Open Access
REVIEW
Accounting for Quadratic and Cubic Invariants in Continuum Mechanics–An Overview
1 Department of Physics, National Research Nuclear University (MEPhI), Kashirskoye Shosse 31, Moscow, 115409, Russian Federation
2 Department of Thermal Engineering and Department of Waterways, Ports and Hydraulic Structures, Russian University of Transport (MIIT), Obraztsova Street 9, Moscow, 127994, Russian Federation
* Corresponding Authors: Artur V. Dmitrenko. Email: ,
Fluid Dynamics & Materials Processing 2024, 20(9), 1925-1939. https://doi.org/10.32604/fdmp.2024.048389
Received 06 December 2023; Accepted 11 March 2024; Issue published 23 August 2024
Abstract
The differential equations of continuum mechanics are the basis of an uncountable variety of phenomena and technological processes in fluid-dynamics and related fields. These equations contain derivatives of the first order with respect to time. The derivation of the equations of continuum mechanics uses the limit transitions of the tendency of the volume increment and the time increment to zero. Derivatives are used to derive the wave equation. The differential wave equation is second order in time. Therefore, increments of volume and increments of time in continuum mechanics should be considered as small but finite quantities for problems of wave formation. This is important for calculating the generation of sound waves and water hammer waves. Therefore, the Euler continuity equation with finite time increments is of interest. The finiteness of the time increment makes it possible to take into account the quadratic and cubic invariants of the strain rate tensor. This is a new branch in hydrodynamics. Quadratic and cubic invariants will be used in differential wave equations of the second and third order in time.Keywords
Nomenclature
| Linear invariant |
| Quadratic invariant |
| Cubic invariant |
| the “n” order Jacobian of the velocity field |
| u is the fluid velocity component along the |
| |
| w |
P | is the fluid pressure |
e.g. | Index refers to the start time of the deformation |
The article is devoted to the continuity equation in fluid and gas mechanics. Classical fluid mechanics is based on the continuity hypothesis. This is expressed in the tendency of infinitely small elementary volumes dV and time intervals dt to zero. The idea of using the conservation law in a differential form belongs to d’Alembert. He used it in his work on the cause of the wind.
Euler [1–4] derived a differential continuity equation, which has a general form and is not related to an applied problem. There are materials on the derivation by some researchers of additional terms in the continuity equation. In this regard, it is of interest to consider the effect of the finiteness of processes in time on the Euler equation of conservation.
The terms of the high order of smallness of the continuity equation describe the occurrence of self-oscillations, vibrations, sound and water hammer. The terms of the high order of smallness of the continuity equation describe the possible occurrence of the initial stage of turbulent fluctuations. This can be used to study turbulent stochastic and statistical processes when averaged over time (Taylor) or mass (Favre).
Here, in the approximation of the finiteness of an infinitesimal time interval dt, a number of applied problems are considered. As a result, all three invariants of the strain rate tensor are taken into account. Quadratic and cubic invariants will be used in a second-order differential wave equation in time and in a third-order differential wave equation in time.
The deformation theory of continuum mechanics [5,6] gives a formula for the cubic expansion coefficient, which contains three invariants. These are the linear invariant, the quadratic invariant and the cubic invariant. The formula for the cubic expansion coefficient provides the basis for deriving the equation for the conservation of the amount of matter. However, the solution of hydrodynamic problems contains only a linear invariant. In this, one can see the incompleteness of the solution to the problem. This is an asymmetry between the formulation of the problem and its solution. We can expect the existence of new solutions that reflect the geometric properties of the quadratic and cubic invariants. Restoration of symmetry can give new physical properties of the flow and new regimes.
V.M. Bubnov in 1998 pointed out the presence of terms of a high order of smallness in the continuity equation for an incompressible fluid, which was derived by N.E. Zhukovsky. In 2006, Ovsyannikov [7] found terms of a high order of smallness in the derivation of the continuity equation in Euler’s work “Principien motus fluidorum” 1752. Euler calculated the terms of a high order of smallness in terms of the deformation time of the control figure
Therefore, in 2006, the continuity equation with terms of a high order of smallness was written for a compressible fluid [7]. The physical meaning of the terms of a high order of smallness in the continuity equation for a compressible fluid was understood. These terms generate density waves and pressure waves against the background of a stationary flow of a compressible fluid. They generate self-oscillations, vibrations and a solitary pressure wave.
