Session R31: General Fluid Dynamics: General II

High Reynolds number incompressible crossflow on a rectangular cylinder near a plane wall boundary

Classical literature on different types of cylinders (rectangular or circular) in crossflow have well established the relationship between the drag coefficient versus the Reynolds number; where in the drag coefficient is generally independent of viscous effects beyond a critical Reynolds number.

This study takes the classical cylinder problem and introduces a single plane wall boundary near the rectangular cylinder resulting in a gap between the cylinder and the plane wall. This model is investigated under high Reynolds number incompressible flow. Because of the proximity to a plane wall boundary, the classical symmetric von Kármán sheet becomes asymmetric. Additionally, the characteristics of the flow are no longer independent of the Reynolds number. The geometric setup can be described as semi-confined, with the cylinder confined by a free slip “atmospheric” boundary on one end, and a no slip plane wall boundary on the other end.

This work has applications in crosswind with road vehicles and trains, aerodynamics of an airplane on a runway, engineered debris structures, mechanical fluid systems, and marine and coastal systems to name a few examples.   

Eigenfunction expansions for sixth-order boundary value problems arising in elastic-plated thin-film dynamics

Thin-film flows with a surface that has elastic bending resistance are governed by a sixth-order parabolic long-wave equation for the film height. Linearizing for small deflections about the equilibrium film height leads to a sixth-order boundary value problem (BVP). We discuss the boundary conditions (BCs) under which such sixth-order BVPs relevant to thin-film dynamics are self-adjoint. For a particular set of BCs, corresponding to an elastic-plated thin film in a closed trough, we explicitly derive a complete set of odd and even orthonormal eigenfunctions, which resemble trigonometric sines and cosines, as well as the so-called ``beam'' functions. Further, we derive explicitly the formulae for expressing derivatives of these eigenfunctions back into the same basis. Based on these novel, explicitly-constructed eigenfunctions and their derivative expansions, we propose a Galerkin spectral method for sixth-order BVPs relevant to thin-film dynamics. Importantly, due to the higher-order nature of the BVP, the coefficients of the spectral series decay rapidly in an algebraic manner, making the proposed expansions a highly-efficient computational tool. The proposed Galerkin spectral method and its convergence are demonstrated by solving model sixth-order problems.   

A Minimization Principle for Incompressible Fluid Mechanics

Fluid mechanics is a branch of mechanics concerned with the motion of fluid flows. Mechanics is typically classified into Newtonian and variational mechanics. The great victories in mechanics achieved by physicists in the past century were mainly enabled by variational approaches. While classical mechanics fails at small scales (e.g., atoms) and large scales (e.g., planets), variational mechanics has provided a fundamental approach in both Einstein’s general relativity and quantum mechanics. On the other hand, when we focus our attention on the mechanics of fluids, we find little success has been achieved beyond the Newtonian approach which was culminated in the Navier-Stokes equations of motion. Certainly, the distribution of research efforts between Newtonian and variational formulations in fluid mechanics is exorbitantly uneven; the overwhelming majority of research efforts are concerned with the Newtonian-mechanics formulation of Navier-Stokes' equations, which may be stalling. Indeed, there is a real need to seriously consider other branches in mehcanics and what they may offer to the motions of fluids. 

Most variational principles in classical mechanics are based on the principle of least action, which is only a stationary principle; the “least action” is a misnomer. In contrast, Gauss’ prin-

ciple of least constraint is a true minimum principle. In this paper, we apply Gauss’ principle to the mechanics of incompressible flows, thereby discovering the fundamental quantity that Nature minimizes in most flows encountered in everyday life. We show that the magnitude of the pressure gradient over the domain is minimum at every instant of time. We call it the principle of minimum pressure gradient (PMPG). The PMPG is expected to be of paramount importance for theoretical modeling of fluid mechanics as it encodes a complicated nonlinear partial differential equation into a minimization problem. It even transcends Navier-Stokes’ equations in its applicability to non-Newtonian fluids with arbitrary constitutive relations and fluids subject to arbitrary forcing (e.g. electromagnetic).   

Magnus Force Estimation using the principle of minimum pressure gradient

The flow around a rotating cylinder is one of the fundamental problems that piqued the interests of many venerable aerodynamicists and fluid mechanicians since the time of Lord Rayleigh. The force caused by the rotation of the cylinder has always been considered as an immediate consequence of viscosity, since the potential flow model failed entirely to predict the value of the circulation, due to the lack of a Kutta-like condition. On the other hand, Glauert modeled the flow outside the boundary layer of a rotating cylinder as a potential flow with an unknown circulation. He then obtained an approximate solution of Prandtl’s boundary layer equations and applied the no-slip condition to estimate the circulation in the outer flow. Interestingly, for rapidly rotating cylinders (α = ωR/U >> 1), up to fourth-order in the small parameter ϵ = 1/α, the obtained circulation is independent of viscosity. In this work, we use Glauert’s model of the outer flow over a rotating cylinder (i.e., a potential flow with an unknown circulation). However, instead of the tedious boundary layer calculations, we rely on the recently developed Principle of Minimum Pressure Gradient to obtain the unknown circulation. Perfect matching with Glauert’s solution is found. Moreover, our solution, in contrast to Glauerts’, points to the existence of different physics at small rotational speeds. The obtained results, given their perfect matching with Glauert’s solution (relying on the no-slip condition), points to a potential equivalence between the no-slip condition and fluid body forces.   

Pressure Unit Conversion in Lattice Boltzmann Method

The lattice Boltzmann method (LBM) is a class of computational fluid dynamics techniques that operates on mesoscale to simulate complex flows. It is a special discrete representation of the Boltzmann equation on lattice nodes/cells, solving the time evolution of particle distribution functions through a set of collision and propagation operations. The fluid particles propagate from one lattice node/cell to another and collide with each other at each time step, resulting in the redistribution of particle distribution functions according to local fluid properties. The macroscopic variables such as density and velocity are calculated through the zeroth and 1^st order moments of the particle distribution functions. One challenge in LBM is that pressure values are typically represented on the mesoscopic level, which may not directly correspond to macroscopic units like Pascals or atmospheres used in real-world flow systems. Therefore, it is crucial to establish an appropriate conversion between lattice units (mesoscale) and physical units (macroscale) to ensure the accuracy and relevance of computed pressure. In this study, we systematically explore pressure unit conversion in two specific scenarios: Stokes flow and laminar or turbulent pipe flow, considering viscous and inertial effects, respectively, to characterize the pressure field. The simulated pressure fields are rigorously validated against analytical solutions and experimental measurements. We also address pertinent issues concerning the influence of lattice models and LBM models on the accuracy of pressure quantification, providing practical recommendations to enhance the precision of pressure computations.