Computational aerothermodynamics

Aerothermodynamics is defined1 as “the study of the relationship of heat and mechanical energy in gases, especially air”. To those familiar with fluid dynamics (the study of the flow properties of liquids and gases) this means that we must consider thermodynamic and chemical processes as they are coupled to the fluid motion. Computational fluid dynamics involves the numerical simulation of the equations of motion for an ideal gas; these equations are the conservation of mass, momentum and energy, and in their most general form are the compressible Navier-Stokes equations. Computational aerothermodynamics concerns the coupling of real gas effects with these equations of motion to include thermochemical rate process for chemical and energy exchange phenomena. These processes concern the creation and destruction of gas species by chemical reactions and the transfer of energy between the various species and between the various energy modes (e.g. translation, rotation, vibration, ionization, dissociation/recombination, etc.) of the species. To gain some insight into when such phenomena occur for current and future aerospace flight vehicles Fig. 1 shows the flight regimes of some typical vehicles (e.g. Concord, aerospace plane, Space Shuttle, aeroassisted space transfer vehicles, Apollo entry vehicle, etc.) in terms of flight altitude and flight speed. Also indicated in the figure are regimes where chemical reactions such as dissociation and ionization are important and where nonequilibrium thermochemical phenomena are important. To account for chemical reactions equations for the conservation of each chemical species must be added to the flow field equation set. There are 5 flow field equations; one continuity, three momentum and one energy equation. For a simple model of dissociating and ionizing air there are typically 11 major species (N2, 2, N, , NO, N+2, +2, +, N+, NO+, e-). The inclusion of conservation equations for each of these species nearly triples the number of equations to be solved. When there are combustion processes or gas/surface interactions or ablation products, the number of species increases dramatically. To account for thermal non-equilibrium there are additional energy conservation equations to describe the energy exchange between the various energy modes (translational, rotational, vibrational, electronic, etc.). The full set of these conservation equations have been derived by Lee2. Under the following assumptions, namely: continuum and no slip at solid boundaries (Kn≪≪1)the thermal state of the gas can be described by separate, independent temperaturesthe rotational state of the gas is in equilibrium with the translational stateweak ionization (≪1%) the governing equations are written in Cartesian coordinates as3: n mass conservation equations where s is the chemical species @@@@s / @@@@t + @@@@ / @@@@j (suj) = - @@@@ / @@@@j (svsj) + ws (1)d momentum conservation equations, where d is the number of spatial dimensions @@@@ / @@@@t (ui) + @@@@ / @@@@j (uiuj + pδij) = - @@@@τij / @@@@j - @@@@ (Ns / Ne) Zs (@@@@pe / @@@@i) (2)conservation of vibrational energy for each of the m diatomic species @@@@Evs / @@@@t + @@@@ / @@@@j (Evsuj) = - @@@@ / @@@@j (Evsvsj) - @@@@qvsj / @@@@j + QT - vs + Qv - vs + Qe - vs + wsevs (3)electron energy is conservation @@@@Ee / @@@@t + @@@@ / @@@@j (Eeuj) = - @@@@ / @@@@j (Eevej) - pe@@@@uj / @@@@j - @@@@qej / @@@@j + QT-e - @@@@ Qe-vs + weee (4)And conservation of total energy @@@@E / @@@@t + @@@@ / @@@@j ((E + p)uj) = - @@@@ / @@@@≫j (qj + quj + qej) - @@@@ / @@@@j (uiTij) - @@@@ @@@@ / @@@@jvsjhs - @@@@ Ns / NeZs@@@@pe / @@@@iui (5)