Variational Projection of Navier-Stokes: Fluid Mechanics as a Quadratic Programming Problem

TL;DR

The paper reframes incompressible fluid dynamics by applying Gauss's principle of least constraint through the Principle of Minimum Pressure Gradient, turning Navier–Stokes evolution into a convex quadratic programming problem. It derives an explicit projected dynamics equation (VPNS) via Moore–Penrose inverses, removing the need to solve a pressure Poisson equation at each time step. The approach yields a direct, time-marchable ODE for the discretized velocity field whose divergence is preserved by construction, and is demonstrated on a lid‑driven cavity with strong agreement to OpenFOAM and exceptionally small continuity errors. The work provides a rigorous variational underpinning for a projection-based incompressible flow solver, with potential benefits for stability analysis and flow-control design, and outlines directions for performance comparisons with standard projection methods.

Abstract

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. 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 L2-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. 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. 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 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 lid-driven cavity problem.

Figures (6)

• Figure 1: Comparison between the VPNS simulation over four meshes, and OpenFoam using $125\times125$ in terms of the flow field $u$, $v$ at the final time $T$ along the vertical and horizontal lines, respectively, passing through the mid point. The comparison shows excellent mesh convergence and matching with OpenFOAM.
• Figure 2: Contours of the nondimensional vorticity $\omega T_{ref}$ at $T/4$, $T/2$, $3T/4$, and $T$ from the VPNS and OpenFOAM simulations.
• Figure 3: Comparison between the VPNS and OpenFOAM in terms of the RMS and mean continuity residuals. The VPNS formulation is far more accurate in enforcing the continuity constraint.
• Figure 4: Variation of the normalized cost $\hat{\mathcal{A}}=\frac{\mathcal{A}}{\rho U_{lid}^4}$ with the size $\epsilon$ of perturbation from the true evolution $\dot{\bm{U}}^*$. The pressure gradient cost attaints its minimum precisely at $\epsilon=0$. That is, the VPNS evolution $\dot{\bm{U}}^*$ minimizes the cost $\mathcal{A}$ over all kinematically admissible evolutions $\dot{\bm{U}} = \dot{\bm{U}}^* + \epsilon \bm\eta$.
• Figure 5: Contours of the magnitude of the nondimensional evolution $\frac{\partial \bm{u}}{\partial t}\frac{T_{ref}}{U_{lid}}$ corresponding to (a, b) the perturbations $\bm\eta_1$, $\bm\eta_2$, and (c) the true evolution $\dot{\bm{U}}^*$. The subfigure (d) shows contours of the nondimensional cost $\hat{\mathcal{A}}$ in the $\epsilon_1$-$\epsilon_2$ plane. The pressure gradient cost attains its minimum precisely at $\epsilon_1=\epsilon_2=0$, confirming the optimality of $\dot{\bm{U}}^*$ over the family $\dot{\bm{U}} = \dot{\bm{U}}^* + \epsilon_1 \bm\eta_1 + \epsilon_2 \bm\eta_2$.