Homotopy between exact coherent structures in shear flows

It is found that mirror-symmetric exact coherent structures exist throughout a homotopy continuation path between two canonical flows without linear instability; plane Couette flow and Hagen–Poiseuille flow . Mirror-symmetric exact coherent structures are also identified in plane Poiseuille flow.


Introduction
Study of subcritical transition from laminar to turbulent states in shear flows has attracted significant attention in fluid mechanics in recent years. Among uni-directional shear flows, plane Couette flow (PCF) and Hagen-Poiseuille flow (HPF), are special in that no linear instability of their laminar states exists at any finite Reynolds numbers ( [1] for PCF and [2] for HPF), and therefore no smooth transition to other nontrivial equilibrium states is possible according to the implicit function theorem. Yet, instabilities are observed in both flows ( [3] in PCF and [4] in HPF). This puzzling mathematical property has been presenting fundamental difficulties in understanding how transition to turbulence occurs in these flows from the early 20th century.
In 1990, [5] made a breakthrough by numerically finding nonlinear equilibrium solutions of the Navier-Stokes equations for PCF. The key to the breakthrough was the fact that the linear stability of the laminar flow does not exclude the possibility of nonlinear states that do not have any connection to the laminar flow. This work showed for the first time that nonlinear solutions could abruptly show up via a saddle-node bifurcation at a certain finite Reynolds number. More specifically, in [5], the solution branch is continued from a rotating flow problem where the linear instability of the laminar state exists and thus nonlinear solution branches can be followed successively by bifurcation analysis. Then, one of the nonlinear solution branches has been successfully continued to the original norotation problem. This strategy to find a nonlinear solution of one system through a continuous deformation of the known solution of another system, is called homotopy method. About a decade later, nonlinear solutions on such disconnected branches were found in HPF as well [6,7].
As various nonlinear states in various shear flows became available (e.g. [8][9][10] in PCF, [11][12][13] in plane Poiseuille flow (PPF), and [14] in HPF), the study of such nonlinear solutions, called exact coherent structures (ECS), has gathered considerable attention because it may offer a new way to explain subcritical transition to turbulence in shear flows. Our particular attention in the current article is focused on seeking possible nonlinear solution links between the available ECS's in PCF , PPF and HPF.

Formulation of the problem
We consider the motion of an incompressible viscous fluid with the constant density q and kinematic viscosity m between two horizontal parallel plates of infinite extent separated by the distance 2d Ã , subject to a system rotation of constant rate X Ã about a spanwise axis. The motion is induced by moving the plates in the opposite directions with a constant speed U Ã (PCF) and/or applying a constant pressure gradient G Ã in the direction of the motion of the bottom plate (PPF). Adopting a Cartesian coordinate with origin on the mid-plane between the plates, we non-dimensionalise the Navier-Stokes equations and incompressibility condition expressed in the rotating frame of reference: In these expression velocity u has components u i in the and wall-normal directions, respectively, t denotes time and p is a modified pressure that includes the centrifugal force term. All lengths and time have been scaled by the half-gap d Ã and the viscous diffusion time d Ã2 =m, respectively, while velocities has been scaled by m=d Ã . The rotation number X :¼ 2X Ã d Ã2 =m has been introduced, where X ¼ ½0; X; 0 denotes the system rotation vector. Subject to no-slip boundary condition at the plates a laminar basic flow solution is given by u ¼ U B ¼ ½U B ðzÞ; 0; 0, in which U B ¼ ÀRe z þ R P ð1 À z 2 Þ where Re :¼ U Ã d Ã =m is the Reynolds number induced by the motion of the plates, and R P : ¼ G Ã d Ã3 =2qm 2 is the Reynolds number induced by the pressure gradient.
The total flow field is decomposed into the basic state and a disturbance flow field fũ;pg, which satisfies subject to the boundary condition,ũ ¼ 0 at z ¼ AE1 .
Following [15] we expressũ as u ¼ũî þ r Â r Â ð/kÞ þ r Â ðwkÞ; ð5Þ where Á ab Á d x d y denotes the average over the periodic domain given by the streamwise and spanwise wavenumbers, a and b. Note that the expression (5) ensures the incompressible condition (4). The equations for the scalar variables, /; w andũ, are obtained by operations,k Á r Â rÂ(3),k Á rÂ (3) andî Á ð3Þ, respectively. The boundary condition (4) becomesũ In order to solve the resulting three partial differential equations numerically we expand the scalar variables /; w andũ by the following truncated sums: 'mn U ' ðzÞ exp½iðmaðx À ctÞ þ nbyÞ; ð6Þ wðx; y; z; tÞ ¼ X M m ¼ ÀM ðm; nÞ 6 ¼ ð0; 0Þ 'mn W ' ðzÞ exp½iðmaðx À ctÞ þ nbyÞ; '00 U ' ðzÞ; ð8Þ where the basis functions, U ' ¼ ð1 À z 2 Þ 2 T ' ðzÞ and W ' ¼ U ' ¼ ð1 À z 2 ÞT ' ðzÞ with T ' ðxÞ being the '-th Chebyshev polynomial of the first kind, satisfy no-slip condition at z ¼ AE1. Here, we have assumed thatũ ¼ ðũ;ṽ;wÞ andp take a travelling-wave form propagating in the streamwise direction with the wave speed c, After substituting (6), (7) and (8) into the three partial differential equations, we discretize the equations by using a standard Chebyshev-collocation and Fourier-Galerkin technique, namely the equations projected on the x-and y-directions, are evaluated at the collocation points, LÞ. Then, we solve the resulting quadratic algebraic equations for the spectral coefficients X ðjÞ 'mn ; ðj ¼ 1; 2; 3Þ by the Newton-Raphson iteration method. The wave speed c is unknown, but it can be swapped with the real or imaginary part of an appropriate X ðjÞ 'mn by resetting the origin of time t.
The momentum transport on the top plate, where the prime denotes the differentiation with respect to z, and the friction factor, are chosen as nonlinear measures (see [16] for further computational details).

