Variational projection of Navier-Stokes: Fluid mechanics as a quadratic programming problem

Gauss's principle of least constraint transforms a dynamics problem into a pure minimization problem, where the total magnitude of the constraint force is the cost function, minimized at each instant. A candidate motion whose evolution minimizes the Gaussian cost at each instant is guaranteed to satisfy Newton's equation. In other words, Newton's equation is the first-order necessary condition for minimizing the Gaussian cost, subject to the given kinematic constraints. The principle of minimum pressure gradient (PMPG) is to incompressible fluid mechanics what Gauss's principle is to particle mechanics. The PMPG asserts that an incompressible flow evolves from one instant to another by minimizing the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msup> <a:mi>L</a:mi> <a:mn>2</a:mn> </a:msup> </a:math> -norm of the pressure gradient force. A candidate flow field whose evolution minimizes the pressure gradient cost at each instant is guaranteed to satisfy the Navier-Stokes equation. Consequently, the PMPG transforms the incompressible fluid mechanics problem into a pure minimization framework, allowing one to determine the evolution of the flow field by solely focusing on minimizing the cost—without directly invoking the Navier-Stokes equation. In this paper, we show that the resulting minimization problem is a convex quadratic programming (QP) problem—one of the most computationally tractable classes in nonlinear optimization, which has a rich literature with many efficient algorithms. Moreover, leveraging tools from analytical mechanics and the Moore-Penrose theory of generalized inverses, we derive an analytical solution for this QP problem. As a result, we present an explicit formula for the projected dynamics of the spatially discretized Navier-Stokes equation on the space of divergence-free fields. The resulting ordinary differential equation (ODE) is ready for direct time integration, eliminating the need for solving the Poisson equation in pressure at each time step. It is typically an explicit nonlinear ODE with constant coefficients. This compact form is expected to be highly valuable for both simulation and theoretical studies, including stability analysis and flow control design. We demonstrate the framework on the benchmark problem of unsteady flow in a lid-driven cavity.