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Publication date: Available online 17 July 2021
Source: Computers & Fluids
Author(s): Carina Nogueira Sondermann, Raphael Viggiano Neves de Freitas, Felipe Bastos de Freitas Rachid, Gustavo C.R. Bodstein
Publication date: Available online 15 July 2021
Source: Computers & Fluids
Author(s): Wagner J. Gonçalves S. Pinto, Florent Margnat
Publication date: Available online 15 July 2021
Source: Computers & Fluids
Author(s): Pratikkumar Raje, Krishnendu Sinha
Publication date: Available online 22 July 2021
Source: Computers & Fluids
Author(s): Zhipeng Qin, Amir Riaz
Publication date: 15 September 2021
Source: Computers & Fluids, Volume 227
Author(s): Gregorio Gerardo Spinelli, Bayram Celik
Publication date: 15 September 2021
Source: Computers & Fluids, Volume 227
Author(s): Lijun Hu, Li Yuan
Publication date: 15 September 2021
Source: Computers & Fluids, Volume 227
Author(s): Saleh Rezaeiravesh, Ricardo Vinuesa, Philipp Schlatter
Publication date: 15 September 2021
Source: Computers & Fluids, Volume 227
Author(s): Alexandra Metallinou Log, Svend Tollak Munkejord, Morten Hammer
Publication date: Available online 10 July 2021
Source: Computers & Fluids
Author(s): Eslam Ezzatneshan, Ali Asghar, Khosroabadi
Publication date: Available online 6 July 2021
Source: Computers & Fluids
Author(s): Xiangda Cui, Wagdi G. Habashi
The simulation of geometrically resolved rigid particles in a fluid relies on coupling algorithms to transfer momentum both ways between the particles and the fluid. In this article, the fluid flow is modeled with a parallel lattice Boltzmann method using adaptive grid refinement to improve numerical efficiency. The coupling with the particles is realized with the momentum exchange method. When implemented in plain form, instabilities may arise in the coupling when the particles are lighter than the fluid. The algorithm can be stabilized with a virtual mass correction specifically developed for the lattice Boltzmann method. The method is analyzed for a wide set of physically relevant regimes, varying independently the body-to-fluid density ratio and the relative magnitude of inertial and viscous effects. These studies of a single rising particle exhibit periodic regimes of particle motion as well as chaotic behavior, as previously reported in the literature. The new coupled lattice Boltzmann method is compared with available experimental and numerical results. This serves to validate the presented method and additionally it leads to new physical insight for some of the parameter settings.
The Gay-Lussac (GL) approach is an incompressible-based strategy for non-Boussinesq treatment of the governing equations for free convection problems that is established based on extending the density variations beyond the gravity term. Such a strategy leads to emerging the GL parameter as a non-Boussinesq pre-factor of different terms in the governing equations. In this paper, the GL approach is expressed/discussed in terms of the secondary variables, i.e. vorticity and stream-function, for the first time and a simplified version of this approach is proposed by removing density variations from the continuity equation. The difference of results under the simplified and traditional GL approach ranges within a maximum of 1% for different parameters. The lower computational cost of numerical solution of governing equations in the secondary variables formula and the corresponding convergence rate is scrutinized for the simplified GL approach showing around 25% lower computational cost. The performance of this approach is evaluated at high relative temperature differences against the low Mach number scheme and the Boussinesq approximations. In this respect, natural convection in an annulus cavity is numerically simulated using a CVFEM solver under the aforementioned approximations up to Rayleigh number Ra = 105 at Prandtl number Pr = 1 and high relative temperature differences (ε = 0.15 and 0.3). The largest deviations found for either the simplified GL or Boussinesq methods from the low Mach number scheme solution are less than 20% for velocity magnitude, 14% for stream function, 6% for vorticity and 5% for temperature. Results under the three approximations are also analysed in terms of the skin friction and local and average Nusselt number, indicating that the Gay-Lussac approach requires some revisions to act more accurately than the classical Boussinesq approximation at high relative temperature differences in natural convection problems, especially within the convection dominated regime.
This article is protected by copyright. All rights reserved.
This article presents a numerical scheme for two-phase flows consisting of compressible gas and incompressible liquid. Assuming that the gas density is determined by pressure only and that the liquid density is constant, we develop a new set of governing equations for compressible–incompressible two-phase flows at low Mach numbers. The governing equations are simpler than the previous ones and gives better results for one-dimensional cases and rational results for two-dimensional cases.
