| Abstract: |
| The projection of Navier-Stokes on the space of divergence-free vector fields is associated with a minimization problem. We recently formulated such a minimization problem in what we call The Principle of Minimum Pressure Gradient (PMPG). The principle asserts that an incompressible flow evolves from one instant to another in order to minimize the L2 norm of the pressure gradient. We proved that Navier-Stokes equation is the necessary condition for minimizing the pressure gradient cost subject to the divergence-free condition on the local acceleration. In the discretized domain, this problem is a convex quadratic programming problem, which has a closed-form solution. The resulting necessary condition is a quadratic ODE in velocity, directly without the need to solve for pressure, avoiding the need to solve the Poisson equation in pressure at every instant of time. This approach is expected to provide significant savings in computations. Moreover, it should facilitate the mathematical analysis of incompressible flows by exploiting tools from nonlinear systems theory to analyze the resulting quadratic ODE. Based on this view, we provide a conjecture for a necessary condition for nonlinear hydrodynamic stability: If an equilibrium solution is stable, it must minimize the L2 norm of the convective (and viscous) acceleration among all equilibrium, divergence-free solutions. This conjecture applies successfully to the ideal flow over an airfoil. |
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