Decay of honeycomb-generated turbulence in a duct with a static transverse magnetic field is studied via direct numerical simulations. The simulations follow the revealing experimental study of Sukoriansky et al. (1986), in particular the paradoxical observation of high-amplitude velocity fluctuations, which exist in the downstream portion of the flow when the strong transverse magnetic field is imposed in the entire duct including the honeycomb exit, but not in other configurations. It is shown that the fluctuations are caused by the large-scale quasi-two-dimensional structures forming in the flow at the initial stages of the decay and surviving the magnetic suppression. Statistical turbulence properties, such as the energy decay curves, two-point correlations and typical length scales are computed. The study demonstrates that turbulence decay in the presence of a magnetic field is a complex phenomenon critically depending on the state of the flow at the moment the field is introduced.
Decay of turbulence in a duct with transverse magnetic field]Decay of turbulence in a duct with transverse magnetic field Zikanov, Krasnov, Boeck and Sukoriansky]Oleg Zikanov, Dmitry Krasnov, Thomas Boeck and Semion Sukoriansky
This paper addresses decay of turbulence in an electrically conducting fluid in the presence of an imposed static magnetic field. The parameters typical for technological and laboratory flows of liquid metals are considered, so the quasi-static approximation, according to which the magnetohydrodynamic flow-field interaction is reduced to the effect of the imposed field on a flow, is adopted (see, e.g., Davidson 2016, for the derivation and a discussion of validity of the approximation).
In any three-dimensional flow of an electrically conducting fluid, an imposed magnetic field suppresses turbulent fluctuations via the Joule dissipation of induced electric currents. Unlike its viscous counterpart, the Joule dissipation is active irrespective of the length scale, and anisotropic in the sense that its rate is proportional to the square of the gradient of velocity along the magnetic field lines. As described by Moffatt (1967), this transforms an initially isotropic flow into a form with reduced or even zero velocity gradients along the magnetic field lines. In flows with walls, the picture is more complex due to the effect of walls on the velocity and electric currents. In particular, in the case of an MHD duct, the mean flow is changed by the Lorentz force, and special boundary layers appear (see, e.g., Branover 1978; Müller & Bühler 2001). The principal features of the transformation of turbulence still remain (1) suppression of fluctuations, so the MHD flows are found in a laminar or transitional state at much higher Reynolds numbers than their hydrodynamic counterparts (see, e.g., Zikanov et al. 2014a, for a review), and (2) dimensional anisotropy with weaker velocity gradients along the field lines than across them (see, e.g., Moffatt 1967; Davidson 1997; Zikanov & Thess 1998; Vorobev et al. 2005; Krasnov et al. 2008; Reddy & Verma 2014; Verma 2017). The anisotropy may reach the asymptotic state of flow’s quasi- (i.e. to the degree allowed by the boundary conditions) two-dimensionality if the magnetic field is sufficiently strong to suppress three-dimensional instabilities inherently present in such a flow (Thess & Zikanov 2007).
The term anisotropy is used in this paper with the meaning commonly employed in the research of MHD turbulence (see, e.g., Zikanov & Thess 1998; Vorobev et al. 2005; Knaepen & Moin 2004; Knaepen et al. 2004)–as the persistent inequality of the typical length scales of the flow structures in the directions along and across the magnetic field. The anisotropy of the Reynolds stress tensor (the inequality of velocity components) is, as discussed, e.g., by Burattini et al. (2010); Favier et al. (2010, 2011); Verma & Reddy (2015) not caused directly by the magnetic field and strongly affected by the presence of walls and other features of a particular flow, as well as the typical length scale at which the velocity is considered.
It must be stressed that while the picture outlined above is generally correct for any transformation of conventional three-dimensional turbulence, the behaviours of MHD flows are more complex and, often, counterintuitive. Good examples are the flow regimes with spatially localized or intermittent turbulence reported by Zikanov et al. (2014a); Boeck et al. (2008); Krasnov et al. (2013); Brethouwer et al. (2012); Krasnov et al. (2012); Zikanov et al. (2014b) and the experimental demonstration by Pothérat & Klein (2014, 2017) that under certain circumstances the magnetic field can, in fact, enhance turbulence.
A starting point of the modern understanding of the decay of turbulence in the presence of a magnetic field is the theoretical analysis of Moffatt (1967). A linearized model based on the assumption of a very strong magnetic field acting on an initially isotropic flow was used. The main result was the establishment of the concept of the magnetically induced anisotropy, that largely formed the basis of the future work. The other results of Moffatt (1967), such as the power law of the energy decay and the asymptotically reached energy partition such that the energy of the field-parallel velocity component becomes two times larger than in the transverse components, have been less influential and later found to be non-universal (see, e.g., discussion in Burattini et al. 2010).
We should mention the theoretical model of decaying homogeneous turbulence by Davidson (1997) (see also Davidson 2016). Estimates of the rates of viscous and Joule dissipation in terms of the integral length scales along and across the magnetic field have led to a simple model of the decay. The model shows that the power law scaling of energy with time is only possible when one dissipation mechanism is much stronger than the other. In the other circumstances, the decay rate varies with the flow’s anisotropy in the course of the process.
Numerical analysis of homogeneous decaying turbulence in the framework of the periodic box model was performed by Schumann (1976); Knaepen & Moin (2004); Burattini et al. (2010); Favier et al. (2011). The results of simulations of Schumann (1976) and, to a lesser degree, of Favier et al. (2011) were limited to the behavior at small Reynolds numbers due to the DNS accuracy requirements and the rapid magnetic suppression of turbulence. The limitations were avoided by Knaepen & Moin (2004) and Burattini et al. (2010) via the use of the dynamic Smagorinsky LES model, which was demonstrated to be reliably accurate for the MHD quasi-static turbulence by Knaepen & Moin (2004); Vorobev et al. (2005); Vorobev & Zikanov (2007). It was confirmed by Burattini et al. (2010) that the linear model of Moffatt (1967) is only valid at very strong magnetic suppression and during short (less than one turnover time) transformation of the flow. Otherwise, the evolution is complex and strongly influenced by the large-scale anisotropic structures forming in the flow during the initial decay period. This implies inevitable influence of the boundaries and, in general, lack of universality of the decay behavior.
