A Non-Perturbative Approximation for the Moderate Navier-Stokes Equations Marcus Roper ∗ † and Michael P. Brenner ∗ ∗School of Engineering and Applied Sciences, Harvard University, and †Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA, 02138

Submitted to Proceedings of the National Academy of Sciences of the United States of America

The nonlinearity of the Navier-Stokes equations makes predicting is the typical size of the body (defined herein as the cubed the flow of fluid around rapidly moving small bodies highly resistant root of the body volume). If we use the bare fluid , to all approaches save careful experiments or brute force computa- ν˜(Re) = ν, in the right hand side, then Equation (2) be- tion. Here we show how a novel linearization of the Navier-Stokes comes the Oseen equation, originally proposed [8] to capture equations captures the -determining features of the flow and al- weak inertial effects upon the flow as a perturbation to the lows simplified or analytical computation of the drag upon bodies up to Reynolds number of order 100. We illustrate the utility of this lin- slow viscous flow solutions. The Oseen approximation is al- earization in two practical problems that normally can only be tackled ways valid far from the body, where the velocity disturbance with sophisticated numerical methods: understanding flow separa- due to the body is small. On the other hand, near the body, tion in the flow around a bluff body and finding drag-minimizing the approximation is known to be reasonably accurate only shapes. when Re is less than 1 [9, 10]. Kaplun and Lagerstrom [11] and Proudman and Pearson[12] extended Oseen’s approxima- Many methods that are currently used for finding ap- tion to higher order expansions in Re; these series expansions proximate solutions of nonlinear partial differential equations are known to quantitatively capture experiments when Re is were inspired by studies of fluid flows. The most success- small [13, 14], but do not converge and are not accurate for ful methods can be divided into two classes. The first aims Re much larger than unity. Attempts to enlarge the domain to approximate the flow field by a perturbative expansion in of validity of these series expansions have focused upon com- a small parameter. Examples include Stokes’ equations for pleting the series, either by matching to a postulated high viscous flow, or Prandtl’s theory of laminar boundary lay- Reynolds number asymptote (see Proudman’s appendix to ers [1]. The second class of approximations is fundamentally [15]), or, more recently, using the renormalisation group to non-perturbative; these arise when instead of detailed solu- reorganize terms in the expansion [16]. Recent renormaliza- tions, it is sufficient to consider averaged quantities, such as tion group approaches have yielded apparently more accurate the flux of momentum or of a passive scalar. These approx- approximations for drag on a sphere for 1 ≤ Re ≤ 10 [16]. imations work by exploiting scale separation and averaging Nonetheless, generally applicable approximate equations for over unresolved degrees of freedom. Prominent examples in- calculating drag of moving bodies at Reynolds numbers in clude eddy viscosity closure relations in turbulent shear flows the tens or hundreds have remained elusive. Brute force nu- [2], effective dispersion of solute under the combined action merical solution of the equations of motion therefore remains of shearing flows and molecular diffusion [3], the renormaliza- the only viable method for computing the drag upon bodies tion group in condensed matter physics [4], and anti-aliasing in moderate Reynolds number flows. methods in spectral numerical solvers [5] as well as recent ad- Here we demonstrate that with an appropriately chosen ef- vances in constructing linear effective equations to describe fective viscosityν ˜(Re), Equation (2) accurately captures the the spatially averaged solutions of the non-linear Poisson and drag on a wide range of shapes for bodies for Re as large as advection-diffusion equations [6, 7]. Approximate solutions several hundred. The idea for this approximation dates back to nonlinear partial differential equations have two-fold value: to a paper of Carrier [17], which, while unpublished to this providing asymptotic solutions or closure relations in cases day, was nonetheless cited by Goldstein