This article presents a numerical scheme for two-phase flows consisting of compressible gas and incompressible liquid. Assuming that the gas density is determined by pressure only and that the liquid density is constant, we develop a new set of governing equations for compressible–incompressible two-phase flows, which can be seen as a simplification of the previous unified governing equations. The volume fraction of gas, velocity, and pressure are the primary variables. Given the liquid's usual leading role in the flow motion, a pressure-based method is employed to solve the equations. A fifth-order accurate weighted essentially nonoscillatory (WENO) scheme is applied to discretize the advection terms, and the tangent of hyperbola for interface capturing (THINC) scheme is utilized to capture the interface. The numerical scheme proposed is validated against the generalized Bagnold model and the free drop of a water patch in a closed tank. The results show that the scheme can tackle low Mach number compressible–incompressible two-phase flows. For the one-dimensional Bagnold model, the present numerical model achieves first-order convergence and gives better results than that based on the previous unified governing equations with the same numerical methods. In the simulation of the free drop of a water patch in a closed tank, a similar pressure curve with other reported work is given.
In this article we present a numerical approach to compute conformal mappings that requires lower computational time than the Schwarz–Christoffel mapping. The method can be easily extended to solve time-dependent water wave problems with large topographic obstacles. Besides, we present an alternative approach to compute the inverse conformal mapping.
An iterative numerical method to compute the conformal mapping in the context of propagating water waves over uneven topographies is investigated. The map flattens the fluid domain onto a canonical strip in which computations are performed. The accuracy of the method is tested by using the MATLAB Schwarz–Christoffel toolbox mapping as a benchmark. Besides, we give a numerical alternative to compute the inverse of the conformal map.
In the standard implementation of Borgnakke-Larsen (BL) rotation and translation energy exchange in DSMC not every rotational mode of each molecule is active in every collision. In some collisions, the rotation mode of one molecule is ‘frozen’ (cannot be changed by the collision); in other collisions the rotation mode of both molecules is frozen. Different relaxation rates are obtained by varying the proportion of frozen-rotation collisions. In 1974 Larsen and Borgnakke proposed a ‘restricted exchange’ version of BL in which a portion of the energy of each mode is frozen in each collision; the method seemed physically plausible but did not satisfy the detailed balance condition. In 1978 Pullin proposed a general restricted exchange of BL exchange which did satisfy the detailed balance condition; it was more CPU intensive than the original BL scheme. It was seen as ‘laborius’ and ‘cumbersome’ and it seemed to be incompatible with vibration energy-exchange models. Here a simplified version, a special case, of Pullin's restricted exchange scheme is described. The new method is only 12-17% more CPU-intensive than standard BL methods and can be used with the standard DSMC quantum vibration exchange model. It is shown that the elimination of frozen-rotation collisions changes the high energy distribution of vibration energy in non-equilibrium conditions which may slightly decrease the non-equilibrium dissociation rate when restricted exchange is used with the Quantum-Kinetic method for dissociation reactions.
We present a new least squares based diamond scheme for anisotropic diffusion problems on polygonal meshes. This scheme introduces both cell-centered unknowns and vertex unknowns. The vertex unknowns are intermediate ones and are expressed as linear combinations with the surrounding cell-centered unknowns by a new vertex interpolation algorithm which is also derived in least squares approach. Both the new scheme and the vertex interpolation algorithm are applicable to diffusion problems with arbitrary diffusion tensors and do not depend on the location of discontinuity. Benefitting from the flexibility of least squares approach, the new scheme and vertex interpolation algorithm can also be extended to 3D cases naturally. The new scheme and vertex interpolation algorithm are linearity-preserving under given assumptions and the numerical results show that they maintain nearly optimal convergence rates for both L 2 error and H 1 error in general cases. More interesting is that a very robust performance of the new vertex interpolation algorithm on random meshes compared with the algorithm LPEW2 in the work of Gao and Wu (2011) is found from the numerical tests.
The self-propelled fish maneuvering for avoiding obstacles under intelligent control is investigated by numerical simulation. Three cases are tested to validate the novel approach, including the fish model maneuvering to avoid a single obstacle and double or multiple obstacles. The results indicate that the fish model can avoid obstacles in a complex environment under intelligent control. This work illustrates the possibility of producing navigation algorithms by DRL and brings potential applications of bionic robotic swarms in engineering.
