Effect of normal magnetic field on the exact coherent state of channel flow at large Reynolds number

Laminar-turbulent subcritical transition has been always a hot issue in fluid mechanics. Exact coherent states are important for predicting the path of transition and understanding the cycle of turbulent self-sustaining process. One of the most common methods of investigating subcritical transition is dynamical system method and many different forms of invariant solutions have been obtained in many shear flows. In order to study the effect of magnetic field on the exact coherent state of channel flow at large Reynolds number, the direct numerical simulation combined with bisection method is used to calculate the exact coherent state-periodic orbit solution with different magnetic field strength at Re ⩾ 9000, and the structure and morphology in the flow field are compared. The results show that the magnetic field strength does not change the structure and shape of the solution and the scaling law between the transition threshold and Reynolds number does not change significantly, which shows obviously self-similarity. As the magnetic field strength increases, the period of the exact coherent state decreases and the perturbation energy in each direction exhibits periodical oscillation changes. In addition the shape of the amplitude curve also changes due to the magnetic field effect. The above results show that the magnetic field has a certain inhibition effect on the disturbance at large Reynolds number, and the flow field remains relatively stable.


Introduction
The three-dimensional nonlinear solution of Navier-Stokes equation plays an important role in the study of laminar-turbulent transition and turbulent flow, which has been a hot topic in fluid mechanics.The N-S equation is completely determined, but it is very difficult to accurately predict the evolution process of turbulent flow with time and the transition is usually caused by the disturbance of finite amplitude, which generates the new flow state to fluctuate in space and time.Coherent structures are widely present in most shear flows and have a long lifetime which can capture many characteristics of transition flows.Therefore, identifying and tracking the time-space evolution of coherent structures provides a new path to understand turbulence and predict laminar -turbulent transition [1].
In most shear flows, invariant solutions of various forms of the N-S equation have been observed, including but not limited to periodic orbital solutions, traveling wave solutions, and equilibrium states [2].The main components of common coherent structures are streaks and vortices.These structures have been observed in pipe flow [3], boundary-layer flow [4] and channel flow [5,6], and then streaks structures of different scales have been observed in various shear flows.There are two general methods to obtain these structures.One is through vortex identification methods [7], analyze the direct numerical simulation (DNS) or experimental data directly.The other one is to use DNS combined with the dynamical system method.
Waleffe [8] obtained for the first time the invariant solution of the N-S equation in the plane Poiseuille flow, which is composed of streamwise streaks and vortices on both sides and propagates downstream at a fixed speed.The structure is similar to the coherent structure in the flow field obtained by Jeong [9] through direct numerical simulation.Waleffe named them exact coherent states (ECS) and the structures in the flow were called exact coherent structures.Nagata [10] transformed the solution of Taylor-Couette flow into the solution of channel Couette flow and obtained the ECS and its flow field structure by using homotopy transformation method.ECS are usually unstable and difficult to be observed in experiments, but Hof [11] captured ECS in tubular flow for the first time through experiments, indicating the universality of ECS in various shear flows and relevant results were published in Science.
Toh and Itano [12] adopted the bisection method to obtain the periodic orbit solution of the Poiseuille flow.Kreilos [13] also found similar ECS in the asymptotic suction boundary layer, and a common feature of them is the presence of a sudden process [14].
The above studies mainly focus on low Reynolds number flows.With the improvement of experimental techniques and computational accuracy, the research on large Reynolds number has been gradually increased.ECS are not only useful for understanding subcritical transition and turbulence, but also important for the study of self-sustaining turbulent motion at high Reynolds numbers.Recently, it has been shown that there is a relationship between the nonlinear solution of the N-S equation and the solution of the filter equation used by LES (Large Eddy Simulation) [15], suggesting that the coherent structure may be related to the large-scale motion of turbulence which is an important characteristic of fluid flow at large Reynolds numbers.
Magnetohydrodynamics is a discipline that studies the interaction between flow field and magnetic field.Under the magnetic field, the components of physical variables in the flow field perpendicular to the direction of field strength are suppressed, while the components parallel to the direction of field strength are basically unaffected.Therefore, turbulence under the magnetic field mostly presents certain anisotropy, which is a typical feature.It is found that the subcritical transition of channel flow changes little after magnetic field is applied, and the basic structure of periodic orbit solution obtained at a small Reynolds number is not affected by magnetic field.It is proved that periodic orbit bifurcation is related to the chaotic state transition in magnetohydrodynamics [13,16].So, are ECS affected by magnetic fields at large Reynolds numbers?For this reason, high precision direct numerical simulation is used to search for ECS at high Reynolds number in channel flow, and the influence of magnetic field on ECS is investigated in detail.
Section 1 describes the physical model and equation.Section 2 analyzes ECS under the action of magnetic field and compares them with ECS without magnetic field to explore the common rules.Section 3 summarizes and discusses.

