Inviscid Incompressible Flow Research Articles

Stability of Symmetric and Asymmetric Vortices Pairs over SlenderConical Wing-Body Combinations

Theoretical analyses of the stability of symmetric and asymmetric vortex pairs over slender conical wing-body combinations consisting of a slender circular or elliptic cone and a flat-plate delta wing under small perturbations in an inviscid incompressible steady flow at high angles of attack with or without sideslip are presented. The three-dimensional flow problem is reduced to a problem in two dimensions, for which a general stability condition for vortex pairs can be applied. The stationary positions of symmetric and asymmetric vortex pairs and their stabilities are examined for various geometric configurations, angles of attack, and sideslip. Results of the analyses are compared with available experimental data and used to help gain insight into the flow behavior.

Cavitating vortex generation by a submerged jet

The surface geometry of a cavitating vortex is determined in the limit of inviscid incompressible flow. The limit surface is an ovaloid of revolution with an axis ratio of 5: 3. It is shown that a cavitating vortex ring cannot develop if the cavitation number is lower than a certain critical value. Experiments conducted at various liquid pressures and several jet exit velocities confirm the existence of a critical cavitation number close to 3. At cavitation numbers higher than the critical one, the cavitating vortex ring does not develop. At substantially lower cavitation numbers (k ⩽ 0.1), an elongated asymmetric cavitation bubble is generated, with an axial reentrant jet whose length can exceed the initial jet length by several times. This flow structure is called an asymmetric cavitating vortex, even though steady motion of this structure has not been observed.

Resolving small-scale structures in Boussinesq convection by adaptive grid methods

Inviscid Boussinesq convection is a challenging problem both analytically and numerically. Due to the complex dynamic development of small scales and the rapid loss of solution regularity, the Boussinesq convection pushes any numerical strategy to the limit. In E and Shu (Phys. Fluids 6 (1994) 49), a detailed numerical study of the Boussinesq convection in the absence of viscous effects is carried out using filtered pseudospectral method and a high-order accurate ENO schemes. In their computations, very fine grids have to be used in order to resolve the small-structures of the Boussinesq fluid. In this work, we will develop an efficient adaptive grid method for solving the inviscid incompressible flows, which can be useful in resolving extremely small-structures with reasonably small number of grid points. To demonstrate the effectiveness of the proposed method, the Boussinesq convection problem will be computed using the adaptive grid method.

A stable moving-particle semi-implicit method for free surface flows

In this paper, a mesh-less numerical approach is utilized to solve Euler's equation that is the governing equation of the irrotational flow of ideal fluids. A fractional step method of discritization is applied which consists to split each time step in two steps. This numerical method is based on moving-particle semi-implicit method (MPS) for simulating incompressible inviscid flows with free surfaces. The motion of each particle is calculated through interactions with neighboring particles covered with the kernel function. There are limitations for getting a stable solution by MPS method. In this paper, various kernel functions are considered and applied to improve the stability of MPS method. Based on these studies a kernel function is introduced that improves the stability of MPS method. The numerical results of the model are in good agreement with experimental results. The applicability of this model to simulate hydraulic problems with free surface is shown through the solution of dam break problem. The present method is a very useful utility for solving problems with irregular free surface in hydraulic and coastal engineering when an accurate prediction of free water surface is required.

Numerical solution of steady and unsteady flow over a vibrating airfoil in a channel

AbstractThe work deals with a numerical solution of 2D steady and unsteady inviscid incompressible flow over the profile NACA 0012 in a channel. The finite volume method (FVM) in a form of cell‐centered explicit schemes at quadrilateral C‐mesh is used. Governing system of equations is the system of incompressible Euler equations. The method of artificial compressibility and time dependent method is applied to steady computations. The small disturbance theory (SDT) applied to a numerical solution of flow over a rotated profile by a small angle only is mentioned. Brief introduction is given to the Arbitrary (Semi) Lagrangian‐Eulerian (ALE) method used for unsteady computations. Some numerical results of unsteady flow over a vibrating profile achieved by both SDT and ALE method are presented. Unsteady flow is caused by prescribed oscillations of the profile (one degree of freedom) fixed in an elastic axis. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Finite size vortices in the wake of an accelerated bent lamina

