On the Infinite-Reynolds number Limit of Navier-Stokes Solutions: Insights from the Principle of Minimum Pressure

The convergence of Navier-Stokes solutions to one of Euler's solution as Reynolds number goes to infinity (i.e., viscosity goes to zero) is an open problem in mathematics. It is known that if such a convegence holds, it will be to a weak solution of Euler (i.e., a non-smooth solution which does not satisfy the PDE at every point, but satisfies an integral version of the equation). However, weak solutions of Euler are non-unique; and a selection criterion is needed. That is, which weak solution in Euler's family does match the limit of Navier-Stokes' solutions when viscosity goes to zero?

This problem is not only mathematically elegant, but also of great practical value because infinite-Reynolds number limits may serve as good approximations for flows at very high Reynolds numbers, ubiquitously encountered in industry (over airplanes, wind turbines, submarines, cars, etc) and are elusive to find from Navier-Stokes alone without ad-hoc modeling (e.g., RANS models and wall models in LES). So, if a successful selection criterion is developed from first principles, it will be game changing. In this case, one may be able to compute a good approximation of the flow at a very high Reynolds number only by solving Euler's equation and using such a selection criterion to determine the special flow that matches the zero-viscosity limit of Navier-Stokes, i.e., without the need to resolve small scales in the boundary layer.

In an earlier effort, we presented the principle of minimum pressure gradient (PMPG), which asserts that the magnitude of the pressure gradient over the domain is minimum at every instant of time. We proved mathematically that Navier-Stokes' equation represents the necessary condition for minimization of the pressure gradient. Unlike typical variational formulations, the PMPG turns the fluid mechanics problem into a pure minimization one.

Here, we test the conjecture: the PMPG provides a selection criterion for zero-viscosity limits of Navier-Stokes. That is, the unique solution in Euler's family that minimizes the pressure gradient cost is expected to be the zero-viscosity limit of Navier-Stokes. We test this conjecture on the airfoil problem, the flow over a rotating cylinder, and the separating flow over a cylinder.