Lighthill [8,9] proposed a new method for deriving the wave equation. This is the method of acoustic analogy. It uses the time derivative of the continuity equation for a compressible medium. Such a derivative can be easily obtained from the continuity equation, which was derived by Euler in 1752. The continuity equation with terms of a high order of smallness can be used in Lighthill’s acoustic analogy method.
The inhomogeneous wave equation was derived by Ovsyannikov [10] in 2007. The inhomogeneous part of the wave equation contains quadratic and cubic invariants of the strain rate tensor. These invariants describe the generation of density and pressure waves during the flow of a compressible fluid.
Purpose of the review shows that flow regions with different Lagrangian laws of motion of a liquid particle have different additional terms of a high order of smallness in time. It was realized that quadratic invariant terms generate density and pressure waves that are close to harmonic oscillations [11,12]. Reviews [11,12] describe solutions to problems of the formation of sound waves with potential air flowing around a right angle, cylinder. Monograph [10] contains a description of the formation of waves when air flows into narrow gaps. These papers investigate the formation of waves by incorporating a quadratic invariant into the continuity equation. The calculation results are consistent with the experiments.
The main contribution of this work is in the study of the physical meaning of the cubic invariant. There are few works that take into account the cubic invariant [13,14].
Most of the works take into account only the quadratic invariant, which describes harmonic waves [10,11,12,15]. The quadratic invariant of the strain rate tensor is used in the method of regularization of the system of hydrodynamic equations [15] to increase the stability of iterations of T.G. Elizarova’s numerical method.
A.V. Dmitrenko noted that these harmonic waves can be used in stochastic methods for calculating turbulent flows.
Thus, calculations taking into account quadratic and cubic invariants may give an extension of the area of application of hydrodynamic equations. The final answer can be given by comparing the calculations with the experimental results.
Section 2 presents simple geometric constructions that show the reason for the appearance of a high-order term of smallness in the continuity equation for a plane two-dimensional flow with a linear law of deformation of the control figure in time. Here is the reason why mathematicians are divided into two camps. Newton’s followers considered the differential to be an infinitesimal quantity. They had to replace it with zero. Leibniz’s followers considered the differential to be a small but finite value. They were required to perform arithmetic operations with differentials according to the rules for quantities of finite size. This is discussed in Carnot’s book.
There are two main derivations of the differential continuity equation for an incompressible fluid. This is Euler’s derivation and Ostrogradsky’s derivation. Both conclusions are based on geometric constructions. If Euler’s geometrical derivation is used in exact form, then it gives terms of a high order of smallness. This article studies the physical meaning of terms of a high order of smallness.
The reason for the appearance of terms of a high order of smallness in the equation of continuity lies in the use by Euler, Zhukovsky, Ostrogradsky of the Lagrangian law of motion of a liquid particle linear in time. We will demonstrate this by deforming the control square. The side of this square is equal to one. We will take into account only compression and tension deformations.
The side of the square, which is parallel to the
Here
Here
Let us equate the area of the deformed control figure and the area of the initial square
A simplification of this formula gives the continuity equation with a term of high order of smallness in time
Euler made similar calculations for a three-dimensional flow with allowance for the shear deformation in 1752.
The terms of the high order of smallness reflect the simultaneity of deformations that occur in perpendicular directions. The terms of the high order of smallness reflect the volume that is obtained due to the deformation of the deformations. Hydrodynamicists must calculate the basic problems of wave formation, the results of which can be verified experimentally.
When using the exponential Lagrangian law of motion of a liquid particle, terms of a high order of smallness do not arise. In this case, the continuity equation contains three terms of the velocity vector divergence. This conclusion is contained in Section 5.