Physical models and equations.
The incompressible viscous fluid with density ρ and kinematic viscosity υ was driven to flow by pressure gradient in a channel with height of 2h and a normal uniform magnetic field B was applied in the z direction perpendicular to the flow for numerical simulation.It is worth noting that the strength of induced magnetic field caused by flow is much smaller and can be ignored compared with the applied uniform normal magnetic field, which satisfies the low magnetic Reynolds number hypothesis [17,18].
Where Re=Uh/v, Ha=Bh(σ/v), x, y, z corresponding to the direction of streamwise, spanwise and normal respectively.There were non-slip boundary conditions for the upper and lower walls, and periodic boundary conditions were set for the streamwise direction and spanwise direction.The initial condition is set as the laminar Poiseuille flow, which is disturbed by a vortex with amplitude A 1 superimposed by a spanwise wave with amplitude A 2 .
For the convenience of discussion and comparison, the initial velocity field U has the same mass flowrate as the laminar Poiseuille flow [19].The flow field is excited by the initial perturbation and turns to be turbulence when A 1 is large; the flow field grows transiently and eventually falls back to laminar flow when A 1 is small.The relative periodic orbit solution of the flow field can be obtained when A 1 is set to a suitable value.The above search process can be carried out by bisection method until A 1 meets certain accuracy requirements, and this paper requires |Δ |≤1e-10.

Results
At first, exact coherent states of channel flows with different magnetic field strength have been obtained by direct numerical simulation.In previous studies [20] periodic orbit solutions of Reynolds numbers ranging from Re=3000 to 8000 have been calculated respectively before and after the application of normal magnetic field in the domain of 2π×2.4×2 and grid resolution of 32×32×64.In order to further explore the influence of normal magnetic field on subcritical transition process under large Re, the nonlinear solutions under different magnetic field strengths with Re≥9000 are calculated in the same computing domain.Due to the suppression effect of the normal magnetic field, the flow may be in laminar, transitional or weak turbulent state at a large Re when the magnetic field intensity is large enough.After application of normal magnetic field, the more appropriate dimensionless parameter to judge the flow field state is R=Re/Ha [17], and in general it is determined that R>400, the channel flow is turbulent; R<400, the flow is laminar [13].For higher Re, the computation requires a higher resolution to ensure that the solution can convergent.After several verifications, when Re>10000, the resolution used for the calculation needs to be increased to 32×32×128 to ensure the convergence of the solution.By dichotomously optimizing the amplitude A 1 of the initial perturbation, the exact coherent state of the channel flow and its corresponding flow field structure can be obtained near a suitable value of A 1 .The result of bisection optimization at Re=9000 is shown in Figure 1.where K'= E p /E, E is the total energy of the flow field and E p is the total disturbance energy.The flow field rapidly transits to turbulence (blue line) when A is higher than the threshold value A 1 and the flow field falls back to laminar flow (green line) when A is lower than the threshold value A 1 .But after a long enough time at a suitable threshold A 1 , the edge state (red lines about 3E-4) of laminar and turbulent flows will be obtained, and it corresponds to the periodic orbital solutions of the N-S equation.
The evolution of the flow field corresponding to the periodic orbit is shown in Fig. 2, taking the plane of x=0, z=0 at Re=3000(top) [20] and Re=9000(bottom) respectively.The flow velocity component u exhibits an obvious periodic variation, and it is worth noting that after half period u translates 1/2L y in the y direction and returns to the original position after one period.A complete period corresponds to the two wave fluctuations in Figure 1.The shape of the flow velocity component at Re=9000 is more ambiguous compared to the case at Re=3000, due to its corresponding longer period for larger Re.The periodic orbital solutions calculated in the ||ʹ||-||ʹ|| plane for different magnetic field strengths are illustrated in Figure 3, where ||ʹ||2=(1/V(∫ V (ս-u ) 2 dV) denotes the amplitude of the streak and ||ʹ||=(1/V) ∫ V ν 2 dV denotes the amplitude of the vortex.After applying the normal magnetic field, the shape of the amplitude curve is unchanged when Ha is small.When Ha is large, the shape of the amplitude curve changes significantly with the number of Ha, and the global amplitude increases with the increase of the magnetic field strength.This is due to the suppression of the flow field by the magnetic field, which suppresses the occurrence of the laminar-turbulent transition process and requires a larger disturbance energy to trigger the transition.In addition, the magnetic field suppresses the velocity components (u and v) perpendicular to the magnetic field direction, while the velocity (w) parallel to the magnetic field direction is unaffected.Therefore, the general amplitude increases and the shape of the curve changes as the magnetic field strength increases.3 respectively.A clear vortex can be seen in Fig. 4(a) and the vortex amplitude is larger when Ha=0.The blue and red colours represent ±50% of the flow vorticity w max respectively.The green colour in Fig. 4(c) shows the streak, which represents ±74% of the streamwise direction velocity u max but the vortex is not obvious which is due to the low vortex amplitude.After experiencing the cycle from (a) to (d), the vortex moves L y /2 in the expansion direction and then repeats the cycle back to the original position in the same way.It is a complete cycle which consistent with the phenomenon observed in Figure 2. It is similar to the case without magnetic field when Ha= 6, but the vortex structure in the flow field is more obvious, which may be due to the magnetic field effect.