as one of the great ad- where fine spatial scales or short time scales prohibit direct vances in fluid mechanics in the twentieth century [18]. Car- numerical solutions, and giving qualitative and quantitative rier observed that if the functional form of Oseen’s equations insights into why solutions take the forms that they do. is retained, but the Reynolds number Re replaced by a new In this paper, we describe a non-perturbative approxima- (geometry-independent) parameter (equivalent to ν/ν˜(Re) in tion for the moderate Reynolds number steady Navier-Stokes our formulation), then Oseen’s approximation can be made equations, to give empirically good agreement to the drag at moderate 1 Reynolds numbers. By inspecting the experimental drag-to- u · ∇u = ν∇2u − ∇p, [1] ρ speed curves for three standard bodies–a sphere, flat plate and 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 de- Reserved for Publication Footnotes termining features of a body with almost any shape translat- ing through a fluid with velocity U. The approximate equation replaces the nonlinear term on the left-hand side with a linear approximation 1 U∂ u =ν ˜(Re)∇2u − ∇p, [2] z ρ where hereν ˜(Re) is an effective viscosity that depends upon the Reynolds number of the flow: Re ≡ UL/ν, where L www.pnas.org — — PNAS Issue Date Volume Issue Number 1–5 a cylinder, Carrier hypothesizedν ˜(Re) ≈ 2.3ν. In honor of ponents by quadratic finite elements, and the field Carrier’s insight, we henceforth refer to equation (2) as Car- by linear elements, and solve the associated finite element rier’s equation. model using Comsol Multiphysics. The point forcing is han- We were led to revisit Carrier’s equation from our previous dled by subtracting off from either flow field the analytical studies of the shapes of drag-minimizing bodies [19]. There forms of the corresponding singular solutions for a stationary we found that the bodies exhibit a surprising degree of fore aft point force. Figure 2 shows the resulting ν(Re), and Figure 3 symmetry up to Re ∼ 100, despite sitting in a flow field that shows the shapes of the separatrices for the two equations for becomes markedly asymmetric above Re ≈ 1. In contrast Ref = 0.13, 5.3, 34.8 and 101.3. The agreement between the to the Navier-Stokes equations (1), the drag of equation (2) separatrix shapes is especially striking given (a) the complete on an asymmetric body is exactly identical whether a body disagreement in the near field and far field behaviors of the translates forward or backwards [20]; and hence minimal drag solutions, and (b) the fact that at each Re we are tuning only shapes are symmetrical [19]. This inspired us to hypothe- a single parameterν ˜(Re) but are capturing the shape of an size that the drag determining features of the flow might be entire curve. well approximated by a linear equation. In what follows we We highlight the intermediate Reynolds number behavior revisit Carrier’s observation, using a combination of modern of the renormalized viscosity – a numerical fit to our data gives fast numerical solvers for the fully nonlinear Navier-Stokes ν/ν˜ ∼ Re1/4 at large Reynolds numbers (Fig. 2). If we write equations and insights from singularity methods. We first the effective viscosity asν ˜ ∼ U`, where ` is a (dynamically determine the effective viscosityν ˜(Re) by requiring that the determined) length scale, our numerical fitting is then consis- shape of the fluid separatrix surrounding a point force in the tent with ` = (ν3L/U 3)1/4: such a length scale can not be steady Navier Stokes equation matches that for the approxi- arrived at by any geometrical averaging of the viscous bound- mate equation. We then compare the drag and the flow fields 1/2 ary layer thickness `visc ∼ (νL/U) and body scales, and predicted by the Navier-Stokes equation with the approximate suggests that at this range of Reynolds numbers the drag is equation over a wide range of body shapes and demonstrate controlled by different effective physics. quantitative agreement. To demonstrate the utility of the ap- On inputting this value ofν ˜(Re), Carrier’s equation ac- proximation, we consider two hard fluid mechanics problems. curately predicts the moderate Re drag for a wide range of First we demonstrate that the linear approximation quantita- different body shapes. We show this by considering ellipsoids tively captures the