A theory of an unsteady incompressible inviscid flow past an accelerated lamina, moving across the fluid, is extended to address a bent (rather than flat) lamina and finite size (rather than point) vortices in its wake. It is shown that when the velocity of the lamina varies with time t as t m , the distance of the vortices from the lamina and the size of their cores grow, initially, as t ( 2 m + 2 ) / 3 and t ( 2 m + 2 ) / 3 exp ( - Ct - ( 8 m + 2 ) / 3 ) , respectively, where C is a strictly positive constant. The growth rate of the core dimensions with time seems to provide a plausible energy argument for the experimentally observed wake asymmetry that culminates the initial symmetric vortex arrangement.

Re-entrant corner flows of Oldroyd-B fluids

The method of matched asymptotic expansions is used to construct solutions for the planar steady flow of Oldroyd-B fluids around re-entrant corners of angles π / α (1/2≤ α <1). Two types of similarity solutions are described for the core flow away from the walls. These correspond to the two main dominant balances of the constitutive equation, where the upper convected derivative of stress either dominates or is balanced by the upper convected derivative of the rate of strain. The former balance gives the incompressible Euler or inviscid flow equations and the latter balance the incompressible Navier–Stokes equations. The inviscid flow similarity solution for the core is that first derived by Hinch (Hinch 1993 J. Non-Newtonian Fluid Mech. 50 , 161–171) with a core stress singularity that depends upon the corner angle and radial distance as O ( r −2(1− α ) ) and a velocity behaviour that vanishes as O ( r α (3− α )−1 ). Extending the analysis of Renardy (Renardy 1995 J. Non-Newtonian Fluid Mech. 58 , 83–39), this outer solution is matched to viscometric wall behaviour for both upstream and downstream boundary layers. This structure is shown to hold for the majority of the retardation parameter range. In contrast, the similarity solution associated with the Navier–Stokes equations has a velocity behaviour O ( r λ ) where λ ∈(0,1) satisfies a nonlinear eigenvalue problem, dependent upon the corner angle and an associated Reynolds number defined in terms of the ratio of the retardation and relaxation times. This similarity solution is shown to hold as an outer solution and is matched into stress boundary layers at the walls which recover viscometric behaviour. However, the matching is restricted to values of the retardation parameter close to the relaxation parameter. In this case the leading order core stress is Newtonian with behaviour O ( r −(1− λ ) ).

Numerical Simulation of Vortex Crystals and Merging inN-Point Vortex Systems with Circular Boundary

In two-dimensional (2D) inviscid incompressible flow, low background vorticity distribution accelerates intense vortices (clumps) to merge each other and to array in the symmetric pattern which is called ``vortex crystals''; they are observed in the experiments on pure electron plasma and the simulations of Euler fluid. Vortex merger is thought to be a result of negative ``temperature'' introduced by L. Onsager. Slight difference in the initial distribution from this leads to ``vortex crystals''. We study these phenomena by examining N-point vortex systems governed by the Hamilton equations of motion. First, we study a three-point vortex system without background distribution. It is known that a N-point vortex system with boundary exhibits chaotic behavior for N\geq 3. In order to investigate the properties of the phase space structure of this three-point vortex system with circular boundary, we examine the Poincar\'e plot of this system. Then we show that topology of the Poincar\'e plot of this system drastically changes when the parameters, which are concerned with the sign of ``temperature'', are varied. Next, we introduce a formula for energy spectrum of a N-point vortex system with circular boundary. Further, carrying out numerical computation, we reproduce a vortex crystal and a vortex merger in a few hundred point vortices system. We confirm that the energy of vortices is transferred from the clumps to the background in the course of vortex crystallization. In the vortex merging process, we numerically calculate the energy spectrum introduced above and confirm that it behaves as k^{-\alpha},(\alpha\approx 2.2-2.8) at the region 10^0<k<10^1 after the merging.

A NEW NUMERICAL SOLVER FOR A2-D NON-LINEAR-SHALLOW WATER EQUATION USING A SOROBAN GRID SYSTEM.

