S. Goto

National Institute for Fusion Science (NIFS-Repository)

To predict and control high-Reynolds-number tur-bulence with active scalars, we need to understand thestatistical properties and ﬂow structures of such highReynolds number turbulence. Direct numerical simula-tion (DNS) of turbulence is one of the most eﬀective toolsfor such purposes because we can obtain detailed infor-mation of the statistical properties and ﬂow structuresof such complicated phenomena. Recent development ofsupercomputers enables us to perform DNS of turbulentﬂows with huge degrees of freedom.In the last year, we made several presentations sum-ming up our ﬁndings1) and published a journal paper2).Also, by performing the DNS of the autoignition processof n-heptane pre-mixture in three-dimensional homoge-neous compressible turbulence using a reduced chemicalkinetic mechanism, we remarkably developed our under-standing of the mutual interaction between turbulenceand chemical reactions (with heat release). In the fol-lowing, we summarize our recent studies based on theDNS of the autoignition process.To understand the mutual interaction between tur-bulence and chemical reactions, we performed the DNSof several autoignition process cases of n-heptane pre-mixture. The turbulent ﬂow with chemical reactions wascompared with the turbulent ﬂow without chemical re-actions. Also, the chemical reactions with turbulenceas the initial condition, were compared with the chem-ical reactions without turbulence. We studied not onlyseveral statistical quantities including the energy spectraand the spectra of temperature ﬂuctuation, but also thetime-development of the membranal structures of the re-gions of high rate of heat release. Our ﬁndings are asfollows.1. In the case of turbulence with chemical reactions,energy spectrum at high wave numbers is small ascompared with the case of turbulence without chem-ical reactions, which indicates that the chemical re-actions with heat release suppress the growth ofsmall scale ﬂuctuations. Visualization of the tem-perature as well as the second invariant of velocityderivatives suggested that the growth of small scaleﬂuctuations is suppressed at high temperature re-gions where viscosity is high.(a) (b)Fig. 1: (a) Membranal structure of the regions ofstrong exothermal reaction in the autoignition processof n-heptane pre-mixture in 3D homogeneous compress-ible turbulence.(b)Iso-surface of temperature (870K).2. In the autoignition process of n-heptane pre-mixture, the regions of strong exothermal reactionform membranal structures, and travel from hightemperature regions to low temperature regions (seeFig.1(a)). Detailed visualizations suggest that thisfact is true regardless of the presence or absence ofturbulence. Also, the detailed analyses showed thatthe peak positions of heat release agree well withthe positions at which temperature is approximately870K (see Fig.1(b)).3. A time history of average temperature in the turbu-lence with chemical reactions is similar to that ob-tained in zero-dimensional case, i.e. homogeneousreactions. This fact indicates that turbulence haseﬀects of making temperature ﬂuctuations not onlycomplex but also homogeneous.1) Teraji, A., Ishihara, T., Kaneda, Y.: Proc. CO-MODIA2012 (2012) 152-157.2) Teraji, A., Ikuta, H., Miki, T., Ishihara, T.: JSME79 799 (2013) 431-438.A rotation of the spin axis of a rotating object iscalled precession. It is well-known (see Ref. 1 and ref-erences therein) that ﬂuid motion in a precessing con-tainer becomes turbulent even if the precession is veryweak. Since the spin axis of the Earth is precessing witha period of 26000 years, geophysicists have extensively in-vestigated whether or not the Earth’s precession aﬀectsﬂow of melted ion in its outer core which generates geo-magnetic ﬁelds. However, ﬂow in a precessing containeris not interesting only from geophysical viewpoints, butalso from an engineering point of view. Because strongturbulence may be sustained by a weak precession, thissystem is likely to have wide variety of applications suchas a new type of mixer without impellers, for