LLinear stability of flow in a 90-degree bend
Alexander ProskurinAltai state technical university656038, Russian Federation, Barnaul, Lenin prospect, 46 [email protected]
February 23, 2021
Abstract
The paper considers a two-dimensional flow in a channel, which con-sists of straight inlet and outlet branches and a circularly curved bend. Anincompressible viscous fluid flows through the elbow under the action ofa constant pressure gradient between the inlet and outlet. Navier-Stokesequations were solved numerically using a high-fidelity spectral/hp ele-ment method. In a range of Reynolds numbers, an adaptive selective fre-quency dumping method was used to get a steady-state flow. It was foundthat separation bubbles and vortex shedding can exist in the bend. Theinstability of the three-dimensional linear perturbations is investigated. Itwas found that the critical numbers of the two-dimensional disturbancesaccording to the linear approach are much greater than those of three-dimensional ones, but the two-dimensional flow is unstable with respectto the perturbations of finite amplitude. White noise can be amplified bytwo-dimensional nonlinear mechanisms at relatively small Reynolds num-bers, and a nonlinear instability scenario is realized that competes with thelinear instability of three-dimensional perturbations. This way of insta- a r X i v : . [ phy s i c s . f l u - dyn ] F e b ility development is characterized by the formation of a two-dimensionalvortices sequence, where the amplitude varies along the channel in a quasi-periodic manner. keywords: instability, separated flows, channel flow Complex shape channel flows, when there are recirculation bubbles, flow sepa-ration, and periodic motion, are often observed in nature and are widely used inindustry. Depending on the Reynolds number, laminar, periodic, turbulent, andlocally turbulent patterns can be realized. In the past 30 years, a generation ofnonstationary regimes in such flows has been studied using the global stabilitytheory. A review of such research is given in [1].An example of flow with separation bubbles is the flow over a backward-facing step. The stability of this flow has been examined extensively and isdiscussed in several publications [2, 3, 4]. Kaikitis et al. [4] investigated two-dimensional stability and found that the flow was globally-stable according tothe linear analysis if Re ≤ . They show, that a large part of the flow domainis convectively unstable to the sustained upstream-generated finite-amplitudedisturbances for Reynolds numbers ≤ Re ≤ . Also, Kaikitis et al. as-sumed that the computations may show global unsteadiness due to discretizationerrors that mask the convective instability of the flow.Barkley at al. [2] have presented the results of a study of the two- andthree-dimensional linear stability of this flow. They found the critical Reynoldsnumber for the three-dimensional perturbations( Re ∗ = 748 ) and have shownthat the lower boundary of the critical Reynolds number for two-dimensionalperturbations is much higher( Re ∗ D > ). It proves the leading role of thethree-dimensional perturbations. It also explains that the instability, according2o the linear theory, is associated with the first recirculation bubble, whichoccurs immediately after the step, and the centrifugal instability is responsiblefor generating secondary flow with the separation zone.Blackburn et al. [3] examined optimal disturbances growth. They found thatthree-dimensional optimal disturbances have slightly larger growth in compari-son to the two-dimensional. The three-dimensional mode appears near the stepand gains energy downstream by the inviscid Orr mechanism and interaction ofthe Kelvin–Helmholtz instabilities of the two separated bubbles. Also, Black-burn et al. investigated of the three-dimensional motion arising under the actionof the random inflow noise. They observed a flow with a strong dominance ofthe two-dimensional dynamics.Zhang&Poth´erat [5] investigated flow in a 180-degree sharp bend. In theoutput branch of the bend, as the Reynolds number increases, the followingpatterns can exist: laminar, the appearance of the first recirculation bubblenear the inner wall, and the appearance of the second recirculation bubble nearthe outer wall downstream of the first bubble. If the Reynolds number increasesfurther, vortex shedding can occur in the outlet branch. It was also establishedthat the flow patterns