Exact coherent structures in plane Couette flow and plane Poiseuille flow
In this section we first establish nonlinear solution branches in the rotating problems. Then, by treating the rotation number X as a homotopy parameter, solutions on the solution branches are followed as X ! 0 so as to recover ECS's in PCF and PPF.

Rotating plane Couette flow
No linear instabilities are known to PCF. However, the flow can be linearly unstable subject to a system rotation about a spanwise axis. The linear stability of rotating plane Couette flow to a streamwise-independent perturbation is determined by a single parameter, called the Taylor number Ta :¼ XðRe À XÞ. Figure 1a shows two loci of the critical Taylor numbers, Ta ¼ Ta ð1Þ c ¼ 106:735 and Ta ¼ Ta ð2Þ c ¼ 1100:650, on the ðX; ReÞ-plane, at which nonlinear streamwise-independent flows bifurcate. The corresponding critical spanwise wavenumbers are b ð1Þ c ¼ 1:5582 and b ð2Þ c ¼ 2:6823, respectively. Typical nonlinear states, referred to as Taylor vortex flow type I (TV 1 ) and type II (TV 2 ), are presented in Fig. 1b. It is found that TV 1 is characterised by a single-layered array of vortex rolls, in comparison with a doublelayered array for TV 2 . Three-dimensional tertiary flows bifurcate from these secondary flows. With a suitable choice of parameters, Re; X; a and b, tertiary solution branches reach the line of X ¼ 0, creating nonlinear solutions of PCF, as exemplified in Fig. 2.
The tertiary solution branch, denoted by WVF in Fig. 2, corresponds to the wavy vortex flow in the Taylor-Couette system. The solution at X ¼ 0 on WVF, called Nagata solution [5], is discovered for the first time as described in Introduction. The solution on the other tertiary solution branch, denoted by Ribbon in the figure, is characterised by the spanwise mirrorsymmetry [17]. Although the existence of Ribbon had been recognized by [18], the mirror-symmetric solutions of PCF themselves were found recently by means of different types of homotopy approaches [8,9].
When X ¼ 0, these two types of solutions appear via a saddle-node bifurcation as shown in Fig. 3. Nagata solution in Fig. 3a is known to possess the following symmetries, the shift-reflection symmetry, S : ½u; v; wðx; y; zÞ ¼ ½u; Àv; wðx þ p=a; Ày þ p=b; zÞ; ð11Þ and the shift-rotation symmetry, X : ½u; v; wðx; y; zÞ ¼ ½Àu; v; ÀwðÀx; y þ p=b; ÀzÞ: The mirror-symmetric solutions in blue in Fig. 3b possess an additional symmetry, called the mirrorsymmetry, Z y : ½u; v; wðx; y; zÞ ¼ ½u; Àv; wðx; Ày; zÞ: We denote these solutions by P Ã pcf and M Ã pcf (an asterisk indicating the steady state), respectively. Also Meccanica (2016) 51: 3015-3023 3017 shown in red in Fig. 3b is the travelling-wave solution branch, which is found during the continuation approach from plane Couette flow to sliding Couette flow to be discussed in Sect. 5.1 below. This solution is denoted by M pcf since the mirror-symmetry is preserved.