The self-propelled fish maneuvering for avoiding obstacles under intelligent control is investigated by numerical simulation. The NACA0012 airfoil is adopted as the two-dimensional fish model. To achieve autonomous cruising of the fish model in a complex environment with obstacles, a hydrodynamics/kinematics coupling simulation method is developed with artificial intelligence (AI) control based on deep reinforcement learning (DRL). The Navier–Stokes (NS) equations in the arbitrary Lagrangian–Eulerian (ALE) framework are solved by the dual-time stepping approach, which is coupled with the kinematics equations in an implicit strong coupling way. Moreover, the moving mesh based on radial basis function and overset grid technology is taken to achieve a wide range of maneuvering. DRL is introduced into the coupling simulation platform for intelligent control of obstacle avoidance when the self-propelled fish swimming. Three cases are tested to validate the novel approach, including the fish model maneuvering to avoid a single obstacle and double or multiple obstacles. The results indicate that the fish model can avoid obstacles in a complex environment under intelligent control. This work illustrates the possibility of producing navigation algorithms by DRL and brings potential applications of bionic robotic swarms in engineering.
In this paper, we clarify reconstruction-based discretization schemes for unstructured grids and discuss their economically high-order versions, which can achieve high-order accuracy under certain conditions at little extra cost. The clarification leads to one of the most economical approaches: the flux-and-solution-reconstruction (FSR) approach, where highly economical schemes can be constructed based on an extended κ-scheme combined with economical flux reconstruction formulas, achieving up to fifth-order accuracy (sixth-order with zero dissipation) when a grid is regular. Various economical FSR schemes are presented and their formal orders of accuracy are verified by numerical experiments.
A p-multigrid spectral element method is proposed for water wave simulation. The p-multigrid method is an iterative solver designed to efficiently solve the Laplace problem involved in solving the fully nonlinear potential flow equations. Through numerical test cases it is demonstrated to work well for nonlinear and dispersive water wave simulations addressing both pure wave propagation and a more advanced case for wave-body interaction with a fixed floating production storage and offloading where experimental results are available to validate the simulations.
In marine offshore engineering, cost-efficient simulation of unsteady water waves and their nonlinear interaction with bodies are important to address a broad range of engineering applications at increasing fidelity and scale. We consider a fully nonlinear potential flow (FNPF) model discretized using a Galerkin spectral element method to serve as a basis for handling both wave propagation and wave-body interaction with high computational efficiency within a single modeling approach. We design and propose an efficient 𝒪(n)-scalable computational procedure based on geometric p-multigrid for solving the Laplace problem in the numerical scheme. The fluid volume and the geometric features of complex bodies is represented accurately using high-order polynomial basis functions and unstructured meshes with curvilinear prism elements. The new p-multigrid spectral element model can take advantage of the high-order polynomial basis and thereby avoid generating a hierarchy of geometric meshes with changing number of elements as required in geometric h-multigrid approaches. We provide numerical benchmarks for the algorithmic and numerical efficiency of the iterative geometric p-multigrid solver. Results of numerical experiments are presented for wave propagation and for wave-body interaction in an advanced case for focusing design waves interacting with a floating production storage and offloading. Our study shows, that the use of iterative geometric p-multigrid methods for the Laplace problem can significantly improve run-time efficiency of FNPF simulators.
A new class of explicit second-order schemes is proposed for solving time-dependent partial differential equations. This class of proposed schemes is constructed on three-time levels. Stability is found for the scalar two-dimensional heat equation and the system of time-dependent partial differential equations. This partial differential equations system comprises a non-dimensional set of equations obtained from the governing equations of natural convection chemically reactive fluid flow in a rectangular enclosure with thermal radiations. Flow is generated by applying the force of pressure. Graphs of streamlines, contours plots of velocity, temperature and concentration profiles, Local Nusselt number, and Local Sherwood number are displayed with the variation of time and parameters in the considered partial differential equations. Results are shown in the form of streamlines and contour plots. It is found that local Nusselt number has dual behavior by enhancing radiation parameter whereas local Sherwood number de-escalates by upraising the reaction rate parameter. It is hoped that the results in this pagination will serve as a valuable resource for future fluid-flow studies in an enclosed industrial environment.
