Tag - High Reynolds number flow

In this talk, we consider the 3D incompressible Navier-Stokes equation in a bounded domain, with a canonical example of Poiseuille flow in mind. We provide an unconditional upper bound for the boundary layer separation and energy dissipation of Leray-Hopf weak solutions, uniformly in high Reynolds numbers. We estimate layer separation by measuring the energy norm of the discrepancy between a (turbulent) low-viscosity Leray-Hopf solution and a fixed (laminar) regular Euler solution with similar initial conditions and body force. This is accomplished by a new nonlinear boundary vorticity estimate.

The Euler equation does not possess a unique solution for the flow over a 2-dimensional object. This problem has serious repercussions in aerodynamics; it implies that the inviscid aero-hydrodynamic lift force over a 2-dimensional object cannot be determined from first principles; a closure condition must be provided. The Kutta condition has been ubiquitously considered for such a closure in the literature, even in cases where it is not applicable (e.g. unsteady). In this talk, I will present a special variational principle that we revived from the history of analytical mechanics: Hertz’s principle of least curvature. Using this principle, we developed a novel variational formulation of Euler’s dynamics of ideal fluids that is fundamentally different from the previously developed variational formulations based on Hamilton’s principle of least action. Applying this new variational formulation to the century-old problem of the ideal flow over an airfoil, we developed a general (dynamical) closure condition that is, unlike the Kutta condition, derived from first principles. In contrast to the classical theory, the proposed variational theory is not confined to sharp edged airfoils; i.e., it allows, for the first time, theoretical computation of lift over arbitrarily smooth shapes, thereby generalizing the century-old lift theory of Kutta and Zhukovsky. Moreover, the new variational condition reduces to the Kutta condition in the special case of a sharp-edged airfoil, which challenges the widely accepted wisdom about the viscous nature of the Kutta condition. We also generalized this variational principle to Navier-Stokes’s via Gauss’s principle of least constraint, thereby discovering the fundamental quantity that Nature minimizes in every incompressible flow. We proved that the magnitude of the pressure gradient over the field is minimum at every instant! We call it the Principle of Minimum Pressure Gradient (PMPG). We proved that the Navier-Stokes equation is the necessary condition for minimizing the pressure gradient subject to the continuity constraint. Hence, the PMPG turns any fluid mechanics problem into a minimization one where fluid mechanicians need not to apply Navier-Stokes equations, but merely need to minimize the proposed action.

We develop a new theory of circulation statistics in strong turbulence (ν → 0 in the Navier-Stokes equation), treated as a degenerate fixed point of a Hopf equation. We use spherical Clebsch variables to parametrize vorticity in the stationary singular Euler flow. This flow has a tangent velocity gap due to the phase gap in the angular Clebsch variable across a discontinuity surface bounded by a stationary loop C in space. We find a circular vortex with a singular core on this loop, regularized as a limit of the Burgers vortex. We compute anomalous contributions to the Euler Hamiltonian, helicity, and the energy flow, staying finite in the vanishing viscosity limit. The normalization constant in the spherical Clebsch variables is determined from the energy balance between incoming flow and anomalous dissipation. The randomness (spontaneous stochastization) comes from the Gaussian fluctuations of a background velocity due to random locations of remote vortex structures. Assuming weak fluctuations of the background velocity field, we compute the probability distribution of velocity circulation Γ, which decays exponentially with pre-exponential factor 1/√Γ in perfect match with numerical simulations of conventional forced Navier-Stokes equations on periodic lattice 8K³. We also compute effective multifractal indexes for the tails of velocity circulation probability density as a function of conditional probability below that tail. The anomalous dimensions are independent of this probability and decrease as inverse powers of the logarithm of the size of the loop.