Moderate Rotation Rates Research Articles

Rotating free-shear flows. Part 2. Numerical simulations

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.

Numerical Simulations of Coherent Vortices in Turbulence

After presenting the general features of turbulent flows and coherent vortices, we discuss the major progress brought by Computational Fluid Dynamics (CFD) to the understanding of coherent vortices in turbulence. Afterwards, we present some simple vortex dynamics arguments allowing us to understand qualitatively the formation of coherent vortices during the transition to turbulence in shear flows. Of particular interest are the following elementary vortex interactions: roll up of a vortex sheet, pairing, dipole, even longitudinal hairpin, and odd longitudinal hairpin. Then direct numerical or large-eddy simulations of free-shear flows (mixing layers, backstep, jets, wakes), isotropic turbulence, and spatially-developing boundary-layers on a flat plate are presented. Following the editor’s request, these simulations focus mainly on the work done in France in Grenoble, which is however discussed within a broader numerical and experimental context. We show for instance that helical pairings may occur in plane mixing layers. The paper also presents in details the formalism of large-eddy simulations (LES) of turbulence, with the various models developed since Smagorinsky. We see for instance how longitudinal hairpin vortices are taken into account within these LES. Effects of compressibility upon turbulence are also considered: we study in particular mixing layers (where it is shown that helical pairing is inhibited above a certain convective Mach number), strongly heated boundary layers at low Mach number, and supersonic compression ramps within the frame of the HERMES European space-shuttle reentry project. Finally, we look at the influence of solid-body rotation on incompressible turbulence. In the case of a free-shear layer, we study shear/Coriolis linear instability, which, in anticyclonic conditions and at moderate rotation rates, yields a purely longitudinal mode. Numerical simulations show how this mode evolves non-linearly into concentrated longitudinal hairpin vortices of absolute vorticity. We also consider the case of initially isotropic turbulence subject to rotation.

Winds from rotating Wolf-Rayet stars: the wind-compressed zone model

Theoretical models for Wolf-Rayet winds that include the effects of rotation and magnetic fields are discussed. These are of two types: magnetic rotator models and wind compression models. The magnetic rotator models are examples of equatorial expulsion models, typically requiring large rotation rates to produce significant equatorial density enhancements. Wind- compression models are introduced here as an application of the two-dimensional Wind-Compressed Disk (WCD) model that has recently been developed for Be stars. An equatorial disk forms because of the supersonic confluence of the flow from the upper and lower hemispheres of the star. In the application to WR stars, we suggest that disk formation is not required, and instead a less extreme example that we call a wind-compressed zone (WCZ) may suffice. Because the winds of WR stars have a geometrically extended acceleration region, or a “slow” velocity law, there can be compressions by an order of magnitude, even with moderate stellar rotation rates of about 16% critical. Magnetic flux conservation is then used to estimate the enhancement of the field strength that occurs because of the equatorial wind-compression. We consider an application of the model to explain the occurrence of polarization and dust formation among only some WR stars. Finally, we consider the WR + compact companion system WR 147, and suggest that the enhanced accretion in a WCZ could help explain the large X-ray emission.

Radiative wind models with azimuthal symmetry - II. A latitude-dependent wind for Be and B[e] stars

We present an axisymmetric radiative wind model in which we explore different contributions of thin and thick lines as a function of star latitude. The finite disc correction is incorporated in the line force. We also consider a viscosity parameter, which causes the wind to deviate from angular momentum conservation, and the effects of rapid rotation in the star itself. Numerical calculations for early Be stars give density contrasts between the equator and the pole that may be as high as 150 if a rotation rate of 90 per cent of the critical speed is assumed. If a lower rate is used (70 per cent), this density ratio decreases to about 15–20. Global mass losses, obtained from mass fluxes per unit solid angle in different directions, are in the range |$5 \times 10^{-9}-5\times 10^{-8} \text M _\odot\enspace \text {yr}^{-1}$|⁠. High velocities are still present in the equatorial plane. Terminal velocities in this region are about 1000–1300 km s−1, while polar values are 2300 km s−1. Application to B[e] supergiants leads to |$\upsilon_\infty \approx 650 \enspace\ \text {km} \enspace \text s ^{-1}$| (pole) and to |$\upsilon_\infty \approx 450-500 \enspace\ \text {km} \enspace \text s ^{-1}$| (equator). A significant equator–pole density contrast (about 30) is obtained even with moderate rotation rates, which could increase the degree of linear polarization in these objects.

Some effects of crystal rotation on large-scale Czochralski oxide growth: analysis via a hydrodynamic thermal-capillary model

A hydrodynamic thermal-capillary model (HTCM) for heat transfer in Czochralski crystal growth systems is used to calculate steady-state, axisymmetric solutions for heat transfer and fluid mechanics while incorporating a self-consistent description of the free boundaries of the melt/crystal interface, the melt meniscus, and the crystal diameter. The model employs a Galerkin finite-element method to discretize the model equations, and solutions are obtained using a Newton-Raphson iterative scheme. Sample results are presented for the growth of a large-dimension oxide crystal with thermophysical properties similar to those of gadolinium gallium garnet (GGG). Calculations with the HTCM show the effects of crystal rotation on heat transfer, flow in the melt, and melt/crystal interface shape. Severe deflections of the melt/crystal interface are calculated for moderate rotation rates, and limit points in the steady-state solutions are found with respect to crystal rotation.

