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 Navier–Stokes 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.

Many 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 ∇·u = 0, ρ is the liquid density, and ν 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 approximation

where here ν̃(Re) is an effective viscosity that depends on the Reynolds number of the flow: Re≡ UL/ν, where L is the typical size of the body (defined herein as the radius of a sphere of equivalent volume). If we use the bare fluid viscosity, ν̃(Re) = ν, in the right hand side, then Eq. 2 becomes the Oseen equation, originally proposed to capture weak inertial effects on the flow as a perturbation to the slow viscous flow solutions (8). The Oseen approximation is always valid far from the body, where the velocity disturbance, due to the body, is small. However, near the body, the approximation is known to be reasonably accurate only when Re is <1 (9, 10). Kaplun and Lagerstrom (11) and Proudman and Pearson (12) extended Oseen's approximation to higher order expansions in Re; these series expansions are known to quantitatively capture experiments when Re is small (13, 14), but do not converge and are not accurate for Re much larger than unity. Attempts to enlarge the domain of validity of these series expansions have focused on completing the series, either by matching to a postulated high Reynolds number asymptote (see Proudman's appendix to ref. 15), or, more recently, using the renormalization group to reorganize terms in the expansion (16). Recent renormalization group approaches have yielded apparently more accurate approximations for drag on a sphere for 1 ≤ Re ≤ 10 (16). Nonetheless, generally applicable approximate equations for calculating drag of moving bodies at Reynolds numbers in the tens or hundreds have remained elusive. Brute force numerical solution of the equations of motion therefore remains the only viable method for computing the drag on bodies in moderate Reynolds number flows.

Here, we demonstrate that with an appropriately chosen effective viscosity ν̃(Re), Eq. 2 accurately captures the drag on a wide range of body shapes for Re as large as several hundred. The idea for this approximation dates back to a article of Carrier (17), which, although unpublished to this day, was nonetheless cited by Goldstein as one of the great advances in fluid mechanics in the twentieth century (18). Carrier observed that if the functional form of Oseen's equations is retained but the advective velocity U is rescaled, then Oseen's approximation can be made to give empirically good agreement to the drag at moderate Reynolds numbers. Carrier's effective equation is equivalent to ours on simultaneous rescaling of the viscosity and dimensionless pressure. By inspecting the experimental drag-to-speed curves for 3 standard bodies—a sphere, flat plate, and cylinder, Carrier hypothesized that (in our notation) ν̃(Re) ≈ 2.3ν. In honor of Carrier's insight, we henceforth refer to Eq. 2 as Carrier's equation.

We were led to revisit Carrier's equation from our previous studies of the shapes of drag-minimizing bodies (19). There we found that the bodies exhibit a surprising degree of fore aft symmetry up to Re ∼ 100, despite sitting in a flow field that becomes markedly asymmetric above Re ≈ 1. In contrast to the Navier–Stokes equations (Eq. 1), the drag of Eq. 2 on an asymmetric body is exactly identical whether the body translates forward or backwards (20); and hence minimal drag shapes are symmetrical (19). This inspired us to hypothesize that the drag determining features of the flow might be well approximated by a linear equation. In what follows we revisit Carrier's observation, using a combination of modern fast numerical solvers for the fully nonlinear Navier–Stokes equations and insights from singularity methods. We first determine the effective viscosity ν̃(Re) by requiring that the shape of the fluid separatrix surrounding a point force in the steady Navier–Stokes equation matches that for the approximate equation. We then compare the drag and the flow fields predicted by the Navier–Stokes equation with the approximate equation over a wide range of body shapes and demonstrate quantitative agreement. To demonstrate the utility of the approximation, we consider 2 hard fluid mechanics problems. First, we demonstrate that the linear approximation quantitatively captures the flow separation around a steadily translating bluff body (21). Second, we consider the practical problem of design of lowest-drag micro-projectiles (19).

Results

Finding the Renormalized Viscosity ν̃(Re).

We find the effective viscosity ν̃(Re) empirically. Eq. 2 must capture the flow, independently of the shape of the translating body. Thus, we determine ν̃(Re) by focusing on the characteristics of the fundamental solution to the governing equations, the flow created by a translating point force.