Rotating plane Poiseuille flow
In contrast to PCF, PPF can be linearly unstable. The critical Reynolds number R Pc is 5772.22 for a spanwise-independent perturbation and the corresponding critical streamwise wavenumber is a c ¼ 1:02.
With the inclusion of a spanwise system rotation X, the critical Reynolds number is reduced drastically. Streamwise-independent perturbations cause this instability. [19] calculated ðR Pc ; b c Þ ¼ ð66:40; 2:45Þ when X ¼ 22:133. Figure 4 shows that the marginal stability curve corresponding to ða; bÞ ¼ ð0; 2:5Þ attains the smallest Reynolds number among other combinations of wavenumbers in the figure. Secondary flows bifurcate from these marginal curves. Figure 5a shows the bifurcations of three-dimensional tertiary flows, G 1 ; . . .; G 5 , from the two-dimensional secondary flow D 1 with b ¼ 2:5 when X ¼ 22:1325. Three-dimensional flows can also bifurcate direct from the basic laminar state as exemplified by G 11 ; . . .; G 15 in Fig. 5b.
The shift-rotation symmetry (12) is broken for all of these three solutions because of the symmetric flow profile, U B ðÀzÞ ¼ U B ðzÞ, for PPF. TW1 possesses the shift-reflection symmetry S only, while TW2 possesses the symmetry S and the mirror symmetry Z y . TW3 possesses the symmetry S and a new symmetry Z yz , Z yz : ½u; v; wðx; y; zÞ ¼ ½u; Àv; Àwðx; Ày þ p=b; ÀzÞ: The saddle-node bifurcations of these nonlinear states are shown in Fig. 6.

Homotopy from plane Couette flow to plane Poiseuille flow
Homotopy continuation from PCF to PPF is straightforward by adjusting two Reynolds numbers, Re and R P . As soon as the Poiseuille component is added the shift-rotation symmetry (12) is broken. Nagata solution, P Ã pcf , in PCF was used as a starting solution for the homotopy by [11], who considered the combination of the two Reynolds numbers as where PCF is given when the homotopy parameter l ¼ 0 with R 0 ¼ Re. The half channel PPF is retrieved in z 2 ½À1; 1 with dU B dz j z¼1 ¼ 0 when l ¼ 1 with R 0 ¼ R P . The flow which reached the PPF limit is referred to as W03 here. W03 inherits the shiftreflection symmetry of Nagata solution. With the half channel restriction, W03 is bound to possess the topbottom symmetry, Z z , additionally: Z z : ½u; v; wðx; y; zÞ ¼ ½u; v; Àwðx; y; ÀzÞ: ð16Þ The steady and travelling-wave mirror-symmetric states, M Ã pcf and M pcf , in PCF were continued to PPF by [12], where the basic flow, is considered. The solution that reached the PPF limit was referred to as MS-S in [12] and had symmetries, S, Z y and Z z . [12] also found another mirror-symmetric state, called MS-A, which bifurcated from MS-S. The top-bottom symmetry Z z was broken for MS-A. The saddle-node bifurcations of these solutions are presented in Fig. 7.

Homotopy continuation from plane Couette flow to Hagen-Poiseuille flow
A nonlinear solution link between PCF and HPF is sought in this section. We choose sliding Couette flow (SCF), i.e. flow between concentric cylinders in mutual relative motions in the axial directions, as an intermediate between these two flows that are very different geometrically.

Homotopy from plane Couette flow to sliding Couette flow
As in the previous section we consider an incompressible viscous motion of fluid with the density q and kinematic viscosity m. Fluid occupies the gap between two infinitely long concentric cylinders with radii a Ã and b Ã ð¼a Ã þ 2d Ã Þ. The flow is driven by moving the cylinders in the opposite axial directions with the same speed U Ã and/or applying an axial pressure gradient G Ã . Subject to no-slip on the cylinders, the basic flow in non-dimensional form is given by where g ¼ r i =r 0 is the radius ratio with r i and r 0 being the inner and outer cylinder radii, respectively. Re is the Reynolds number induced by the sliding motion of the cylinders and R P is the Reynolds number induced by the axial pressure gradient, defined by Re :¼ U Ã d=m and R P :¼ G Ã d 3 =2qm 2 , respectively, as in the planer case. U B takes its maximum value at r ¼ r p ¼ r o ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðg 2 À 1Þ=ð2 ln gÞ p when Re ¼ 0. As in the last section we superimpose disturbances on the basic flow. Disturbances are decomposed into a mean part and residuals with toroidal and poloidal potentials. They are expressed by a Fourier-Chebyshev spectral expansion. To discretize the disturbance equations, collocation method is used. The resultant algebraic equations for the expansion coefficients are solved numerically by Newton's method. The momentum transport M :¼ Àðr 0 =r m ÞðU B þũÞ 0 j r¼r 0 on the outer cylinder is chosen as a nonlinear measure of the solution where r m ¼ ð1 þ gÞ=ð1 À gÞ is the mean radius.
A continuation to SCF has been made by using the steady mirror-symmetric solution M Ã pcf in PCF. The upper and the lower branch M Ã pcf solutions, indicated by black dots in Fig. 8 are the starting points. The orange arrows in the figure indicate the continuation directions taken, where the radius ratio g is a homotopy parameter. In the annulus geometry, M Ã pcf solution in the plane geometry turns into a travellingwave form. It possesses the shift-reflection symmetry, S : ½u; v; wðx; h; rÞ ¼ ½u; Àv; wðx þ p=a; Àh þ p; rÞ; ð19Þ and the mirror-symmetry, Z h : ½u; v; wðx; h; rÞ ¼ ½u; Àv; wðx; Àh; rÞ: The shift-rotation symmetry is broken because the reflection about the mean radius r ¼ r m corresponding to z ¼ 0 in (12) becomes asymmetric due to the curvature in the cylindrical coordinates. We denote this travelling-wave solution by M scf . As can be seen in Fig. 8, this particular attempt was not successful in the sense that the continuation routes did not reach the wide gap limit, g ! 0. Instead, the routes turned back and ended up at the blue dot in the PCF limit, producing the travelling-wave form of the mirror-symmetric solution M pcf as a by-product [10].
The homotopy continuation using Nagata solution P Ã pcf was attempted by [20]. It is shown there that the continued solution branch P scf terminates on a mirrorsymmetric travelling-wave solution branch M scf . This mirror-symmetric branch separates into two solution branches with different flow structures in a complicated manner in the wavenumber space: one of them preserves the same single-layered vortical structure as Nagata solution and the other exhibits a doublelayered vortical structure. It turns out that the latter  belongs to the branch that approached g % 0:2 in Fig. 8.