Thus, we propose to use the completely geometric characteristics of the flow, which are contained in the quadratic and cubic invariants.
Similar terms of a high order of smallness can be seen in the geometric constructions of Ostrogradsky before making the transition from geometric objects of finite size to infinitely small objects. It is also necessary to point out the historical reasons for the elimination by Euler of the terms of the high order of smallness of the continuity equation. Modern integral and differential calculus appeared as a result of the merging of Leibniz’s theory of infinitesimals and Newton’s theory of vanishingly small quantities. They were created independently at the same time. There are slight differences in these theories.
The analysis of the differences was made by Lazar Carnot. Leibniz considers the differential to be a small quantity with which mathematical operations can be carried out according to the rules of arithmetic of finite quantities.
Newton in his theory considers the analogue of the differential as a vanishingly small value, which is equal to zero. A mathematician must consider the result of multiplying a number by such a differential equal to zero.
L. Carnot divided the mathematicians and mechanics of his era into two lists. Euler is written on the list of mathematicians who supported Newton’s mathematical theory. This position of Euler can explain the reason for his exclusion of terms of the high order of smallness of the continuity equation by passing to the limit
In engineering calculations, it is necessary to choose the most dangerous situation. For example, a seismologist must predict the possibility of an earthquake using the continuity equation, which contains terms of a high order of smallness.
3.1 Euler’s Continuity Equation with Terms of High Order of Smallness
Euler’s 1752 derivation of the continuity equation for the three-dimensional flow of an incompressible fluid, taking into account tensile and shear deformations, gave the following result:
Let us analyze a similar continuity equation for a compressible fluid, published in 2006. Methods for deriving the continuity equation in hydrodynamics and elasticity theory are similar in geometric constructions.
The theory of deformations studies the general approach to the construction of equations for the theory of elasticity and for hydrodynamics. The textbook by Sedov [5] considers these disciplines at the same time. In this textbook, the theory of deformations for a solid elastic body and for an incompressible fluid is given from the standpoint of the general properties of the strain tensor and the strain rate tensor.
These properties are the same since the components of the strain tensor and the strain rate tensor are related to each other. When deriving the expression for the coefficient of cubic expansion
In this formula,
Let us pay attention to the inexact equal sign in this formula. In the sections that study Hydrodynamics, Sedov considers the strain rate tensor
To solve problems of non-stationary hydrodynamics, taking into account quadratic and cubic invariants, it is necessary to use differential equations of the second and third order in time. Before excluding terms with higher invariants, it is necessary to understand their physical meaning. This became possible, when Lighthill [8,9] proposed the method of acoustic analogy for deriving the wave equation from the system of equations of motion and continuity.
In the same years, Truesdell [2] translated from Latin into English the first version of Euler’s classic work Principia motus fluidorum [1]. There is now a translation of the first version of Euler’s work Principia motus fluidorum 1752 [1] into various languages [3]. A detailed translation of Euler’s work into English [4] was made in 2008.
C. Truesdell drew attention to Euler’s intermediate result in the form of a complete continuity equation, which contains terms of a high order of smallness in time
Here
Lectures on hydroaeromechanics by Wallander [6] contain such formulas for the invariants of the strain rate tensor
Linear invariant is
Cubic invariant
If we use the expressions for the invariants of the strain rate tensor (3), (4), then the Euler continuity equation for an incompressible fluid (2) takes the form
After deriving the continuity Eq. (2) or (5), Euler made the following passage to the limit
The change in density can be taken into account by replacing the velocity components
Factors
or
Here
The continuity Eqs. (6) and (7) for a compressible fluid are more accurate.
Must have a finite time increment value
Large values of additional terms in the continuity Eq. (7) can arise with large values of the invariants
3.2 Derivation of the Wave Equation by Lighthill’s Acoustic Analogy Method
Lighthill’s method consists in creating a d’Alembert operator on the left side of the wave equation. To preserve the square and cubic invariants in the system of fluid dynamics equations, it is necessary to take the time derivative of the right and left sides of the continuity Eq. (7).
