain single valued beyond the point at which waveform steepening renders them multivalued in physical coordinates. Presented here is a fast simulation approach based on intrinsic coordinates with shocks inserted into the multivalued pressure waveform using the equal area rule from weak shock theory, yielding a single-valued waveform in physical coordinates without requiring fine discretization of the shock. A two-stage approach is developed by first solving the modified Burgers equation in intrinsic coordinates out to distances where shocks are no longer expected to be thin. Beyond that distance, the solution is propagated using a time-domain algorithm in physical coordinates without restrictively small discretization in time. The two-stage approach achieves high accuracy and significantly reduced computational cost when compared to time-domain numerical solutions using exclusively physical coordinates. Computational efficiency and accuracy are demonstrated with simulations of N waves propagating in air and shocks with exponential tails in seawater, each involving two relaxation mechanisms.

Numerical simulations of intense sound fields are employed in research involving therapeutic ultrasound, underwater acoustics, and air acoustics. Nonlinear evolution equations such as the Burgers and Khokhlov-Zabolotskaya-Kuznetsov equations generalized for biological tissue and relaxing media, and the one-way Westervelt equation, have been used for simulating strongly nonlinear sound fields with shocks. Marching schemes based on operator splitting enable modeling of different wave phenomena, e.g., nonlinearity, diffraction, attenuation, and dispersion, using the most effective algorithm for each effect. Simulation of strong nonlinear propagation effects leading to formation of steep shocks has been a challenging problem. Time-domain modeling has proven to be most effective because the number of operations over each propagation step is proportional to the number of grid points in the time window, whereas frequency-domain schemes require the number of harmonics squared. This talk will describe and compare specific features, accuracy, and effectiveness of various time-domain algorithms that have been developed over the years to simulate nonlinear effects in shock-wave propagation regimes. Algorithms based on the exact solution of the lossless Burgers equation with an interpolation procedure, an intrinsic coordinates algorithm, conventional conservative finite-difference schemes, and the Godunov-type shock-capturing method will be discussed. [Work partially supported by RSF No. 25-12-00157.]

Nonlinear elastic metamaterials (NLEMs) support a variety of dynamic phenomena that enable the manipulation of large-deformation elastic waves. Full-scale dynamic simulation of NLEMs is often prohibitively expensive due to the importance of complex, sub-wavelength geometry. Low-order effective medium models based on mass-spring lattices [Wallen et al., arXiv:2407.20434 (2024)] have recently been developed to capture history-dependent effects of plasticity for 1-D simulation of nonlinear wave propagation in NLEMs. However, the model developed therein requires significant preparatory effort to obtain empirical constitutive relations and their derivatives via a complex, ad-hoc curve-fitting procedure. Here, an alternative method is proposed whereby trained deep neural networks provide the constitutive relations, allowing for application of automatic differentiation methods to obtain derivatives for implicit solution of the differential-algebraicequations of motion. The networks are trained using cyclical force–displacement data from a finite-element model of a unit cell of interest, which exhibits buckling under elasto-plastic deformation. The trained neural networks are then incorporated into the discrete-element framework of Wallen et al. to simulate wave propagation in a chain of unit cells.

Discretization of thin shocks requires high computational cost for conventional time-domain algorithms that solve the augmented Burgers equation describing nonlinear propagation in a relaxing fluid. The Burgers equation can be expressed using intrinsic coordinates [Hammerton and Crighton, J. Fluid Mech. 252, 585–599 (1993)], in which waveforms remain single valued beyond the point at which waveform steepening renders them multivalued in physical coordinates. This coordinate transformation underlies a simulation approach in which shocks are inserted into the multivalued pressure waveform using the equal area rule from weak shock theory, yielding a single-valued waveform in physical coordinates without requiring fine discretization of the shock. A two-stage approach is developed by first solving the Burgers equation in intrinsic coordinates out to distances where shocks are no longer expected to be thin. Beyond that distance, the solution can be propagated in physical coordinates using a time-domain algorithm without restrictively small discretization. The two-stage approach significantly reduces computational cost compared to time-domain numerical solutions using exclusively physical coordinates and maintains high accuracy. This advantage is demonstrated with simulations of N waves propagating in air and shocks with exponential tails in seawater, each involving two relaxation mechanisms [Willis et al., JASA 157, 3824–3835 (2025)].