flow separation around a steadily translat- of many different aspect ratios, both oblate and prolate, and ing bluff body [21]. Second we consider the practical problem compare the Navier-Stokes drag with the drag predicted from of design of lowest-drag micro-projectiles [19]. Carrier’s equation (4). Equivalently, for each ellipsoid, we can again obtain an expression forν ˜(Re) by matching the drag predicted from Carrier’s equation with the drag predicted by Results the Navier Stokes equations. For every possible aspect ratio Finding the renormalized viscosity ν˜(Re). We find the effec- of ellipsoid, this procedure generates a new mapping of Re to tive viscosityν ˜(Re) empirically. The equation (2) must cap- ν˜(Re). Remarkably, over a wide range of Reynolds numbers, ture the flow, independently of the shape of the translating the mapping is independent of body shape (Figure 5). body. Thus we determineν ˜(Re) by focusing on the character- istics of the fundamental solution to the governing equations, the flow created by a translating point force. Applications We consider flow around a translating point force in a We now turn to two applications demonstrating the utility frame comoving with the force (Fig. 1). This flow can be of Carrier’s equation. Our first example is a classically dif- divided into three regions. Far from the point force, the solu- ficult problem from fundamental fluid mechanics, the onset tion to the Navier-Stokes equation asymptotes to the classical of separation in flow around a steadily moving body. We Oseen solution. Close to the point force, the Navier-Stokes demonstrate that Carrier’s equation quantitatively captures solution balances nonlinear inertia with viscous forces, result- the onset and early growth of separated flow. Secondly we ing in the so-called Landau-Squires jet for a point momentum consider an example from biology, the design of minimal drag source [1]. Both far and near field behaviors disagree with the shapes, inspired by forcibly ejected fungal spores. solution to Carrier’s equation. The far field solution in the Carrier equation disagrees with that of the Oseen equation Flow separation around bluff bodies. It has long been known since, in general,ν ˜(Re) 6= ν. The near field solution to the that the flow field around a steadily moving body develops Carrier equation disagrees with the Landau-Squires solution, a ring of recirculating fluid in its wake [1] above a critical because it is dominated by viscous forces, and is given by the Reynolds number. The prediction of the characteristics of classical Stokes solution for a moving point force [1]. this transition have long been thought to be theoretically in- In between these two regions of asymptotic disagreement, tractable; predicting the characteristics of the region of re- both approximate and Navier-Stokes flows produce a separa- circulating fluid (the so-called separation bubble) around a trix, a fluid surface that delimits the compact region of fluid body has been thought to require full simulations of the steady that is entrained by or “attached” to the moving point force Navier Stokes equations. (Fig. 1). The shape of this separatrix is dynamically de- We already saw that Carrier’s equation captures the termined, and depends upon the strength of the point force, shapes of the separated flows around point forces, where the F , and its speed of motion U through the effective Reynolds largest compact streamtube is the union of the boundary of 2 number Ref = F/6πρν . We scale our computational domain the body and of a bubble of separated flow clinging to the so that a length scale defined by L = F/6πρνU is kept equal rear of the body. We now demonstrate that Carrier’s equa- to 1. tion quantitatively captures the flow separation transition. We chooseν ˜(Re) so that the shapes of the separatrix Figure 6 shows the development from creeping, unsepa- given by Carrier’s equation match as closely as possible to rated flow into rapid separated flow for spheroids of three dif- the separatrix obtained from the Navier-Stokes equation (see ferent aspect ratios, a/b = 0.4, 1, 2. (The length scales a and b Figure 3). We solve the axisymmetric steady Navier-Stokes are defined are in Fig. 7A). The left panel of the figure shows flow and Carrier flows by representing the velocity field com- the solution to the steady Navier-Stokes equation, whereas