We consider flow around a translating point force in a frame comoving with the force (Fig. 1). This flow can be divided into 3 regions. Far from the point force, the solution to the Navier–Stokes equation asymptotes to the classical Oseen solution. Close to the point force, the Navier–Stokes solution balances nonlinear inertia with viscous forces, resulting in the so-called Landau-Squires jet for a point momentum source (1). Both far and near field behaviors disagree with the solution to Carrier's equation. The far field solution in the Carrier equation disagrees with that of the Oseen equation because, in general, ν̃(Re) ≠ ν. The near field solution to the Carrier equation disagrees with the Landau-Squires solution, because it is dominated by viscous forces, and is given by the classical Stokes solution for a moving point force (1).

Fig. 1.

A moving point force, F, entrains a compact region of fluid. In the rest frame of the point force, in which the flow approaches the force from the negative z direction at speed U, this region is bounded by the separatrix, the streamtube (gray surface) separating closed and open streamlines.

In between these two regions of asymptotic disagreement, both approximate and Navier–Stokes flows produce a separatrix, a fluid surface that delimits the compact region of fluid that is entrained by or “attached” to the moving point force (Fig. 1). The shape of this separatrix is dynamically determined, and depends on the strength of the point force, F, and its speed of motion U through the effective Reynolds number Re_f = F/6πρν². We scale our computational domain so that a length scale defined by L = F/6πρνU is kept equal to 1.

We choose ν̃(Re) so that the shapes of the separatrix given by Carrier's equation match as closely as possible to the separatrix obtained from the Navier–Stokes equation (see Fig. 3). We solve the axisymmetric steady Navier–Stokes flow and Carrier flows by representing the velocity field components by quadratic finite elements, and the pressure field by linear elements, and solve the associated finite element model using Comsol Multiphysics. The point forcing is handled by subtracting off from either flow field the analytical forms of the corresponding singular solutions for a stationary point force. Fig. 2 shows the resulting ν̃(Re), and Fig. 3 shows the shapes of the separatrices for the 2 equations for Re = 0.13, 5.3, 34.8 and 101.3. The agreement between the separatrix shapes is especially striking given (a) the complete disagreement in the near field and far field behaviors of the solutions, and (b) the fact that at each Re we are tuning only a single parameter ν̃(Re) but are capturing the shape of an entire curve.

Fig. 3.

Comparison of separatrices of flow around point forces according to the Navier–Stokes equations (solid curves) and Carrier's equation (dashed curves) for: Re_f = 0.13 (A) 5.25 (B), 34.8 (C), and 101.3 (D). Separatrices are shifted in Carrier's equation, compared with the Navier–Stokes equations—we align the two families of shapes by shifting the point force. Crosses denote the location of the point force in Carrier's equations. The point force is always located at the origin in the Navier–Stokes representation of the flow.

Fig. 2.

Finding the function ν̃(Re) by matching the shape of a separatrix for a point force in Carrier's equation to the nonlinear Navier–Stokes equations. ν(Re) for best fit of separatrix shape is obtained by minimizing either the unsigned stream function; I₁ ≡ ∫_S|ψ|dS or the unsigned geometrical distance between the 2 separatrices; $l_2\equiv {\displaystyle \int s\frac{\left|\psi \right|}{\left|\partial \psi /\partial n\right|}}$. Minimization of I₂ gives the dependence of the effective viscosity ν̃(Re) on Re, best power law fit: ν̃(Re)/ν ∼ Re^1/4. (Inset) I₁ and I₂ have very close functional minima ν̃(Re) (representative data for Re = 101.30: I₁; solid curve, left axis, I₂; dashed curve, right axis).

We highlight the intermediate Reynolds number behavior of the renormalized viscosity—a numerical fit to our data gives ν̃/ν ∼ Re^1/4 at large Reynolds numbers (Fig. 2). If we write the effective viscosity as ν̃ ∼ Uℓ, where ℓ is a (dynamically determined) length scale, our numerical fitting is then consistent with ℓ = (ν³ L/U³)^1/4: Such a length scale cannot be arrived at by any geometrical averaging of the viscous boundary layer thickness ℓ_visc ∼ (νL/U)^1/2 and body scales, and suggests that at this range of Reynolds numbers the drag is controlled by different effective physics.