The falling-grid experiment of Alemany et al. (1979) was the only experimental study, in which reproduction of decay of homogeneous MHD turbulence was attempted. At moderate distances from the grid, the fluctuation energy of the field-parallel velocity component was found to fall as . Farther from the grid, the decay accelerated to approximately . This change of behaviour was attributed by Alemany et al. (1979) to the increase of the effective local magnetic interaction parameter (we define the parameter in section 2.1). An interesting result was found for the energy power spectra, whose slope gradually approached indicating strong anisotropy or even approximate two-dimensionality. It is pertinent to mention in view of our following discussion that in the experiment of Alemany et al. (1979) turbulence was generated entirely within the zone of the applied magnetic field. Furthermore, we note that the energy spectrum is difficult to ascertain in strongly suppressed flows at high N. The scaling other than , for example, is found to be also consistent with the experimental and computational data (Verma 2017).
An extensive series of experimental studies of the mercury flows in ducts with imposed transverse magnetic fields was carried out from the late 1960s to 1980s in Riga (see e.g. Branover et al. 1970; Kolesnikov & Tsinober 1974; Votsish & Kolesnikov 1976a, b; Kljukin & Kolesnikov 1989). A major motivation of the experiments was the desire to explain the so-called residual fluctuations of velocity found in the flows with strong magnetic fields when the measurements of pressure drop indicated full laminarization. It was hypothesized that the fluctuations were manifestations of nearly two-dimensional flow structures forming in the flow. It was argued that the decay rate would be reduced by the presence of such structures in two ways. Their quasi-two-dimensionality would mean that they are only weakly suppressed by the magnetic field. Furthermore, the strong anisotropy would imply reduction of the energy cascade to small length scales or inversion of the cascade, thus leading to reduction of the viscous dissipation rate.
The existence of quasi-two-dimensional structures was confirmed in the experiments. The flow behaviour was also found to be strongly influenced by the mechanism of turbulence generation. A particularly interesting example was the experiment of Kljukin & Kolesnikov (1989). Turbulence in a duct was generated by a grid combining two sets of cylindrical bars, one parallel and one perpendicular to the magnetic field. Two experiments were performed: with the bars parallel to the magnetic field located on the downstream or the upstream side of the grid. No significant difference between the two flows was found at weak magnetic fields. In the strong field case, however, the flow with the field-parallel bars on the downstream side of the grid demonstrated residual fluctuations with intensity decreasing very slowly along the duct. No such behavior was found in the flow with the field-parallel bars located on the upstream side of the grid. The effect was attributed by Kljukin & Kolesnikov (1989) to formation of strong quasi-two-dimensional vortical structures in the former case.
The recent numerical simulations of the MHD duct flow by Zikanov et al. (2014a); Krasnov et al. (2013, 2012); Zikanov et al. (2014b) have shown that the presence of velocity perturbations at apparently laminar pressure drop along the duct can also be caused by turbulence in the sidewall (parallel to the magnetic field) boundary layers, which survives at much stronger magnetic fields than the turbulence in the core of the duct and in the Hartmann boundary layers normal to the field. Such turbulence cannot be registered in the measurements of pressure during the experiments such as those of Branover et al. (1970); Kolesnikov & Tsinober (1974); Votsish & Kolesnikov (1976a, b), since the pressure drop is dominated by the friction in the thin Hartmann layers. At the same time, the alternative explanation proposed by the Riga researchers certainly had substantial experimental support.
The present work follows more closely the experiments of Sukoriansky et al. (1986), in which the phenomenon of turbulent fluctuations persisting along the duct in the presence of a strong magnetic field was revisited on a higher level of accuracy and technical sophistication. Flows of mercury in a duct of cm cross-section were studied. Magnetic field of strength up to 1.1 T with the main component transverse to the flow’s direction and parallel to the shorter side of the duct was imposed in the test section by a long (pole length about cm) electromagnet. In the experiments particularly interesting for us, the inlet into the test section was equipped with a honeycomb consisting of densely packed round tubes of diameter 2.4 mm with electrically insulating 0.5 mm thick walls (common drinking straws). The purpose of the honeycomb was two-fold. It generated approximately isotropic and uniform field of velocity fluctuations and prevented the M-shaped mean velocity profile normally forming at the entrance into the magnetic field (see e.g. Branover 1978). The Reynolds and Hartmann numbers were
where was the duct’s hydraulic diameter, was the mean velocity, and , , and were the kinematic viscosity, electric conductivity, and density of the fluid. The experimental setup and the key results are shown in figures 1a and b, respectively.
The striking and, at first glance, paradoxical results were obtained in the hot-film measurements of velocity fluctuations 43 cm downstream of the honeycomb’s exit. The measurements showed completely different signals for the two distributions of the magnetic field illustrated in figure 1a. In the situation identified in Sukoriansky et al. (1986) and this paper as Case 1, the entire length (27 cm) of the honeycomb was located between the magnet poles (see the upper schematic illustration in figure 1b), and turbulence was generated and decayed entirely within the practically uniform transverse magnetic field. In the situation identified as Case 2, the magnet poles were shifted downstream so that the axial distance between the honeycomb’s exit and the nearest corner of the pole was 15.5 cm (see the lower schematic illustration in figure 1b). In this case, turbulence was generated at negligible magnetic field and traveled about 5.5 convective times before entering the space between the poles and thus experiencing the full magnetic suppression effect.
