Using helical-wave decomposition (HWD), a solenoidal vector field can be decomposed into helical modes with different wavenumbers and polarities. Here, we first review the general formulation of HWD in an arbitrary single-connected domain, along with some new development. We then apply the theory to a viscous incompressible turbulent channel flow with system rotation, including a derivation of helical bases for a channel domain. By these helical bases, we construct the inviscid inertial-wave (IW) solutions in a rotating channel and derive their existing condition. The condition determines the specific wavenumber and polarity of the IW. For a set of channel turbulent flows rotating about a streamwise axis, this channel-domain HWD is used to decompose the flow data obtained by direct numerical simulation. The numerical results indicate that the streamwise rotation induces a polarity-asymmetry and concentrates the fluctuating energy to particular helical modes. At large rotation rates, the energy spectra of opposite polarities exhibit different scaling laws. The nonlinear energy transfer between different helical modes is also discussed. Further investigation reveals that the IWs do exist when the streamwise rotation is strong enough, for which the theoretical predictions and numerical results are in perfect agreement in the core region. The wavenumber and polarity of the IW coincide with that of the most energetic helical modes in the energy spectra. The flow visualizations show that away from the channel walls, the small vortical structures are clustered to form very long columns, which move in the wall-parallel plane and serve as the carrier of the IW. These discoveries also help clarify certain puzzling problems raised in previous studies of streamwise-rotating channel turbulence.
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.
It is shown that the two remarkable properties of turbulence, the Extended Self-Similarity (ESS) [R. Benzi et al., Phy. Rev. E 48, R29, (1993)] and the She-Leveque Hierarchical Structure (SLHS) [Z.S. She and E. Leveque, Phy. Rev. Lett. 72, 336, (1994)] are related to each other. In particular, we have shown that a generalized hierarchical structure together with the most intense structures being shock-like give rise to ESS. Our analysis thus suggests that the ESS measured in turbulent flows is an indication of the shock-like intense structures. Results of analysis of velocity measurements in a pipe-flow turbulence support our conjecture.PACS numbers: 47.27.-i Fully developed turbulence is characterized by powerlaw dependence of the moments of velocity fluctuations. It was suggested by Kolmogorov in 1941 (K41)[1] that there is a constant rate of energy transfer from large to small scales and that the statistical properties of the velocity difference across a separation r, δv r ≡ v(x+r)−v(x), depend only on the mean energy transfer or equivalently the mean energy dissipation rate ǫ and the scale r when r is within an inertial range. Dimensional considerations then lead to the prediction that the velocity structure functions, which are moments of the magnitude of the velocity difference, have simple power-law dependence on r within the inertial range:(1)Experiments [2] have indicated that there is indeed power law scaling in the inertial range but the scaling exponents are different from p/3:where ζ p has a nonlinear dependence on p. Such a deviation implies that the functional form of the probability density function (pdf) of δv r depends on r, that is, the velocity fluctuations have scale-dependent statistics. Understanding this deviation from K41 is essential to our fundamental understanding of the small scale statistical properties of turbulence.Recently, Benzi et al. [3] have discovered a remarkable new scaling property: S p (r) has a power-law dependence on S 3 (r) over a range substantially longer than the scaling range obtained by plotting S p (r) as a function of r. This behavior was named Extended Self-Similarity (ESS); its discovery has enabled more accurate determination of the scaling exponents ζ p , particularly at moderately high Reynolds numbers assessible experimentally and numerically, It was later reported that ESS is invalid for anisotropic turbulent flows such as atmospheric boundary layer and channel flow [4,5,6]. This inspires the study of a generalized ESS (GESS), a scaling behavior of the normalized structure functions when plotted against each other [7,8], which is still valid in these anisotropic flows. The validity of ESS suggests that the different order structure functions have the same dependence on r when r is near the dissipative range [9,10,11]. Very recently, Yakhot argued that some mean-field approximation of the pressure contributions in the Navier-Stokes equation would lead to ESS [12].A number of phenomenological models have been proposed to explain the anomalous scaling expon...
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