Numerical Simulation Of Rotating Flow And Heat Transfer In A Cylinder Under The Effect Of A Magnetic Field

Symmetrical rotating flow in a cavity has always been a subject that many scholars have devoted to research.Among them,the most widely used cavity is a cylindrical cavity with a rotating end wall.Because even under stable laminar flow conditions,an important flow phenomenon occurs in this cavity,and the vortex breaks.Applying a uniform magnetic field or temperature gradient outside the cylindrical cavity will affect the vortex rupture and heat transfer generated when the fluid flows in the cavity due to the Lorentz force and buoyancy force.In order to analyze the influence of uniform magnetic field and temperature gradient on the fluid flow in the cavity,this paper simulated the fluid flow and heat transfer in an insulated cylindrical cavity driven by the upper end wall to rotate the fluid under the effect of a magnetic field.After listing the governing equations based on the physical model,this paper firstly made them dimensionless,and then solved the dimensionless governing equations by the spectral projection method.For the discretization of the momentum equation,Chebyshev-Fourier polynomials were used for spatial discretization,where the r direction and z direction were discretized with Chebyshev-Gauss-Lobatto configuration points according to the boundary conditions.Then combined with the Improved Projection Scheme(IPS)to decouple the velocity and pressure in the momentum equation into the Poisson equation,and then converted the Poisson equation to the matrix equation,and solved each governing equation(except continuity equation)matrix equations.The simulation calculation of the rotating flow and heat transfer in a cylindrical cavity with a fixed aspect ratio of 1.5 under the effect of a magnetic field was realized by programming a program in MATLAB.Through analyzed of the results,the specific effects of temperature gradient and uniform magnetic field on the pressure,velocity,and the position and size of the vortex rupture were obtained,and the Reynolds number,Richardson number,Hartman number,Prandtl number and Nusselt number were specified.Analyzed the changes in rotating flow and heat transfer.The results obtained were as follows:When the temperature of the top lid of the cylinder is higher than the temperature of the bottom lid,with the range of 0<Re<2500,the buoyancy force generated by the temperature difference can inhibit the vortex breakdown and stabilize the flow,and as the Richardson number increases,the vortex breakdown's size and central position of the restraint are getting bigger and bigger.The Prandtl number is very small,the increase of Richardson number has little effect on temperature,and the temperature gradient in the fluid is close to linear distribution in the vertical direction.When Pr=0.7,an almost uniform temperature is generated in the core area of the fluid.When r<0.52,the local Nusselt number decreases as the Richardson number increases;when r>0.52,the opposite is true.The local Nusselt number of the bottom lid is near the axis,and decreases as the Richardson number first increases,while near the side wall,the local Nusselt number increases along with the Richardson number.The Lorentz force generated by the magnetic field also has an inhibitory effect on the vortex breakdown.With the change of Hartmann number,the Lorentz force mainly suppresses the generation of vortex breakdown by suppressing the axial and radial speeds.The size of the vortex breakdown is getting smaller and smaller,and the central position is getting lower and lower.When the Hartmann number increases to a certain value,the vortex rupture disappears.As a result,the flow tends to be stable.When r<0.7,the Nusselt number of the top lid gradually decreases with the increase of Hartmann number.Compared with Ha=0 when Ha=60,it decreases by about 6.0%at the axis;when r>0.7,on the contrary,compared with Ha=0 when Ha=60,the increase at the side wall is about 5.4%.When the bottom lid Nusselt number is r<0.6,the Nusselt number first decreases and then increases as the Hartmann number increases;when r>0.6,the opposite is true,the Nusselt number first increases and then decreases.With the increase of Hartmann number,the change of Nusselt number in the radial direction gradually tends to be stable,and the change of temperature in the radial direction gradually tends to be linear.