## Numerical Method

The Reynolds-Averaged Navier-Stokes equations in conjunction with a near wall Reynolds stress turbulence model are solved using the chimera RANS method together with near-wall second-moment turbulence closure. The flow is considered to be incompressible since the Mach number is quite low. However, the density in the centrifugal force terms is approximated by p=poTo/T to account for the density variations caused by the temperature differences. po and To are the density and temperature at the inlet of the cooling channel. In general, the density is also a function of the rotating speed because the centrifugal force creates a pressure gradient along the duct. In the planned experiments to be conducted later, the maximum pressure variation between the channel inlet and the exit is approximately 0.0113 atm at the highest rotating speed of W = 550 rpm. This gives a maximum density variation of only about 1.1% from the inlet to the exit of the duct at the highest rotation number. It is therefore reasonable to omit the density variation caused by the pressure gradients induced by the channel rotation.

The present method solves the mean flow and turbulence quantities in arbitrary combinations of embedded, overlapped, or matched grids using a chimera domain decomposition approach. In this approach, the solution domain was first decomposed into a number of smaller blocks to facilitate efficient adaptation of different block geometries, flow solvers, and boundary conditions for calculations involving complex configurations and flow conditions. Within each computational block, the finite-analytic numerical method was employed to solve the unsteady RANS equations on a general curvilinear, body-fitted coordinate system. The coupling between the pressure and velocity was accomplished using the hybrid PISO/SIMPLER algorithm. The method satisfied continuity of mass by requiring the contravariant velocities to have a vanishing divergence at each time step. Pressure was s°lved using the c°ncept °f pseud°-vel°cities and, when c°mbined with the finite-analytic discretizati°n gives the P°iss°n equati°n f°r pressure.

A unif°rm vel°city profile was used at the inlet °f the duct (Z = 0). The unheated length (Lj) was l°ng en°ugh f°r the vel°city profile t° be fully devel°ped turbulent prefile bef°re the heating start-p°int (Z = Lj). Zero-gradient b°undary c°nditi°ns were used at the exit °f the duct f°r mean vel°city and all turbulent quantities, while linear extrap°lati°n was used f°r the pressure field. The c°°lant fluid at the inlet °f the duct is air at unif°rm temperature T = To (i.e., q = (T - To) / (Tw - To) = 0). The wall temperature °f the unheated secti°ns is kept c°nstant at T = To (6 = 0) while the wall temperature °f the heated secti°n is kept c°nstant at T = Tw (6= 1).

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