The augmented Burgers equation that describes nonlinear propagation in a relaxing fluid can be expressed using intrinsic coordinates [Hammerton and Crighton, JFM 252, 585 (1993)]. A multivalued pressure waveform in physical coordinates can be represented as single-valued with intrinsic coordinates. Shocks can then be inserted into the multivalued pressure waveform using the equal area rule from weak shock theory to obtain a single-valued waveform solution, thus avoiding the high computational cost associated with discretizing thin shocks with conventional algorithms. By first solving the evolution equation in intrinsic coordinates, and then using the solution as the input to a conventional algorithm based on the augmented Burgers equation, a two-stage approach can be developed that achieves both high accuracy and significantly reduced computational cost when compared to solving the Burgers equation with conventional numerical algorithms alone. Approaches based on intrinsic coordinates are applied to realistic problems in relaxing media, which include air and seawater, to allow for comparisons of computational efficiency and accuracy. [WAW is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

High-speed schlieren images of the field surrounding a supersonic jet provide a rich database for testing convolutional neural network (CNN) classification methods due to the presence of many waveform structures of interest. One such structure is waveform coalescence, where waves intersect at small angles, which can lead to increased steepening and nonlinear distortion in the jet near field [Willis et al., AIAA Journal (2023)]. Although waves exhibiting coalescence behavior have been identified in narrow field-of-view (FOV) schlieren images, new methods are needed for large FOV images to improve computational efficiency. This presentation will explore methods to track and classify waves observed in large FOV schlieren images as coalescing or noncoalescing using CNNs. Using transfer learning, pretrained networks can be retrained for this problem, with training data obtained from narrow FOV schlieren or from two-dimensional simulated waveforms using the Khokhlov–Zabolotskaya–Kuznetzov (KZK) equation. The impact of model hyperparameters, choice of pretrained network, image scaling, and other factors on the results of waveform classification by the CNN will be explored. [WAW is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

View Video Presentation: https://doi.org/10.2514/6.2023-0021.vid The coalescence of intersecting Mach waves has been proposed as a significant contributor to acoustic waveform steepening in the near field of a jet, thus providing a potential cause for the observation of more steepened waves in a laboratory-scale jet than predicted by effective Gol'dberg numbers [Baars et al., J. Fluid Mechanics 749 (2014); Fiévet et al., AIAA Journal 54, 254 (2016)]. Recent numerical simulations have demonstrated that the coalescence process can lead to increased steepening [Willis et al., AIAA SciTech Forum (2022)]. Schlieren imaging of a laboratory-scale, Mach 3 jet flow has been used for comparison with simulations based on reduced-order models, but additional examples of coalescing waves were desired that did not depend on turbulence for wave generation. Thus, two experiments have been designed using spark generated waveforms that intersect, either after reflection from a rigid surface or after emanating from a 3D-printed enclosure. Schlieren images and microphone measurements allow for analysis of these waveforms to examine steepening behavior. Additionally, large-eddy simulation (LES) of the same Mach 3 jet flow presents an opportunity to compare behavior of intersecting Mach waves with prior simulations and experimental results. A machine learning algorithm has been trained using transfer learning and then applied to the LES pressure data to identify waveforms of interest for further analysis of coalescence.

