A computational method for low mach number unsteady compressible free convective flows

Optimal control of stationary, low Mach number, highly nonisothermal, viscous flows

“…The essential differences between the model (1)(2)(3)(4)(5)(6)(7)(8) and the Navier-Stokes system are evident. Not only is there coupling between the conservation of energy equation equation (4) and the conservation of momentum and mass equations (1) and (2), respectively, but we also have the appearance of the density function ρ.…”

“…Not only is there coupling between the conservation of energy equation equation (4) and the conservation of momentum and mass equations (1) and (2), respectively, but we also have the appearance of the density function ρ. Note, in particular, that from (2), the velocity field u itself is not solenoidal.…”

“…Also, T actually denotes a temperature difference with respect to a reference temperature. A detailed derivation of the model (1)(2)(3)(4)(5)(6)(7)(8) is given in [6]; see also [4]. Here, we only mention that the model is derived from the equations for viscous, compressible flows (see, e.g.…”

“…In Section 2, we study the existence of solutions of the boundary value problem, i.e., of the system (1)(2)(3)(4)(5)(6)(7)(8) where θ is a prescribed function. In Section 3, we study the existence of solutions of the optimal control problem, i.e., the problem of finding θ and u such that the functional of (9) is minimized subject to (1)(2)(3)(4)(5)(6)(7)(8). Then, in Section 4, we derive an optimality system, i.e., a boundary value problem from which the optimal control θ and optimal state (u, T, p) may be determined.…”

“…In the nondimensional model (1)(2)(3)(4)(5)(6)(7)(8), the variables and data functions have been appropriately rescaled. Also, T actually denotes a temperature difference with respect to a reference temperature.…”