are similar to the flow near a round cylinder, exceptfor the symmetry, and that the two-dimensional dynamics determine the mainfeatures of the three-dimensional flow. Sapardi et al. [6] studied the stability ofthis flow. It was shown that the flow can be unstable concerning to the three-dimensional periodic perturbations. Unstable modes are associated with thefirst recirculation bubble. A two-dimensional nonlinear stability analysis wasalso made and the hysteresis of the critical Reynolds numbers was found.Matsumoto et al. [7] systematically studied systematically two-dimensionalflow patterns in a sharply bent channel for angles less than 90 degrees anddescribe recirculation bubbles in the outer angle of the bend and near the inner3all of the outlet branch right after the bend. They also observed the periodicvortex emission in the outlet branch and have described conditions for eachpattern. Matsumoto et al. [7] especially notes that the intermediate regimesbetween the turbulent and the laminar in the bend even are poorly studied.Park&Park [8] studied nonlinear stability of the smooth bent flow in the caseof small bent angles which were less than 60 degrees and a large bending radius.For such small curvature stability modes shape close to the case of the straightPouiseuille flow. Park&Park describe the nonlinear interaction of the obliquewaves, and the amplification of streamwise streaks and vortices. The amplitudeof the streaks increases in the bend and the outlet branch, but the streamwisevortices increases only in the bend.From this bibliographic review, one can conclude that 90-degree bent flowstability has not been studied before. This research closes this gap in the knowl-edge. The study was carried out at one intermediate value of the bending radius,except the three-dimensional stability analysis, when the critical Reynolds num-bers were obtained for a set of the bent radii. Figure 1 shows a bent flow. The geometry consists of two parallel impermeableplanes, curved so that the bending radius on the centre line is equal to R . Thedistance between the planes is constant and is equal to d . A bending parameteris δ = R/ d . The paper considers the case of δ = 1 . The incompressible viscousfluid flows under the action of a constant pressure difference between the inflowand the outflow. The Reynolds number Re = Udν was introduced, where U is themaximum velocity of the laminar Pouiseuille flow, and ν is the fluid viscosity.4 nf low o u t f l o w xyR v d Figure 1: The bent channelThe Navier-Stokes equation is ∂ V ∂t + ( V ∇ ) V = −∇ p + 1 Re ∆ V + F ,div V = 0 , (1)where V is the velocity, F is the external force, and p is the pressure. At thechannel walls, the no-slip condition boldsymbolV = 0 holds true. The Pouiseuilleparabolic profile V y = 1 − x is imposed at the inflow. At the outflow, I imposethe standard boundary condition [2, 6] ∂ V ∂x = 0 , p = 0 . (2)On all other boundaries the pressure satisfies the high-order Newmann boundarycondition ∂p∂ n = 0 [9].If U ( x, y ) is a steady-state solution of (1), this equation can take a linearizedform ∂ v ∂t + ( U ∇ ) v + ( v ∇ ) U = −∇ p + 1 Re ∆ v ,div V = 0 , (3)5here v ( x, y, z, t ) and p ( x, y, z, t ) are the small disturbances. Since the baseflow U ( x, y ) is two-dimensional, the disturbances may be written as v ( x, y, z, t ) = v ( x, y ) e ( σ + iω ) t + iβz + c.c. ,p ( x, y, z, t ) = p ( x, y ) e ( σ + iω ) t + iβz + c.c. , (4)because the base flow does not depend on the z -coordinate. Here σ is thedisturbance growth, ω is the disturbance frequency, and the spanwise wavenumber is β = πλ , where λ is the spanwise wavelength. The substitution (4)allows decoupling from the three-dimensional stability problem (3) to a series oftwo-dimensional problems for each value of β , which varies continuously. Thisanalytical transformation was used by [1, 2, 6] and many others.Boundary conditions for the disturbance have the form v = 0 at the walls and the inflow ,∂ v ∂x = 0 , p = 0 at the outflow (5) The baseflow and stability calculations were made by the open source spec-tral/hp element framework Nektar++ [10]. The multi-element formulation givesgeometric flexibility by comparison to the single-domain spectral methods, andallows to implement the high-order approach to complex shape flows. High-ordermethods can improve the reliability, accuracy, and computational efficiency ofcalculations.Figure 2 shows the mesh that