On inputting this value of ν̃ (Re) Carrier's equation accurately predicts the moderate Re drag for a wide range of different body shapes. We show this by considering ellipsoids of many different aspect ratios, both oblate and prolate, and compare the Navier–Stokes drag with the drag predicted from Carrier's equation (Fig. 4). Equivalently, for each ellipsoid, we can again obtain an expression for ν̃(Re) by matching the drag predicted from Carrier's equation with the drag predicted by the Navier–Stokes equations. For every possible aspect ratio of ellipsoid, this procedure generates a new mapping of Re to ν̃ (Re). Remarkably, over a wide range of Reynolds numbers, the mapping is independent of body shape (Fig. 5).

Fig. 4.

Predicting drag using Carrier's equation for spheroids of different aspect ratios for 0< Re < 200. In each frame the solid curve is the (real) Navier–Stokes drag, and the dotted curve the drag predicted by Carrier's equation. Different frames correspond to different aspect ratios in the order: 0.667 (A), 1.0 (B), 3.0 (C), and 5.0 (D). All drags are nondimensionalized by the Stokes (Re = 0) limiting drag, so tend to 1 as Re → 0.

Fig. 5.

One-to-one mapping of Re to ν̃(Re) for spheroids of aspect ratios 2/3 (orange stars), 1 (yellow circles), 4/3 (green squares), 2 (blue plus signs), 3 (blue inverted triangles), 4 (purple triangles), and 5 (red diamonds). The black curve gives the fit to the separatrix of a moving point force, from Fig. 2.

Applications.

We now turn to 2 applications demonstrating the utility of Carrier's equation. Our first example is a classically difficult problem from fundamental fluid mechanics, the onset of separation in flow around a steadily moving body. We demonstrate that Carrier's equation quantitatively captures the onset and early growth of separated flow. Second we consider an example from biology, the design of minimal drag shapes, inspired by forcibly ejected fungal spores.

Flow Separation Around Bluff Bodies.

It has long been known that a steadily moving body develops a ring of recirculating fluid in its wake (1) above a critical Reynolds number. Theoretical prediction of the characteristics of the transition from open to closed streamlines has long been thought to be intractable, so that the shape of the region of recirculating fluid (the so-called separation bubble) can be obtained only by full simulations of the steady Navier–Stokes equations.

We already saw that Carrier's equation captures the shapes of the separated flows around point forces, and we now demonstrate that they quantitatively capture the separated flow for rigid bodies also by matching, near to the transition to separated flow, the shape of the largest compact streamtube produced by the union of the body boundary and a bubble of separated flow clinging to the rear of the body.

Fig. 6 shows the development from creeping, unseparated flow into rapid separated flow for spheroids of 3 different aspect ratios, a/b = 0.4,1,2. (The length scales a and b are defined are in Fig. 7A). The left half of each frame in Fig. 6 shows the solution to the steady Navier–Stokes equation, and the right half of each frame shows the solution to Carrier's equation. The streamline plots visually affirm that Carrier's equation reproduces the shape of the separation bubble around translating solid bodies. Indeed for the sphere, which gives the worst agreement of the three, the maximum discrepancies between the width and length of the separation bubble predicted by Carrier's equation are never worse than 15% and 18% respectively, for Re < 100; beyond this value, experiments show that the separation bubble starts to oscillate irregularly in time (21). For further comparison of the flow fields, Fig. 7B compares the growth in length, ℓ(Re), of the separation bubble predicted by Carrier's equations for an ellipsoid of aspect ratio 0.4.

Fig. 6.

Comparison of solutions of Navier–Stokes equations (Left) and Carrier's equation (Right) for the flow field around a steadily translating spheroids of aspect ratios 0.4, 1 and 2 (rows), at Reynolds numbers 1, 10, 25, 50 and 100 (columns).

Fig. 7.

Carrier's equation predicts the onset of separation behind bluff bodies. We analyze separation for ellipsoidal bodies of streamwise length 2a and diameter 2b (A). Other frames show (B) growth in length (measured from stagnation point to stagnation point) of the separated zone behind a spheroid of aspect ratio a/b = 0.4, and (C) the critical Reynolds number, Re_c, (D) length coefficient l₀ and (E) width coefficient, w₀, describing the near onset growth of the separated zone for spheroids of aspect ratios a/b = 0.4, 1, 2, 4. The solid curves are the results of direct numerical simulations, and dashed curves are the coefficients predicted by Carrier's equation.

