A nonperturbative approximation for the moderate Reynolds number Navier–Stokes equations

Abstract

The nonlinearity of the Navier-Stokes equations makes predicting the flow of fluid around rapidly moving small bodies highly resistant to all approaches save careful experiments or brute force computation. Here, we show how a linearization of the NavierStokes equations captures the drag-determining features of the flow and allows simplified or analytical computation of the drag on bodies up to Reynolds number of order 100. We illustrate the utility of this linearization in 2 practical problems that normally can only be tackled with sophisticated numerical methods: understanding flow separation in the flow around a bluff body and finding drag-minimizing shapes.analytical approximation ͉ drag minimization ͉ flow separation ͉ nonlinear PDEs M any methods that are currently used for finding approximate solutions of nonlinear partial differential equations were inspired by studies of fluid flows. The most successful methods can be divided into 2 classes. The first aims to approximate the flow field by a perturbative expansion in a small parameter. Examples include Stokes's equations for viscous flow, or Prandtl's theory of laminar boundary layers (1). The second class of approximations is fundamentally nonperturbative; these arise when instead of detailed solutions, it is sufficient to consider averaged quantities, such as the flux of momentum or of a passive scalar. These approximations work by exploiting scale separation and averaging over unresolved degrees of freedom. Prominent examples include eddy viscosity closure relations in turbulent shear flows (2), the effective dispersion of solute under the combined action of shearing flows and molecular diffusion (3), the renormalization group in condensed matter physics (4), and anti-aliasing methods in spectral numerical solvers (5) and recent advances in constructing linear effective equations to describe the spatially averaged solutions of the nonlinear Poisson and advection-diffusion equations (6, 7). Approximate solutions to nonlinear partial differential equations have 3-fold value: providing asymptotic solutions or closure relations in cases where fine spatial scales or short time scales prohibit direct numerical solutions, giving qualitative and quantitative insights into why solutions take the forms that they do and allowing accelerated optimization of the reduced set of quantities modeled by the approximate equations.In this article, we describe a nonperturbative approximation for the moderate Reynolds number steady Navier-Stokes equations,where u is an incompressible velocity field, satisfying ٌ⅐u ϭ 0, is the liquid density, and is the kinematic viscosity. Our approximation allows accurate determination of the drag determining features of a body with almost any shape translating through a fluid with velocity U.The approximate equation replaces the nonlinear term on the left-hand side with a linear approximationwhere here ˜(Re) is an effective viscosity that depends on the Reynolds number of the flow: Re § UL/ , where L is the typical size of the body (defin...

Fluid dynamics of moving fish in a two-dimensional multiparticle collision dynamics model

“…In sum, our results indicate the importance of several theoretical and empirical follow-up studies. For low-Reynolds simulations it may be possible to model organisms in a simpler manner, by linearizing the Navier-Stokes equations [73]. However, it remains to be tested whether this is also possible for unsteady motion such as the organisms in this study perform.…”

Solving the aerodynamics of fungal flight: how air viscosity slows spore motion

“…Roper and Brenner (2009) developed a novel approach for calculating fluid flow around microscopic bodies of various shapes based upon an earlier recommendation by Carrier (1953). They used Oseen's approximation for drag (C D = 24/Re + 3/16), which is a simple modification of Stokes (C D = 24/Re), and a renormalized or Re-dependant viscosity term.…”

“…It is encouraging, from the standpoint of the results of the present paper, that this model was successful in modeling fluid flow because Oseen and Stokes are very similar. But despite the apparent utility of the Ossen model, it is not clear whether the conclusions reached by Roper and Breener (2009) are fully applicable to the analysis of spore trajectories. In common with many previous studies on the drag on tiny projectiles, Roper and Brenner (2009) were concerned with steady-state conditions.…”

“…But despite the apparent utility of the Ossen model, it is not clear whether the conclusions reached by Roper and Breener (2009) are fully applicable to the analysis of spore trajectories. In common with many previous studies on the drag on tiny projectiles, Roper and Brenner (2009) were concerned with steady-state conditions. The high-speed movements addressed in our work are far from steady-state, being characterized by an initial high acceleration from rest followed by fast deceleration.…”

A tree swaying in a turbulent wind: a scaling analysis

“…At intermediate Reynolds numbers, a quadratic nonlinearity may be retained in the Navier-Stokes equation to set up a convenient nonlinear version of the Brinkman screening theory [21]. Recently, however, a modification of the Oseen approximation [22] has been proposed to increase its range of validity well beyond Re = O(1) [23]. The ruse is to use a higher renormalized viscosity (here denoted by ν).…”