was used for the calculations. This meshcontains elements and is convenient for flow at Reynolds number Re =500 ∼ . For higher Reynolds numbers, meshes with more elements were6able 1: The convergence of the growth σ and the horizontal velocity v x at thecontrol points, Re = 600 p Point 1 Point 2 Point 3 Point 4 σ Re , less detailed meshes are optimal.The steady-state flow was found by integrating equations (1) in time until aconstant velocity was observed at the selected points and the time convergencewas achieved in at least − digits. Some of these points are marked withcrosses in figure 3, the numbers increase from left (No 1) to right (No 4). Table1 shows the values of the horizontal velocity and the growth depending on theorder of the approximation polynomials p . Velocity converges up to digits or . of the velocity scale U . Table 1 also shows the growth convergence forthe leading (real) mode. Figure 2: The meshFigure 3: The control points, Re = 600 ˚Akervik et al. [11] describe a selective frequency damping method that can7onverge to an unstable flow. This is an equivalent of the Newton method,but it is easier to use since the method does not require a high quality initialcondition. However, the selective frequency damping method requiires a verylarge computational cost. To apply the method let us write (1) in the form ˙ q = E ( q ) , (6)where q are the flow variables and the dot over stands for the time derivation.Equation (6) can be modified for the stabilization purpose ˙ q = E ( q ) − χ ( q − q s ) , (7)where q s is the steady-state solution, and χ is the real positive value (it is thecontrol coefficient). Since the steady-state flow is unknown, it is replaced withthe function q ( t ) = (cid:90) t −∞ exp (cid:18) τ − t ∆ (cid:19) q ( τ ) dτ, (8)which is the filtered version of q . Expression (8) is a well-known exponentialfilter. ∆ is the filter width. This filter suppresses high frequency fluctuations inthe flow. Using an equivalent differential form of (8) it is possible to write ˙ q = E ( q ) − χ ( q − q ) , ˙ q = q − q ∆ . (9)Equations (9) were solved from the zero initial conditions until | q − q | be-comes less than some given tolerance. The parameters χ и ∆ determine themethod convergence. Jordy et al. in their article [12] describe an adaptive selec-tive frequency dumping method, which applies one-dimensional reduced modelto find these parameters. The adaptive selective frequency damping method8s implemented in the Nektar++ framework [13, 12]. The complete numericalcalculations set up and the convergence analysis are presented in a preprint [14].A nonlinear form of the advection terms ( V ∇ ) V = ( U ∇ ) v + ( v ∇ ) U + ( v ∇ ) v (10)can be used to split the base flow and the disturbance dynamics calculations.Appropriate modification of the Nektar++ code was reported in [15, 16]. Thisapproach allows the study of the nonlinear disturbances near the stabilized baseflow. Figure 4 shows streamlines at δ = 1 and Re = 20 , , , . The flow islaminar at small Reynolds numbers and its streamlines are parallel (see 4(a)).When the Reynolds number increases, vortices V , V and V (see 4(b,c,d))appear. Figure 5 explains the vortex indexing. For relatively high Reynoldsnumbers it is possible for two different flow patterns to exist: the steady-stateand the pulsating. The pulsating pattern is shown in figure 4(e). One of thetwo patterns can exist depending on the conditions that are described later inthe article.Figure 6 presents a vortex diagram in the bend. V vortex area is markedby circles, V by squares, and V by diamonds. To calculate the end positionof the vortices, a curved coordinate system is introduced, the axis x a which isthe axis of the channel. The ends of the vortices are projected onto it, as shownin figure 5 using dotted lines.Matsumoto at al. [7] found the vortices V , V in the sharp bend. Zhang&Poth´eratand Sapardi et al. [5, 6] found vortices V , V , and V in the 180-degree sharp9end, where V vortex has a great influence on the flow’s characteristics. Arti-cles [7, 5, 6] also describe the appearance of the pulsating flow. The dashed linein figure 6 is the lower limit for the pulsations. (a)(b)(c)(d)(e) Figure 4: The base flow patterns: Re = 20 (a), Re = 200 (b), Re = 500 (c), Re =1300 (d)(the steady-state flow), Re = 1300 (e)(the pulsating flow) x a − V V V Figure 5: The vortex measurements10 −
