Influence of oblique magnetic fields on the periodicity of large-scale intermittent states in channel flow

Further investigation of the large-scale intermittent states in channel flow reveals that the addition of a streamwise magnetic field and the original spanwise magnetic field has an effect on the cycle of the large-scale intermittent states. We also studied the effect of oblique magnetic field directions on the large-scale intermittent state. It is found that the introduction of a streamwise magnetic field increases the cycle time of the circulation.


Introduction
Large-scale intermittency (LSI) refers to a particular state observed in conducting fluids under a uniform magnetic field [1].In a channel flow, the conducting fluid evolved between laminar and fully turbulent states due to the damping effect of the magnetic field.When it is laminar state, there is a uniform spanwise structure and no Joule dissipation is generated by the spanwise magnetic field [2].However, as the flow develops and produces finite amplitude three-dimensional perturbations, it cyclically oscillates between the laminar and three-dimensional turbulent state [3].In channel flow, the damping effect of the spanwise and vertical channel magnetic fields on the streamwise velocity component is often observed, while the streamwise magnetic field usually increases the streamwise velocity component [2].The suppression effect of the spanwise and vertical channel magnetic fields is exerted on the fluid through Joule dissipation generated by conducting fluids flowing through the magnetic field [2,3].On the other hand, under the influence of the streamwise magnetic field the streamwise velocity component does not generate Lorentz forces, and instead, it increases the stability of the streamwise streaks, thereby increasing the streamwise velocity component [4].
The periodic cycle of large-scale intermittency is similar to Poiseuille laminar flow for most of the time, providing the lowest frictional resistance for fluid flow in the channel [1,5].Extend the duration of the large-scale intermittent growth phase, this paper continues to study the effect of oblique magnetic fields effect of large-scale intermittent periodic time [2,6].

Model
The present study employs pressure-driven, incompressible, conducting fluid flow that fully develops in a channel under in the presence of a magnetic field.The magnetic field is uniform in the streamwise, spanwise, or vertical channel directions [7,8].The magnetic Reynolds number ( is assumed to be very small, which is applicable to almost all industrial and laboratory liquid metal flows [9,10].Compared to the external magnetic field, the include magnetic field is negligibly small and adjusts instantaneously with velocity fluctuations, which satisfies the quasi-static criterion.The governing equations and boundary conditions are [1]: Here, x, y, and z respectively denote the streamwise, spanwise, and vertical channel directions; u, v, and w represent the velocity components in the x, y, and z directions [11]., , and B denote the pressure field, electric potential, and magnetic field, respectively.The Reynolds number is defined as  /, and the Hartmann number is   / , where L and U are the half-width and centerline velocity of the channel Poiseuille flow.We assume the channel walls are no-slip and perfectly electrically insulating [2].The effect of the magnetic field on the conductive fluid is Joule dissipation, which converts kinetic energy into heat and suppresses the flow.The Fourier velocity is given by: Here,  is the angle between the external magnetic field B and the wave vector k, and  and  are the electrical conductivity and density of the liquid.[13] The computational domain has dimensions of 2π×4π×2 in the x, y and z directions, respectively, with a grid resolution of 64×64×64 [1,2].The Fourier-Chebyshev method is used with periodic boundary conditions for direct numerical simulation (DNS).The Reynolds number Re is above the threshold of linear instability,  5772.The perturbation velocity  ,
In the following discussion of the LSI cycle, the amplitude of ′ is approaching to the round off of the machice.The initial condition corresponds to a fully developed turbulent flow [5,12].

Result
In order to investigate the effect of oblique magnetic field on the large-scale intermittency period, we introduced streamwise and vertical channel magnetic fields under the influence of the spanwise magnetic field [2], and investigated the effects of streamwise and vertical channel magnetic fields on the streamwise and spanwise structures [14].We found that the introduction of the streamwise magnetic field can greatly increase the large-scale intermittent period, achieving a good drag reduction effect.
The LSI cycle can be divided into four stages: (a) growth stage, where the perturbation kinetic energy is almost entirely in the spanwise direction; (b) three-dimensional turbulent stage, where the T-S mode experiences secondary instability of three-dimensional perturbations after reaching finite amplitude, and the mode breaks down into a turbulent state; (c) streamwise streaks attenuation stage, as the perturbation energy increases, Joule dissipation also sharply increases, suppressing fluid fluctuations, and the streamwise streaks are more likely to survive and decay under the influence of magnetic and viscous dissipation; (d) streamwise streaks transform into spanwise waves stage, where the streamwise streaks are suppressed to nearly laminar state, and under the action of the fluid, they will return to the growth stage to complete the cycle, as shown in Figure 1.