The key results are illustrated by the curves in figure 1b reproduced from figure 5 of Sukoriansky et al. (1986). The curves show the turbulence intensity based on the streamwise velocity fluctuations measured at the duct axis 43 cm downstream the honeycomb axis, i.e. well in the zone of the uniform magnetic field. The signals measured in the two cases are about the same at weak magnetic fields, approximately at . As a manifestation of turbulence suppression by the magnetic field, the intensities decrease with growing reaching at . At stronger magnetic fields, however, the signals show entirely different trends. In the case 2, the intensity continues to decrease to about at high . In the case 1, the intensity grows rapidly with growing and reaches 0.09 (almost twice the intensity in the flow without magnetic field) at .
The appearance of high-amplitude fluctuations at strong magnetic fields in the case 1 configuration was explained in Sukoriansky et al. (1986) by the effect described above, i.e by development of quasi-two-dimensional flow structures with weak gradients along the magnetic field lines. Such structures would experience weak magnetic suppression and reduced energy cascade to small length scales thus preserving the strength of the associated velocity fluctuations as the fluid moved downstream. The explanation is consistent with the conclusions of the other experiments, e.g. of Kljukin & Kolesnikov (1989). No direct evidence of this scenario has, however, been obtained. The type of the flow structures and the degree of their anisotropy could also not be determined in the experiments and has not been a subject of numerical analysis.
In this paper, we present high-resolution numerical simulations designed to explore validity of the hypothesized scenario leading to the residual velocity fluctuations and to produce a detailed description of the flow. The numerical model reproduces the geometry and parameters of the experiment of Sukoriansky et al. (1986) with one adjustment. For the purpose of understanding the effect of walls on decaying turbulence, two orientations of the transverse magnetic field, along the shorter (as in Sukoriansky et al. (1986)) and longer sides of the duct are considered. The role of the anisotropy introduced by the honeycomb is also addressed. The problem formulation, parameters and numerical procedure are described in section 2. The structure and statistical properties of the computed flows are presented in section 3 . The concluding remarks are provided in section 4.
2 Problem formulation, method and parameters
2.1 Problem formulation
An isothermal flow of an incompressible electrically conducting Newtonian fluid in a duct of rectangular cross-section is considered. A transverse magnetic field specified below is imposed. Assuming the asymptotic limit of low magnetic Reynolds and Prandtl numbers, the quasi-static approximation of the magnetohydrodynamic interactions (see e.g. Davidson 2016) is used. The non-dimensional governing equations are
where , , and are the fields of velocity, pressure and electric potential and is the non-dimensionalized magnetic field. The typical scales used to derive (2.1)–(2.1) are the mean streamwise velocity for velocity, shorter half-width of the duct for length, for time, for pressure, the maximum strength of the transverse component for the magnetic field, and for electric potential. The non-dimensional parameters are the Reynolds number
and the Hartmann number
related to the parameters (1) based on the hydraulic diameter as and .
We will also use the magnetic interaction parameter
Further settings of the problem are illustrated in figure 2. The computational domain reproduces the test section of the experiment of Sukoriansky et al. (1986). It is a duct segment of length and cross-section , with , and .
The sidewalls are of zero slip and perfect electric insulation:
At the inlet , we require that
A velocity distribution imitating the flow exiting the honeycomb is applied. In the experiment, the tubes of the honeycomb are densely packed and have the inner diameter mm and wall thickness about 0.5 mm. The parameters for the flow in a single tube are and and the non-dimensional pipe length is . At such parameters, the flow is expected to be weakly turbulent in the case 2. In the case 1, the magnetic field suppresses turbulence and slightly deforms the streamwise velocity profile (see Zikanov et al. 2014a; Müller & Bühler 2001; Li & Zikanov 2013). The numerical model ignores the differences and uses the same velocity distribution in the two cases (see figure 2b). To compute the distribution, the inlet plane is covered by hexagons, into which circles of inner diameters and wall thickness corresponding to those of the honeycomb tubes are fitted. The axisymmetric parabolic profile of streamwise velocity is imposed within each tube. At each time step, random three-dimensional velocity perturbations of relative amplitude are added, after which the entire distribution is rescaled so that the mean streamwise velocity is equal to 1.0.
As discussed in section 3.2, the presence of the magnetic field implies that essentially different flows are generated by the two packing arrangements of the honeycomb tubes shown in figure 2b. The tubes can be packed so that they form straight rows along the longer (the honeycomb type A in the following discussion) or shorter (type B) walls of the duct.
Soft boundary conditions
are applied at the exit of the computational domain.
Two orientations of the main component of the magnetic field, parallel to the longer () or shorter () walls of the duct are used. In each case, the distribution of the magnetic field is approximated in the simulations using the model suggested by Votyakov et al. (2009). The model provides simple formulas for divergence-free, two-dimensional, bi-component field created by a magnet with two infinitely wide rectangular pole-pieces. The accuracy of the model was verified in comparison with measurements in Zikanov et al. (2013). The input parameters of the model are the coordinates of the corners of the pole-pieces, for which we take (for ) or (for ), and , in the case 1 and , in the case 2. The resulting magnetic field has the main component illustrated in figure 2a and the component , which is much weaker and only significant within the flow domain around the entrance into the magnetic field in the case 2.
The problem is solved numerically using the finite-difference scheme first described as the scheme B in Krasnov et al. (2011) and extended to spatially evolving flows in a duct e.g. in Zikanov et al. (2014b). The solver has been successfully applied in numerous simulations of turbulent and transitional MHD flows at high Re and Ha (see e.g. Zikanov et al. 2014a; Krasnov et al. 2013, 2012; Zikanov et al. 2014b; Li & Zikanov 2013). The scheme is explicit and of the second order in time and space. The discretization is on the structured collocated grid built along the lines of the Cartesian coordinate system. The exact conservation of mass, momentum, and electric charge, as well as near-conservation of kinetic energy are achieved by using the velocity and current fluxes obtained by interpolation to staggered grid points. The standard projection technique is applied to compute pressure and enforce incompressibility. The numerical algorithm is parallelized using the hybrid MPI-OpenMP approach.
