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Rotating free-shear flows. Part 2. Numerical simulations
- Olivier Métais, Carlos Flores, Shinichiro Yanase, James J. Riley, Marcel Lesieur
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- Journal:
- Journal of Fluid Mechanics / Volume 293 / 25 June 1995
- Published online by Cambridge University Press:
- 26 April 2006, pp. 47-80
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The three-dimensional dynamics of the coherent vortices in periodic planar mixing layers and in wakes subjected to solid-body rotation of axis parallel to the basic vorticity are investigated through direct (DNS) and large-eddy simulations (LES). Initially, the flow is forced by a weak random perturbation superposed on the basic shear, the perturbation being either quasi-two-dimensional (forced transition) or three-dimensional (natural transition). For an initial Rossby number Ro(i), based on the vorticity at the inflexion point, of small modulus, the effect of rotation is to always make the flow more two-dimensional, whatever the sense of rotation (cyclonic or anticyclonic). This is in agreement with the Taylor–Proudman theorem. In this case, the longitudinal vortices found in forced transition without rotation are suppressed.
It is shown that, in a cyclonic mixing layer, rotation inhibits the growth of three-dimensional perturbations, whatever the value of the Rossby number. This inhibition exists also in the anticyclonic case for |Ro(i)| ≤ 1. At moderate anticyclonic rotation rates (Ro(i) < −1), the flow is strongly destabilized. Maximum destabilization is achieved for |Ro(i) ≈ 2.5, in good agreement with the linear-stability analysis performed by Yanase et al. (1993). The layer is then composed of strong longitudinal alternate absolute vortex tubes which are stretched by the flow and slightly inclined with respect to the streamwise direction. The vorticity thus generated is larger than in the nonrotating case. The Kelvin–Helmholtz vortices have been suppressed. The background velocity profile exhibits a long range of nearly constant shear whose vorticity exactly compensates the solid-body rotation vorticity. This is in agreement with the phenomenological theory proposed by Lesieur, Yanase & Métais (1991). As expected, the stretching is more efficient in the LES than in the DNS.
A rotating wake has one side cyclonic and the other anticyclonic. For |Ro(i)| ≤ 1, the effect of rotation is to make the wake more two-dimensional. At moderate rotation rates (|Ro(i)| > 1), the cyclonic side is composed of Kármán vortices without longitudinal hairpin vortices. Karman vortices have disappeared from the anticyclonic side, which behaves like the mixing layer, with intense longitudinal absolute hairpin vortices. Thus, a moderate rotation has produced a dramatic symmetry breaking in the wake topology. Maximum destabilization is still observed for |Ro(i)| ≈ 2.5, as in the linear theory.
The paper also analyses the effect of rotation on the energy transfers between the mean flow and the two-dimensional and three-dimensional components of the field.
The modified cuinulant expansion for two-dimensional isotropic turbulence
- Tomomasa Tatsumi, Shinichiro Yanase
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- Journal:
- Journal of Fluid Mechanics / Volume 110 / September 1981
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- 20 April 2006, pp. 475-496
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The two-dimensional isotropic turbulence in an incompressible fluid is investigated using the modified zero fourth-order cumulant approximation. The dynamical equation for the energy spectrum obtained under this approximation is solved numerically and the similarity laws governing the solution in the energy-containing and enstrophy-dissipation ranges are derived analytically. At large Reynolds numbers the numerical solutions yield the k−3 inertial subrange spectrum which was predicted by Kraichnan (1967), Leith (1968) and Batchelor (1969) assuming a finite enstrophy dissipation in the inviscid limit. The energy-containing range is found to satisfy an inviscid similarity while the enstrophy-dissipation range is governed by the quasi-equilibrium similarity with respect to the enstrophy dissipation as proposed by Batchelor (1969). There exists a critical time tc which separates the initial period (t < tc) and the similarity period (t > tc) in which the enstrophy dissipation vanishes and remains non-zero respectively in the inviscid limit. Unlike the case of three-dimensional turbulence, tc is not fixed but increases indefinitely as the viscosity tends to zero.
Wrap, tilt and stretch of vorticity lines around a strong thin straight vortex tube in a simple shear flow
- GENTA KAWAHARA, SHIGEO KIDA, MITSURU TANAKA, SHINICHIRO YANASE
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- Journal:
- Journal of Fluid Mechanics / Volume 353 / 25 December 1997
- Published online by Cambridge University Press:
- 25 December 1997, pp. 115-162
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The mechanism of wrap, tilt and stretch of vorticity lines around a strong thin straight vortex tube of circulation Γ starting with a vortex filament in a simple shear flow (U=SX2Xˆ1, S being a shear rate) is investigated analytically. An asymptotic expression for the vorticity field is obtained at a large Reynolds number Γ/ν[Gt ]1, ν being the kinematic viscosity of fluid, and during the initial time St[Lt ]1 of evolution as well as St[Lt ](Γ/ν)1/2. The vortex tube, which is inclined from the streamwise (X1) direction both in the vertical (X2) and spanwise (X3) directions, is tilted, stretched and diffused under the action of the uniform shear and viscosity. The simple shear vorticity is on the other hand, wrapped and stretched around the vortex tube by a swirling motion, induced by it to form double spiral vortex layers of high azimuthal vorticity of alternating sign. The magnitude of the azimuthal vorticity increases up to O((Γ/ν)1/3S) at distance r=O((Γ/ν)1/3 (νt)1/2) from the vortex tube. The spirals induce axial flows of the same spiral shape with alternate sign in adjacent spirals which in turn tilt the simple shear vorticity toward the axial direction. As a result, the vorticity lines wind helically around the vortex tube accompanied by conversion of vorticity of the simple shear to the axial direction. The axial vorticity increases in time as S2t, the direction of which is opposite to that of the vortex tube at r=O((Γ/ν)1/2 (νt)1/2) where the vorticity magnitude is strongest. In the near region r[Lt ](Γ/ν)1/3 (νt)1/2, on the other hand, a viscous cancellation takes place in tightly wrapped vorticity of alternate sign, which leads to the disappearance of the vorticity normal to the vortex tube. Only the axial component of the simple shear vorticity is left there, which is stretched by the simple shear flow itself. As a consequence, the vortex tube inclined toward the direction of the simple shear vorticity (a cyclonic vortex) is intensified, while the one oriented in the opposite direction (an anticyclonic vortex) is weakened. The growth rate of vorticity due to this effect attains a maximum (or minimum) value of ±S2/33/2 when the vortex tube is oriented in the direction of Xˆ1+Xˆ2∓ Xˆ3. The present asymptotic solutions are expected to be closely related to the flow structures around intense vortex tubes observed in various kinds of turbulence such as helical winding of vorticity lines around a vortex tube, the dominance of cyclonic vortex tubes, the appearance of opposite-signed vorticity around streamwise vortices and a zig-zag arrangement of streamwise vortices in homogeneous isotropic turbulence, homogeneous shear turbulence and near-wall turbulence.