We present exact solutions of the incompressible Navier-Stokes equations in a
background linear shear flow. The method of construction is based on Kelvin's
investigations into linearized disturbances in an unbounded Couette flow. We
obtain explicit formulae for all three components of a Kelvin mode in terms of
elementary functions. We then prove that Kelvin modes with parallel (though
time-dependent) wave vectors can be superposed to construct the most general
plane transverse shearing wave. An explicit solution is given, with any
specified initial orientation, profile and polarization structure, with either
unbounded or shear-periodic boundary conditions.Comment: 6 pages, 2 figures; version published in the European Physical
  Journal Plu

A. Lifschitz

A.D.D. Craik

B. Eckhardt

K.K. Tung

P.S. Marcus

S. Sridhar

S.A. Balbus

W. Thomson (Lord Kelvin) Thomson (Lord Kelvin)

English

arXiv

Singh, N.

Sridhar, S.

MPG.PuRe

Plane shearing waves of arbitrary form: Exact solutions of the Navier-Stokes equations

We present exact solutions of the incompressible Navier-Stokes equations in a background linear shear flow. The method of construction is based on Kelvin's investigations into linearized disturbances in an unbounded Couette flow. We obtain explicit formulae for all three components of a Kelvin mode in terms of elementary functions. We then prove that Kelvin modes with parallel (though time-dependent) wave vectors can be superposed to construct the most general plane transverse shearing wave. An explicit solution is given, with any specified initial orientation, profile and polarization structure, with either unbounded or shear-periodic boundary conditions.</p

Singh, Nishant K.,

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Plane shearing waves of arbitrary form : Exact solutions of the Navier-Stokes equations

