The turbulent channel flow with streamwise rotation has been investigated by means of several different analytical, numerical, and modelling approaches. Lie group analysis of the two-point correlation equations led to linear scaling laws for the streamwise mean velocity. In addition it was found that a cross-flow in the spanwise direction is induced, which may also exhibit a linear region. By further analysis of the two-point correlation equation, it is shown that all six components of the Reynolds stress tensor are non-zero. In addition certain symmetries and skew-symmetries about the centreline have been established for all flow quantities. All these findings of the analysis have been verified very well by means of direct numerical simulations (DNS). The flow has also been calculated with large-eddy simulations (LES) and second-moment closure models. The dynamic LES captured most of the theoretical and DNS findings quantitatively. Except for one stress component the second-moment closure model was able to capture most of the basic trends, but no quantitative agreement could be achieved.

Turbulent convection may have played a major role in determining the structure and evolution of the primordial solar nebula, but current, incomplete models of convection and turbulence give very different results and remain largely untested in the absence of detailed astronomical observations. Numerical simulations provide an 'experimental' database for comparison with these models, and, to this end, direct numerical simulations of turbulent convection were performed with modifications intended to mimic some of the unique physical features of thin accretion disks, such as the primordial solar nebula: internal heating, a gravitational acceleration that is linearly proportional to the distance from midplane of the nebula, and rapid rotation. Peclet numbers in the simulations are comparable to those in solar nebula models; Rossby numbers in the simulations are an order of magnitude larger than those in solar nebula models because of the unrealistically high Prandtl and low, Reynolds numbers required to resolve all scales of the convective flow. We find that, despite the loss of buoyancy at midplane, turbulent motions easily penetrate the midplane region with little loss of intensity, providing efficient transport of heat and turbulent kinetic energy throughout the interior. A simple mixing length model modified to include rotation is found to give convective heat fluxes for the interior flow in rough agreement with the numerical simulations. We discuss these preliminary results with regard to assumptions about heating distributions and convective heat fluxes made in standard solar nebula models. More definitive comparisons with solar nebula modelling will become possible when more realistic effects of shear, density, stratification, and compression are included.

The cost of large eddy simulation (LES) in the near-wall region of attached turbulent boundary layers scales as the square of the friction Reynolds number, thus limiting LES to moderate Reynolds numbers. Wall stress boundary conditions are frequently used to alleviate this resolution requirement, but commonly used models are shown to perform poorly at high Reynolds numbers even in turbulent channel flow. Techniques from optimal control theory are used to find wall stresses that yield much better results in turbulent channel flow at high Reynolds numbers than existing models even on extremely coarse grids. In this approach, a suboptimal control strategy is used in which the objective is to force the outer LES towards a desired solution by using the wall stress boundary conditions as control. The suboptimal wall stresses are not necessarily physical, rather they are whatever is necessary to overcome the numerical and modeling errors present in the near-wall region to yield the correct mean velocity profile. Furthermore, the suboptimal control strategy generates reference data for comparing and deriving new wall models. Using linear stochastic estimation it is shown that the dynamically relevant part of the suboptimal wall stresses can be predicted from the local velocity field. A wall model derived from linear stochastic estimation yields good mean flow predictions in LES of turbulent channel flow on a 323 uniform grid for friction velocity Reynolds numbers from 640 to 20 000.