10 0 10 20 30 40 50 60 70 80 R e x a Figure 6: Flow diagram, δ = 1 , the circles indicate the appearance of V ,rectangles – V , diamonds – V , the dashed line suggests to the pulsating floworigin This section summarizes the results of studying three-dimensional linear sta-bility in the bent channel. Let us consider the dependencies of the growth σ on the Reynolds number Re , and the spanwise wave number β for the bendingparameter δ = 1 . Figure 7 shows the growth dependence from the spanwisewavenumber β for Re = 300 , , ..., . At Re = 900 and in a cer-tain range of β , σ becomes greater than zero and the flow is unstable. Thecurve Re = 300 has a maximum at β ≈ . and a minimum at β ≈ . . Asthe Reynolds number increases, this minimum becomes less noticeable, and themaximum goes to the σ > . At β → , σ remains negative.Figure 8 shows graphs of the real (a) and imaginary (b) parts of the eigen-values for the most dangerous real and the second and third complex modes at Re = 1100 . The growth graphs are similar to those shown in figure 7. For thereal mode the growth tends to a two-dimensional value if β → . The growth11able 2: The critical Reynolds numbers and the corresponding spanwisewavenumbers δ Re ∗ , ± β ∗ , ± . . < β < , thefrequencies of the complex modes have maxima and minima.Figure 9 presents the eigenvalues for Re = 1100 and δ = 1 . . The emptycircles are the modes shown in figure 8. From the figure, it is possible to concludethat the monotonic mode leads to instability of three-dimensional perturbations,while the periodic modes are stable.Figure 10 shows the streamwise vorticity isosurfaces (a, b, c) and the trans-verse velocity amplitude (aa, bb, cc) for the three modes previously mentioned.The unstable mode is located near the recirculation bubble V . The secondoscillating mode is also associated with this region, as indicated by the largeamplitude of the transverse velocity. In contrast, the amplitude of the thirdmode is greater downstream, where the oblique structures are observed.The dependences of the critical Reynolds numbers on the bending radius aregiven in table 2. The critical Reynolds number increases almost ten times byincreasing the bending parameter, and when δ = 2 . it has the same order ofmagnitude as for the plane flow. 12 . − . − . − . − . − . − . − . − . . .
01 0 . σ βRe = 300 Re = 500 Re = 700 Re = 900 Re = 1100 Figure 7: The dependencies of the growth σ from the spanwise wavenumber β , δ = 1 − . − . − . − . − . − . . .
001 0 .
01 0 . σ β . . . . . . . . . .
001 0 .
01 0 . ω β (a) (b)Figure 8: The dependences of the growth σ (a) and the frequency ω (b) from thespanwise wavenumber β for the free leading eigenmodes, δ = 1 . Diamonds arefor the real mode and circles and squares for the complex modes. Re = 1100 , δ = 1 This section considers the stability of the two-dimensional perturbations at δ = 1 . The base steady-state flow was calculated up to Re ∼ by time-13 . − . − . − . − . − . − . . − . − . − .
05 0 0 .
05 0 . . σ ω Figure 9: The eigenvalues at Re = 1100 , β = 1 . , δ = 1 integrating the equations (1). For larger Reynolds numbers, the adaptive selec-tive frequency dumping method described above was applied. With this method,it was possible to calculate the steady-state flow up to Re = 1900 .The most dangerous is the monotonic mode. Figure 11 shows the dependen-cies of σ from Re . In this graph, all σ is less than zero, so the two-dimensionalperturbations are stable up to Re = 1900 . A dashed line represents a depen-dency σ ( Re ) for ≤ Re ≤ using the least squares method. The appro-priate critical Reynolds number is Re ∗ = 3495 . Figure 12 shows the streamlinesof the base flow(a) and the most dangerous mode(b) at Re = 1600 , and thevertical velocity amplitude(c). Thus, the most dangerous mode is localized atdownstream the recirculation bubble V . This section presents the results of a study of the pulsations that are observedin the outlet branch at
Re > . Matsumoto et al. [7] have found such non-14a)(aa)(b)(bb)(c)(cc)Figure 10: The three leading eigenmodes at Re = 1100 : the streamwise vortic-ity(a,b,c) and the spanwise velocity(aa,bb,cc)stationary patterns in the sharp bend. Zang&Poth´erat and Sapardi et al. [5, 6]found such vortex shedding in the sharp 180-degree bend. At the same Reynoldsnumber Re > at δ = 1 , there can be either the steady-state flow or theperiodic flow. At the present time, it is not entirely clear under what conditions15 . − . − . − . − . − . − . − . − .