Nishant K. Singh

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DOI 10.1140/epjp/i2017-11659-5Regular ArticleEur. Phys. J. Plus (2017) 132: 403 THE EUROPEANPHYSICAL JOURNAL PLUSPlane shearing waves of arbitrary form: Exact solutions of theNavier-Stokes equationsNishant K. Singh1,2,a and S. Sridhar3,b1 Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden2 Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, D-37077 Go¨ttingen, Germany3 Raman Research Institute, Sadashivanagar, Bangalore 560 080, IndiaReceived: 20 July 2017Published online: 28 September 2017c© The Author(s) 2017. This article is published with open access at Springerlink.comAbstract. We present exact solutions of the incompressible Navier-Stokes equations in a background linearshear ﬂow. The method of construction is based on Kelvin’s investigations into linearized disturbances inan unbounded Couette ﬂow. We obtain explicit formulae for all three components of a Kelvin mode interms of elementary functions. We then prove that Kelvin modes with parallel (though time-dependent)wave vectors can be superposed to construct the most general plane transverse shearing wave. An explicitsolution is given, with any speciﬁed initial orientation, proﬁle and polarization structure, with eitherunbounded or shear-periodic boundary conditions.1 IntroductionIn 1986 Craik and Criminale [1] presented a class of exact solutions of the Navier-Stokes equations which werewavelike disturbances in background shear ﬂows. Since then these solutions have proved extremely useful in thestudy of astrophysical and atmospheric ﬂuid dynamics; a very useful collection of exact solutions can be found in [2].The approach taken in [1] was a generalization of a century-old method invented by Kelvin [3] to study linearizedperturbations of Couette ﬂows; see also [4]. These shearing wave solutions, also referred to as Kelvin modes, havetime-dependent wave vectors and amplitudes. This feature makes them extremely useful in local stability analysis [5,6]. Although a single Kelvin mode is an exact solution of the full Navier-Stokes (NS) equations, it has been remarked [1]that until about 1965 there seems to be no evidence that this was so recognized; in fact, the ﬁrst published mentionis as late as 1983 [7]. Moreover, an explicit formula has been published [3,1] for only one of the three components ofthe disturbance.In this paper we present exact solutions for all three components of the velocity ﬁeld of a Kelvin mode, in closedform using only elementary mathematical functions. We identify a subset of these modes whose wave vectors —thoughtime-dependent— remain parallel to each other for all time. These are used to synthesize the most general planetransverse shearing wave, which can have any speciﬁed initial orientation, proﬁle and polarization structure, witheither unbounded or shear-periodic boundary conditions.Let (eˆ1, eˆ2, eˆ3) be the unit basis vectors of a Cartesian coordinate system in the laboratory frame. Using notationx = (x1, x2, x3) for the position vector and t for time, we write the total ﬂuid velocity as (Sx1eˆ2 + v), where S is therate of shear parameter and v(x, t) is the incompressible disturbance (∇ · v = 0), which obeys the NS equations(∂t + Sx1∂2)v + Sv1eˆ2 + (v · ∇)v = −∇p + ν∇2v,∇2p = −∇ · [(v · ∇)v]− 2S∂2v1. (1)a e-mail: singh@mps.mpg.deb e-mail: ssridhar@rri.res.inPage 2 of 5 Eur. Phys. J. Plus (2017) 132: 403We seek a solution in the form of a single Kelvin modevk(x, t) = Re{A(k, t) exp[iksh(t) · x]} ,pk(x, t) = Re{ψ(k, t) exp[iksh(t) · x]} , (2)where the time-dependent sheared wave vector, ksh(t), has componentsksh1 = k1 − Stk2, ksh2 = k2, ksh3 = k3, (3)with k ≡ (k1, k2, k3) being a constant wave vector. Our task now is to determine the amplitudes A(k, t). Incompress-ibility requires that ksh(t) · A(k, t) = 0. Therefore, when eqs. (2) and (3) are substituted in eqs. (1), the nonlinearterm, (v · ∇)v vanishes because(A · ∇) exp [iksh(t) · x] = (iksh(t) · A) exp [iksh(t) · x] = 0. (4)The pressure can be eliminated by using the second of eqs. (1): |ksh(t)|2ψ = 2iSk2A1, where |ksh(t)|2 = [(k1−Stk2)2+k22 + k23]. Then A satisﬁes∂tA + SA1eˆ2 = 2S(k2ksh(t)|ksh(t)|2)A1 − ν∣∣ksh(t)∣∣2 A. (5)We now obtain explicit solutions for A. To do this, deﬁne a new amplitude variable, a(k, t), byA(k, t) = G˜ν(k, t)a(k, t), (6)where G˜ν(k, t) is a Fourier-space viscous Green’s function,G˜ν(k, t) = exp[−ν∫ t0ds∣∣ksh(s)∣∣2]= exp[−ν(k2t− Sk1k2t2 + S23k22t3)]. (7)When eqs. (6) and (7) are substituted in eq. (5), we obtain the following equations for the three components of a(k, t):∂ta1 − 2S[(k1 − Stk2)k2(k1 − Stk2)2 + k22 + k23]a1 = 0, (8)∂ta2 − 2S[k22(k1 − Stk2)2 + k22 + k23− 12]a1 = 0, (9)∂ta3 − 2S[k2k3(k1 − Stk2)2 + k22 + k23]a1 = 0. (10)Equation (8) can be solved to get an explicit expression for a1(k, t),a1(k, t) =k2(k1 − Stk2)2 + k22 + k23a1(k, 0), (11)which is given in [3]. When this is substituted in eqs. (9) and (10), the latter can be integrated to obtain expressionsfor a2(k, t) and a3(k, t). However, neither Kelvin nor anyone else, to the best of our knowledge, have published explicitformulae for these two components1. Thus