We also test whether Carrier's equations give quantitatively correct flow features near the critical Reynolds number at which separation commences. Because near onset the separation bubble is parabolic, this behavior can be expressed in terms of 2 critical coefficients, for the length, ℓ, and width, w, of the separation bubble.

Fig. 7 C–E compares the fitted values of the 3 parameters w₀, l₀ and Re_c as a function of aspect ratio for the 2 equations. The Carrier equation quantitatively captures the growth of the separation bubble (and hence the region of entrained fluid around a moving solid body) over a wide range of aspect ratios. Although analytical prediction of these coefficients is impossible for the fully nonlinear Navier–Stokes equations, for high symmetry bodies such as ellipsoids, Carrier's equations may be solved analytically at all Reynolds numbers by regular series expansions (22–24).

Design of Drag-Minimizing Shapes.

We now turn to another application of Carrier's equation, to the practical problem of self-assembling or growing perfect projectiles—bodies of prescribed volume that are engineered to experience the minimum possible drag in flight. We draw inspiration here from the explosively launched meiospores of many species of fungi (25, 26). These propagules must eject through a thin boundary layer of nearly still air that clings to the originating fruiting body to reach dispersive air flows beyond. Previously we presented evidence that spores of ascomycete fungi have drag-minimizing shapes (27).

For the full Navier–Stokes equation, calculating minimal drag shapes is a complicated exercise: The criterion for a perfect projectile (a body that suffers the smallest possible drag for its size and speed) (28, 19), follows from requiring that the drag on the projectile cannot be improved by any small, volume preserving, perturbation of its shape. This condition is equivalent to requiring that the drag variational $J\equiv \frac{\partial u}{\partial n}·\frac{\partial w}{\partial n}$ is uniform over the entire of the boundary of the projectile, where ∂/∂n is the derivative in the body-normal direction. The function w is an adjoint field, which must be solved for independently: It solves a second order linear partial differential equation of mixed type whose coefficients depend on the components of the physical velocity field (28, 19). Determining J for a given projectile shape therefore requires knowledge of the velocity field throughout the entire of the fluid-filled domain.

In contrast, the optimality criterion for Carrier's equation does not require independent calculation of w, and only uses local information about the fluid velocity field. Flows governed by Carrier's equation also require that the drag variational be uniform over the shape boundary for optimality, but now the equation for w simplifies to:

with an incompressibility constraint ∇ · w = 0, and boundary conditions w = 0 on the surface of the body and w → − e_z in the far-field. w is thus the reversed flow that the projectile would encounter if it were to travel tail-first rather than nose-first through the fluid, and q the corresponding pressure field. For a fore-aft symmetric test projectile, the drag-variational J is then equal to the fore-aft symmetrized shear stress on the surface of the body. Namely, if a body is fore-aft symmetrical, and x = (x, y, z) is any point on the body, and x_r ≡ (x, y, −z) the symmetrically opposite point on the body then the drag variational can be written as

and can be computed directly from the local velocity field. We can apply the approximate criterion for drag minimization to generate ‘almost perfect’ projectiles in either of two ways. First we solve the exact (Navier–Stokes) equations and apply the approximate drag minimization criterion. Second we solve the approximate (Carrier) equations and apply the approximate drag minimization criterion. We obtain these shapes by an iterative algorithm: at each iteration, the Navier–Stokes or respectively Carrier's equations are solved for the flow around a steadily translating fore-aft symmetric test projectile, the symmetrized-shear stress is computed, the optimum fore-aft symmetric shape perturbation is then determined and used to update the shape.*

Approximate shapes obtained by applying the approximate drag variational (Eq. 5) to the exact Navier–Stokes flow are nearly identical in shape and in drag to the exact optimal shapes up to Re = 100 (Fig. 8 and Fig. S1). The approximate variational even reproduces features that can be shown to contribute only weakly to the total drag on the body, such as the cones on the front and rear of the shape (27). The excess drag (percentage increase in drag compared with the optimal shape of the same size) for these approximate shapes remains <0.13% for Re < 100. Symmetrized shear stress therefore provides an equation-free method for growing or assembling drag minimizing shapes with only local knowledge of the flow.

Fig. 8.