Homotopy from sliding Couette flow
to Hagen-Poiseuille flow The following two steps are required for the homotopy from SCF to HPF: (I) To add an axial pressure gradient so that the basic velocity profile becomes a parabola-like shape. (II) To eliminate the inner cylinder.
For the requirement (I), we gradually increase the pressure-based Reynolds number, R P , until the constraint is met for a fixed Re. With this constraint the basic flow profile (18) becomes a parabola-like shape with vanishing shear on the inner cylinder, i:e: d U B ðrÞ= d rj r¼r i ¼ 0. This choice of the basic flow is convenient because the profile coincides with that of HPF, U B ðrÞ ¼ R P ðr 2 o À r 2 Þ=2 þ Re, in the limit g ! 0. Recall that the pipe is moving in the axial direction with a constant speed Re.
Regarding the requirement (II), we remove the noslip boundary effect on the disturbances and ensure the regularity of the flow at the pipe centre. In order for a function, expressed by a truncated series of the '-th polynomial basis functions as to be regular at r ¼ 0, the following two conditions must be satisfied by P m ' ðrÞ [21]. (a) Parity condition: If m is an odd/even integer, P m ' ðrÞ must be an odd/even function. (b) Order condition: The minimum order of P m ' ðrÞ ! jmj. For the disturbance flow field (5), it can be shown that /=r; w=r andũ must satisfy the regularity conditions. This leads to new basis functions U ' ; W ' and U ' . Accordingly, we introduce another homotopy parameter which bridges the two sets of basis functions, one for no-slip and the other for regularity at r ¼ 0 (see [22] for detail). Now, we complete our homotopy continuation to HPF by increasing from zero to 1 for the mirrorsymmetric solution M scf obtained in the annular geometry with vanishingly thin inner cylinder, g ¼ 0 [23]. We find that a ¼ 0:73 for m ¼ 1 optimises the bulk Reynolds number, R b ¼ R r o r i ðU B þũÞrdr=2 ¼ 192:85. These values agree well with those of the M1 solution (mirror-symmetric solution with m =1) in HPF found by [14]: R PK ¼ 773ð¼ 4 Â 193:25Þ at a PK ¼ 1:44ð¼ 2 Â 0:72Þ. Note that their length scale is different from ours: R PK ¼ 4R b and a PK ¼ 2a. We claim that our mirror-symmetric solution corresponds to M1.

Concluding remarks
A homotopy of exact coherent structures between PCF, PPF and HPF is established by developing continuous deformations between planar and circular cross-sectional shear flows. It is found that mirror- symmetric solutions exist continuously throughout the homotopy route (see Fig. 9).
The TW2 solution provides, to our knowledge, the lowest Reynolds number at which PPF solutions other than the basic flow are known to exist (the solution appears in a saddle-node bifurcation at R P ¼ 665). It is confirmed by [13] that the upper TW2 solution branch terminates on the MS-S branch of [12]. However, it still remains to be checked whether TW2 and MS-A are the same. The mirror-symmetric solution of HPF found by our homotopy continuation corresponds to that of [14].
We believe that the mirror-symmetric disturbances play an important role in transition to turbulence in shear flows, in general.