The intersection of Mach waves in the near field of a jet can lead to a phenomenon referred to as coalescence, which is believed to increase acoustic waveform steepening more than predicted by effective Gol'dberg numbers [Baars et al., J. Fluid Mechanics 749, 331 (2014); Fiévet et al., AIAA Journal 54, 254 (2016)]. Recent studies have constructed algorithms to identify coalescing waveforms in narrow field-of-view (FOV) schlieren images, then compared the detected coalescence events to simulations by using reduced-order models [Willis et al., AIAA Journal 61, 2022 (2023)]. Large-FOV images can capture a larger region of the sound field but require a decrease in the imaging frame rate that decreases the effectiveness of previous coalescence-detection algorithms. A new method for coalescence detection is thus desired. Convolutional neural networks are trained using transfer learning and then applied to large-FOV schlieren intensity data to identify waveforms of interest for further analysis of coalescence. Two approaches are compared for network training, one using pressure data and the other using pressure gradient data, both simulated using the Khokhlov-Zabolotskaya-Kuznetzov (KZK) equation. Interacting waves classified as coalescing are examined both in single image frames and in translating coordinates that follow the waves as they propagate.

Prior measurements of the sound field produced by a laboratory-scale, Mach 3 jet flow (Baars and Tinney, Journal of Sound and Vibration, Vol. 333, No. 12, 2014, pp. 2539–2553; Fiévet et al., AIAA Journal, Vol. 54, No. 1, 2016, pp. 254–265) suggest that acoustic waveforms steepen early on in their development. This explained the discrepancy between theoretical predictions, based on effective Gol’dberg numbers, that shocks should not form, and observations of steepened Mach waves close to laboratory-scale jets. The present work continues studying this phenomenon by exploring coalescence processes that occur when neighboring waveforms intersect, forming larger-amplitude waveforms with increased cumulative nonlinear distortion. A numerical model based on the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation is developed to show that coalescence-induced steepening is sensitive to the intersection angle between adjacent waveforms, waveform duration, and cylindrical spreading effects. High frame-rate schlieren images of sound waves propagating from the post-potential core region of a laboratory-scale Mach 3 jet are then captured along an angle following the ridge of most intense noise to study the development and evolution of coalescence. A shock detection algorithm isolates shock-like events, which are tracked using a translating coordinate system and decomposed using proper orthogonal decomposition. Reduced-order reconstructions of both schlieren images and the KZK model identify common patterns that characterize the shock coalescence process.

Mach waves generated by turbulent structures in supersonic jet flows have the potential to intersect and coalesce, a process proposed as a significant contributor to acoustic waveform steepening in the near field of a jet and to the noise referred to as crackle [Baars et al., AIAA (2013); Fiévet et al., AIAA (2016)]. Numerical simulations of intersecting waveforms have demonstrated that coalescence can lead to increased steepening that is dependent on intersection angle, waveform duration, and geometrical spreading [Willis et al., AIAA (2022)]. In this study, two simplified experiments involving a spark source in air are examined. The first involves grazing incidence on a rigid plane to model symmetric intersection between waveforms. The second involves intersecting waveforms emanating from side-by-side openings in a 3D-printed enclosure. Measurements of nonlinear evolution were made for the intersecting waveforms produced with each experimental setup and compared with measurements made for one waveform alone. Schlieren images of the coalescing waves allow for comparison of waveform steepening between the two cases without the limitations imposed by microphones. The measurements permit assessment of the extent to which coalescence enhances waveform steepening. [WAW is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

Prior measurements of the sound field produced by a laboratory-scale, Mach 3 jet flow (Baars et al. 2013; Fiévet et al. 2016) suggest that acoustic waves steepen early on in their development. This explained the discrepancy between the theoretical prediction, based on expressions for effective Gol’dberg numbers, that shocks should not form in most laboratory scale facilities, and the apparent observation of steepened Mach waves close to laboratory-scale jets. The present work serves to continue our understanding of this phenomenon by exploring the coalescence process that occurs when neighboring waveforms intersect to form large amplitude waveforms capable of undergoing cumulative nonlinear distortion. A numerical model based on the Khokhlov–Zabolotskaya–Kuznetsov equation is first developed to show that coalescence-induced steepening is most sensitive to the intersection angle between adjacent waveforms, while increasing waveform duration decreases steepening overall. The model is expanded to include cylindrical spreading effects, which are pronounced within the peripheries of axisymmetric jets. High frame-rate schlieren images of sound waves propagating from the post potential core region of a laboratory scale Mach 3 jet are then captured along an angle following the ridge of most intense noise in order to study the development and evolution of coalescence. A new shock detection algorithm isolates shock-like features in the images at both upstream and downstream points along the propagation path. The isolated events are tracked using a translating coordinate system and decomposed using a Lagrangian form of the proper orthogonal decomposition. Reduced-order reconstructions of both the schlieren images and the KZK model identify common patterns that characterize the shock coalescence process as it leads to cumulative nonlinear waveform distortions

