Effect of wall electrical conductivity and transverse magnetic field on turbulent mercury flows in nonuniformly heated pipes

. The problem of mixed convection of mercury in a horizontal circular tube heated from below under the effect of a transverse magnetic field is examined. A numerical study was performed using the LES method in a conjugate formulation including the wall which may be with or without a fouling layer (deposits) with low electrical conductivity on its inner surface. To simulate the subgrid viscosity, the CSM model was used. In the case with a Reynolds numbers of Re 0 = 5000, Hartmann number of Ha 0 = 150, Grashof number of Gr 0 = 2.75×10 6 , which is based on the inner radius of the pipe, and Prandtl number of Pr = 0.024, a strong influence of the fouling layer electrical conductivity on the fluid flow structure and the amplitude and frequency of fluctuations of temperature, velocity, and electric potential has been found.


Introduction
Over the past decade, many experimental and computational studies into the turbulent flows of mercury in pipes and channels of various orientations in space and exposed to nonuniform thermal load and a transverse magnetic field (MF) have been carried out [1].The combined effect of electromagnetic forces and natural convection on the flow induces in some cases low-frequency fluctuations of the temperature of the liquid and the channel wall inner surface with abnormally high amplitudes.At the same time, the temperature fluctuations inside the solid wall and the effect of the thermophysical and electrical properties of the wall material on the fluctuations and the flow structure have hardly been investigated at all.Such effects can be very dangerous in liquid metal cooling systems of fusion reactors.
In this paper, numerical simulation is carried out of turbulent flows of mercury in non-uniformly heated pipes under the influence of transverse MF and buoyancy forces in a wall-conjugate formulation considering a thin fouling layer of deposits (such as oxides or other contaminants) with low electrical conductivity on the wall inner surface.Calculations are performed for both vertical and horizontal pipes.Within the scope of this paper, a horizontal pipe is considered in detail.

Problem formulation
The flow of mercury in a horizontal steel pipe with a inner radius of R * = 9.5 mm and a wall thickness of * w  = 0.5 mm is considered (hereinafter dimensional values are indicated by the superscript "*").
The computation domain along the pipe axis z * is divided into three sections: the inlet turbulence generator (IPG) with a length of 10R * , the working section with the heating zone and the MF area with a length of 90R * , and the adiabatic section (see Fig. 1).The total length of the computation domain is 126R * .The lower half of the pipe outer surface (semicircle in cross section) of the working section is heated by an attached heater with a constant heat flux * w q .The transverse magnetic field acts in the negative direction of the x * axis and varies along the z * axis according to the curve shown in Fig. 1.
where * 0 u = 0.059 m/s is the bulk velocity, * 0 B = 0.6 T is the "plateau" magnetic field induction (Fig. 1), kinematic viscosity, thermal conductivity, density, electrical conductivity, specific heat capacity, coefficient of volume expansion, g * is module of the gravitational acceleration vector, * * 0.5 av w q q  .
The calculations were performed using the in-house CFD-code ANES [2].The mathematical description of the problem in dimensionless form includes the equations of conservation of mass (1), momentum (2), energy for a liquid (3) and a solid wall with potential deposits layer (4), Ohm's law for calculating the electric current density (5), as well as the Poisson equation for the electric potential (6).The properties of the liquid, wall, and fouling layer are considered constant.The electromagnetic force in the momentum equation is calculated by the equation (7).To account for the buoyancy forces, the Boussinesq approximation (7) is used.The subgrid viscosity is calculated using the CSM model proposed in [3], and the subgrid Prandtl number is Pr sgs = 0.85.(5) T is the inlet temperature, subscript "w" stands for the solid wall.
The boundary conditions prescribed on the outer surface of the solid wall were as follows: , 10 100 and 0 0, else At the inlet ("In" in Fig. 1 ) to the heating section of the pipe the fields of instantaneous turbulent velocity ux, uy, uz from IPG and in = 0 were used as the inlet boundary conditions.The velocity vector u was zero in the wall.At the outlet boundary ("Out" in Fig. 1), the The predictions by the k- RANS model [4] for a steady fully developed isothermal turbulent flow with imposed random disturbances were used as the initial conditions.
The unstructured Cartesian computational grid within the solid wall and the near-wall region in the liquid was constructed in the form of hexagonal layers of cells adapted in shape to the cylindrical surface of the wall (Fig. 2).The fouling film on the inner surface of the solid wall was simulated by singling out a thin area with a thickness of * f  (highlighted in yellow in Fig. 2) with  whose values were assumed to be 10, 30, 90 and 10000.
The grid was uniform along the longitudinal z axis.The total number of cells in the computational grid is 13.1 -13.5 million.