Many high Reynolds number, wall-bounded turbulent flow applications are still too expensive to compute with well resolved large-eddy simulation (LES), which requires resolutions nearly as fine as direct numerical simulation (DNS) near walls. Baggett et al. (1997) estimated that well resolved LES of channel flow would require a number of grid points scaling as Reτ , where Reτ is the friction Reynolds number, based on the requirement that the mesh capture most of the energy-containing scales, which become very small approaching the walls. Thus high Reynolds number flows, especially in complex geometries, become exorbitantly expensive to compute. Approximate wall boundary conditions are needed to avoid computing the finescale, near-wall regions and to allow the LES mesh to be determined solely by large outer flow scales and geometry. In many engineering applications involving turbulence in complex geometries, low-order numerical differencing schemes are often employed because they are easy to implement and, for central differencing schemes, have good conservation properties. The subgrid-scale (SGS) model and the wall model must also be fairly simple to implement and computationally inexpensive if they are to gain widespread acceptance. It is therefore of practical interest to examine the performance of fairly simple wall models applied to the standard second-order central finite difference codes employed in many codes at CTR, despite the fact that the numerical errors seriously degrade the effective resolution of the simulation and the accuracy of the SGS model (Lund & Kaltenbach 1995, Ghosal 1996, Kravchenko & Moin 1997). When wall stress models are used in conjunction with very coarsely meshed LES of channel flow, the near-wall points in the outer LES flow are observed to be poorly predicted (e.g., Nicoud et al. 1999, Nicoud & Baggett in this volume), with the mean flow moving too rapidly with respect to the core flow. In the channel this means that the skin friction is overpredicted for a given bulk mass flux. This problem also occurs in the attached boundary layers and can lead to poor predictions of separation in flows experiencing adverse pressure gradients farther downstream. Another anomalous feature of these coarse simulations is the lack of any significant “wake” region in the core of the flow, which is observed in experiments and in well resolved DNS and LES.

The near-wall regions of high Reynolds numbers turbulent flows must be modelled to treat many practical engineering and aeronautical applications. In this review we examine results from simulations of both attached and separated flows on coarse grids in which the near-wall regions are not resolved and are instead represented by approximate wall boundary conditions. The simulations use the dynamic Smagorinsky subgrid-scale model and a second-order finite-difference method. Typical results are found to be mixed, with acceptable results found in many cases in the core of the flow far from the walls, provided there is adequate numerical resolution, but with poorer results generally found near the wall. Deficiencies in this approach are caused in part by both inaccuracies in subgrid-scale modelling and numerical errors in the low-order finite-difference method on coarse near-wall grids, which should be taken into account when constructing models and performing large-eddy simulation on coarse grids. A promising new method for developing wall models from optimal control theory is also discussed.

Artificial boundary conditions for LES away from the wall have been developed with the hope of avoiding the problem of grid refinement in the wall region of the LES. In the particular example of channel flow, the main idea is to replace the natural no-slip boundary conditions (at y = 0) by artificial boundary conditions at y = y1 > 0. The one-point statistics (mean velocity and turbulence intensities) of the flow at y1 are supposed to be provided externally. In practice, this information could be obtained from a RANS for the same flow. However, it is known that supplying only the one-point statistics of the velocity field is not sufficient for obtaining a reasonable core flow. The method developed here consists of building two-point statistics at the artificial boundary by using information from the core flow at y = y2 > y1. In particular, the time evolution of the velocity fields at y = y1 and y = y2 are assumed to be self-similar with a time scale ratio determined dynamically during the simulation. Encouraging results for the channel flow at Reτ = 1000 have been obtained when the domain removed from the simulation (0 < y < y1) contains half of the grid points used in “full domain” LES of the channel flow.

Three-dimensional, fully compressible hydrodynamic simulations of thermal convection in an idealized circumstellar disk environment have been performed, featuring a linear vertical gravity, Keplerian rotation, and an imposed internal heat source. The simulations use an unrealistically low Reynolds number in order to resolve all relevant scales of motion. The rapid rotational shear is found to remove all azimuthal variations. The resulting two-dimensional structure causes the large scales to extract kinetic energy from small scales and causes angular momentum to flow radially inward, opposite to standard convective disk models. These results imply that one cannot have quasi-steady, self-sustaining disk convection. More realistic simulations at much higher Reynolds numbers are needed to confirm these results.

A general discussion on the structure of the eddy viscosity tensor in anisotropic flows is presented. The systematic use of tensor symmetries and flow symmetries is shown to reduce drastically the number of independent parameters needed to describe the rank 4 eddy viscosity tensor. The possibility of using Onsager symmetries for simplifying further the eddy viscosity is discussed explicitly for the axisymmetric geometry.