Publication date: 15 October 2021
Source: Journal of Computational Physics, Volume 443
Author(s): Armin Shahmardi, Marco Edoardo Rosti, Outi Tammisola, Luca Brandt
Publication date: 15 October 2021
Source: Journal of Computational Physics, Volume 443
Author(s): Will Pazner, Terry Haut
Publication date: 15 October 2021
Source: Journal of Computational Physics, Volume 443
Author(s): Boxiao Zhou, Feng Qu, Di Sun, Zirui Wang, Junqiang Bai
Publication date: Available online 21 July 2021
Source: Journal of Computational Physics
Author(s): Yakun Li, Wenkai Yu, Jia Zhao, Qi Wang
Publication date: Available online 21 July 2021
Source: Journal of Computational Physics
Author(s): Miguel Uh Zapata, Reymundo Itzá Balam
Publication date: Available online 21 July 2021
Source: Journal of Computational Physics
Author(s): Joaquín López, Adolfo Esteban, Julio Hernández, Pablo Gómez, Rosendo Zamora, Claudio Zanzi, Félix Faura
Publication date: Available online 21 July 2021
Source: Journal of Computational Physics
Author(s): Yashraj Bhosale, Tejaswin Parthasarathy, Mattia Gazzola
Publication date: 15 October 2021
Source: Journal of Computational Physics, Volume 443
Author(s): Yiwei Feng, Tiegang Liu
Publication date: 15 October 2021
Source: Journal of Computational Physics, Volume 443
Author(s): E.M. Kolahdouz, A.P.S. Bhalla, L.N. Scotten, B.A. Craven, B.E. Griffith
Publication date: 15 October 2021
Source: Journal of Computational Physics, Volume 443
Author(s): Nicolas Monrolin, Franck Plouraboué
The results of the numerical study of oil recovery enhancement using nanosuspension are presented. The research was carried out using the volume of fluid method (VOF) for a 2D microporous core model. Experimentally measured values of interfacial tension (IFT) and the contact angle (CA) were used for numerical modeling. An aqueous suspension with silicon oxide nanoparticles (5 nm) is used. It was shown that when 1 wt% of nanoparticles are added to the displacing liquid, its density increases by about 1%, the viscosity increases by 10%, the IFT decreases by 10%, and the contact angle increases from 70 to 145 \(^\circ \) . The results of numerical study showed that the injection of nanoparticles has a significant effect on the displacement front movement in the microporous model. It has been shown that the nanosuspension can increase the oil recovery factor (ORF) almost twice as compared to water. To clarify the mechanisms of increasing the oil recovery during the reservoir flooding with nanofluid, a systematic study of factors affecting the displacement process efficiency was carried out. Viscosity, interfacial tension and the wetting angle of the displacing fluid were considered as such factors. As a result of systematic research, it has been shown that the main factor affecting the increase in the oil recovery flooding of nanosuspensions is a variation of wettability.
Using data from numerical simulations, we show that the lift experienced by both impulsively started and surging airfoils correlates well with the sum of the circulation of the leading-edge vortices truncated at the trailing edge. Therefore, we suggest that reasonable estimates of the lift can be obtained using only two vortex parameters, i.e., its circulation and its position. In addition to being convenient for non-intrusive estimation of forces from PIV measurements, we show that this approach can be used to derive low-order models for the analysis of vortex-lift configurations. In particular, we apply this correlation to model high-amplitude surging, which allows us to quantify the effect of wake-capture mechanisms and to determine the flow parameters that drive optimal lift.
This work establishes a procedure to accurately compute heat transfer between an Eulerian fluid and Lagrangian point-particles. Recent work has focused on accurately computing momentum transfer between fluid and particles. The coupling term for momentum involves the undisturbed fluid velocity at the particle location which is not directly accessible in the simulation. Analogously, in the context of thermal coupling, the undisturbed fluid temperature at the particle location is not directly accessible in simulations and must be estimated. In this paper, we develop a scheme to accurately estimate the undisturbed fluid temperature of a point-particle exchanging thermal energy with a surrounding fluid. The temperature disturbance is correlated with the enhanced temperature curvature in the vicinity of the particle and is formally valid in the low heating, low convection limit. We conduct extensive verification of the correction procedure for a settling particle subject to radiation. This setup allows the simultaneous testing of thermal and momentum corrections. By considering equations of drag and Nusselt number extended to finite Péclet and Boussinesq numbers, we establish a large range over which the correction procedure can be applied.
