In a recent paper, Liu, Zhu & Wu (2015, J. Fluid Mech. 784: 304; LZW for short) present a far-field theory for the aerodynamic force experienced by a body in a two-dimensional, viscous, compressible and steady flow. In this companion theoretical paper we do the same for three-dimensional flow. By a rigorous fundamental solution method of the linearized Navier-Stokes equations, we not only improve the far-field force formula for incompressible flow originally derived by Goldstein in 1931 and summarized by Milne-Thomson in 1968, both being far from complete, to its perfect final form, but also prove that this final form holds universally true in a wide range of compressible flow, from subsonic to supersonic flows. We call this result the unified force theorem (UF theorem for short) and state it as a theorem, which is exactly the counterpart of the two-dimensional compressible Joukowski-Filon theorem obtained by LZW. Thus, the steady lift and drag are always exactly determined by the values of vector circulation Γ φ due to the longitudinal velocity and inflow Q ψ due to the transversal velocity, respectively, no matter how complicated the near-field viscous flow surrounding the body might be. However, velocity potentials are not directly observable either experimentally or computationally, and hence neither is the UF theorem. Thus, a testable version of it is also derived, which holds only in the linear far field and is exactly the counterpart of the testable compressible Joukowski-Filon formula in two dimensions. We call it the testable unified force formula (TUF formula for short). Due to its linear dependence on the vorticity, TUF formula is also valid for statistically stationary flow, including time-averaged turbulent flow.Key words: Authors should not enter keywords on the manuscript, as these must be chosen by the author during the online submission process and will then be added during the typesetting process (see http://journals.cambridge.org/data/relatedlink/jfmkeywords.pdf for the full list)

We extend the impulse theory for unsteady aerodynamics from its classic global form to finite-domain formulation then to minimum-domain form and from incompressible to compressible flows. For incompressible flow, the minimum-domain impulse theory raises the finding of Li and Lu [“Force and power of flapping plates in a fluid,” J. Fluid Mech. 712, 598–613 (2012)] to a theorem: The entire force with discrete wake is completely determined by only the time rate of impulse of those vortical structures still connecting to the body, along with the Lamb-vector integral thereof that captures the contribution of all the rest disconnected vortical structures. For compressible flows, we find that the global form in terms of the curl of momentum ∇ × (ρu), obtained by Huang [Unsteady Vortical Aerodynamics (Shanghai Jiaotong University Press, 1994)], can be generalized to having an arbitrary finite domain, but the formula is cumbersome and in general ∇ × (ρu) no longer has discrete structures and hence no minimum-domain theory exists. Nevertheless, as the measure of transverse process only, the unsteady field of vorticity ω or ρω may still have a discrete wake. This leads to a minimum-domain compressible vorticity-moment theory in terms of ρω (but it is beyond the classic concept of impulse). These new findings and applications have been confirmed by our numerical experiments. The results not only open an avenue to combine the theory with computation-experiment in wide applications but also reveal a physical truth that it is no longer necessary to account for all wake vortical structures in computing the force and moment.

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For steady flow, one usually decomposes the total drag into different components by wake-plane integrals and seeks their reduction strategies separately. Unlike the body-surface stress integral, the induced drag as well as the profile drag has been found to depend on the streamwise location of the wake plane used for drag estimate. It gradually diminishes as the wake plane moves downstream, which was often attributed to numerical dissipation. In this paper, we present an exact general force-breakdown theory and its numerical demonstrations for viscous incompressible flow over an arbitrary aircraft to address this puzzling issue. Based on the theory, the induced and profile drags do depend inherently on the wake-plane location rather than being merely caused by numerical dissipation. The underlying mechanisms are identified in terms of the components, moments, and physical dissipation of the Lamb-vector field produced by the aircraft motion. This theoretical prediction is fully consistent with the linear far-field force theory that the induced drag finally vanishes and the profile drag increases to the total drag at an infinitely far field for viscous flow. Moreover, as a product of this exact theory, a new compact midwake approximation for the induced drag is proposed for the convenience of routine wake survey in industry. Its prediction is similar to conventional formulas for attached flow but behaves much better for separated flow.

This paper starts from the far-field behaviours of velocity field in externally-unbounded flow. We find that the well-known algebraic decay of disturbance velocity as derived kinematically is too conservative. Once the kinetics is taken into account by working on the fundamental solutions of far-field linearized Navier-Stokes equations, it is proven that the furthest far-field zone adjacent to the uniform fluid at infinity must be unsteady, viscous and compressible, where all disturbances degenerate to sound waves that decay exponentially. But this optimal rate does not exist in some commonly used simplified flow models, such as steady flow, incompressible flow and inviscid flow, because they actually work in true subspaces of the unbounded free space, which are surrounded by further far fields of different nature. This finding naturally leads to a zonal structure of externally-unbounded flow field. The significance of the zonal structure is demonstrated by its close relevance to existing theories of aerodynamic force and moment in external flows, including the removal of the difficulties or paradoxes inherent in the simplified models.