When taking the time derivative, the time increment
The time derivative is taken from the continuity equation for an unsteady flow of a compressible fluid. Taking the derivative creates a second-order derivative of the density with respect to time
Lighthill’s acoustic analogy method yields an inhomogeneous second-order differential wave equation in time
or
Here
3.3 Physical Meaning of the Quadratic Invariant
We will use the method of successive approximations to calculate the intensity of the generation of density waves and pressure waves for a stationary velocity field. We will use in the inhomogeneous part of the wave equation the invariants
described in articles [11,12]. The right side of the wave equation with the d’Alembert operator describes the process of transmission through the elementary volume of the waves existing in the fluid. The quadratic invariant
The intensity
The situation will be different when the cubic invariant
4 Physical Meaning of the Cubic Invariant
In recent years, reference [13] have been carried out to study the physical meaning of a term with a cubic invariant
Let us assume that the solution to the wave Eq. (10) is the sum of the solution to the inhomogeneous Eq. (11) and the solution to Eq. (12)
or
or
The solution of Eq. (11) has a complex form and is obtained using retarded potential method. But this solution does not tend to strongly increase in time. The solution of Eq. (12) will contain the deformation time of the control figure
The wave Eq. (12) describes the possibility of increasing the pressure as a cubic function of time
The solution of the differential Eq. (12) depends on the value of the deformation time of the control figure
It is necessary to solve the system of four differential Eqs. (8), (9), (11), (12) in order to use the three invariants of the strain rate tensor. In this case, a complete solution to the problem is obtained. We use the method of successive approximations. We will find the velocity and pressure field from the system of the classical continuity Eq. (8)
and the differential Eq. (12) of the third order in time. The solution of Eq. (12) has the form of a power function of time.
This function increases as the third power of time. This deformation time
Mathematicians must give the equation of continuity with terms of high order of smallness to seismologists and acousticians. For rigorous quantitative calculations of the intensity of generated waves, engineers need to obtain from mathematicians various solutions to the wave equation using the retarded potential method. We pose this problem to mathematicians in this review.
5 A Flow for Which the Continuity Equation Does Not Contain Terms of a High Order of Smallness
Let us show that the continuity equation can have different forms in flow regions with different Lagrangian laws of motion of a liquid particle and with different Euler velocity fields. Demonstration of the differences in the continuity equation for different places in the velocity field is the main goal of this review.
There are two ways to describe the motion of a fluid [13]. This is Lagrange’s law of motion of a liquid particle. This is the Euler velocity field. We know the Lagrange variables
We know the Euler variables
Euler used the linear-in-time Lagrange law of motion of a fluid particle in deriving the 1752 continuity equation. These are the Cauchy-Helmholtz formulas with a linear dependence of the coordinates
Here
Let us calculate the velocity components along the coordinate axes
The linearity in time of the Lagrange law of motion of a liquid particle [1–3] gives terms of a high order of smallness in the continuity Eq. (2). The exponential law of motion of a liquid particle gives the exact fulfillment of the continuity equation without additional terms.
The exponential law of motion of a liquid particle can be obtained by replacing the initial values
Initial conditions
The solution of the system of Eq. (18) with initial conditions (19) is contained in the book [10] and gives the exponential Lagrange law of motion of a liquid particle with time.
Here
Let us derive the continuity equation for Lagrange’s exponential law of motion (20), (21). Let the control figure have the shape of a unit square with the following coordinates of the corner points (0, 0), (0, 1), (1, 1), (1, 0). The control figure will have the shape of a parallelogram after being deformed for time t. Parallelogram corner points 2 and 4 will have these coordinates.
The area of a parallelogram can be calculated using this formula.
Equation of conservation of area in time
The area conservation equation after simplifying the notation has the following form:
or
or
We get the continuity equation after dividing by time
This continuity equation is realized in the case of a linear field of velocity components (18) along the coordinates
Liquid particles have an accelerated Lagrange law of motion. There will be no turbulent pulsations in the place of the accelerated Lagrange law of motion. In place of the accelerated law of motion of Lagrange, there will be no generation of sound vibrations.