An objective function based on symmetrized shear stress reproduces the drag-minimizing shapes without needing either an adjoint equation or any nonlocal information about the flow field. The black curves are the drag minimizing shapes for Re = 0.1 (A), 10 (B), 50 (C), 100 (D) calculated by Roper et al. (19), red curves show the approximate drag minimizing shapes obtained by applying the approximate optimization condition (Eq. 5) to the Navier–Stokes equations, and the blue curves are the optimal shapes generated by Carrier's equation.

The second class of approximately optimal shapes uses Carrier's equations both to determine the flow around the test projectile and to approximate the drag variational, J. This approximation produces optimal shapes that match significantly less closely to the exact drag minimizing shapes, tending to be narrower than optimal, and have significantly larger drag (Fig. 7, Fig. S1); the excess drag reaches 3.1% at Re = 100. The discrepancy in shapes is nonetheless small over the range of sizes and speeds relevant to forcibly launched ascomycete spores: At Re = 10, the largest Reynolds number for a forcibly launched ascomycete spore (27), the excess drag is just 0.21%.

Although it does not produce exactly optimal shapes, the simplicity and linearity of Carrier's equation suggest a practical algorithm for growing or building very low drag projectiles. For instance the forcibly launched spores of ascomycete fungi well approximate perfect projectile shapes over a large range of spore sizes (27). It is natural to ask how the dynamical processes that guide spore ontogeny reliably “find” this drag-minimizing shape. The sexual spores of ascomycete fungi are delimited by membranes that form within the ascus after the completion of meiosis (29). Although the precise mechanisms that control the shapes of these membranes are unknown, it is likely that their ontogeny requires either detailed programming of the time sequence of growth processes, or else a robust dynamical process that reliably sculpts drag minimizing shapes. This second class of processes include use of physical stresses to guide growth.

Our calculations show that bodies in moving fluids could be directed to grow into drag minimizing shapes simply by requiring that the symmetrized shear stress be maintained constant over the boundary of the shape. Cell growth is known to be sensitively actuated by physical stresses (30–32), and fluid stresses have already been implicated in multiple embryogenetic processes, from the first breaking of left right symmetry (33) to the creation of the intricate tubing of the growing heart (34). However, the fluid stresses that spores are exposed to after launch are very large, and could not plausibly be generated within the developing ascus.

The structure of Carrier's approximation means that the fluid stress field could be mimicked within an incompressible and isotropic but spatially heterogeneous elastic medium. Specifically, if the immature ascospore were taken to be a rigid inclusion and the surrounding cell matrix is an incompressible elastic medium with shear modulus G = G₀ exp[−2Uz/ν̃(Re)], then any small displacement of the projectile would engender elastic deformations that duplicate the flow field that it would encounter in flight. Because of the linearity of the equations it is then possible for the relative pattern of elastic stresses to match the relative pattern of fluid stresses but yet be many orders of magnitude smaller in absolute value. A spatially heterogeneous shear modulus could be created in the ascus through modulation of the density of the scaffold microtubules within the cell. Experimental tests of our hypothesized mechanism should first measure the elasticity of the ascus and determine whether there are significant variations in elastic moduli on the scale of developing spores, and second determine the effect of perturbations to the stress field, e.g., from other inclusions introduced into the ascus, on the shape of mature spores.

Discussion

We have found a linear and reciprocal approximation to the Navier–Stokes equations that well-captures the drag-determining features of flows around steadily translating bodies. The success of this approximation derives from its (incompletely understood) capacity to capture the shape of the entrained compact region fluid that moves with a moving body: This was demonstrated for the fundamental problem of a moving point force. We note that the close concordance between approximate and exact representations for the drag can be violated by careful choice of sufficiently pathological projectile shapes and slip boundary conditions (24).

We applied the approximation first to the classic problem of predicting the onset of separation in steady flow around a sphere, and of finding a functional form for the separated streamlines. Second, we devised both an equation-free model for the growth or self-assembly of perfect projectiles, that using nothing more than the fluid stress on the boundary updates the shape of a nondrag minimizing projectile until a drag minimizing shape is achieved, and an algorithm that grows very low drag bodies using arbitrarily weak elastic stresses.

We have focused on solutions of the Navier–Stokes equations because, quite apart from their undoubted practical significance, perturbative approximations for these equations have been thoroughly explored. It is likely nonetheless that the procedure by which we have constructed an approximation to the flow equations may be applied to a more general class of nonlinear field equations.