Hierarchical formulations for numerical flow simulations

A new hierarchical formulation for the equations of fluid motion is developed. The hierarchical nature of the new formulation is due to its ability to simulate all levels of fluid flow approximations, namely, inviscid irrotational isentropic flows (potential flow formulation), inviscid rotational non-isentropic flows (Euler formulation) and viscous heat conducting flows (Navier-Stokes formulation). The new formulation uses a potential flow solver as a base solver which is evaluated everywhere in the flow field, while convection/diffusion equations for entropy, vorticity and total enthalpy are only evaluated within limited domains of the flow field where rotational effects are present such as in regions containing shocks, boundary layers and/or wakes. This is accomplished by using a Helmholtz decomposition of the velocity vector into the gradient of a potential function plus a rotational component. The density and pressure are reformulated in terms of the speed and entropy. The new formulation identifies an acoustic mode, governed by the potential equation, from the convection/diffusion mode governing the entropy, vorticity and total enthalpy. This identification of modes together with the ability to restrict the evaluation of entropy, vorticity and total enthalpy to relatively small domains within the flow field offers several advantages over the traditional Euler and Navier-Stokes formulations from the point of view of upwinding, multigrid and the incompressible flow limit. To test the robustness, efficiency and accuracy of this new approach, several flow problems are simulated. These problems include 2-D shock wave/boundary layer interaction, 2-D inviscid and viscous flows over cylinders, 2-D inviscid and viscous flows over airfoils and 3-D inviscid and viscous flows over wings. The results obtained using the new formulation agree well with both experimental results and numerical results obtained from traditional Euler and Navier-Stokes formulations. Fast convergence rates are also achieved through the implementation of multigrid to the augmented potential equation which results in an order of magnitude reduction in work units as compared to single grid computations.