The real velocity field in problems of hydrodynamics cannot be linear in the entire flow region. Terms of a high order of smallness will appear in the equation of continuity at the point of violation of the linear field of the velocity components. The terms of a high order of smallness will generate sound waves and disturbances at the point of violation of the linear field of the velocity components.
Koppel et al.’s experiments [16] on fluid flow with acceleration demonstrate an increase in the critical Reynolds number for the transition from laminar to turbulent flow by a factor of 30–100. This is an experimental confirmation of the disappearance of terms of a high order of smallness in the continuity equation for a flow with acceleration. In the place of the linear field of the velocity components along the coordinates, there will be no generation of sound vibrations, self-oscillations.
Here we also emphasize that for stochastic processes in hydrodynamics, the question of finiteness in space and time of infinitesimal perturbations plays a key role [17–21] in the statistical theory and [22–26] in nonline theory. This leads to the appearance of additional fluctuation terms in the right part of the equations of conservation laws [27–31] in the chaos theory and in the theory of attractors [32–34]. As a result, on the basis of the law of interaction between deterministic and random motion [35–39], the transition from deterministic motion to chaotic, turbulent is realized [40–44]. The solutions considered in the article, albeit in the first approximation, but raise questions for numerical methods such as RANS [45–50], LES [51–53] and DNS methods for bifurcation of periodic solutions [54–58] and methods for simulation of a complete transition to turbulence [59–61]. Note that at the same time, the numerical TDNS method is emerging [36,37,62]. In it, differential equations with a random term on the right side have the ability to take into account the influence of various invariants.
1. A review of articles that take into account the quadratic and cubic invariants of the strain rate tensor in the continuity equation for a compressible fluid is presented.
2. The quadratic invariant describes self-oscillations, which are similar to harmonic oscillations. These are vibrations, sound generation, and the initial stage of turbulence.
3. The cubic invariant describes the occurrence of pressure waves that rapidly increase in time. They are similar to a soliton or Zhukovsky hydraulic shock.
4. These invariants can be taken into account in solving the wave equations, which are derived by the method of acoustic analogy, taking into account the terms of the high order of smallness of the continuity equation.
5. Taking into account quadratic and cubic invariants is important when analyzing emergency situations.
Acknowledgement: Not applicable.
Funding Statement: This research received no external funding.
Author Contributions: Conceptualization, methodology and investigation, original draft preparation and writing—review and editing, Artur V. Dmitrenko and Vladislav M. Ovsyannikov; validation and formal analysis, Vladislav M. Ovsyannikov. All authors have read and agreed to the published version of the manuscript.
Availability of Data and Materials: The datasets analyzed and used during the current study are available from the corresponding author as mentioned in the references.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
References
1. Euler L. Principia motus fluidorum. Novi Commentarii Academiae Imperialis Scientiarum Petropolitanae. 1761;6:271–311. [Google Scholar]