2 www.pnas.org — — Footline Author the right panel shows the solution to Carrier’s equation. The with an incompressibility constraint ∇ · w = 0, and boundary streamline plots visually affirm that Carrier’s equation repro- conditions w = 0 on the surface of the body and w → −ez duces the shape of the separation bubble around translating in the far-field. w is thus the reversed flow that the projec- solid bodies. Indeed for the sphere, which gives the worst tile would encounter if it were to travel tail-first rather than agreement of the three, the maximum discrepancies between nose-first through the fluid. For a fore-aft symmetric test the width and length of the separation bubble predicted by projectile, the drag-variational J is then equal to the fore- Carrier’s equation are never worse than 15% and 18% respec- aft symmetrized shear stress upon the surface of the body. tively, for Re < 100; beyond this value, experiments show Namely, if a body is fore-aft symmetrical, and x = (x, y, z) is that the separation bubble starts to oscillate irregularly in any point on the body, and xr ≡ (x, y, −z) the symmetrically time [21]. For further comparison of the flow fields, Figure 7B opposite point on the body then the drag variational can be compares the growth in length, `(Re), of the separation bub- written as ble predicted by Carrier’s equations for an ellipsoid of aspect ¯ ¯ ∂u ¯ ∂u ¯ ratio 0.4. J(x) ≡ ¯ · ¯ , [5] We also test whether Carrier’s equations give quantita- ∂n ¯ ∂n ¯ x xr tively correct flow features near the critical Reynolds number at which separation commences. Since near onset the sepa- ration bubble is parabolic, this behavior can be expressed in and can be computed directly from the local velocity field. terms of two critical coefficients, for the length, `, and width, We can apply the approximate criterion for drag mini- w, of the separation bubble. mization to generate ‘almost perfect’ projectiles in either of two ways. First we solve the exact (Navier-Stokes) equa- 1/2 `(Re) = l0(Re − Rec) and w(Re) = w0(Re − Rec) . tions and apply the approximate drag minimization criterion. [3] Second we solve the approximate (Carrier) equations and ap- Figure 7C-E compares the fitted values of the three parame- ply the approximate drag minimization criterion. We obtain ters w0, l0 and Rec as a function of aspect ratio for the two these shapes by an iterative algorithm: at each iteration, the equations. The Carrier equation quantitatively captures the Navier-Stokes or respectively Carrier’s equations are solved growth of the separation bubble (and hence the region of en- for the flow around a steadily translating fore-aft symmetric trained fluid around a moving solid body) over a wide range of test projectile, the symmetrized-shear stress is computed, the aspect ratios. While analytical prediction of these coefficients optimum fore-aft symmetric shape perturbation then deter- is impossible for the fully nonlinear Navier-Stokes equations, mined and used to update the shape. 1 for high symmetry bodies such as ellipsoids, Carrier’s equa- Approximate shapes obtained by applying the approxi- tions may be solved analytically at all Reynolds numbers by mate drag variational (5) to the exact Navier-Stokes flow are regular series expansions [22, 23, 24]. nearly identical in shape and in drag to the exact optimal shapes up to Re = 100 (Fig. 8, Supplementary Fig. S2). The Application II. Design of drag-minimising shapes . We now approximate variational even reproduces features that can be turn to another application of Carrier’s equation, to the prac- shown to contribute only weakly to the total drag upon the tical problem of self-assembling or growing perfect projectiles body, such as the cones on the front and rear of the shape [27]. – bodies of prescribed volume that are engineered to experi- The excess drag (per cent increase in drag compared to the ence the minimum possible drag in flight. We draw inspira- optimal shape of the same size) for these approximate shapes tion here from the explosively launched meiospores of many remains less than 0.13% for Re < 100. Symmetrized shear species of fungi [25, 26]. These propagules must eject through stress therefore provides an equation-free method for growing a thin boundary layer of nearly still air that clings to the or assembling drag minimizing shapes with only local knowl- originating fruiting body in order to reach dispersive air flows edge of the flow. beyond. Previously we have presented evidence that spores of The second class of approximately optimal shapes uses ascomycete fungi have drag-minimizing shapes [27]. Carrier’s equations both to determine the flow around the For the full Navier Stokes equation, calculating minimal test projectile and to approximate the drag variational, J. drag shapes is a complicated exercise: The criterion for a per- This approximation produces optimal shapes that match sig- fect projectile (a body that suffers the smallest possible drag nificantly less closely to the exact drag minimizing shapes, for its size and speed) [28, 19], follows from requiring that the tending to be narrower than optimal, and have significantly drag on the projectile cannot be improved by any small, vol- larger drag (Supplementary Figs. S1 and S2); the excess drag ume preserving, perturbation of its shape. This condition is reaches 3.1% at Re = 100. The discrepancy in shapes is ∂u ∂w equivalent to requiring that the drag variational J ≡ ∂n · ∂n nonetheless small over the range of sizes and speeds relevant to is uniform over the entire of the boundary of the projectile, forcibly launched ascomycete spores: at Re = 10, the largest where ∂/∂n is the derivative in the body-normal direction. Reynolds number for a forcibly launched ascomycete spore The functionw is an adjoint field, which must be solved for [27], the excess drag is just 0.21%. independently: it solves a second order linear partial differ- Although it does not produce exactly optimal shapes, the ential equation of mixed type whose coefficients depend upon simplicity and linearity of Carrier’s equation suggest a practi- the components of the physical velocity field [28, 19]. De- cal algorithm for growing or building very low drag projectiles. termining J for a given projectile shape therefore requires For instance the forcibly launched spores of ascomycete fungi knowledge of the velocity field throughout the entire of the well approximate perfect projectile shapes over a large range fluid-filled domain. of spore sizes [27]. It is natural to ask how the dynamical In contrast, the optimality criterion for Carrier’s equation processes that guide spore ontogeny reliably “find” this drag- does not require independent calculation of w, and only uses minimizing shape. The sexual spores of ascomycete fungi are local information about the fluid velocity field. Flows gov- delimited by membranes that form within the ascus after the erned by Carrier’s equation also require that the drag varia- tional be uniform over the shape boundary for uniformity, but now the equation for w simplifies to: 1Just as in our previous study [19], the locus of shapes between the initial test projectile and the ∂w ∇q almost perfect projectile is continuously parametrized by a shape deformation parameter, and an − = − + ν(Re)∇2w [4] adaptive step-size integration routine used to advance the deformation parameter through interme- ∂z ρ diate shapes.