Mesoscale vortex shedding from large islands: A comparison with laboratory experiments of rotating stratified flows

Mesoscale vortex shedding in the wake of large islands has been analysed with respect to the influence of stratification and background rotation. Comparison with results from laboratory experiments on rotating-stratified flows past isolated obstacles based on Rossby and Froude numbers have shown that forFr<0.4 vortex shedding can be expected for moderate rotation rates. Rossby numbers for cases of atmospheric vortex streets were found to be much larger than in laboratory experiments. Depending on the Froude number different flow pattern in the wake of three-dimensional obstacles have been identified in both, laboratory experiments and atmospheric observations.

Laminar forced convection heat transfer in rotating rectangular ducts. (1st report. Numerical analysis in the case of positive rotation).

Fully developed laminar forced convection heat transfer in curved rectangular ducts rotating at a constant angular velocity about an axis through the center of curvature of ducts is studied. Thermal boundary conditions of fluid flow are assumed to be subject to uniform wall heat flux. Navier-Stokes and energy equations are solved by the finite-difference method for Pr=0.7. The case for the positive rotation of a duct is considered. Velocity and temperature fields are obtained for various rotation rates and Reynolds numbers. A double-vortex secondary flow appears at a low to moderate rotation rate and Re number. It is found that an additional pair of vortices appeared and disappeared for a higher range of Reynolds numbers under the influence of centrifugal and Coriolis forces. The friction factors and Nusselt numbers are also obtained. The numerical results in regard to the friction factors are in good agreement with experimental results obtained by other researchers.

Super‐rotation and diffusion of axial angular momentum: I. ‘Speed limits’ for axisymmetric flow in a rotating cylindrical fluid annulus

Abstract‘Super‐rotation’ can be defined with respect to the limitations on the magnitude of specific angular momentum, m, in an inviscid fluid. In a viscous fluid, a steady, super‐rotating axisymmetric flow is shown to require diffusion of m against its gradient ∇m in the meridional plane. The conditions under which m can be diffused in an incompressible fluid by molecular viscosity against ∇m in a cylindrical system are shown to be consistent with the normal properties of isotropic Newtonian viscosity (i.e. appropriate for the laminar flow of a viscous liquid in the laboratory) when cylindrical curvature is fully represented. A series of numerical simulations of thermally‐driven axisymmetric circulations in a cylindrical fluid annulus, subject to various mechanical boundary conditions, is then presented. The role of diffusion in the angular momentum budget of the simulations is examined by diagnosing m, its diffusive flux F (due to molecular viscosity) and divergence ∇·F from the final equilibrium (steady) flow. Up‐gradient diffusion (with respect to ∇m) is found to be particularly important for flows in a system with stress‐free top and side boundaries and a non‐slip base. Similar up‐gradient diffusion can also result in angular momentum expulsion effects in a system confined entirely by stress‐free boundaries. The characteristic dynamics of super‐rotation in a viscous fluid, and the associated limitations upon its magnitude and dependence upon the external conditions and fluid properties, are then explored in a scale analysis for the cylindrical annulus with a non‐slip base (based on a scheme due to Hignett, Ibbetson and Killworth). The most rapid super‐rotation (with zonal Rossby number &gt; 1) is found to be favoured at moderate rotation rates, with strong cylindrical curvature, and a large meridional aspect ratio and Prandtl number. The results of the scale analysis are verified by means of further numerical experiments.

Laminar forced convection heat transfer in rectangular ducts rotating about an axis perpendicular to the duct axis. (2nd report Numerical analysis in case of rectangular ducts with aspect rations 0.2, 0.5, 2 and 5)

Numerical analysis is conduced on the forced convection heat transfer in rectangular ducts with aspect ratios 0.2, 0.5, 2.0 and 5, 0, rotating at a constant angular velocity about an axis perpendicular to its axis. Navier-Stokes and energy equations are solved by a finite-difference method in the fully developed region of laminar flow. Velocity and temperature fields are obtained for various rotation rates and Reynolds numbers at each aspect ratio of the duct. A double-vortex secondary flow appears at low to moderate rotation rates. An additional pair of vortices appeared and disappeared for a higher range of Reynolds numbers due to aspect ratios of the ducts. The friction factor and Nusselt number are influenced by the Coriolis force and aspect ratio. Numerical results of the friction factor for aspect ratios equal to 0.5 and 2.0 are in good agreement with experimental results by other researchers.

Numerical study of viscous flow in rotating rectangular ducts

A numerical study of the laminar flow of an incompressible viscous fluid in rotating ducts of rectangular cross-section is conducted. The full time-dependent nonlinear equations of motion are solved by finite-difference techniques for moderate to relatively rapid rotation rates where both the convective and viscous terms play an important role. At weak to moderate rotation rates, a double-vortex secondary flow appears in the transverse planes of the duct whose structure is relatively independent of the aspect ratio of the duct. For Rossby numbers Ro c 100 this secondary flow is shown to lead to substantial distortions of the axial velocity profiles. For more rapid rotations (Ro c l), the Secondary flow (in a duct with an aspect ratio of two) is shown to split into an asymmetric configuration of four counter-rotating vortices similar to that which appears in curved ducts. It is demonstrated mathematically that this effect could result from a disparity in the symmetry of the convective and Coriolis terms in the equations of motion. If the rotation rates are increased further, the secondary flow restabilizes to a slightly asymmetric double-vortex configuration and the axial velocity wumes a Taylor-Proudman configuration in the interior of the duct. Comparisons with existing experimental results are quite favourable.