2. Euler L. Commentationes mechanicae ad theoriam corporum pertinentes. In: Truesdell CA, editor. Volumen Prius. Stwitzerland: Societas Scientiarum Naturalium Helveticae; 1954. [Google Scholar]
3. Euler L. Principles of the Motion of Fluids. In: Ivanova EV, Ovsyannikov VM, editors. Translators. 4th updates edition, Moscow, Russia: Sputnik+; 2020. p. 203. [Google Scholar]
4. Euler L. Principia motus fluidorum. arXiv:0804.4802. 2008 (Translated by Pauls, L. W.). [Google Scholar]
5. Sedov LI. Mechanics of continuous media. In: Series in theoretical and applied mechanics. Singapore: World Scientific; 1997. vol. 2, p. 1368. [Google Scholar]
6. Wallander SV. Lectures on hydroaeromechanics. Leningrad: Leningrad University Press; 1978. p. 296. [Google Scholar]
7. Ovsyannikov VM. The introduction to axiomatic Hydrodynamics founded on the basic experiments with the liquid. In: Axiomatical problems in hydro-gasdynamics. No 15. Russia, Moscow: Wek knigi; 2006. p. 19–51 (In Russian, English, German, French). [Google Scholar]
8. Lighthill MJ. On sound generated aerodynamically. Part I: general theory. Proc R Soc Lond. 1952;A221:564–87. [Google Scholar]
9. Lighthill MJ. On sound generated aerodynamically. Part II: turbulence as a source of sound. Proc R Soc Lond. 1954;A222:1–32. [Google Scholar]
10. Ovsyannikov VM. Wave formation and the finite-difference continuity equation of Leonhard Euler. Moscow: Sputnik+; 2016. p. 457 (In Russian, English, German, French). [Google Scholar]
11. Ovsyannikov VM. Euler continuity equation with high-order terms in time. Journal of Mathematical Sciences. November 2023;276(4):798–803. [Google Scholar]
12. Ovsyannikov VM. Euler’s equation of continuity: additional terms of high order of smallness—an overview. Fluids. 2021;6:162. doi:10.3390/fluids6040162. [Google Scholar] [CrossRef]
13. Ovsyannikov VM. The appearance of periodic waves and a solitary wave in the turbine of a hydroelectric power plant. In: Int Sci Conf Fundamental and Appl Problems of Mech (FAPM-20202020 Dec 2–4; Moscow, Russia. doi:10.18698/2308-6033-2021-4-2068. [Google Scholar] [CrossRef]
14. Kulikovsky AG, Sveshnikova EI. Nonlinear waves in elastic media. Moscow: Moscow Lyceum; 1998. p. 412. [Google Scholar]
15. Ovsyannikov VM. Comparison of additional second-order terms in finite-difference Euler equations and regularized fluid dynamics equations. J Comput Math Math Phys. 2017;57(5):876–80. doi:10.1134/S0965542517050098. [Google Scholar] [CrossRef]
16. Koppel TA, Liiv UR. Experimental study of the occurrence of fluid movement in pipelines.—Izv. USSR Acad Sci. MZhG. 1977;6:79–85. [Google Scholar]
17. Landau LD. Toward the problem of turbulence. Dokl Akad Nauk SSSR. 1944;44:339–42. [Google Scholar]
18. Landau LD, Lifshits EF. Fluid mechanics. London, UK: Perg. Press Oxford; 1959. [Google Scholar]
19. Kolmogorov AN. Dissipation of energy in locally isotropic turbulence. Dokl Akad Nauk SSSR. 1941;32(1):16–8. [Google Scholar]
20. Kolmogorov AN. A new metric invariant of transitive dynamic sets and automorphisms of the Lebesgue spaces. Dokl Akad Nauk SSSR. 1958;119(5):861–4. [Google Scholar]
21. Kolmogorov AN. Mathematical models of turbulent motion of an incompressible viscous fluid. Usp Mat Nauk. 2004;59(1):5–10. [Google Scholar]
22. Kolmogorov AN. About the entropy per time unit as a metric invariant of automorphisms. Dokl Akad Nauk SSSR. 1958;124(4):754–5. [Google Scholar]
23. Lorenz EN. Deterministic nonperiodic flow. J Atmos Sci. 1963;20:130–41. doi:10.1175/1520-0469. [Google Scholar] [CrossRef]
24. Feigenbaum M. The transition to aperiodic behavior in turbulent sets. Commun Math Phys. 1980;77(1):65–86. [Google Scholar]