example.Towards engineering applications of this system toa mixer, in the present study we have conducted directnumerical simulations (DNS) of turbulence of an incom-pressible ﬂuid in a precessing sphere, and have estimatedits mixing ability. Here, we restrict ourselves within thecase that the two rotation axes are perpendicular to eachother.The scheme of the present DNS is as follows.Thanks to the incompressiblity of ﬂuid, its velocity ﬁeldmay be expressed by two scalar functions. Then, gov-erning equations of these scalar functions, which are de-rived from the Navier-Stokes equations, are numericallyintegrated by the Crank-Nicolson and Adams-Bashforthmethods. In this DNS, spatial derivatives are evaluatedby a spectral method, where the scalar functions are ex-panded in terms of the spherical harmonic functions andthe Zernike spherical polynomials. See Ref. 2 for detailsof this spectral method. In the present study, we tracka number of ﬂuid particles advected by the velocity ﬁeldthus simulated. In this particle tracking, ﬂuid velocityat the position of particles is linearly interpolated, andthe temporal integration of particle motion is made bythe Adams-Bashforth method.It is readily shown from the Navier-Stokes equa-tions that ﬂow in a precessing sphere is controlled bytwo non-dimensional parameters: the Reynolds numberRe = a2Ωs/ν and the Poincare´ number Γ = Ωp/Ωs,where a, Ωs, Ωp, and ν are the radius of sphere, themagnitude of spin and precession angular velocities, andthe kinematic viscosity of ﬂuid, respectively. Our lab-oratory experiments have shown that turbulence statis-tics in a precessing sphere drastically change depend-ing on the Poincare´ number even for a ﬁxed Reynoldsnumber. Therefore, here we ﬁx the latter parameter atRe = 40000 and conducted DNS changing the Poincare´number systematically in a range that 0.01 ≤ Γ ≤ 0.4.A temporal evolution of ﬂuid particles in the pre-cessing sphere (Re = 40000 and Γ = 0.1) is shown inFig. 1, where only the particles in a thin central layerperpendicular both to the spin and the precession axesare plotted. Colors (black and white) of particles aredetermined due to their initial positions. It is observedthat the initially separated two ﬂuids are quite eﬀectivelymixed by the turbulence induced by this rather weak pre-cession of the container.Since the DNS provides us with precise trajectoryof each ﬂuid particle, it is possible to estimate mixingeﬃciency of turbulence. For this quantiﬁcation, we haveemployed and modiﬁed a simple method originally pro-posed in Ref. 3. This quantiﬁcation is based on mixtureratio ρi (i = 1, 2, · · · ,M) of two ﬂuids in the i-th sub-domain of the container. Here, we divide the sphere intoM sub-domains with an approximately same (and suﬃ-ciently small) volume. Note that the standard deviationσ of ρi tends to be zero as the mixing of two ﬂuids pro-ceeds. We may therefore quantify the degree of mixingin terms of the standard deviation σ.It is then shown by the present DNS that for theinvestigated Reynolds number (Re = 40000) the mosteﬀective mixing takes place when the Poincare´ numberΓ ≈ 0.07, and that about 10 spins are suﬃcient to com-plete the mixing in the precessing sphere for this combi-nation of parameters.1) S. Goto, et al., Phys. Fluids 19 (2007) 061705.2) S. Kida & M. Shimizu, J. Phys.: Conf. Ser. 318(2011) 072031.3) P. V. Danckwerts, App. Sci. Res. A 3 (1952) 279.(a) (b)(c) (d)Fig. 1: Turbulent mixing in a precessing sphere.Reynolds number Re = 40000 and Poincare´ numberΓ = 0.1. A number of ﬂuid particles ﬁlled in the sphereare tracked numerically, and only the particles in thecentral thin layer are visualized. Horizontal and verti-cal directions are parallel to the spin and the precessionaxes, respectively. (a) Initial time t = 0, (b) t = 5Ts (Tsis the spin period), (c) t = 10Ts, (d) 15Ts.431§28. Turbulent Mixing in a Precessing SphereGoto, S. (Osaka Univ.)

Turbulent Mixing in a Precessing Sphere

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Turbulent Mixing in a Precessing Sphere

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