Charles G. Speziale

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.

Numerical study of secondary flows and roll-cell instabilities in rotating channel flow

A numerical study is conducted on the pressure-driven laminar flow of an incompressible viscous fluid through a rectangular channel subjected to a spanwise rotation. The full nonlinear time-dependent Navier–Stokes equations are solved by a finite-difference technique for various rotation rates and Reynolds numbers in the laminar regime. At weak rotation rates, a double-vortex secondary flow appears in the transverse planes of the channel. For more rapid rotation rates, an instability occurs in the form of longitudinal roll cells in the interior of the channel. Further increases in the rotation rate leads to a restabilization of the flow to a Taylor–Proudman regime. It is found that the roll-cell and Taylor–Proudman regimes lead to a substantial distortion of the axial-velocity profiles. The specific numerical results obtained are shown to be in excellent agreement with previously obtained experimental measurements and theoretical predictions.

Invariance of turbulent closure models

The invariance of second moment turbulent closure under a change of frame is examined. It is shown that the Reynolds stresses and the higher turbulence correlations based on an ensemble mean are frame‐indifferent while the Reynolds stress transport equations are frame dependent. As a result of this incompatibility, second moment closure cannot form a general foundation for the study of turbulence. Alternative approaches that are properly invariant are discussed.

Some interesting properties of two‐dimensional turbulence

The effect of superimposed rigid body motions on the structure of two‐dimensional turbulence is examined. It is found that with regard to the fluctuation dynamics of the flow, the rotational behavior of two‐dimensional turbulence is quite different from its three‐dimensional counterpart. The implications that this has on turbulence modeling are discussed briefly.

Closure models for rotating two-dimensional turbulence

Abstract The effect of rigid body rotations on the structure of turbulence correlations for incompressible two-dimensional turbulent flow is examined. It is proven that the usual turbulence correlations that are constructed from the fluctuating velocity are invariant under rigid body rotations of the fluid while those that are constructed from the fluctuating pressure are not. An explicit transformation rule for rigid body rotations is developed for an important class of turbulence correlations that are built up from the fluctuating pressure. It is shown how these results can serve as a powerful tool in the development of turbulent closure models that are suitable for the study of atmospheric turbulence in the stratosphere.

On turbulent secondary flows in pipes of noncircular cross-section

Abstract The origin of turbulent secondary flow in pipes of noncircular cross section is examined from a theoretical standpoint. It is proven mathematically that secondary flows result from a nonzero difference in the normal Reynolds stresses on planes perpendicular to the axial flow direction. Furthermore, it is shown that the K-ϵ model of turbulence has no natural mechanism for the development of secondary flow while the currently popular second-order closure models do. The implications that this has on turbulence modeling are discussed briefly.

Modeling the pressure gradient–velocity correlation of turbulence

The modeling of the pressure gradient–velocity correlation of turbulence is considered. Two distinctly different approaches have been proposed in the turbulence literature: one in which the pressure gradient–velocity correlation is decomposed into a pressure‐strain correlation and a pressure‐diffusion correlation, and another in which the pressure gradient–velocity correlation is split into its deviatoric and isotropic parts. By examining the limit of two‐dimensional turbulence, it is demonstrated that the models obtained from the former approach are inconsistent with the Navier–Stokes equations in a fundamental way, whereas the models obtained from the latter approach are not. Consequently, it appears that the direct modeling of the pressure gradient–velocity correlation in its deviatoric and isotropic parts should be favored. The implications that this result has on turbulence modeling are discussed briefly.

Subgrid scale stress models for the large-eddy simulation of rotating turbulent flows

Abstract The modeling of the subgrid scale stresses is considered from a theoretical standpoint with a view toward developing models that are more suitable for the large-eddy simulation of rotating turbulent flows. It is proven, as a rigorous consequence of the Navier-Stokes equations, that such models must be generally invariant under the extended Galilean group and must be frame-indifferent in the limit of two-dimensional turbulence which can be approached in a rapidly rotating framework. Furthermore, it is shown that a significant increase in the rotation rate must be accompanied by a substantial reduction in the energy dissipation rate of the turbulence. Vorticity subgrid scale stress models as well as several other commonly used models are shown to be in serious violation of one or more of these constraints and, hence, are not generally suitable for the description of rotating flows. Alternative models with the correct physical properties are discussed and compared.

On frame-indifference and iterative procedures in the kinetic theory of gases

Abstract The frame-dependence of higher order approximations to the response functions in the kinetic theory of gases is examined. It is shown that this frame-dependence is in no way a characteristic of the kinetic theory but results from defects in the Chapman-Enskog and Maxwellian iterative procedures. A concise proof is given to show that the kinetic theory of gases is consistent with the Principle of Material Frame-Indifference of modern continuum mechanics.

Closure relations for the pressure‐strain correlation of turbulence

The modeling of the pressure‐strain correlation of turbulence is examined. It is found that most of the closure relations that have recently been proposed are not form invariant under a change of frame while the pressure‐strain correlation can be proven to be invariant under this group of transformations. On this basis, it is highly unlikely that any of these expressions will be of general use since they ignore basic invariance requirements. The most complete representation for the pressure‐strain correlation that is properly invariant is obtained for the general class of closures where the correlation is taken to be a function of the Reynolds stresses, mean velocity gradients, and a master length scale of turbulence. Rational approximations to this form are discussed.