25. Ruelle D, Takens F. On the nature of turbulence. Commun Math Phys. 1971;20:167–92. doi:10.1007/bf01646553 also23, 343–344. [Google Scholar] [CrossRef]
26. Samarskii AA, Mazhukin VI, Matus PP, Mikhailik IA. Z/2 conservative schemes for the Korteweg–de Vries equations. Dokl Akad Nauk. 1997;357(4):458–61. [Google Scholar]
27. Haller G. Chaos near resonance. Berlin: Springer; 1999. doi:10.1007/978-1-4612-1508-0. [Google Scholar] [CrossRef]
28. Orzag SA, Kells LC. Transition to turbulence in plane Poiseuille and plane Couette flow. J Fluid Mech. 1980;96(1):159–205. doi:10.1017/s0022112080002066/. [Google Scholar] [CrossRef]
29. Ladyzhenskaya OA. On a dynamical system generated by Navier-Stokes equations. J Sov Math. 1975;3:458–79. [Google Scholar]
30. Struminskii VV. Origination of turbulence. Dokl Akad Nauk SSSR. 1989;307(3):564–7. [Google Scholar]
31. Klimontovich YL. Problems of the statistical theory of open sets: criteria of the relative degree of the ordering of states in the self-organization processes. Usp Fiz Nauk. 1989;158(1):59–91. doi:10.1070/pu1999v042n01abeh000445. [Google Scholar] [CrossRef]
32. Grassberger P, Procaccia I. Dimensions and entropies of strange attractors from a fluctuating dynamics approach. Phys D Nonlinear Phenom. 1984;13(1–2):34–54. doi:10.1016/0167-2789(84)90269-0. [Google Scholar] [CrossRef]
33. Rabinovich MI, Reiman AM, Sushchik M, Kozlov V, Ramazanov M. Correlation dimension of the flow and spatial development of dynamic chaos in the boundary layer. JETP Lett. 1987;13(16):987. [Google Scholar]
34. Dmitrenko AV. The correlation dimension of an attractor determined on the base of the theory equivalence of measures and stochastic equations for continuum. Contin Mechan Thermod. 2020;32(2):63–74. doi:10.1007/s00161-019-00784-0. [Google Scholar] [CrossRef]
35. Dmitrenko AV. Theoretical solutions for spectral function of the turbulent medium based on the stochastic equations and equivalence of measures. Contin Mech Thermod. 2021;33:603–10. doi:10.1007/s00161-020-00890-4. [Google Scholar] [CrossRef]
36. Dmitrenko AV. Uncertainty relation in turbulent shear flow based on stochastic equations of the continuum and the equivalence of measures. Contin Mech and Thermod. 2020;32(1):161–71. doi:10.1007/s00161-019-0566-x. [Google Scholar] [CrossRef]
37. Dmitrenko AV. Formation of the turbulence spectrum in the inertial interval on the basis of the theory of stochastic equations and equivalence of measures. J Eng Phys Thermophys. 2020;93(1):122–7. doi:10.1007/s10891-020-02098-4. [Google Scholar] [CrossRef]
38. Dmitrenko AV. The stochastic theory of the turbulence. IOP Conf Ser Mater Sci Eng. 2018;468:012021. doi:10.1088/1757-899X/468/1/01202. [Google Scholar] [CrossRef]
39. Dmitrenko AV. Regular coupling between deterministic (Laminar) and random (Turbulent) motions-equivalence of measures; 2013. Scientific Discovery. Diploma No. 458, Registration No. 583 of December 2. [Google Scholar]
40. Benyahia A, Castillon L, Houdeville R. Prediction of separation-induced transition on highlift low pressure turbine blade. In: Proceedings of the ASME 2011 Turbo Expo: Turbine Tech. Conf. and Exposit, 2011 Jun 6–10; Vancouver, Canada: Am. Soc. Mech. Eng.; 2011; vol. 5, p. 1835–46. [Google Scholar]
41. Goldstein ME. Effect of free-stream turbulence on boundary layer transition. Philos Trans R Soc A. 2014;372:20130354. doi:10.1098/rsta.2013.0354. [Google Scholar] [PubMed] [CrossRef]
42. Lele SK. Compact finite difference schemes with spectral-like resolution. J Comput Phys. 1992;102:16–42. [Google Scholar]