The Rossiter modes of an open cavity were studied using bi-global linear analysis, local instability analysis and nonlinear numerical simulations. Rossiter modes are normally seen only for short cavities; hence, in the study, the length over depth ratio was two. We focus on the critical region; hence, the Reynolds numbers based on cavity depth were close to 1000. We investigated the effect of the ratio boundary layer thickness to cavity depth, a parameter often overlooked in the literature. Increasing this ratio is destabilizing and increases the number of unstable Rossiter modes. Local instability analysis revealed that the hierarchy of unstable modes was governed by the mixing in the cavity opening. The effect of Mach number was also studied for thin and thick boundary layers. Compressibility had a very destabilizing effect at low Mach numbers. Analysis of the Rossiter mode eigenfunctions indicated that the acoustic feedback scaled to \(\mathrm{Ma}^3\) and explained the strong destabilizing effect of compressibility at low Mach numbers. At moderate Mach numbers, the instability either saturated with Mach number or had an irregular dependence on it. This was associated with resonances between Rossiter modes and acoustic cavity modes. The analysis explained why this irregular dependence occurred only for higher-order Rossiter modes. In this parameter region, three-dimensional modes are either stable or marginally unstable. Two-dimensional simulations were performed to evaluate how much of the nonlinear regime could be captured by the linear stability results. The instability was triggered by the \(10^{-13}\) flow solver noise floor. The simulations initially agreed with linear theory and later became nonlinearly saturated. The simulations showed that, as the flow becomes more unstable, an increasingly more complex final stage is reached. Yet, the spectra present distinct tones that are not far from linear predictions, with the thin boundary layer cases being closer to empirical predictions. The final stage, in general, was dominated by first Rossiter mode, even though the second one was the most unstable linearly. It seems this may be associated with nonlinear boundary layer thickening, which favors lower frequency in the mixing layer, or vortex pairing of the second Rossiter mode. The spectra in the final stages are well described by the mode R1 and a cascade of nonlinearly generated harmonics, with little reminiscence of the linear instability.
A modified formulation of the interfacial boundary condition for the coupling of the Stokes and Darcy models describing the incompressible fluid flow in the free space and porous medium domains is proposed using the dimension analysis procedure. The case is considered for the porous media formed by circular or square cylinders located in the centers of rectangular cells. The vorticity is derived as a linear combination of the tangential velocity components in the free space and porous medium. The proposed condition is potentially directly applicable for a class of 2D problems with an arbitrary shaped boundary for the boundary element method. The fluid flow problems are solved numerically using the Stokes flow model and analytically for the Stokes–Darcy flow model to determine the coefficients in the introduced linear dependence for the vorticity. As a result, the corresponding coefficients in the boundary condition are found as a porosity function for two types of the porous medium configuration. The approximate analytical estimation of the coefficients confirms the numerical dependencies. The verifications of the found coefficients were made by solving two 2D fluid flow problems. It is shown that the fluid flow calculated on the basis of the Stokes–Darcy flow model with modified boundary condition agrees well with the results of the microscopic Stokes flow model. The advantages of the proposed boundary condition are discussed.
Hydroelastic responses of floating elastic surfaces to incident nonlinear waves of solitary and cnoidal type are studied. There are N number of the deformable surfaces, and these are represented by thin elastic plates of variable properties and different sizes and rigidity. The coupled motion of the elastic surfaces and the fluid are solved simultaneously within the framework of linear beam theory for the structures and the nonlinear Level I Green–Naghdi theory for the fluid. The water surface elevation, deformations of the elastic surfaces, velocity and pressure fields, wave reflection and transmission coefficients are calculated and presented. Results of the model are compared with existing laboratory measurements and other numerical solutions. In the absence of any restriction on the nonlinearity of the wave field, number of surfaces, their sizes and rigidities, a wide range of wave–structure conditions are considered. It is found that wave reflection from an elastic surface changes significantly with the rigidity, and the highest reflection is observed when the plate is rigid (not elastic). It is also found that due to the wave–structure interaction, local wave fields with different length and celerity are formed under the plates. In the case of multiple floating surfaces, it is observed that the spacing between plates has more significant effect on the wave field than their lengths. Also, the presence of relatively smaller floating plates upwave modifies remarkably the deformation and response of the downwave floating surface.