The modification of the algorithm in comparison with the original version of Krasnov et al. (2011) concerns the solution of the Poisson equations for pressure and electric potential. The fast cosine decomposition is used in the streamwise direction, for which the right-hand side of the equation is modified to achieve homogeneous Neumann conditions at and . The direct cyclic reduction solver by subroutines of the library FishPack (Adams et al. 1999) is used in the -plane.
The computational results reported below are obtained on the grid consisting of points. The points are clustered towards the duct’s walls using the coordinate transformation
where and are the transformed coordinates, in which the grid is uniform.
A grid sensitivity study was performed to determine that the grid sufficiently accurately reproduces the essential features of the flow, such as mixing and instabilities of the honeycomb jets, generation of turbulence, and its decay in the presence of the magnetic field. Additional simulations for the case 1 and case 2 configurations with the magnetic field parallel to the longer sides of the duct on the smaller grid with and the same clustering scheme were carried out. The results were qualitatively the same as on the larger grid with minor quantitative differences. In particular, the time-averaged wall friction coefficients computed for the entire flow domain changed by less than 1%. The effect of the numerical resolution on the results is further discussed in section 4 of this paper.
The parameters of the simulations are listed in Table 1. Each simulation starts with a laminar initial state and continues for non-dimensional time units, which safely guarantees establishing a fully developed flow. After that, the simulation continues for another (in the runs 1.L to 2.H) or 50 (in the runs 1.V.A to 2.V.B) time units of a “production phase” used to study the flow’s behavior. The turbulence statistics reported later in this paper are based on the, respectively, 1000 or 500 flow samples collected during this phase with the time interval 0.1.
The simulations 1.L and 2.L are for , i.e., for the parameters in the range of moderate magnetic fields where strong (about two-fold in comparison to the non-magnetic flow) reduction of turbulence intensity was detected in the experiment of Sukoriansky et al. (1986) for both the configurations (see figure 1b). The simulations 1.H, 2.H, and 1.V.A to 2.V.B are for . At such a much stronger magnetic field, the experiment showed the anomalous behavior with the turbulence intensity in the case 1 remaining low, but the intensity in the case 2 growing to the level about 50% higher than in the flow without magnetic field.
We start the discussion with the main results summarized in Table 2. The time-averaged amplitudes of the velocity fluctuations computed at , and two values of are shown. The values for correspond to the experimental measurements of Sukoriansky et al. (1986) (see figure 1b) and show that the seemingly paradoxical dependence of the fluctuation amplitude on the strength of the magnetic field and magnet’s location is reproduced by the simulations. Weak fluctuations of all the velocity components are found in the runs 1.L and 2.L performed at . Equally weak fluctuations are found in the runs 2.H, 2.V.A., and 2.V.B performed at when the poles of the magnet shifted downstream (the case 2 configuration in figure 2). Anomalously high fluctuation amplitudes are found in the runs 1.H, 1.V.A and 1.V.B, i.e. in the flows with and the honeycomb exit located within the zone of uniform magnetic field (the case 1 configuration in figure 2). The amplitudes of two velocity components are increased: the streamwise component and the component orthogonal to the magnetic field ( in the run 1.H and in the runs 1.V.A and 1.V.B). The increase in comparison to the other cases is about four-fold in the runs 1.H and 1.V.B and two-fold in the run 1.V.A.
The data in Table 2 show that the flow’s behaviour is affected by the magnetic field strength, magnet location, orientation of the magnetic field with respect to the duct walls, and the honeycomb arrangement. The following discussion is separated into two parts. The mechanism of the generation of high-amplitude fluctuations is explained and illustrated in section 3.1 on the basis of the results obtained in the runs 1.L to 2.H. Further investigation of the fluctuations is presented in section 3.2, where the influence of the magnetic field orientation and honeycomb arrangement is explored on the basis of the material of the runs 1.V.A to 2.V.B.
A comment is in order concerning the comparison between the simulations and the experiments of Sukoriansky et al. (1986). As we discuss in detail below, the qualitative agreement is quite satisfactory. At the same time, the quantitative agreement is poor. Comparing the data in table 2 with those in figure 1b and figure 6 of Sukoriansky et al. (1986) we see that in all the simulations the computed rms fluctuations are about five times lower than in the experiment. The possible reasons for this are discussed in section 4.
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3.1 Effect of magnetic field on turbulence decay
The following discussion is primarily based on the results obtained in the simulations 1.L to 2.H.
3.1.1 Velocity fluctuations
Figure 3 shows the time signals of the velocity components computed at the point , corresponding to the point of velocity measurements in the experiment of Sukoriansky et al. (1986) (see figure 1b). The rms amplitudes listed in table 2 are calculated using these signals and the similar signals recorded at , , . We see that the behaviour indicated by the rms data is not subject to significant variations at long time scales. Consistent anomalously high fluctuation amplitudes of streamwise () and field-normal transverse () velocity components are found in the run 1.H when the magnetic field is strong and has the case 1 configuration.
3.1.2 Flow structure
The spatial structures of the fully developed flows in the simulations 1.L to 2.H are illustrated in figures 4-7. We see that at (runs 1.L and 2.L in figures 4–6) the flows remain turbulent, although the velocity fields are significantly modified by the magnetic fields. The modifications include development of the mean flow profile with a nearly flat core and characteristic Hartmann and sidewall boundary layers (see figure 4) and reduction of turbulence intensity (see figures 5 and 6). Since the Hartmann thickness-based Reynolds number , this result is in agreement with the earlier studies of the flow in a long duct with uniform transverse magnetic field. As discussed, for example, in the review by Zikanov et al. (2014a), fully laminar and fully turbulent flows are typically found at, respectively, and , with the transitional range at . We also note that at no substantial differences are observed between the flows in the case 1 and case 2 configurations except for the fully expected effect that the flow modification becomes visible farther downstream in the case 2 than in the case 1.