Conclusion
The discovery of ECS connects two long-standing separate branches of fluid mechanics, turbulence and transitions.The exact coherent structure in subcritical transitions is similar to that in fully developed turbulence [21], and both consist of flow streaks and vortices.In this paper, the ECS of channel flow at large Reynolds numbers before and after the application of a normal magnetic field is studied by using DNS combined with dynamical system methods, and the following conclusions are obtained: (1) For Re ≥ 9000, the calculation obtains a periodic orbital solution, whose period increases with increasing Re.After applying a normal magnetic field, the period decreases with increasing magnetic field strength for a certain Re.Due to the suppression of the magnetic field, the larger the Hartmann number, the larger the perturbation energy required to activate the transition.
(2) The normal magnetic field has a suppressive effect on the velocity component in the streamwise and spanwise directions, but has no effect on the normal velocity component.The global amplitude of the perturbation energy increases with the increase of the magnetic field strength, and the amplitude curve changes due to the magnetic field effect.
(3) In contrast, it is found that the magnetic field has little effect on the self-sustaining mechanism and the basic structure in the flow field, but the vortices in the flow field are more obvious due to the magnetic field when the magnetic field is stronger.

Figure 1 .
Figure 1.The disturbance evolution process of flow field at different A 1 when Re=9000.

Figure 2 .
Figure 2. The periodic variation of the u on the axis when x=0 z=0 at Re=3000(top) and Re=9000(bottom).

Fig. 4
Fig.4visualizes the ECS at Ha=0(left) and Ha=6(right) for Re=9000.The exact coherent state consists of "wavy" streaks and vortices on both sides, which explains the self-sustaining dynamical circulation process in Figure3.Pictures (a)-(d) show the flow field distribution during the cycle, corresponding to points a-d and a'-d' on the Ha=0 and Ha=6 curves in Fig.3respectively.A clear vortex can be seen in Fig.4(a) and the vortex amplitude is larger when Ha=0.The blue and red colours represent ±50% of the flow vorticity w max respectively.The green colour in Fig.4(c) shows the streak, which represents ±74% of the streamwise direction velocity u max but the vortex is not obvious which is due to the low vortex amplitude.After experiencing the cycle from (a) to (d), the vortex moves L y /2 in the expansion direction and then repeats the cycle back to the original position in the same way.It is a complete cycle which consistent with the phenomenon observed in Figure2.It is similar to the case without magnetic field when Ha= 6, but the vortex structure in the flow field is more obvious, which may be due to the magnetic field effect.