010 500 1000 1500 2000 σ Re Figure 11: The growth σ form Re for the 2D disturbances, the dashed line isthe least squares approximation for Re > (a)(b)(c)Figure 12: The stabilized flow streamlines at Re = 1600 (a), its leading eigen-function (b) and the vertical velocity amplitude (c)the vortex shedding occurs in the two-dimensional bent flow. Sapardi et al.have found that the large hysteresis for the critical Reynolds number dependsupon the initial conditions. They found for a special case, that Re (cid:48)∗ ≈ for calculations from rest, or the lower Reynolds number steady-state velocityfield, and Re (cid:48)∗ = 743 for calculations from the unsteady initial condition. Theapostrophe here means that they calculated the critical Reynolds number bythe channel width and Re = 0 . Re (cid:48) . 16uring routine calculations, I have observed that an increase in the qualityof the approximation, i.e., a decrease in the time step, an increase in the or-der of approximation p , or the number of mesh elements, suppresses the vortexshedding. I also have found out that the flow is linearly stable for Re < .Therefore, it is possible to assume that these pulsations arise due to the insta-bility of the flow under the influence of finite amplitude perturbations imposedby the numerical scheme noise.To prove this hypothesis, the perturbation was imposed as the force F (see(1)) of the random amplitude A at each point in the physical space. The valuesof this force were updated at each time step. Figure 13 shows the noise responseas the dependencies of the maximum vertical velocity amplitude on the centreline of the outlet branch from the noise amplitude. The curve U = 0 shows theresponse in the case of the rest fluid, when there is no gain due to the instability.In the case of Re = 200 this response is smaller than for the rest fluid, and thusthe perturbations do not grow. In the cases Re = 500 and Re = 800 theperturbations are amplified, and when Re = 800 , the motion amplitude is times greater compared to the rest fluid.Figure 14 shows graphs of the kinetic energy of the perturbation under thenoise with the amplitude A = 10 − . At Re = 10 , the perturbation energy isslightly lower than at Re = 500 , and for these cases the energy of the forcedmotion remains relatively small. At Re = 800 , the energy of the perturbedmotion at the initial step has the same level as at Re = 500 . At t ≈ thisenergy increases at least times and the oscillatory motion continues at thisenergy level. If the noise was been switched off at time t = 200 , the pulsatingmotion faded. Thus, the pulsations must be supported by permanent externalimpulsion.Figure 15 presents the vertical velocity on the outlet branch centreline at17everal consecutive moments in time after the noise is turned on. Each periodof the graph corresponds to one vortex. Vortices are arising near the trailingedge of V . The velocity amplitude of each vortex increases downstream andreaches a maximum at x > . Then each vortex moves with its own amplitudeuntil it leaves the channel. The amplitude of the vortices along the longitudinalcoordinate x fluctuates in some way, not quite exactly periodically. This meansthat the vortex path is modulated in amplitude in the longitudinal direction, inother words, a sequence of vortex packets forms in the channel. − − − − − − − − − − − m a x ( v y ) A U = 0 Re = 200 Re = 500 Re = 800 Figure 13: The response for the random noise as the vertical velocity v y maximalamplitude in the outlet branch centre line for the motionless fluid and Re = 200 , , Two-dimensional flow regimes in a curved channel have been investigated. It wasfound that as the Reynolds number increases, three recirculation bubbles appearsequentially. With a further increase in the Reynolds number, a regime withnon-stationary vortex shedding occurs. Such vortex generation was observed18 − − − − − − − E time Re = 10 Re = 500 Re = 800 Re = 800 , T = 200 Figure 14: The disturbance energy for Re = 10 , , for the permanentnoise amplitude A = 10 − and, for the case Re = 800 , the noise had beenswitched off at t = 200 − . − . − . − . . . . . .
05 0 20 40 60 80 100 120 V y x t = 50 − . − . − . . . .
15 0 20 40 60 80 100 120 V y x t = 100 − . − . − . . . .
15 0 20 40 60 80 100 120 V y x t = 500 − . − . − . . . .