In the simulations 1.H and 2.H performed at , we have , which is below the laminar-turbulent transition range. It is, therefore, not surprising that turbulence is suppressed (albeit not completely, as we will see in the following analysis) as the fluid moves through the magnetic field (see figures 4–6). The flows obtained for the two configurations of the magnetic field are, however, clearly different.
In the case 2 configuration, there is a distance between the honeycomb and the beginning of the zone of full-amplitude magnetic field. The plots for the run 2.H in figures 4–7 clearly show that the distance is sufficient for the instability and mixing of the jets generated by the honeycomb. Three-dimensional turbulence develops. Upon entering the magnetic field zone, the turbulent fluctuations are quickly suppressed, which is reflected by the strong reduction of the rms velocity fluctuations at shown in table 2.
In the case 1 configuration, there is no such distance. The formation of turbulence near the honeycomb exit occurs in the presence of a full-amplitude magnetic field. As shown in figures 4–7, the result is that the velocity field in the flow 1.H quickly becomes strongly anisotropic. The instability of the honeycomb jets leads not to a three-dimensional turbulent state, but to a quasi-two-dimensional flow dominated by structures aligned with the magnetic field. The structures are superimposed on the plug-like profile of the streamwise velocity and can be characterized as planar jets of moderate amplitude quickly rolling into quasi-two-dimensional vortices. The quasi-two-dimensionality (weak variation in the direction of the main magnetic field component) of the surviving fluctuations is well illustrated by the plots in figures 4, 5, and 7. Their vortex-like structure is especially clearly seen in the plot of the vorticity component parallel to the magnetic field in the cross-section of the duct shown in figure 6.
This is the key observation. It provides basis to the explanation suggested earlier for the anomalously strong velocity fluctuations observed in the experiments of Sukoriansky et al. (1986) and, likely, other experiments such as those of Kljukin & Kolesnikov (1989). Due to their weak gradients along the magnetic field lines, the quasi-two-dimensional vortices do not generate strong Joule dissipation. Furthermore, the quasi-two-dimensionality reduces the energy flux from large to small length scales, which implies weaker viscous dissipation. The flow structures are still suppressed by the Joule and viscous dissipation in the boundary layers, but the effect is not strong. The quasi-two-dimensional vortices are visible till the end of the flow domain (see figures 4–7), and are responsible for the generation of high-amplitude velocity fluctuations at far downstream locations.
3.1.3 Turbulence decay along the duct
The distributions of the turbulent kinetic energy are computed as functions of along the lines and , . This is done separately for each velocity component as
where stands for time averaging over the entire stage of developed flow.
The turbulence decay curves obtained at are shown in figures 8 and 9. The segment in figure 8 and segment in figure 9 are excluded to highlight the decay stage of the flow evolution and to eliminate the initial stage of jet instability and mixing, at which the data are strongly influenced by the position of the point with respect to the honeycomb pattern. The slope lines are plotted to illustrate the decay rate rather than to suggest a specific scaling.
For the flows 1.L and 2.L, the energy decay curves obtained at two locations of the magnet are not very different from each other. This suggests weak influence of the magnetic field, and is in agreement with the low value of the magnetic interaction parameter . At small , the magnetic damping causes somewhat more rapid decay in the case 1.L than in the case 2.L. At larger , approximately at where the strength of the magnetic field is about the same in the two flows, turbulence decays faster in the case L.2. We attribute that to the stronger Joule dissipation caused by the stronger velocity gradients in the field direction retained by the flow. At the end of the duct, the turbulent kinetic energy in the two flows decreases to approximately the same level.
The curves in figure 9a,b show significant level of fluctuations in all three velocity components. This is in agreement with the three-dimensional fully turbulent nature of the flow visualized in figures 4-6. At the same time, the Reynolds stress tensor is not isotropic. At small , . At larger , approximately at in the flow 1.L and in the flow 2.L, we see , which indicates the presence of anisotropic flow structures elongated in the direction of the magnetic field.
The effect of the magnetic field is much more pronounced in the flows 1.H and 2.H. In the case of 2.H, the energy decay curves is practically indistinguishable from the 2.L curve at (see figure 8). Downstream of this point, the strong magnetic field imposed on three-dimensional small-scale turbulence results in rapid decay and the lowest value of the turbulent kinetic energy at the duct exit among all four cases. Interestingly, at the initial stages of this decay, approximately at , the fluctuation of the velocity component parallel to the magnetic field remain stronger than the fluctuations of the other two components (see figure 9d). We do not have data that would allow us to identify the specific flow structures responsible for this effect. At the same time, the behaviour is consistent with the known picture of the evolution of isotropic turbulence after sudden application of a strong magnetic field. As predicted by Moffatt (1967) and confirmed in simulations of homogeneous turbulence by Burattini et al. (2010) and Favier et al. (2010), the initial stages of the decay are characterized by the energy of field-parallel velocity fluctuation component substantially larger (two times larger in the asymptotic limit ) than the energy of the field-perpendicular components. In the far-downstream portion of the duct, approximately at , the remaining fluctuations and decay very slowly, with the rate approaching .
For the most interesting for us case 1.H, figures 8 and 9 show very strong effect of the magnetic field. In the entrance portion of the duct, the generation of turbulence is inhibited. As a result, the turbulent kinetic energy is an order of magnitude smaller than in the other three cases. The energy grows slightly at and then decays, but much slower than in the other cases. The energy becomes larger than in the other cases at (see figure 8).