A new numerical scheme to solve a water flow with a free water surface and arbitrary curved ground surface is proposed. In the proposed method, by using a soroban grid system proposed by one of the authors, computational mesh is reconstruct easily at each time step so that some grid points locate on a free water surface exactly. Time development of velocity components and water surface are calculated according to an incompressible inviscid flow equation without any transformation of the equation. Advection effects are calculated by direct interpolation of the CIP method in 3rd order accuracy in spatial. After the advection is solved, acceleration of velocity components due to the pressure gradient is solved using a FDM approach on a soroban grid system. The proposed method is applied to some problems and computational results are examined through comparisons with results of other FDM scheme and theoretical result.

Efficient second-order analytical solutions for airfoils in subsonic flows

This paper presents simple and efficient analytical solutions for the velocity and pressure distributions on airfoils of arbitrary shapes, which are obtained considering the rigorous boundary conditions, without the use of the small perturbation assumption. A second-order accurate method using special singularities in the expression of the fluid velocity is first developed for airfoils in inviscid incompressible flows, by simultaneously solving the symmetric and anti-symmetric flow components defined by coupled boundary conditions. Accurate analytical solutions in closed form are thus derived and then successfully validated by comparison with the exact solutions for special airfoils obtained by conformal transformation and with numerical inviscid results. The method is then extended to model the main viscous effects on the velocity and pressure distributions, by considering the real velocity behavior at the airfoil trailing edge in viscous flow and the displacement thickness of the boundary layer developed on the airfoil and the wake. The resulting analytical solutions including viscous effects, derived for attached flows past airfoils at moderate angles of attack, were found in good agreement with experimental and numerical viscous results for various incidences, Reynolds and Mach numbers (Karman–Tsien compressibility correction was used to extend the solutions to compressible subsonic flows). In all cases studied (symmetric and cambered thick airfoils of arbitrary shapes, with rounded or pointed leading edges), these analytical solutions in closed form were found to be very efficient and accurate.

Flow over a profile in a channel with dynamical effects

AbstractThe work deals with a numerical solution of 2D inviscid incompressible flow over the profile NACA 0012 in a channel. The finite volume method in a form of cell‐centered scheme at quadrilateral C‐mesh is used. Governing system of equations is the system of Euler equations. Numerical results are partially compared with experimental data. Steady state solutions of the flow as well unsteady flows caused by prescribed oscillation of the profile were computed. The method of artificial compressibility and the time dependent method are used for computation of the steady state solution. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Construction of Lagrangians in continuum theories