we were pleasantly surprised to ﬁnd that a2(k, t) and a3(k, t) could beexpressed entirely in terms of elementary functions:a2(k, t) = a2(k, 0) +{k2k23k2(k22 + k23)3/2[arctan(k1 − Stk2√k22 + k23)− arctan(k1√k22 + k23)]− k2k2k22 + k23[k1 − Stk2(k1 − Stk2)2 + k22 + k23− k1k2]}a1(k, 0), (12)a3(k, t) = a3(k, 0)−{k2k3(k22 + k23)3/2[arctan(k1 − Stk2√k22 + k23)− arctan(k1√k22 + k23)]+k2k3k22 + k23[k1 − Stk2(k1 − Stk2)2 + k22 + k23− k1k2]}a1(k, 0). (13)1 Markus and Press [4] study perturbations of plane Couette ﬂow using Kelvin waves. However, their analysis is limited totwo-dimensional perturbations, whereas the shearing waves we consider here are fully three-dimensional.Eur. Phys. J. Plus (2017) 132: 403 Page 3 of 5Fig. 1. Plots of the three components of the velocity ﬁeld, measured at the origin, as functions of St. We have chosen k = (1, 1, 1)and a(k, 0) = (1, 0,−1). The bold lines are for v1(0, t), the dotted for v2(0, t), and the dashed for v3(0, t). Panel (a) is for the non-viscous case, ν = 0, so the velocity components are identical to the amplitudes, a(k, t), of eqs. (11)–(13). Panel (b) correspondsto (νk2/S) = −0.1, and all three components ultimately suﬀer viscous decay.Incompressibility requires that ksh(t) · a(k, t) = 0, which is guaranteed if the initial conditions are chosen such thatk · a(k, 0) = 0. From eqs. (11)–(13), we can see that, at late times, a1(k, t) → 0, whereas both a2(k, t) and a3(k, t)saturate at non-zero values. This happens because the background ﬂow shears out the a1 component, and generatesthe a2 and a3 components.When eqs. (6), (7), (11)–(13) are substituted in eq. (2), we obtain the full velocity ﬁeld of a single Kelvin mode;it is readily veriﬁed that structure of the mode depends on the dimensionless variable, St, and the dimensionlessparameter, (νk2/S). The spatio-temporal behavior of these modes is brieﬂy explored through ﬁgs. 1 and 2. In orderto understand its time variation, it is convenient to measure the velocity components at the origin, as is done in ﬁg. 1.Then, v(0, t) = G˜ν(k, t)Re{a(k, t)}. Figure 1(a) corresponds to the case of zero viscosity, (νk2/S) = 0. In this caseG˜ν = 1, and the plots give v(0, t) = Re{a(k, t)}, where we can see the decay of a1 and the saturation of a2 and a3discussed above. In ﬁg. 1(b), we have chosen (νk2/S) = −0.1, so that all three components of v(0, t) ultimately suﬀerviscous decay. It can be seen that, before this decay, there is transient ampliﬁcation of v2 and v3, due to competitionbetween shear and viscosity. For larger values of viscosity (not shown here), this transient ampliﬁcation may be absentbecause the damping can overwhelm shear.Until now we have considered an unbounded ﬂow. However, in numerical simulations of the local dynamics of diﬀer-entially rotating discs in astrophysical systems [8,9], it is customary to employ “shear-periodic” boundary conditions.Let us deﬁne sheared coordinates byxsh1 = x1, xsh2 = x2 − Stx1, xsh3 = x3. (14)These may be thought of as the Lagrangian coordinates of ﬂuid elements that are carried along by the backgroundshear ﬂow. A function is said to be shear-periodic when it is a periodic function of (xsh1 , xsh2 , xsh3 ) with periodicities(L1, L2, L3), respectively. The phase of the function vk can be written as ksh(t)·x = k ·xsh. Therefore, a shear-periodicKelvin mode has wave vectors k ∈ (2πm1/L1, 2πm2/L2, 2πm3/L3), where the mi take any integer values.We now use the explicit expressions obtained for the Kelvin modes to construct the most general plane transverseshearing wave. Let us consider two Kelvin modes, vk(x, t) and vk′(x, t) corresponding to wave vectors k and k′, whichare parallel to each other but could diﬀer in magnitudes. Using eqs. (3), we see that the corresponding sheared wavevectors, ksh(t) and k′sh(t), are also parallel to each other for all time. Incompressibility implies that vk(x, t) andvk′(x, t) are perpendicular to ksh(t) and k′sh(t) for all time. So, if we superpose vk(x, t) and vk′(x, t), the nonlinearterm in the NS equations vanishes, because the superposed velocity ﬁeld remains parallel to the wavefronts. Thus thesuperposition of an arbitrary number of Kelvin modes, all with wave vectors parallel to each other, is an exact solutionof the NS equations.Let us choose a unit vector nˆ = (n1, n2, n3), and deﬁne the sheared (non-unit) vector nsh(t) bynsh1 = (n1 − Stn2) , nsh2 = n2, nsh3 = n3. (15)Page 4 of 5 Eur. Phys. J. Plus (2017) 132: 403Fig. 2. Evolution of plane transverse shearing wavepackets. The polarization structure of the velocity ﬁeld is indicated on somesections of the plane wavefronts. The parameters values used are S = −1, ν = 1, W0 = 1, σ = 10 and k = 1. Panels (a) and(b): Linearly polarized (h = 0) wavepackets at times t = 0 and t = 1. Panels (c) and (d): Right circularly polarized (h = 1) attimes t = 0 and t = 1.Superposing all Kelvin modes with wave vectors q = qnˆ, where −∞ < q < ∞, we obtain an exact plane-wave solutionof the NS equations with wavefronts perpendicular to nsh(t):vi(x, t) =∫ ∞−∞dq2πG˜ν(qnˆ, t) W˜i(q) exp[iqnsh(t) · x]+[Fi(nsh(t))− Fi(nˆ)n22 + n23] ∫ ∞−∞dq2πG˜ν(qnˆ, t) W˜1(q) exp[iqnsh(t) · x] , (16)where