Interesting behaviour is observed in the segment between, approximately and . While the fluctuation energy of the field-parallel velocity component continues to decay along the duct, the fluctuation energies of the other two components grow. We see this as a manifestation of the energy transfer from the mean flow to the fluctuations accompanying the formation of the quasi-two-dimensional vortices.
3.1.4 Turbulence statistics
The velocity fields computed in the runs 1.L to 2.H for fully developed flows at are used to accumulate the turbulence statistics discussed in this section. We start with the energy power spectra obtained from the velocity signals at , (see figure 3), which are are shown in figure 10. We see that even at the spectra are continuously populated in a wide range of frequencies , so the flows can be classified as turbulent. The inertial ranges cannot be reliably determined due to their shortness typical for turbulence decay in the presence of MHD suppression. Still, one sees portions of the spectra with the slope at and at . The latter can be viewed as an indication of the quasi-two-dimensional character of the turbulence, although, as argued by Alemany et al. (1979) and Sommeria & Moreau (1982), the same spectrum may appear as a result of the equilibrium between the local angular energy transfer and the Joule dissipation in the core flow or the Hartmann boundary layers.
The spectrum of is particularly convenient for characterization of the anomalous high-amplitude turbulent fluctuations observed in the flow 1.H (see figure 10f). The energy peak at is evidently associated with the characteristic streamwise size of the vortices (see figures 5–7).
The two-point velocity correlations along the and axes were computed using the formulas (here for the velocity component )
where , are the small offsets used to exclude the Hartmann and sidewall boundary layers, thus limiting the estimation of the correlations to the zone of approximately homogeneous turbulence in the core flow. The integrals are calculated at the time moments separated by and time-averaged over the period of fully developed flow. The calculations are preformed for several duct’s cross-sections , namely at and at with a step of .
The correlation curves obtained for the transverse velocity components and are shown, respectively, in figures 11 and 12. We see that long-range correlations remain weak in the 1.L and 2.L flows during the entire decay. This confirms the essentially three-dimensional small-scale structure of turbulence in these flows already indicated by the illustrations in figures 4–7 and the spectra in figures 10. There is practically no difference between the curves corresponding to the 1.L and 2.L flows.
The strong magnetic field imposed on the flows 1.H and 2.H causes development of strong correlations along the magnetic field lines, i.e., in the -direction. This is accompanied by establishment of weaker, but still significant long-range correlations in the -direction. This is consistent with the results of earlier simulations and theoretical models (see e.g. Moffatt 1967; Davidson 1997; Zikanov & Thess 1998; Krasnov et al. 2008; Burattini et al. 2010) of transformation of turbulent flow under the impact of a strong magnetic field. While the growth of the typical scale of the turbulent structures in the field direction is always stronger and the only one caused directly by the Joule dissipation, the growth of the typical transverse size is caused by the enlargement of the quasi-two-dimensional vortices.
In the flows with , the long-range correlations appear at the early stages of the decay in the 1.H case and approximately at , i.e. in the very beginning of the zone of strong magnetic field in the 2.H case. The development at the later stages of decay is different between the two cases. The suppression of turbulence in the 2.H flow is accompanied by further growth of the long-range correlations. In the 1.H flow, the correlations change little and eventually become weaker than in the 2.H flow. We attribute this behavior to persistence of the quasi-two-dimensional vortical structures.
The results obtained for the correlation coefficient in the 2.H flow at (see figure 12f) may appear surprising. The flow has nearly constant significant correlations () over almost the entire duct width. This behavior is unique as it is not observed for any other computed correlation coefficient in any other cross-section (see figures 11 and 12). The reason for this behavior is illustrated in figure 13. We see that, as the fluid moves from to , the streamwise velocity changes its profile in the way typical for a duct flow entering a strong magnetic field (see e.g. Andreev et al. 2006, for a discussion of the flow transformation). Along the -axis parallel to the magnetic field, the Hartmann profile with nearly uniform velocity in the core and thin Hartmann boundary layers develops. Along the -axis, the profiles acquires the typical M-shape. The redistribution of the streamwise velocity is accompanied by a non-zero mean flow toward the walls at (clearly visible in the distribution of at ) and in the -direction (visible in the distribution of at , i.e. slightly upstream of the beginning of full-strength magnetic field, in agreement with the scenario of formation of the M-shaped profile. The elevated correlation coefficient in the 2.H flow at is caused by the flow in the -direction. The flow in the -direction undoubtedly leads to elevated values of at (not computed in our simulations).
The correlation functions (3.1.4), (3.1.4) are used to compute the turbulent integral length scales. The longitudinal () and transverse () length scales along () and across () the magnetic field are computed as:
In isotropic turbulence, we would find . These relationships are, quite expectedly, not satisfied by the flows 1.H and 2.H with strong magnetic field. For the flows 1.L and 2L with weak magnetic field, the relationships hold for and at large distances from the inlet, where the honeycomb-created jets are properly mixed (see figures 14c and d), but not for and (not clearly visible in figures 14a and b, but verified in our analysis. We also see that at weak magnetic field the scales and remain practically constant, while and grow downstream. The outlying point in figure 14c corresponds to the effect of the local flow transformation discussed above and illustrated in figures 12f and 13.
In the flows 1.H and 2.H, the strong magnetic field causes rapid growth of , , and , but not . The most interesting for us are the length scales and computed on the basis of the fluctuations of the velocity component . We see that the length scale along the magnetic field grows monotonically downstream after the full-strength magnetic field is introduced (at in the 1.H flow and at in the 2.H flow) as an indication of flow’s transition into strongly anisotropic form. Interestingly, the large vortices developing in the 1.H flow result in slower growth, so at the end of the domain, is smaller in the 1.H flow than in the 2.H flow. The length scale in the direction perpendicular to the magnetic field grows very rapidly at small in the 1.H flow and stabilizes at about at above approximately . This value as associated with the typical transverse size of the quasi-two-dimensional vortices. On the contrary, in the 2.H flow, where the vortices do not form, grows continuously downstream.
