# Time-domain BEM solution of convection-diffusion-type MHD by Bozkaya N., Tezer-Sezgin M.

By Bozkaya N., Tezer-Sezgin M.

The two-dimensional convection-diffusion-type equations are solved by utilizing the boundary point approach (BEM) in line with the time-dependent basic answer. The emphasis is given at the answer of magnetohydrodynamic (MHD) duct circulation issues of arbitrary wall conductivity. The boundary and time integrals within the BEM formula are computed numerically assuming consistent diversifications of the unknowns on either the boundary parts and the time durations. Then, the answer is complicated to thesteady-state iteratively. therefore, it's attainable to take advantage of fairly huge time increments and balance difficulties are usually not encountered. The time-domain BEM resolution approach is demonstrated on a few convection-diffusion difficulties and the MHD duct stream challenge with insulated partitions to set up the validity of the approach.The numerical effects for those pattern difficulties examine rather well to analytical effects. Then, the BEM formula of the MHD duct move challenge with arbitrary wall conductivity is received for the 1st time in this kind of approach that the equations are solved including the coupled boundary stipulations. using time-dependent primary resolution allows us to acquire numerical strategies for this challenge for the Hartmann quantity values as much as three hundred and for numerous values of conductivity parameter. Copyright q 2007John Wiley & Sons, Ltd.

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SHEN Table I. 1 density, velocity and pressure L 1 errors for the Godunov scheme with the exact Riemann solver and the new solver in different scales of grids. 321964E−3 Table II. 2 density, velocity and pressure L 1 errors for the MUSCL scheme with the exact Riemann solver and the new solver in different scales of grids. 808145 Table III. 3 density, velocity and pressure L 1 errors for the MUSCL scheme with the exact Riemann solver and the new solver in different scales of grids. 182284 Table IV.

These equations express conservation of mass and momentum. In conservative form, they are as follows: Ut + F(U )x = S U= h hu , F(U ) = hu hu 2 + 12 gh 2 , S= 0 (1) gh(s0 − sf ) where U is the conservative variables vector in which, h and u are flow depth and velocity, respectively; F(U ) the flux vector; S the source term vector, consist of bed slope S0 and friction slope Sf . 33 where n is the empirical Manning resistance coefficient. The governing equations are based on the assumptions of hydrostatic pressure distribution, incompressibility of water and a sufficiently small channel slope.

The source term consists of two parts, the bed slope and the friction slope. Treatment of each part will be explained separately. 1. Bed slope treatment The main subject in dealing with the bed slope is to balance it with the convection flux in each cell at stationary flow field. This will be explained in more detail under the C-property preposition [29]. The C-property will be explained and approved for the proposed algorithm later on, but first the details of the source term discretization are presented below.