Streamwise field
The variation of the LSI cycle period after adding streamwise magnetic field is shown in Figure 2. We obtained the same result for Reynolds numbers of 6000, 8000, 10000 and Hartmann numbers of 60, 80, 120.The main effect of streamwise magnetic field in the LSI cycle is to increase the growth stage and thus increase the cycle period.Figures 2a, 2b, and 2c show the range of streamwise magnetic field added at Reynolds number Re=8000, and Hartmann number Ha from 0 to 15, where large-scale intermittent states (LSI) can be continuously found between Ha=0 and 12.5.The trend of increasing cycle period becomes more evident near the critical value.The period of the flow oscillation at a streamwise magnetic field strength of 11.25 is approximately 8 times longer than the case without the streamwise magnetic field, resulting in a longer cycle time and a lower average drag throughout the entire flow time.The flow is also more stable under such conditions.Figure 2d shows the range of streamwise magnetic field added at Reynolds number Re=6000, and Hartmann number Ha from 0 to 10, where Reynolds number Re is close to the threshold of linear instability ( =5772).The suppression effect of the streamwise magnetic field is relatively strong, and a large streamwise magnetic field directly suppresses LSI to a two-dimensional laminar flow structure.Figure 2e shows the range of streamwise magnetic field added at Reynolds number Re=10000, and Hartmann number Ha from 0 to 20.When the streamwise magnetic field is higher than Ha=15, the fluid is not suppressed to a two-dimensional laminar flow structure due to the high Reynolds number, and a low perturbation wave can be observed.In Figure 3, we added a magnetic field along the reverse streamwise direction.Under the same conditions of Ha=80, we obtained results similar to those in Figure 1, indicating that the direction of added magnetic field does not affect LSI.

Vertical channel magnetic field
We also added vertical channel magnetic field under the action of spanwise magnetic field.However, we did not find the LSI circulation structure after adding the normal magnetic field, as shown in Figure 4.The effect of the vertical channel magnetic field is strong, which affects the growth of the streamwise structure.Even under the action of small normal magnetic field, we could not find the large-scale intermittent state LSI.

Streamwise field and vertical channel magnetic field
We further attempted to search for large-scale intermittent (LSI) structures under a general magnetic field in all three directions, as shown in Figure 5.The results showed that LSI structures do not exist when magnetic fields are applied in all three directions.

Conclusion
Under the influence of the transverse magnetic field, large-scale intermittent states can be generated.By adding the streamwise magnetic field under the action of the transverse magnetic field, we found that the superposition of multiple magnetic field directions will increase the period of the large-scale intermittent state.Additionally, it will stretch streamwise streak structure and extend the duration of the large-scale intermittent growth phase.

Figure 1 .
Figure 1.The streamwise velocity perturbation cloud maps exhibit the maximum perturbation state on six iso-surfaces, which correspond to ±30%, ±60%, and ±90% of the maximum perturbation.(a), (b), (c), and (d) respectively represent the growth stage, three-dimensional turbulent stage, decay stage of streamwise streaks, and transition from streamwise streaks to spanwise waves.

Figure 3 .
Figure 3. Perturbation energy varies with time under a magnetic field in reverse streamwise direction

Figure 4 .
Figure 4. Perturbation energy varies with time under a magnetic field in vertical channel direction

Figure 5 . 3 . 4 .
Figure 5. Perturbation energy varies with time under a magnetic field in vertical channel and streamwise direction3.4.Comparison of the effect of the Streamwise magnetic field and increasing the Reynolds numberFigure6shows the variations of the cycle time at different Reynolds numbers, with the addition of the streamwise magnetic field resulting in longer cycle times and a greater overall drag reduction effect compared to Figure2.There are two sets of data in Figure6: ℎ =80.6226, which is compared to ℎ =80 and ℎ =10 in Figure2, and ℎ =85, which is larger than all the Ha values used in Section 3.1.It is clearly evident that increasing the cycle time of LSI with the addition of the streamwise magnetic field yields a better drag reduction effect.

Figure 6 .
Figure 6.Perturbation energy varies with time under a magnetic field in spanwise direction