the dimensionless and scale-invariant functions, Fi(Q), are deﬁned byFi(Q) =Q3√Q22 + Q23[Q3Q2δi2 − δi3]arctan(Q1√Q22 + Q23)− Q1QiQ2. (17)For shear-periodic boundary conditions, the integral over q in eq. (16) should be replaced by an appropriate sum.The W˜ (q) are Fourier-space initial conditions corresponding to the a(k, 0) of eqs. (11)–(13), and must satisfy theincompressibility condition, nˆ · W˜ (q) = 0. They are determined by the initial proﬁle and polarization stucture of theplane wave. At t = 0, the wavefronts are perpendicular to nˆ, so we write v(x, 0) = W (nˆ · x), where nˆ · W = 0. Notethat the only constraint on the initial condition, W , is that it is a vector ﬁeld that is perpendicular everywhere toEur. Phys. J. Plus (2017) 132: 403 Page 5 of 5the unit vector nˆ; otherwise it is a quite arbitrary function of its one argument. Thus, no restriction need be placedon the initial proﬁle and polarization structure of the initial conditions. Given W (y), we can determine W˜ (q) =∫∞−∞ dy W (y) exp[−iqy], and use this in eq. (16) to calculate v(x, t).Equation (16) is a complete solution for a general plane shearing wave, expressed in terms of a Fourier integral.However, it is physically more transparent to rewrite the right side in terms of real-space quantities. To do this,we must introduce the real-space viscous Green’s function, whose natural deﬁnition is with respect to the shearedcoordinates [10]Gν(xsh, t)=∫dk(2π)3G˜ν(k, t) exp[ik · xsh] . (18)The properties of this function are discussed in [11,10], where it is shown that it takes the form of a sheared heatkernel, which is an anisotropic Gaussian function of xsh with time-dependent coeﬃcients; all the principal axes increasewithout bound and rotate against the direction of the background shear. Noting that nsh(t) ·x = nˆ ·xsh, we can writethe general form of the plane shearing wave asvi(x, t) =∫d3ξ Gν(ξ, t)Wi(nˆ · [xsh(t)− ξ]) +[Fi(nsh(t))− Fi(nˆ)n22 + n23] ∫d3ξ Gν(ξ, t)W1(nˆ · [xsh(t)− ξ]) . (19)As an illustrative example let us consider the following initial condition, corresponding to a polarized wavepacketwith wave vector pointing along the x2-axis: nˆ = eˆ2, W1(x2) = W0 exp[−x22/2σ2] sin kx2, W2 = 0, W3(x2) =hW0 exp[−x22/2σ2] cos kx2, where −1 ≤ h ≤ 1. The wavepacket is linearly polarized when h = 0, and right/leftcircularly polarized when h = ±1; other values of h correspond to diﬀerent degrees of elliptical polarizations. At alater time, the wave vector has components nsh1 = −St, nsh2 = 1, nsh3 = 0. Since both Wi and Gν(ξ, t) are Gaussianfunctions, the integrals in eq. (19) can be performed analytically and v(x, t) evaluated explicitly. We present the resultsgraphically in ﬁg. 2 for two cases, one linearly polarized and the other right circularly polarized. As the wavepacketsare sheared, they undergo transient ampliﬁcation due to the combined action of shear and viscosity, and at late timessuﬀer viscous damping.In conclusion, we have constructed exact solutions of the Navier-Stokes equation with a background linear shearﬂow. All three components of the velocity ﬁeld of the Kelvin modes are given in closed form using only elementarymathematical functions. It is demonstrated that, when Kelvin modes with parallel wave vectors are superposed, theyremain exact solutions. We give in explicit form the most general plane transverse shearing waves, with any speciﬁedinitial orientation, proﬁle and polarization structure, with either unbounded or shear-periodic boundary conditions.Of particular interest is the stability of our solutions; if they are stable then they might serve as local representationsof disturbances in simulations of astrophysical ﬂows.Open access funding provided by Max Planck Society.Open Access This is an open access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium,provided the original work is properly cited.References1. A.D.D. Craik, W.O. Criminale, Proc. R. Soc. London A 406, 13 (1986).2. P.G. Drazin, N. Riley, The Navier-Stokes equations: a classiﬁcation of ﬂows and exact solutions, in London MathematicalSociety Lecture Note Series, Vol. 334 (Cambridge University Press, 2006).3. W. Thomson (Lord Kelvin), Philos. Mag. 24, 188 (1887).4. P.S. Marcus, W.H. Press, J. Fluid Mech. 79, 525 (1977).5. A. Lifschitz, E. Hameiri, Phys. Fluids A 3, 2644 (1991).6. B. Eckhardt, D. Yao, Chaos, Solitons Fractals 5, 2073 (1995).7. K.K. Tung, J. Fluid Mech. 133, 443 (1983).8. J. Binney, S. Tremaine, Galactic Dynamics, second edition (Princeton University Press, 2008).9. S.A. Balbus, J.F. Hawley, Rev. Mod. Phys. 70, 1 (1998).10. S. Sridhar, N.K. Singh, J. Fluid Mech. 664, 265 (2010).11. F. Krause, K.-H. Ra¨dler, Elektrodynamik der mittleren Felder in turbulenten leitenden Medien und Dynamotheorie, inErgebnisse der Plasmaphysik und der Gaselektronik, Band 2 (Akademie, 1971) pp. 2–154.

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Plane shearing waves of arbitrary form: exact solutions of the
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Plane shearing waves of arbitrary form: exact solutions of the Navier-Stokes equations

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