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Scholle M. 2004Construction of Lagrangians in continuum theoriesProc. R. Soc. Lond. A.4603241–3260http://doi.org/10.1098/rspa.2004.1354SectionRestricted accessConstruction of Lagrangians in continuum theories M. Scholle M. Scholle Lehrstuhl für Technische Mechanik und Strömungsmechanik, University of Bayreuth, Universitätsstraβe 30, 95440 Bayreuth, Germany Google Scholar Find this author on PubMed Search for more papers by this author M. Scholle M. Scholle Lehrstuhl für Technische Mechanik und Strömungsmechanik, University of Bayreuth, Universitätsstraβe 30, 95440 Bayreuth, Germany Google Scholar Find this author on PubMed Search for more papers by this author Published:08 November 2004https://doi.org/10.1098/rspa.2004.1354AbstractFor physical systems the dynamics of which is formulated within the framework of Lagrange formalism, the dynamics is completely defined by only one function, namely the Lagrangian. For instance, the whole conservative Newtonian mechanics has been successfully embedded into this methodical concept. In continuum theories, however, the situation is different: no generally valid construction rule for the Lagrangian has been established in the past. In this paper general properties of Lagrangians in non–relativistic field theories are derived by considering universal symmetries, namely space– and time–translations, rigid rotations and Galilei boosts. These investigations discover the dual structure, i.e. the coexistence of two complementary representations of the Lagrangian. From the dual structure, relevant restrictions for the analytical form of the Lagrangian are derived which eventually result in a general scheme for Lagrangians. For two examples, namely Schrödinger's theory and the flow of an ideal fluid, the compatibility of the Lagrangian with the general scheme is demonstrated. The dual structure also has consequences for the balances which result from the respective symmetries by Noether's theorem: universally valid constitutive relations between the densities and the flux densities of energy, momentum, mass and centre of mass are derived. By an inverse treatment of these constitutive relations a Lagrangian for a given physical system can be constructed. This procedure is demonstrated for an elastically deforming body. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Scholle M (2022) A weakly nonlinear wave equation for damped acoustic waves with thermodynamic non-equilibrium effects, Wave Motion, 10.1016/j.wavemoti.2021.102876, 109, (102876), Online publication date: 1-Feb-2022. Mellmann M and Scholle M (2021) Symmetries and Related Physical Balances for Discontinuous Flow Phenomena within the Framework of Lagrange Formalism, Symmetry, 10.3390/sym13091662, 13:9, (1662) Scholle M (2020) A discontinuous variational principle implying a non-equilibrium dispersion relation for damped acoustic waves, Wave Motion, 10.1016/j.wavemoti.2020.102636, 98, (102636), Online publication date: 1-Nov-2020. Scholle M, Marner F and Gaskell P (2020) Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances, Water, 10.3390/w12051241, 12:5, (1241) Marner F, Scholle M, Herrmann D and Gaskell P (2019) Competing Lagrangians for incompressible and compressible viscous flow, Royal Society Open Science, 6:1, Online publication date: 1-Jan-2019.Scholle M and Marner F (2017) A non-conventional discontinuous Lagrangian for viscous flow, Royal Society Open Science, 4:2, Online publication date: 1-Feb-2017. Davey K and Darvizeh R (2016) Neglected transport equations: extended Rankine–Hugoniot conditions and J -integrals for fracture, Continuum Mechanics and Thermodynamics, 10.1007/s00161-016-0493-2, 28:5, (1525-1552), Online publication date: 1-Sep-2016. Scholle M (2014) Variational Formulations for Viscous Flow, PAMM, 10.1002/pamm.201410293, 14:1, (611-612), Online publication date: 1-Dec-2014. Prakash J, Lavrenteva O and Nir A (2014) Application of Clebsch variables to fluid-body interaction in presence of non-uniform vorticity, Physics of Fluids, 10.1063/1.4891198, 26:7, (077102), Online publication date: 1-Jul-2014. Scholle M (2011) Variational formulations in continuum mechanics, PAMM, 10.1002/pamm.201110336, 11:1, (693-694), Online publication date: 1-Dec-2011. Scholle M, Haas A and Gaskell P (2010) A first integral of Navier–Stokes equations and its applications, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 467:2125, (127-143), Online publication date: 8-Jan-2011. Zuckerwar A and Ash R (2009) Volume viscosity in fluids with multiple dissipative processes, Physics of Fluids, 10.1063/1.3085814, 21:3, (033105), Online publication date: 1-Mar-2009. Scholle M, Heining C and Haas A (2008) Comment on: “Variational principle for two-dimensional incompressible inviscid flow” [Phys. Lett. A 371 (2007) 39], Physics Letters A, 10.1016/j.physleta.2008.07.015, 372:36, (5857), Online publication date: 1-Sep-2008. This Issue08 November 2004Volume 460Issue 2051 Article InformationDOI:https://doi.org/10.1098/rspa.2004.1354Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/11/2004Published in print08/11/2004 License: Citations and impact Keywordsinverse variational problemsconstitutive relationsuniversal symmetriesgalilei invariancelagrange formalism

Helicity generation in uniform helical flows

Helicity generation conditions are derived for helical flows of Joukowski type with allowance for effects due to viscosity, buoyancy, temperature nonuniformity, and solid-body rotation. The upper and lower limits are determined for the rotation-frequency interval in which helicity can be generated by viscous forces. These conditions correspond to the regime of an isolated tornado-like vortex. An exact solution to the time-independent equations of motion for inviscid incompressible flow is obtained. The solution describes a generalized Kelvin-Helmholtz vortex having the form of a localized cylindrical vortex with nontrivial stable topological vortex-core structure determined by a finite value of helicity. For linear traveling inertia waves, which must have uniform helical structure, a general representation is found that characterizes helical structures of different origin.