Footline Author PNAS Issue Date Volume Issue Number 3 completion of meiosis [29]. Though the precise mechanisms fect of perturbing the stress field, e.g. by introducing other that control the shapes of these membranes are unknown, it inclusions into the ascus, upon the shape of mature spores. is likely that their ontogeny requires either detailed program- ming of the time sequence of growth processes, or else a ro- bust dynamical process that reliably sculpts drag minimizing Discussion shapes. This second class of processes include use of physical We have found a linear and reciprocal approximation to stresses to guide growth. the Navier-Stokes equations that well-captures the drag- Our calculations show that bodies in moving fluids can be determining features of flows around steadily translating bod- directed to grow into drag minimizing shapes simply by requir- ies. The success of this approximation derives from its (incom- ing that the symmetrized shear stress be maintained constant pletely understood) capacity to capture the shape of the en- over the boundary of the shape. Cell growth is known to be trained compact region fluid that moves with a moving body: sensitively actuated by physical stresses [30, 31, 32], and fluid this was demonstrated for the fundamental problem of a mov- stresses have already been implicated in multiple embryoge- ing point force. We note that the close concordance between netic processes, from the first breaking of left right symmetry approximate and exact representations for the drag can be vi- [33] to the creation of the intricate tubing of the growing heart olated by careful choice of sufficiently pathological projectile [34]. However, the fluid stresses that spores are exposed to af- shapes and slip boundary conditions [24]. ter launch are very large, and could not plausibly be generated We applied the approximation first to the classic problem within the developing ascus. of predicting the onset of separation in steady flow around The structure of Carrier’s approximation means that the a sphere, and of finding a functional form for the sepa- fluid stress field can be mimicked within an incompressible rated streamlines. Secondly, we devised both an equation-free and isotropic but spatially heterogeneous elastic medium. model for the growth or self-assembly of perfect projectiles, Specifically if the immature ascospore is taken to be a rigid that using nothing more than the fluid stress on the bound- inclusion and the surrounding cell matrix is an elastic medium ary updates the shape of a non-drag minimizing projectile with shear modulus G = G0 exp(−2Uz/ν˜(Re)), then any until a drag minimizing shape is achieved, and an algorithm small displacement of the projectile will engender elastic defor- that grows very low drag bodies using arbitrarily weak elastic mations that duplicate the flow field that it would encounter stresses. in flight. Because of the linearity of the equations it is pos- We have focused upon solutions of the Navier-Stokes equa- sible for the relative pattern of elastic stresses to match the tions since, quite apart from their undoubted practical signif- relative pattern of fluid stresses but yet be many orders of icance, perturbative approximations for these equations have magnitude smaller in absolute value. A spatially heteroge- been thoroughly explored. It is likely nonetheless that the neous shear modulus could be created in the ascus through procedure by which we have constructed an approximation to modulation of the density of the scaffold microtubules within the flow equations may be applied to a more general class of the cell. Experimental tests of our hypothesized mechanism nonlinear field equations. should first measure the elasticity of the ascus and determine whether there are significant variations in elastic moduli on ACKNOWLEDGMENTS. This research is funded by Eastmann-Kodak Graduate the scale of developing spores, and second determine the ef- and Harvard University Herbaria postdoctoral fellowships to MR, and by the NSF Division of Mathematical Sciences. Discussions with Howard Stone and Todd Squires are gratefully acknowledged.