Fig. 2. Computation grid.
The following cases were simulated: IsolW when the fouling layer has a high electrical resistance (kf =10000), SW without a fouling layer.

Results
Let's take a closer look at the results for 0 Ha = 150.Figure 3a shows temperature oscillations obtained for the case IsolW at the point x * = 0.75R*, y * = 0 at the section z * = 74R * from the starting point of heating.A good agreement between the predictions and the data obtained by the DNS method in [1] is evident.It can be seen that the peak-to-peak amplitude of the fluctuations is as high as 8 0 C. At the same time, the temperature fluctuations in the wall (at x * = 1+δ * w/2, y * =0), indicated by "IsolW wall", are much less at the same section.
Without fouling on the inner surface of the pipe (case SW), fluctuations in the temperature of the wall and the liquid can hardly occur at all (Fig. 3b).This indicates a different flow structure in cases with an isolated or well-conducting wall, and this effect is strongly manifested in the structure of temperature fields obtained in these cases (Fig. 4).In the case IsolW, the field of the axial velocity uz is "compressed" in the vertical direction.At the same time, the maximum value of the dimensionless velocity ranges from 1.2 to 1.3.In the case SW, the velocity field is almost uniform and close to the plug flow pattern (Fig. 5).These differences are explained by the nature of the flow of electric current.In the liquid flow core, the electric current passes in the negative direction of the y axis.Near the walls at y < 0, the current-flow lines turn and become directed along the wall towards the upper generatrix of the pipe.The reverse trend is observed in the upper part of the pipe section (y > 0).In the IsolW case, a thin layer of the upward flow of electric current is observed only in the liquid, while in the SW case, the "reverse" current flows in both the liquid and the wall.At the same time, the density of the upward current in the wall is many times greater than the density of the upward current in the liquid (Fig. 6).Very high currents in the conducting wall of the pipe also increase the current density jy in the flow core (since ).As a result, the braking action of the electromagnetic force fmz is enhanced by a factor of 4 at Ha0 = 150.Profiles of fmz force are shown in Fig. 7.The natural consequence of these effects is a grow in the hydraulic resistance.
The coefficients of hydraulic resistance are presented in Table 1 ( p  is the pressure difference across the section with a length LB, in which the magnetic field induction is constant).It should be noted that with an increase in the Hartmann number, discrete maxima appear in the axial velocity profile near the wall (Robert layers) for SW case, which can hardly be seen at Ha0 = 150 at all.
The calculations were performed using the JIHT RAS and MVS 10P MSC RAS Fisher supercomputers.This work was supported by the Ministry of Science and Higher Education of the Russian Federation under project FSWF-2023-0017.

Fig. 1 .
Fig. 1.Schematic of the model.Parameters of simulated cases: remaining variables were set.

Fig. 4 .
Fig. 4. Instantaneous temperature fields in the symmetry planes of the pipe.

Fig. 5 .
Fig. 5. Fields of instantaneous dimensionless axial velocity (a) and graphs uz along the horizontal and vertical axes of the pipe (b) in the cross section z * = 74R * from the beginning of heating.

Fig. 6 .
Fig. 6.Electric currents in the cross-section z * = 74R * from the beginning of heating.

Fig. 7 .
Fig. 7. Profiles of electromagnetic force fmz along the horizontal and vertical axes of the pipe in the cross section z * = 74R * from the beginning of heating.

Table 1 .
The coefficients of hydraulic resistance.