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Register a new userZonal structure of unbounded external-flow and aerodynamics
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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.
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A cylinder undergoes precession when it rotates around its axis and this axis itself rotates around another direction. In a precessing cylinder full of fluid, a steady and axisymmetric component of the azimuthal flow is generally present. This component is called a zonal flow. Although zonal flows have been often observed in experiments and numerical simulations, their origin has eluded theoretical approaches so far. Here, we develop an asymptotic analysis to calculate the zonal flow forced in a resonant precessing cylinder, that is when the harmonic response is dominated by a single Kelvin mode. We find that the zonal flow originates from three different sources: (1) the nonlinear interaction of the inviscid Kelvin mode with its viscous correction; (2) the steady and axisymmetric response to the nonlinear interaction of the Kelvin mode with itself; and (3) the nonlinear interactions in the end boundary layers. In a precessing cylinder, two additional sources arise due to the equatorial Coriolis force and the forced shear flow. However, they cancel exactly. The study thus generalises to any Kelvin mode, forced by precession or any other mechanism. The present theoretical predictions of the zonal flow are confirmed by comparison with numerical simulations and experimental results. We also show numerically that the zonal flow is always retrograde in a resonant precessing cylinder (m=1) or when it results from resonant Kelvin modes of azimuthal wavenumbers m=2, 3, and presumably higher.
For rapidly rotating turbulent Rayleigh--Benard convection in a slender cylindrical cell, experiments and direct numerical simulations reveal a boundary zonal flow (BZF) that replaces the classical large-scale circulation. The BZF is located near the vertical side wall and enables enhanced heat transport there. Although the azimuthal velocity of the BZF is cyclonic (in the rotating frame), the temperature is an anticyclonic traveling wave of mode one whose signature is a bimodal temperature distribution near the radial boundary. The BZF width is found to scale like $Ra^{1/4}Ek^{2/3}$ where the Ekman number $Ek$ decreases with increasing rotation rate.
We analyse the nonlinear dynamics of the large scale flow in Rayleigh-Benard convection in a two-dimensional, rectangular geometry of aspect ratio $Gamma$. We impose periodic and free-slip boundary conditions in the streamwise and spanwise directions, respectively. As Rayleigh number Ra increases, a large scale zonal flow dominates the dynamics of a moderate Prandtl number fluid. At high Ra, in the turbulent regime, transitions are seen in the probability density function (PDF) of the largest scale mode. For $Gamma = 2$, the PDF first transitions from a Gaussian to a trimodal behaviour, signifying the emergence of reversals of the zonal flow where the flow fluctuates between three distinct turbulent states: two states in which the zonal flow travels in opposite directions and one state with no zonal mean flow. Further increase in Ra leads to a transition from a trimodal to a unimodal PDF which demonstrates the disappearance of the zonal flow reversals. On the other hand, for $Gamma = 1$ the zonal flow reversals are characterised by a bimodal PDF of the largest scale mode, where the flow fluctuates only between two distinct turbulent states with zonal flow travelling in opposite directions.
We extend unsteady thin aerofoil theory to model aerofoils with generalised chordwise porosity distributions. The analysis considers a linearised porosity boundary condition where the seepage velocity through the aerofoil is related to the local pressure jump across the aerofoil surface and to the unsteady characteristics of the porous medium. Application of the Plemelj formulae to the resulting boundary value problem yields a singular Fredholm--Volterra integral equation which does not admit an analytic solution. Accordingly, we develop a numerical solution scheme by expanding the bound vorticity distribution in terms of appropriate basis functions. Asymptotic analysis at the leading- and trailing-edges reveals that the appropriate basis functions are weighted Jacobi polynomials whose parameters are related to the porosity distribution. The Jacobi polynomial basis enables the construction of a numerical scheme that is accurate and rapid, in contrast to the standard choice of Chebyshev basis functions that are shown to {be unsuitable} for porous aerofoils. Applications of the numerical solution scheme to discontinuous porosity profiles, quasi-static problems, and the separation of circulatory and non-circulatory contributions are presented. Analogues to the classical Theodorsen and Sears functions are computed numerically, which show that an effect of trailing-edge porosity is to reduce the amount of vorticity shed into the wake, thereby reducing the magnitude of the circulatory lift. {Fourier transform inversion of these frequency-domain functions produces porous extensions to the Wagner and K{u}ssner functions for transient aerofoil motions or gust encounters, respectively.}
Rayleigh-Benard (RB) convection with free-slip plates and horizontally periodic boundary conditions is investigated using direct numerical simulations. Two configurations are considered, one is two-dimension (2D) RB convection and the other one three-dimension (3D) RB convection with a rotating axis parallel to the plate. We explore the parameter range of Rayleigh numbers Ra from $10^7 to $10^9$ and Prandtl numbers $Pr$ from $1$ to $100$. We show that zonal flow, which was observed, for example, by Goluskin emph{et al}. emph{J. Fluid. Mech.} 759, 360-385 (2014) for $Gamma=2$, is only stable when $Gamma$ is smaller than a critical value, which depends on $Ra$ and $Pr$. With increasing $Gamma$, we find a second regime in which both zonal flow and different convection roll states can be statistically stable. For even larger $Gamma$, in a third regime, only convection roll states are statistically stable and zonal flow is not sustained. For the 3D simulations, we fix $Ra=10^7$ and $Pr=0.71$, and compare the flow for $Gamma=8$ and $Gamma = 16$. We demonstrate that with increasing aspect ratio $Gamma$, zonal flow, which was observed for small $Gamma=2pi$ by von Hardenberg emph{et al}. emph{Phys. Rev. Lett.} 15, 134501 (2015), completely disappears for $Gamma=16$. For such large $Gamma$ only convection roll states are statistically stable. In between, here for medium aspect ratio $Gamma = 8$, the convection roll state and the zonal flow state are both statistically stable. What state is taken depends on the initial conditions, similarly as we found for the 2D case.
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