3.2 Effect of walls and anisotropy of inlet conditions
The discussion of section 3.1 as well as previous works by various authors (see e.g. Moffatt 1967; Sukoriansky et al. 1986; Kljukin & Kolesnikov 1989; Burattini et al. 2010) suggest that the development and persistence of quasi-two-dimensional structures aligned with the strong imposed magnetic field is a general physical phenomenon to be observed, in some form, in all decaying MHD turbulent flows. At the same time, features of the flow’s configuration may strongly affect the realization of the phenomenon in a specific case. For our system, the most important such features are: (i) the location of the duct’s walls non-parallel to the magnetic field, which limit the longitudinal size of the quasi-two-dimensional flow structures and (ii) the design of the honeycomb, which may introduce anisotropy into the initial state of the flow.
We note that the importance of these features is due to the presence of the strong transverse magnetic field. Without the field, approximately homogeneous and isotropic turbulence insensitive to such details of the system’s geometry is expected to form in the core of the duct shortly downstream of the honeycomb’s exit.
The two effects are explored in our study in the simulations 1.V.A, 2.V.A, 1.V.B, and 2.V.B (see table 1 for parameters). The strong magnetic field corresponding to is applied in all the simulations, so we expect the behaviour similar to that observed earlier in the simulations 1.H and 2.H. Unlike the earlier runs, the main component of the magnetic field is oriented along the shorter side of the duct (). The case 1 and case 2 distributions of the magnetic field along the duct are considered. In addition to allowing us to see the effect of the distance between the field-crossing walls, the new simulations provide the possibility of direct comparison with the experiment of Sukoriansky et al. (1986), in which the magnetic field was in the -direction.
Two arrangements of the honeycomb tubes are considered. As illustrated in figure 2b, the tubes are arranged into straight rows along the longer (Type A) or shorter (Type B) sides of the duct. This implies different anisotropies of the flows exiting the honeycomb. The type A (runs 1.V.A and 2.V.A) produces structures with weaker average gradients in the -direction, i.e. perpendicularly to the magnetic field. The type B (runs 1.V.B and 2.V.B) results in the flow structures with weaker gradient in the -direction, i.e. the direction of the magnetic field.
The velocity fluctuations in fully developed flows are presented in figure 15 and, in terms of the rms values, in table 2. We see that the situation is generally similar to that observed earlier in the simulations 1.H and 2.H. The anomalously strong velocity fluctuations appear when the magnetic field has the configuration of case 1 (runs 1.V.A and 1.V.B) but not of case 2 (runs 2.V.A and 2.V.B). Also as before, the strong fluctuations develop in the streamwise velocity component and the transverse component perpendicular to the magnetic field .
The effect of the anisotropy introduced by the honeycomb is clearly visible. The fluctuation amplitude in the run 1.V.B is about the same as in the run 1.H, while it is about two times smaller in the run 1.V.A.
To explain these results, we will analyze the spatial structure of the flows visualized in figures 16–18. As in section 3.1, profiles of the streamwise velocity (figure 16), distributions of the vorticity component parallel to the magnetic field (figure 17), and isosurfaces of the transverse velocity component perpendicular to the magnetic field (figure 18) are considered.
We start with the simulations 2.V.A and 2.V.B, in which the magnet poles are shifted downstream of the honeycomb exit (the case 2 flow, see figures 1b and 2a). One can see that, similarly to the simulation 2.H, three-dimensional turbulence forms before the fluid enters the zone of strong magnetic field. Subsequent effective magnetic damping evidently results in the low amplitude of remaining velocity fluctuations reported in figure 15 and table 2.
The flows of the simulations 2.V.A and 2.V.B also have prominent M-shaped profiles of streamwise velocity (see figure 16). Such a profile is expected when the flow in a duct with electrically insulating walls enters the zone of strong transverse magnetic field (see e.g. Branover 1978; Andreev et al. 2006). The profile can also be noticed in the run 2.H (see figure 4), but it is more pronounced in the runs 2.V.A and 2.V.B due to the larger distance between the sidewalls (the walls parallel to the magnetic field).
The two just discussed flow features are approximately equally observed in the simulations 2.V.A and 2.V.B. The only difference between the two flows is that we see significant velocity fluctuations near the sidewalls in the far downstream portion of the duct in the flow 2.V.A but not 2.V.B (see figures 16 and 18). The plausible explanation of this effect is the shear layer instability of the planar near-wall jets forming the M-shaped profile. Higher level of perturbations is introduced into the jets by the honeycomb of type A than type B.
In the simulations 1.V.A and 1.V.B, the honeycomb exit is located within the zone of strong transverse magnetic field (the case 1 flow, see figures 1b and 2a). Similarly to the case 1.H, the simulations show development of quasi-two-dimensional structures that are poorly suppressed by the magnetic field and have the from of large-scale vortices aligned with the field. Interestingly, the strength of the structures and the amplitude of the associated velocity fluctuations is about the same in the 1.V.B and 1.H runs (see table 2 and figures 3, 4–7, 15, 16–18). We see that the process of formation of the quasi-two-dimensional vortices is practically unaffected by the orientation of the magnetic field.
At the same time, the effect of the initial flow anisotropy introduced by the honeycomb is quite strong. The vortices are noticeably weaker and the fluctuation amplitude is about two times smaller in the run 1.V.A (when the honeycomb produces structures elongated across the magnetic field) than in the runs 1.H and 1.V.B (when the elongation is along the field).