1. Batchelor, G. (1967) Introduction to Dynamics. (Cambridge University Press). 19. Roper, M, Squires, T, & Brenner, M. (2008) Symmetry un-breaking in perfect pro- 2. Barenblatt, G. (1996) Scaling, Self-Similarity, and Intermediate Asymptotics. (Cam- jectiles. accepted for publication in Phys. . bridge University Press). 20. Olmstead, W. (1975) Acta Mech. 21, 289–300. 3. Taylor, G. (1953) Proc. Roy. Soc. Lond. Ser. A 219, 186–203. 21. Taneda, S. (1956) J. Phys. Soc. Jpn. 11, 1104–1108. 4. Kadanoff, L. P. (2000) Statistical Physics Statics, Dynamics and Renormalization. 22. Goldstein, S. (1929) Proc. Roy. Soc. Lond., Ser. A 123, 225–235. (World Scientific). 23. van Dyke, M. (1970) J. Fluid Mech. 44, 365–372. 5. Canuto, C, Hussaini, M, Quarteroni, A, & Zang, T. (1988) Spectral methods in fluid 24. Roper, M. (2007) Ph.D. thesis (Harvard School of Engineering and Applied Sciences). dynamics. 25. Pringle, A, Patek, S. N, Fischer, M, Stolze, J, & Money, N. P. (2005) Mycologia 97, 6. Alexander, S, Chaikin, P, Grant, P, Morales, G, Pincus, P, & Hone, D. (year?) J. 866–871. Chem. Phys. 26. Vogel, S. (2005) J. Biosci. 30, 167–175. 7. Chorin, A. (2003) Proc. Nat. Acad. Sci. USA 100, 9674–9679. 27. Roper, M, Pepper, R, Brenner, M. P, & Pringle, A. (2008) Explosively launched spores 8. Oseen, C. (1927) Neuere Methoden und Ergebnisse in der Hydrodynamik. (Akademis- of ascomycete fungi have drag minimizing shapes. to appear in Proc. Nat. Acad. Sci. che Verlagsgesellschaft). USA. 9. Chester, W. (1962) J. Fluid Mech. 13, 557–569. 28. Pironneau, O. (1974) J. Fluid Mech. 64, 97–110. 10. Brenner, H & Cox, R. G. (1963) J. Fluid Mech. 17, 561–595. 29. Read, N & Beckett, A. (1996) Mycol. Res. 100, 1281–1314. 11. Kaplun, S & Lagerstrom, P. (1957) J. Math. Mech 6, 585–593. 30. Davies, P. (1989) Physiol. 4, 22–25. 12. Proudman, I & Pearson, J. R. A. (1957) J. Fluid Mech. 2, 237–262. 13. Maxworthy, T. (1965) J. Fluid Mech. 23, 369–372. 31. LaBarbera, M. (1990) Science 249, 992–1000. 14. Van Dyke, M. (1975) Perturbation Methods in Fluid Mechanics. (Parabolic Press). 32. Shraiman, B. (2005) Proc. Nat. Acad. Sci. 102, 3318–3323. 15. Chester, W & Breach, D. (1969) J. Fluid Mech. 37, 751–760. 33. Nonaka, S, Tanaka, Y, Okada, Y, Takeda, S, abd Y. Kanai, A. H, Kido, M, & Hi- 16. Veysey, J & Goldenfeld, N. (2007) Rev. Mod. Phys. 79, 883–927. rokawa, N. (1998) Cell 95, 829–837. 17. Carrier, G. (1953) On slow viscous flow, (ONR Mathematical Sciences Division), 34. Hove, J, Koster, R, Forouhar, A, Acevedo-Bolton, G, Fraser, S, & Gharib, M. (2003) Technical Report NR-062-163. Nature 421, 172–177. 18. Goldstein, S. (1969) Ann. Rev. Fluid Mech. 1, 1–29.