In this paper, the mixing and combustion at low-heat release in a turbulent mixing layer are studied numerically using large eddy simulation. The primary aim of this paper is to successfully replicate the flow physics observed in experiments of low-heat release reacting mixing layers, where a duty cycle of hot structures and cool braid regions was observed. The nature of the imposed inflow condition shows a dramatic influence on the mechanisms governing entrainment, and mixing, in the shear layer. An inflow condition perturbed by Gaussian white noise produces a shear layer which entrains fluid through a nibbling mechanism, which has a marching scalar probability density function where the most probable scalar value varies across the layer, and where the mean-temperature rise is substantially over-predicted. A more sophisticated inflow condition produced by a recycling and rescaling method results in a shear layer which entrains fluid through an engulfment mechanism, which has a non-marching scalar probability density function where a preferred scalar concentration is present across the thickness of the layer, and where the mean-temperature rise is predicted to a good degree of accuracy. The latter simulation type replicates all of the flow physics observed in the experiment. Extensive testing of subgrid-scale models, and simple combustion models, shows that the WALE model coupled with the Steady Laminar Flamelet model produces reliable predictions of mixing layer diffusion flames undergoing with fast chemistry.
This paper presents a method to evaluate the near-singular line integrals that solve elliptic boundary value problems in planar and axisymmetric geometries. The integrals are near-singular for target points not on, but near the boundary, and standard quadratures lose accuracy as the distance d to the boundary decreases. The method is based on Taylor series approximations of the integrands that capture the near-singular behaviour and can be integrated in closed form. It amounts to applying the trapezoid rule with meshsize h, and adding a correction for each of the basis functions in the Taylor series. The corrections are computed at a cost of \(O(n_w)\) per target point, where typically, \(n_w\) =10–40. Any desired order of accuracy can be achieved using the appropriate number of terms in the Taylor series expansions. Two explicit versions of order \(O(h^2)\) and \(O(h^3)\) are listed, with errors that decrease as \(d\rightarrow 0\) . The method is applied to compute planar potential flow past a plate and past two cylinders, as well as long-time vortex sheet separation in flow past an inclined plate. These flows illustrate the significant difficulties introduced by inaccurate evaluation of the near-singular integrals and their resolution by the proposed method. The corrected results converge at the analytically predicted rates.
Streamwise vortices and the associated streaks evolve in boundary layers over flat or concave surfaces due to disturbances initiated upstream or triggered by the wall surface. Following the transient growth phase, the fully developed vortex structures become susceptible to inviscid secondary instabilities resulting in early transition to turbulence via ‘bursting’ processes. In high-speed boundary layers, more complications arise due to compressibility and thermal effects, which become more significant for higher Mach numbers. In this paper, we study Görtler vortices developing in high-speed boundary layers using the boundary region equations (BRE) formalism, which we solve using an efficient numerical algorithm. Streaks are excited using a small transpiration velocity at the wall. Our BRE-based algorithm is found to be superior to direct numerical simulation (DNS) and ad hoc nonlinear parabolized stability equation (PSE) models. BRE solutions are less computationally costly than a full DNS and have a more rigorous theoretical foundation than PSE-based models. For example, the full development of a Görtler vortex system in high-speed boundary layers can be predicted in a matter of minutes using a single processor via the BRE approach. This substantial reduction in calculation time is one of the major achievements of this work. We show, among other things, that it allows investigation into feedback control in reasonable total computational times. We investigate the development of the Görtler vortex system via the BRE solution with feedback control parametrically at various freestream Mach numbers \(M_\infty \) and spanwise separations \(\lambda \) of the inflow disturbances.
Mack (1977) criticized methods referring to a single frequency perturbation for correlation of transition prediction because the external disturbance source (like free stream turbulence) should have a broadband spectrum. Delta-correlated perturbations are characterized by the mean square of physical amplitude, which is expressed as a double integral of the power spectral density in frequency and the spanwise wave number. It is suggested to evaluate this integral asymptotically. The results obtained using the asymptotic method and direct numerical integration are compared with ad hoc approaches for high speed and moderate supersonic boundary layers. This allows us to suggest recommendations on rational usage of the amplitude method with avoiding unconfirmed simplifications while reducing the computational effort to the level affordable for engineering practice.