4 Discussion and concluding remarks
We have performed numerical simulations inspired by the experiment of Sukoriansky et al. (1986). The main goal was to understand the mechanisms leading to the anomalous high-amplitude velocity fluctuations detected in the experiment when a strong magnetic field covered the entire test section including the honeycomb. This goal has been largely achieved. The results of the simulations are in good qualitative agreement with the experimental data. The presence or absence of anomalously strong fluctuations is found, respectively, at the same flow parameters as in the experiment (cf. the experimental data in figure 1b and computed data in table 2).
The computed spatial structure and statistical properties of the flow provide the explanation of the experimental observations. The jets forming at the honeycomb exit are unstable and serve as a strong source of small-scale turbulence. When the magnetic field is weak (runs 1.L and 2.L), the kinetic energy injected into the flow is transferred to small length scale in the conventional process of development of three-dimensional turbulence. The turbulence then decays under the combined action of viscous and Joule dissipation as the fluid moves downstream.
Similar formation of three-dimensional turbulence occurs in the flows 2.H, 2.V.A and 2.V.B, in which the magnetic field is strong but begins at a distance from the honeycomb exit. When the fluid enters the strong magnetic field zone, the turbulence experiences strong magnetic suppression. Its subsequent evolution is characterized by low amplitude of velocity fluctuations (see figures 3, 4–7 and table 2) and development of weak quasi-two-dimensional structures (see figures 11, 12, 14).
High-amplitude velocity fluctuations develop in the runs 1.H, 1.V.A and 1.V.B when the strong magnetic field imposed at the exit from the honeycomb leads to rapid development of strongly anisotropic flow structures and prevents the energy cascade to small length scales and formation of conventional three-dimensional turbulence. The dominant flow structures evolve into quasi-two-dimensional vortices, which are aligned with the magnetic field and, therefore, only weakly suppressed and retain their strength and structure till the end of the computational domain, i.e. at the streamwise distance of at least 25 shorter duct widths. It appears highly plausible that the anomalously strong velocity fluctuations recorded in the experiment are caused by such vortices.
The difference in the flow evolution between the cases with weak and strong magnetic fields can be related to the differences in the values of the magnetic interaction parameter (the Stuart number) . This parameter estimates the typical ratio between the Lorentz and inertial forces and, therefore, is often used as a measure of expected transformation of turbulence by an imposed magnetic field (see e.g. Zikanov & Thess 1998; Vorobev et al. 2005; Krasnov et al. 2008; Burattini et al. 2010; Krasnov et al. 2012). The values of N about and higher than 1 are typically required for strong transformation (there are inevitable variations of this rule due to various definitions of the length and velocity scales, various types of the flow, and the variation of the transformation effect with the typical length scale). In our study, in the flows 1.L and 2.L and in the flows 1.H, 2.H, 1.V.A–2.V.B. The fact that the suppression of three-dimensional turbulence and dramatic changes of the flow structure are found in the simulations with strong magnetic field but not with weak one is, therefore, fully consistent with the known trend.
We have explored the effect of the geometric features of the system on the flow’s behaviour at strong magnetic field. It has been found that the role of the orientation of the magnetic field, which can also be interpreted as the role of the wall-to-wall distances across and along the field, is minimal. This is demonstrated by the lack of noticeable differences between the flows in the runs 1.H, 2.H on the one hand and 1.V.B, 2.V.B on the other hand.
On the contrary, the initial anisotropy introduced by the honeycomb has strong effect on the flow with the quasi-two-dimensional vortices. As demonstrated by the simulations 1.H, 1.V.B and 1.V.A, the amplitude of the vortices is substantially reduced when the flow structures formed at the exit of the honeycomb are elongated across rather than along the magnetic field.
As we have already mentioned, the results of the simulations are in good qualitative agreement with the experimental data of Sukoriansky et al. (1986). The high-amplitude fluctuations appear at the same values of Ha. Assuming that the simulation 1.V.B is the closest analogue of the experiment, we notice that the ratios between the fluctuation amplitudes in the case 1 and case 2 configurations of the magnetic field are of the same order of magnitude: about 5 in the experiment and about 2.5 in the simulations (see table 2).
At the same time, we cannot ignore the lack of quantitative agreement between the simulations and the experiment. In the computed flows, the turbulence intensity is about five times lower than measured in the experiment. This is true for both low and high values of Ha and for different orientations and spatial structures of the magnetic field. Several possible explanations can be given, related to both the numerical and experimental procedures. The most likely members of the first group are the insufficient resolution of the shear layers in the jets exiting the honeycomb and the assumption of laminar, with weak random noise, nature of the jets. It is well known (see e.g. Kim & Choi 2009) that, in numerical simulations, the instability and mixing of submerged jets are strongly affected by the resolution and the inlet conditions. This may potentially lead to lower energy injection from the jets into the small-scale turbulent fluctuations. We cannot reliably discuss the possible role of the experimental procedure due to the substantial time that has passed since the experiment was completed.
From the viewpoint of the turbulent decay theory, the results of our work provide a good example of non-universality of decay of MHD turbulence. The curves in figures 8 and 9 show complex behaviour of the fluctuation energy. The decay rate varies with the stage of the process and among the velocity components. The values of the two independent non-dimensional parameters (for example, N and Re) do not determine the decay scenario in a unique way. The process is strongly affected by the development, or lack thereof, of quasi-two-dimensional structures. The appearance and nature of such structures is, in turn, determined not just by the strength of the magnetic field, but also by the features of the flow evolution, most importantly, by the state of the flow at the moment the magnetic field is introduced.
The work is financially supported by the DFG grants KR and SCHU , the Helmholtz Alliance “Liquid metal technologies” (Limtech) and the grants CBET and CBET from the US NSF. Computations were performed on the parallel supercomputers Jureca of the Forschungszentrum Jülich (NIC) and SuperMUC of the Leibniz Rechenzentrum (LRZ), flow visualization was done at the computing center of TU Ilmenau.
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