Prior acoustic measurements of the sound field produced by a laboratory-scale, Mach 3 jet flow [Baars et al., AIAA (2013); Fiévet et al., AIAA (2016)] showed a discrepancy between the theoretical prediction that shocks should not form, based on effective Gol’dberg numbers, and the apparent observation of steepened waves. This suggests an additional mechanism relating to wave coalescence is responsible for increased wave steepening. To better understand this new mechanism, high framerate schlieren images of sound waves propagating from the post potential core of the same Mach 3 jet are studied. A shock detection algorithm is developed to isolate shock-like features in the images and along the propagation path, which are then used to determine occurrences of coalescence-induced steepening in the flow. Numerical models of coalescing waves using the Khokhlov–Zabolotskaya–Kuznetzov (KZK) equation are then leveraged to determine identifiable characteristics of coalescence events to improve detection. POD-based reduced-order representations of the isolated events are then studied in a Lagrangian frame. Ensemble averages of tracked events are used to identify common patterns for coalescence, which is believed to contribute to “crackle” noise in full-scale jet flows. [WAW is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

An efficient two-stage algorithm is presented for simulating a weak shock in an arbitrary waveform propagating in a fluid with multiple relaxation mechanisms. The first stage of the algorithm is based on an evolution equation expressed in intrinsic coordinates for nonlinear propagation in a relaxing fluid [Hammerton and Crighton, JFM (1993)], and it includes effects of spherical spreading. Simulation using intrinsic coordinates allows the pressure waveform to become multivalued, avoiding the need to discretize thin shocks. At selected distances, a shock is inserted into the multivalued pressure waveform according to the equal-area rule, rendering the waveform single-valued. Waveforms calculated in this way agree with corresponding time-domain solutions of a Burgers equation augmented to include relaxation, while simulation with intrinsic coordinates requires orders of magnitude less computation time. An analytical solution for shock evolution [Crighton and Scott, Philos. Trans. R. Soc. A (1979)] is used to estimate shock thickness as a function of propagation distance for determining the appropriate distance at which the second stage, the time-domain solution of the augmented Burgers equation, can be used to continue propagation into the far field. [WAW is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

An algorithm based on intrinsic coordinates is presented for calculating the propagation of a weak shock starting with an arbitrary initial waveform in a fluid with multiple relaxation mechanisms. Use of intrinsic coordinates avoids discretization of thin shocks, allowing for more efficient simulation.

Nonlinear propagation of sound from underwater explosions has been treated empirically and analytically since the work of Cole [Underwater Explosions (1948)]. A semi-analytical solution for propagation of a weak shock followed initially by an exponential tail was obtained by Rogers [J. Acoust. Sci. Am. (1977)] under the assumption of no relaxation. Here, an efficient algorithm is presented for calculating the propagation of a weak shock followed by a tail with arbitrary waveform in a fluid with multiple relaxation mechanisms. The model is based on an evolution equation in intrinsic coordinates for nonlinear propagation in a relaxing fluid [Hammerton and Crighton, JFM (1993)]. Intrinsic coordinates permit the waveform to become multivalued, which avoids having to discretize thin shocks. At distances where output is desired, a shock is inserted according to Landau's equal-area rule, rendering the waveform single-valued. Waveforms calculated in this way agree with numerical solutions of a Burgers equation augmented to include relaxation, with the proposed method requiring less computation time. When an exponential tail following the shock decays faster than the shortest relaxation time, explicit solutions are obtained for the amplitude and arrival time of the shock. [W.A.W. is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]