We present the main ideas behind the derivation and some applications of a theorem that parallels Lorentz’s reciprocal theorem and provides general expressions for the force and torque acting on a rigid body of arbitrary shape moving in an inhomogeneous incompressible flow at arbitrary Reynolds number. We show that this theorem allows a clear physical interpretation of the various contributions to the loads and encompasses all results available in both limits of inviscid and creeping flows

Magnaudet, Jacques

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To cite this version : Magnaudet, Jacques A generalized reciprocal theorem for predicting the force and torque on bodies moving in an unhomogeneous flow at arbitrary reynolds number. In: 23rd International Conference on Theoretical and Applied Mechanics, 19 August 2012 - 23 August 2012 (Beijing, China). Open Archive TOULOUSE Archive Ouverte (OATAO)OATAO is an open access repository that collects the work of Toulouse researchersand makes it freely available over the web where possible.  This is an author-deposited version published in : http://oatao.univ-toulouse.fr/Eprints ID : 10248 Any correspondance concerning this service should be sent to the repositoryadministrator: staff-oatao@listes-diff.inp-toulouse.frA GENERALIZED RECIPROCAL THEOREM FOR PREDICTING THE FORCE AND TORQUEON BODIES MOVING IN AN INHOMOGENEOUS FLOW AT ARBITRARY REYNOLDS NUMBERJacques Magnaudet∗∗Université de Toulouse and CNRS, Institut de Mécanique des Fluides de Toulouse, Allée Camille Soula,31400, Toulouse, FranceSummary We present the main ideas behind the derivation and some applications of a theorem that parallels Lorentz’s reciprocal the-orem and provides general expressions for the force and torque acting on a rigid body of arbitrary shape moving in an inhomogeneousincompressible flow at arbitrary Reynolds number. We show that this theorem allows a clear physical interpretation of the variouscontributions to the loads and encompasses all results available in both limits of inviscid and creeping flows.Predicting the forces and torques acting on bodies moving in arbitrary flow fields has always been a central concern inFluid Mechanics. Specific formulations of the problem have been developed in the two limits where the governing equa-tions become linear, namely Stokes flows and potential flows. However the general case where inertial and viscous effectsare both present poses much greater difficulties, owing to the nonlinear interplay between the various contributions. Ingeneral, no closed-form expression of the loads can be expected in this regime and the best that can be achieved is toexpress them as the sum of various surface and volume integrals involving solely the velocity field U and its derivatives,especially the local vorticity ω = ∇ ×U. The situation may however be more favourable with high-Reynolds-numberbubbles because the boundary layer induced by the shear-free condition at a bubble surface only induces secondarychanges in the flow field.The work described in this communication summarizes the main findings of a recent investigation [1] in which the abovegoal has been pursued with the objective of obtaining general expressions for the loads on a body or arbitrary shape thatobeys either a no-slip or a shear-free condition at its surface SB and moves in an inhomogeneous flow field with a trans-lational velocity V(t) and a rotation rate Ω(t), flow inhomogeneity being either due to velocity gradients in the carryingflow UU(x, t) or to the presence of a bounding wall SW (figure 1). The approach we follow stems directly from thecomplete Navier-Stokes equations an parallels the well-known Lorentz reciprocal theorem widely used in low-Reynolds-number hydrodynamics. It appears to be the proper way to derive rationally the generalized Kelvin-Kirchhoff equationsto predict all components of the force and torque acting on a rigid body moving in an incompressible inhomogeneous flowat arbitrary Reynolds number. The key to this approach, which extends that of [2] and [3], stands in the use of auxiliaryvelocity fields Uˆ which are both incompressible and irrotational and correspond to the flow disturbance induced by a unittranslational velocity iˆ or rotation rate jˆ of the body moving in a fluid at rest. Such auxiliary fields are exact solutions ofthe full Navier-Stokes equations since the viscous stress associated with an irrotational velocity field Uˆ = ∇φˆ has zeroflux. By properly selecting the orientation of iˆ and jˆ, all required components of the loads can be evaluated, irrespectiveof their orientation with respect to the relative motion between the body and fluid.When the fluid is at rest at infinity and the vorticity distribution is compact within the flow domain V , suitable combina-tions of the governing equations and proper use of Green and Stokes theorems provide each component of the hydrody-namic force F in the formiˆ.F =ddt∫SBφˆW.ndS +∫V{(ω + ωB)×U}.