15 0 20 40 60 80 100 120 V y x t = 1000 Figure 15: The graphs of the vertical velocity v y at the outlet branch centrelinefor t = 50 , , , , Re = 1000 earlier in a channel with a back-face step, and in a sharp bend.The non-stationary pulsating flow was observed at Re > . The vortexshedding is being developed by the approximation noise. By reducing the cal-culation error the steady-state flow was found up to Re ≈ in the presentcalculations. At < Re < , the steady-state solution was obtainedusing the selective frequency dumping method. Thus, at < Re < two19ifferent types of flow can evolve: the steady-state and the quasi-periodical.The linear instability occurs due to a monotone mode, which is localized inthe separation bubble V . The greatest value of the growth is observed at thespanwise wave number β ≈ . There are less dangerous oblique periodic modes,they can be localized in the region of the V bubble or located downstream. Thetransition to the three-dimensional motion is caused by centrifugal instabilityin the V vortex, as described in [2].The critical Reynolds numbers by a linear approach in the two-dimensionalcase are at least . times larger than in the three-dimensional case. However,the effect of two-dimensional nonlinear mechanisms of instability amplificationis large at the Reynolds numbers, which correspond to the critical numbers forthree-dimensional perturbations according to the linear theory. For example,Zhang&Poth´erat [5] and Blackburn et al.[3] observed by direct numerical simu-lation three-dimensional flows with a noticeable role of two-dimensional dynam-ics. In some cases, the two-dimensional nonlinear mechanism of perturbationamplification can have a fundamental effect on the three-dimensional stabilityof the flow. References [1] V. Theofilis. Global linear instability.
Annual Review of Fluid Mechanics ,43:319–352, 2011.[2] Dwight Barkley, M Gabriela M Gomes, and Ronald D Henderson. Three-dimensional instability in flow over a backward-facing step.
Journal of FluidMechanics , 473:167–190, 2002.[3] Hugh Maurice Blackburn, Dwight Barkley, and Spencer J Sherwin. Con-vective instability and transient growth in flow over a backward-facing step.20 ournal of Fluid Mechanics , 603:271, 2008.[4] Lambros Kaiktsis, George Em Karniadakis, and Steven A Orszag. Un-steadiness and convective instabilities in two-dimensional flow over abackward-facing step.
Journal of Fluid Mechanics , 321:157–187, 1996.[5] Lintao Zhang and Alban Poth´erat. Influence of the geometry on the two-and three-dimensional dynamics of the flow in a 180 sharp bend.
Physicsof Fluids , 25(5):053605, 2013.[6] Azan M Sapardi, Wisam K Poth´erat, and Gregory J Sheard. Linear sta-bility of confined flow around a 180-degree sharp bend. arXiv preprintarXiv:1708.08896 , 2017.[7] Daichi Matsumoto, Koji Fukudome, and Hirofumi Wada. Two-dimensionalfluid dynamics in a sharply bent channel: Laminar flow, separation bubble,and vortex dynamics.
Physics of Fluids , 28(10):103602, 2016.[8] Donghun Park and Seung O Park. Streamwise streaks and secondary insta-bility in a two-dimensional bent channel.
Theoretical and ComputationalFluid Dynamics , 28(3):267–293, 2014.[9] George Em Karniadakis, Moshe Israeli, and Steven A Orszag. High-ordersplitting methods for the incompressible Navier-Stokes equations.
Journalof computational physics , 97(2):414–443, 1991.[10] David Moxey, Chris D Cantwell, Yan Bao, Andrea Cassinelli, GiacomoCastiglioni, Sehun Chun, Emilia Juda, Ehsan Kazemi, Kilian Lackhove,Julian Marcon, et al. Nektar++: Enhancing the capability and applicationof high-fidelity spectral/hp element methods.
Computer Physics Commu-nications , 249:107110, 2020. 2111] Espen ˚Akervik, Luca Brandt, Dan S Henningson, J´erˆome H œ pffner, OlafMarxen, and Philipp Schlatter. Steady solutions of the Navier-Stokes equa-tions by selective frequency damping. Physics of fluids , 18(6):068102, 2006.[12] Bastien E Jordi, Colin J Cotter, and Spencer J Sherwin. An adaptiveselective frequency damping method.
Physics of Fluids , 27(9):094104, 2015.[13] Bastien E Jordi, Colin J Cotter, and Spencer J Sherwin. Encapsulatedformulation of the selective frequency damping method.
Physics of Fluids ,26(3):034101, 2014.[14] Alexander V Proskurin. Mathematical modelling of an unstable bentflow using the selective frequency damping method. arXiv preprintarXiv:2011.02646 , 2020.[15] Alexander V Proskurin and Anatoly M Sagalakov. A numericalapproach for transient magnetohydrodynamic flows. arXiv preprintarXiv:1911.11909 , 2019.[16] Alexander V Proskurin and Anatoly M Sagalakov. The evolution of non-linear disturbances in magnetohydrodynamic flows. In