Finite Element Adaptive Method for Hypersonic Thermochemical Nonequilibrium Flows

A ® nite element, segregated method is presented for two-dimensional hypersonic, thermochemical nonequilib- riumows, with emphasis on ef® ciently resolving shock waves by mesh adaptation. The governing equations are decoupled into three systems of partial differential equations (PDE), gasdynamic, chemical, and vibrational sys- tems, and then solved in a sequential manner. This approach has theadvantage of reducing reactingow problems to a manageable computational level and offers the possibility of applying the most appropriate scheme for each system to achieve the best global convergence. Each system of PDE is integrated by an implicit time marching technique, with a Galerkin± ® nite element method used in space. Theow solver is coupled to an adaptive proce- dure based on an a posteriori error estimate and a mesh movement strategy with no orthogonality constraints. The overall methodology is validated on nitrogen and air hypersonicows and experimental results are correctly reproduced using relatively coarse, but adapted meshes. NTEREST in the area of hypersonic vehicles has increased the needforadvancedcomputationaluiddynamicscodestoservein the numerical prediction of thermochemical nonequilibriumows. Theow® eldsofsuchproblemsarechemicallyreacting,andmolec- ularspeciesarevibrationallyexcitedsonumericalanalysesbasedon anidealgasassumptionareinappropriate.Attheotherextreme,sim- ulations including ionization and thermochemical nonequilibrium phenomena remain an onerous task even on today's supercomput- ers. Therefore, a cost-effective solution of the problem requires the proper modeling of the phenomena involved, along with an ef® cient solver. An accurate prediction of suchows includes the resolution of very strong detached shocks, followed by extremely fast vibrational relaxationphenomenonandintense chemicalreactions.Theregions containing those important phenomena are characterized by steep directional gradients ofow variables; their limits are unknown a priori and are embedded in regions where theow variables vary more smoothly. Hence, an accurate numerical simulation of such ¯ ows would require a ® ne grid, compounding the dif® culty of the problem.