(Wˆ − Uˆ)dV−1Re∫SB[(Uˆ− Wˆ)× ω].n−∫SW{12(U.U)Wˆ.n+1Re(Uˆ× ω).n}dS , (1)where W = V + Ω × r and Wˆ = iˆ + jˆ × r, r being the local position from the body centroid and n the local unitnormal directed towards the fluid interior. Expression (1) and its counterpart for the torque achieve a clear splitting be-tween the various contributions to the loads: in (1), the two integrals on SB correspond to the added-mass and viscousforce, respectively, while the volume integral is the vortex-induced contribution which involves the vorticity ω in the bulkbut also the bound vorticity ωB corresponding to the difference on SB between the tangential component n ×U of theactual velocity field and that of the virtual irrotational velocity field ∇Φ that satisfies the same kinematic condition on SBand SW . The wall provides an inertial contribution normal to itself and an additional viscous force. When the Reynoldsnumber goes to zero, only contributions proportional to Re−1 remain in (1), possibly supplemented by the added-massterm if unsteadiness is large.As an application of this formulation, we shall discuss the case on a pair of spherical bubbles translating side-by-side athigh Reynolds number. We will show that, by successively selecting iˆ to be along the path and along the line of centres,one recovers the leading O(Re−1) expression for the viscous drag (including corrections due to the interaction with thesecond bubble) and the inertial transverse attractive force. The viscous correction due to the boundary layer will also beshown to be of O(Re−3/2) in the direction of the motion and of O(Re−1) in the transverse direction, a result which couldnot be obtained with available integral approaches but was recently inferred from direct numerical simulations.Figure 1. Sketch of the general flow configuration.When the body moves in a nonuniform carrying flow, new contributions arise, in addition to the net force and torqueexerted by the undisturbed flow on the volume of fluid VB occupied by the body. To make all couplings between theundisturbed velocity gradients and the body translational and rotational velocities explicit, we consider the case of ageneral linear carrying flowUU (x, t) = U0+r.∇U0 in the weakly inhomogeneous limit where the undisturbed velocityvaries over distances much larger than the body size. Setting U˜ = U − U0, ω˜ = ω − ω0, etc. and expanding thevarious contributions involving the undisturbed strain rate S0 and vorticity ω0, we show for instance that the net force ona non-rotating body moving in an unbounded flow domain is the sum of the viscous contribution on SB and of the vorticalcontribution in V as given in (1) plus an extra force F0 due to the underlying linear flow which writesiˆ.F0 = iˆ.{D0U0DtVB −A.(dVdt−D0U0Dt) + [A.S0 − S0.A].V˜ + (I2+AVB).[ω0 × (A.V˜)]−12A.(V˜ × ω0)}−∫V{φˆω0.(ω˜ + ω˜B)0 + 2(U˜0 −∇Φ˜0).S0.Uˆ}dV , (2)where D0/Dt denotes the material derivative following U0, I is the unit tensor and A is the translational added-masstensor. The terms in closed form in the r.h.s. of (2) may receive a clear physical interpretation. In particular the last twoof them provide a contribution to the lift force, the first of them being nonzero only when the principal axes of the bodyare not aligned with those of S0. The volume integral in (2) results from the distortion of the vortical disturbance dueto the body by the underlying strain rate and vorticity. It vanishes in the special case of inviscid two-dimensional flowsand is negligibly small at short time in inviscid three-dimensional flows. The present formulation recovers all availableinviscid predictions of [4] (which themselves encompass the original theoretical findings of [5]), and [6] for irrotationaland vortical undisturbed flows, respectively. Also, all available inviscid predictions for spheres and circular cylindersimmersed in linear flows, such as those of [7], appear as particular cases of (2). As an application of (2) we will discussthe inviscid lift force acting on circular cylinder translating steadily along a plane wall parallel to a linear shear flow, andwill show that if the cylinder lags behind the fluid, there is a critical distance to the wall at which the net lift force vanishes.To conclude we will emphasize that, beyond theoretical predictions with high-Reynolds-number bubbles or inviscid sit-uations, the present theorem may be useful in the context of experimental studies at finite Reynolds number, especiallythose in which the determination of the pressure distribution at the body surface is difficult, such as with flapping wingsor fins. Indeed, once the velocity and vorticity distributions have been determined (using Particle Image Velocimetry forinstance), this theorem provides a direct tool for evaluating the various contributions to the loads. We will also indicatesome directions to generalize this approach, in particular to deformable bodies and flows with density gradients.References[1] Magnaudet J.: A ‘reciprocal’ theorem for the prediction of loads on a body moving in an inhomogeneous flow at arbitrary Reynolds number. J.Fluid Mech. 689:564–604, 2011.[2] Quartapelle L. & Napolitano M.: Force and moment in incompressible flows. AIAA J. 21:911–913, 1983.[3] Howe M. S.: On the force and moment on a body in an incompressible fluid, with application to rigid bodies and bubbles at high and low Reynoldsnumbers. Q. J. Mech. Appl. Maths 48:401–426, 1995.[4] Galper A. & Miloh T.: Dynamic equations of motion for a rigid or deformable body in an arbitrary non-uniform potential flow field. J. FluidMech. 295:91–120, 1995.[5] Taylor G. I.: The forces on a body placed in a curved or converging stream of fluid. Proc. R. Soc. Lond. A 120:260–283, 1928.[6] Miloh T.: The motion of solids in inviscid uniform vortical fields. J. Fluid Mech. 479:279–305, 2003.[7] Auton T. R., Hunt J. C. R. & Prud’homme M.: The force exerted on a body moving in an inviscid unsteady non-uniform rotational flow. J. FluidMech. 197:241–257, 1988.

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A generalized reciprocal theorem for predicting the force and torque on bodies moving in an unhomogeneous flow at arbitrary reynolds number

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A generalized reciprocal theorem for predicting the force and torque on bodies moving in an unhomogeneous flow at arbitrary reynolds number

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