A partition method for the solution of a coupled liquid-structure interaction problem

A numerical code is presented to study the motion of an incompressible inviscid flow in a deformable tank. It is based on a method belonging to the partition treatment class, as the fluid and structural fields are solved by coupling two distinct models. The fluid field is modeled by the Laplace equation and numerically solved by a Finite Volume technique. The computational grid is updated at each time step to take into account the movements of the free surface and the deformations of the vertical walls. An unsteady finite element formulation is used for modeling the tank on a grid discretized by triangular elements and linear shape functions. Results are presented for two different cases: a flow induced by a perturbation on the free surface in a tank motionless; a flow in a tank forced to oscillate periodically in the horizontal direction.

A note on the flow equations under space‐time transformation

AbstractThe equations governing incompressible and compressible inviscid flows and written in the physical frame (t,x,y,z) are known to be linearly well‐posed and exhibit elliptic or hyperbolic nature. The linear well‐posedness is considered here for these equations under a space‐time transformation (t,x,y,z) → (τ,ξ,η,ς), where the pseudo‐time τ and the new space coordinate (ξ,η,ς) all depend on (t,x,y,z). This type of transformation is typical in Computational Fluid Dynamics when local time‐stepping is used and it could be useful for uniformly treating problems such as mixed fast‐slow unsteady flows in which the flow is fast unsteady somewhere and slowly unsteady or steady elsewhere and mixed incompressible‐compressible flows. It is found that the transformation may alter the ellipticity, the hyperbolicity, and even the well‐posedness of the original equations.

Line–shaped objects and their balances related to gauge symmetries in continuum theories

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Scholle Markus and Anthony Karl–Heinz 2004Line–shaped objects and their balances related to gauge symmetries in continuum theoriesProc. R. Soc. Lond. A.460875–896http://doi.org/10.1098/rspa.2003.1198SectionRestricted accessResearch articleLine–shaped objects and their balances related to gauge symmetries in continuum theories Markus Scholle Markus Scholle Lehrstuhl für Technische Mechanik und Strömungsmechanik, University of Bayreuth, Universitätsstraβe 30, 95440 Bayreuth, Germany () Google Scholar Find this author on PubMed Search for more papers by this author and Karl–Heinz Anthony Karl–Heinz Anthony FB6—Theoretische Physik , Universität–GH Paderborn, Warburger Straβe 100, 33095 Paderborn, Germany () Google Scholar Find this author on PubMed Search for more papers by this author Markus Scholle Markus Scholle Lehrstuhl für Technische Mechanik und Strömungsmechanik, University of Bayreuth, Universitätsstraβe 30, 95440 Bayreuth, Germany () Google Scholar Find this author on PubMed Search for more papers by this author and Karl–Heinz Anthony Karl–Heinz Anthony FB6—Theoretische Physik , Universität–GH Paderborn, Warburger Straβe 100, 33095 Paderborn, Germany () Google Scholar Find this author on PubMed Search for more papers by this author Published:08 March 2004https://doi.org/10.1098/rspa.2003.1198AbstractWithin the Lagrange formalism Noether's theorem is a well–known tool for connecting symmetries of a physical system with homogeneous balance equations. In the context of this paper we call them balance equations of the volume type. They are associated with symmetry groups of the Lie type. However, in physics there are a lot of different balance equations which we call balance equations of the area type. Physically, they are associated with the dynamics of line–shaped objects. In this paper a general theory is presented which supplements Noether's theorem in so far as the area–type balances are associated with regauging symmetry groups of the non–Lie type. The theory is demonstrated for three prominent examples: for the Helmholtz laws of the vortex dynamics in an ideal fluid, for the dislocation dynamics in the dynamical eigenstress problem of elastic crystals, and for the homogeneous Maxwell equations. The theory will also be a valuable tool for solving inverse variational problems, i.e. to construct Lagrangians for physical systems. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Scholle M and Marner F (2016) A generalized Clebsch transformation leading to a first integral of Navier–Stokes equations, Physics Letters A, 10.1016/j.physleta.2016.07.066, 380:40, (3258-3261), Online publication date: 1-Sep-2016. Scholle M and Marner F (2015) The Clebsch transformation and its capabilities towards fluid and solid mechanics, PAMM, 10.1002/pamm.201510232, 15:1, (483-484), Online publication date: 1-Oct-2015. Prakash J, Lavrenteva O and Nir A (2014) Application of Clebsch variables to fluid-body interaction in presence of non-uniform vorticity, Physics of Fluids, 10.1063/1.4891198, 26:7, (077102), Online publication date: 1-Jul-2014. Scholle M, Heining C and Haas A (2008) Comment on: “Variational principle for two-dimensional incompressible inviscid flow” [Phys. Lett. A 371 (2007) 39], Physics Letters A, 10.1016/j.physleta.2008.07.015, 372:36, (5857), Online publication date: 1-Sep-2008. Scholle M (2004) Construction of Lagrangians in continuum theories, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 460:2051, (3241-3260), Online publication date: 8-Nov-2004. This Issue08 March 2004Volume 460Issue 2043 Article InformationDOI:https://doi.org/10.1098/rspa.2003.1198Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/03/2004Published in print08/03/2004 License: Citations and impact KeywordsVolume–Type And Area–Type BalancesDislocationsMaxwell's EquationsLagrange FormalismLie And Non–Lie GroupsVortices

