Uni- and bidirectional mixed convection flow regimes described by dual solutions in a vertical duct

When is natural convection completely passive?

“…All the fluid properties are in fact temperature-dependent, but if cross-sectional variations of temperature are small compared to absolute temperatures, the approximation by constant values may be sound. However, depending on the boundary conditions, it is possible to have a fully developed flow with the cross-section mean temperature T m and density ρ m varying linearly in the streamwise direction along the duct [7], provided that the buoyancy force, proportional to T − T m in (12), remains streamwise invariant. [We show below that passive convection is impossible in circumstances where T m and ρ m vary streamwise, but the implications for the Boussinesq approximation are worth discussing for more general cases.]…”

“…from (27); however, if T m is used as the reference temperature, (12) indicates that a boundary condition on second derivatives of W must involve T m if the wall temperature is given. But the cross-section mean temperature T m is not known a priori, whereas the dynamic pressure gradient may be specified (by a mechanical pump in an engineering application); equation (10) relates this pressure gradient to the first derivative of W at the wall.…”

“…The temperature variation (T − T m ) over the cross-section can then be obtained from (12); since this will yield the value of T a − T m , the temperature distribution can finally be recast in terms of the given wall temperature T a . In any less symmetrical geometry, or if there is any spatial variation in wall temperature, the wall shear stress μ∂W/∂n will not be uniform.…”

“…An iterative solution procedure is then required. The mean temperature T m is first estimated, and this estimate is used in (12), together with the known wall temperature and pressure gradient, to yield a boundary condition involving the second normal derivative of W at the wall (∇ 2 h W will not involve derivatives parallel to the wall, since W = 0 at the wall, but it may also involve the first as well as the second normal derivative if the wall is curved). Equation (29) is then solved with this boundary condition and the no-slip condition, and the result is used to compute ∂W/∂n at all points on the wall.…”

“…In the non-symmetric case, the iterative method involves solving (29) with the no-slip condition and (44), with T m in the latter estimated so as to yield a solution satisfying the wall shear-stress condition (10) as closely as possible. With symmetric geometry and boundary conditions, the solutions for W are exactly the same as with prescribed wall temperatures, but the value of T m is finally obtained from (44) rather than by simply substituting T = T a in (12). The appropriate forms of (44) are…”