4 www.pnas.org — — Footline Author 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 (grey surface) separating closed and open streamlines.

Fig. 2. Finding the function ν˜(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 bestR fit of separatrix shape is obtained by minimizing either the unsigned ; I1 ≡ S |ψ| dS or the R |ψ| unsigned geometrical distance between the two separatrices; I2 ≡ S |∂ψ/∂n| dS. Main figure: (black curve) minimization of I2 gives the dependence of the effective viscosity ν˜(Re) 1/4 upon Re, (blue curve) best power law fit: ν˜(Re)/ν = 1.3Re . Inset figure: I1 and I2 have very close functional minima ν˜(Re) (representative data for Re = 101.30: I1; black curve, left axis, I2; red curve, right axis).

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: Ref = 0.13 (A) 5.25 (B), 34.8 (C) and 101.3 (D). Separatrices are shifted in Carrier’s equation, compared to 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. 4. Predicting drag using Carrier’s equation for spheroids of different aspect ratios for 0 < Re < 200. In each panel the solid curve is the (real) Navier-Stokes drag, and the dotted curve the drag predicted by Carrier’s equation. Different panels correspond to different aspect ratios in the order: A, 0.667, B, 1.0, C, 3.0, D, 5.0. All drags are non-dimensionalized by the Stokes (Re = 0) limiting drag, so tend to 1 as Re → 0.

Fig. 5. One-to-one mapping of Re to ν˜(Re) for spheroids of of aspect ratios 2/3 (F), 1 (•), 4/3 (¥), 2 (+), 3 (H), 4 (N), 5 (♦). The black curve gives the fit to the separatrix of a moving point force, from Figure 2.

Fig. 6. Comparison of solutions of Navier-Stokes equations (left panel) and Carrier’s equation (right panel) 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 stream-wise length 2a and diameter 2b (panel A). Other panels show B the 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, Rec, 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.

Fig. 8. An objective function based on symmetrized shear stress reproduces the drag- minimizing shapes without needing either an adjoint equation or any non-local information about the flow field. Solid curves are the drag minimizing shapes for Re = 0.1 (A) , 10 (B), 50 (C), 100 (D) calculated by Roper et al. [19] and the dashed curves are the approximate drag minimizing shapes.

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       A B                      C D              

5.5

5

4.5

4

ν 3.5 (Re)/ ~ 3 ν

2.5

2

1.5

1 −1 0 1 2 10 10 10 10 Re 0.4

1

2 a/b Re 1 10 25 50 100