43. Visbal MR, Gaitonde DV. On the use of high-order finite-difference schemes on curvilinear and deforming meshes. J Comput Phys. 2002;181:155–85. [Google Scholar]
44. Mardsen O, Bogey C, Bailly C. High-order curvilinear simulations of flows around non-cartesian bodies. J Comput Acoust. 2005;13(4):731–48. [Google Scholar]
45. Tam CKW, Ju H. Numerical simulation of the generation of airfoil tones at a moderate Reynolds number. In: 12th AIAA/CEAS Aeroacoustics Conference (27th AIAA Aeroacoustics Conference2006 May 8–10; Cambridge, Massachusetts, USA. [Google Scholar]
46. Sandberg RD, Jones LE, Sandham ND, Joseph PF. Direct numerical simulations of noise generated by airfoil trailing edges. In: 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference2007 May 21–23; Rome, Italy. [Google Scholar]
47. Harten A. ENO schemes with subcell resolution. J Comput Phys. 1989;83:148–84. [Google Scholar]
48. Harten A, Engquist B, Osher S, Chakravarthy SR. Uniformly high order essentially non-oscillatory schemes III. J Comput Phys. 1987;71:231–303. [Google Scholar]
49. Hu C, Shu CW. Weighted essentially non-oscillatory schemes on triangular meshes. J Comput Phys. 1999;150:97–127. [Google Scholar]
50. Dmitrenko AV. Calculation of the boundary layer of a two-phase medium. High Temp. 2002;40(5):706–15. doi:10.1023/A:1020436720213. [Google Scholar] [CrossRef]
51. Dmitrenko AV. Calculation of pressure pulsations for a turbulent heterogeneous medium. Dokl Phys. 2007;52(7):384–7. doi:10.1134/s1028335807120166. [Google Scholar] [CrossRef]
52. Abalakin A, Bachwalow P, Kozubskaya T. Edge-based reconstruction schemes for prediction of near field flow region in complex aeroacoustic problems. Int J Aeroacoust. 2014;13(3–4):207–34. [Google Scholar]
53. Dumbser M, Kaser M, Titarev VA, Toro EF. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys. 2007;226:204–243. [Google Scholar]
54. Chan WM, Sheriff K, Pulliam TH. Instabilities of two-dimensional inviscid compressible vortices. J Fluid Mech. 1993;253:173–209. [Google Scholar]
55. Skubachevskii AL. Bifurcation of periodic solutions for nonlinear parabolic functional differential equations arising in optoelectronics. Nonlinear Anal TMA. 1998;32(2):261–78. [Google Scholar]
56. Cannarsa P, Martinez P, Vancostenoble J. Global Carleman estimates for degenerate parabolic operators with applications. Mem Am Math Soc. 2016;239(1133):1–207. [Google Scholar]
57. Cannarsa P, Tort J, Yamamoto M. Determination of source terms in degenerate parabolic equation. Inverse Probl. 2010;26(10):105003. [Google Scholar]
58. Mayer CSJ, Terzi DAV, Fasel HF. Direct numerical simulation of complete transition to turbulence via oblique breakdown at Mach 3. J Fluid Mech. 2011;674:5–42. [Google Scholar]
59. Mayer CSJ, Terzi DAV, Fasel HF. DNS of Complete Transition to Turbulence Via Oblique Breakdown at Mach 3. In: 38th Fluid Dynamics Conference and Exhibit, 2008 Jun 23–26; Seattle, Washington, USA. [Google Scholar]
60. Istvan MS, Yarusevych S. Effects of free-stream turbulence intensity on transition in a laminar separation bubble formed over an airfoil. Exp Fluids. 2018;59(3):21. [Google Scholar]
61. Chuvakhov PV, Fedorov AV, Obraz AO. Numerical simulation of turbulent spots generated unstable wave packets in a hypersonic boundary layer. Comput Fluids. 2018;162:26–38. doi:10.1016/j.compfluid.2017.12.001. [Google Scholar] [CrossRef]
62. Dmitrenko AV. Prediction of laminar-turbulent transition on flat plate on the basis of stochastic theory of turbulence and equivalence of measures. Contin Mech Thermodyn. 2022;34:601–15. doi:10.1007/s00161-021-01078-0. [Google Scholar] [CrossRef]
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