Stability of symmetric and asymmetric vortex pairs over slender conical wings and bodies

Theoretical analyses are presented for the stability of symmetric and asymmetric vortex pairs over slender conical wings and bodies under small perturbations in an inviscid incompressible flow at high angles of attack and sideslip. The three-dimensional problem of a pair of vortices over slender conical wings and bodies is reduced to a problem in two dimensions by using the conical flow assumption and classical slender-body theory. The stability of symmetric and asymmetric vortex pairs over flat-plate delta wings, slender circular cones, and elliptic cones of various thickness ratios are examined. Results are compared with available experimental data.

Remarks on the Blow-up of the Euler Equations and the Related Equations

We consider the Euler system for inviscid incompressible fluid flows, and its perturbations in ℝn, n≥2. We prove global well-posedness of this perturbed Euler system in the Triebel-Lizorkin spaces for initial vorticity which is small in the critical Besov norms. Comparison type theorems about the blow-up of solutions are proved between the Euler system and its perturbations. We also study the possiblity of the self-similar type of blow-up of solutions to the equations.

Numerical solution of flow through a cascade and over a profile with dynamicaland aeroelastic effects

AbstractThe work deals with a numerical solution of 2D inviscid incompressible flow through a cascade and over a profile in a channel. The finite volume method in a form of cell‐centered scheme at triangular and qudrilateral mesh is used. The composite scheme applied to a numerical solution consists of more dissipative part (Lax‐Friedrichs scheme) and less dissipative part (Lax‐Wendroff scheme). Governing system of equations is the system of Euler equations. Two possibilities are considered.

A theoretical analysis of formation flight as a nonlinear self-organizing phenomenon

This study analyses the existence, stability and self‐organization of formation flight utilized by migrant birds. Air is approximated as an incompressible inviscid flow, while birds are modelled as elliptically loaded lifting‐lines. Application of conventional wing theory leads to newly derived, basic equations that describe the problem as a dynamical system of multiple wings interacting with each other through induced flow field. Formation flight is defined as the steady‐state solution of the basic equations, in particular the solution that all the birds fly at the same speed. In the case of a prescribed thrust, constant transverse interval between adjacent birds, and a flock of physically identical birds, analytical study of the basic equations reveals the facts that (1) formation flight is self‐organized and (2) this formation flight is stable. The new implication is that a configuration of formation emerges as a result of nonlinear dynamical interaction between many birds and that this nonlinear dynamical system does not exhibit chaotic behaviour. Numerical calculation has also been done for cormorant‐type birds with the same transverse interval between flock members. The proposed numerical scheme quickly converges to very accurate results owing to the recently derived, closed‐form expression of induced velocity distribution around an elliptically loaded lifting‐line. Transverse intervals between birds are found to be a more important factor than the number of birds. Configurations of formations are found to be inverted U rather than inverted V. In these formations every bird enjoys the same amount of drag reduction.