Numerical hydrodynamic simulations of three-dimensional, turbulent, fully compressible thermal convection have been performed for an internally heated layer with varying degrees of density and temperature stratification as a preliminary study of realistic convection in the primordial solar nebula and other protostellar disks. Gravity is made to vary linearly as distance from the midplane, approximating the conditions in thin, gaseous protostellar disks. These simulations have been performed in a periodic channel at unrealistically low Reynolds numbers in order to resolve all relevant scales of turbulent motion; and differential rotation, another important feature of protostellar disks, has not been included. In this paper we describe the numerical techniques used to perform the hydrodynamic simulations; we examine the convective now structure and turbulence statistics; and we discuss the effects of compressibility and stratification on the turbulent convection.The turbulence (rms) Mach numbers M(t) in the central convective regions are found to be less than or similar to 0.25, and acoustic terms, scaling as M(t)(2), are found to be negligible except near (unrealistic) solid boundaries. Near these walls, there is enhanced compression and rapid tangential flow, which sometimes becomes supersonic and exhibits weak shocks. Temperature deviations of less than 25% from the mean in horizontal planes are typically observed. Density stratification is shown to have significant effects on the convection, primarily due to increased thermal diffusivity in the outer, more rarefied regions, which (1) reduces the efficiency of convective heat transport, and (2) can stabilize the gas against convection in the outermost regions. Some simulation results are presented in which the convection is realistically bounded by convectively stable regions that effectively buffer the interior convective flow from the physically unrealistic boundaries. Significant overshooting of convective motions into the stable region is observed, but, because of the inefficient nature of this convection, little penetration of convective layer into the stable layer is observed. The stable layer acts as a ''soft wall'' with less enhancement of compression and horizontal velocities, making supersonic flow and shocks unlikely in turbulent now driven solely by convection in protostellar disks.

The primary goal of this work has been to assess the performance of the dynamic SGS model in the large eddy simulation (LES) of channel flows in a variety of situations, viz., in temporal development of channel flow turned by a transverse pressure gradient and especially in buoyancy-driven turbulent flows such as Rayleigh-Benard and internally heated channel convection. For buoyancy-driven flows, there are additional buoyant terms that are possible in the base models, and one objective has been to determine if the dynamic SGS model results are sensitive to such terms. The ultimate goal is to determine the minimal base model needed in the dynamic SGS model to provide accurate results in flows with more complicated physical features. In addition, a program of direct numerical simulation (DNS) of fully compressible channel convection has been undertaken to determine stratification and compressibility effects. These simulations are intended to provide a comparative base for performing the LES of compressible (or highly stratified, pseudo-compressible) convection at high Reynolds number in the future.

Direct numerical simulations of incompressible channel flow have been performed that explore the effects of centrifugally stable differential rotation on thermal convection (with gravity and rotation axes aligned). In order to provide greater correspondence to the interior regions of astrophysical accretion disks, especially to the convective solar nebula, we consider a gravity that varies linearly with distance from midplane and Keplerian rotation. We are restricted, however, to unrealistically low Reynolds numbers. Our findings are: (1) Statistical thermal convective properties depend almost exclusively on Peclet number and epicyclic frequency, regardless of the anisotropy induced by the shear. (2) At low Reynolds numbers, Reynolds stresses show the remarkable behavior of changing sign with increasing rotation rate, going from positive to negative shear production rates. Higher-Reynolds-number simulations tend to retain positive shear production rates to more rapid rotation. (3) At very rapid rotation, independent of Reynolds number, the flow becomes quasi-two-dimensional by losing streamwise variation in one or more of its fluctuating variables (especially the vertical velocity). At this point the simulation results become unreliable. These results suggest that convection in accretion disks is characterized by very long azimuthal wavelengths, and that, in some circumstances, Reynolds stresses can feed turbulence kinetic energy to the mean flow in contradiction to the conventional eddy-viscosity ansatz.

The dynamic subgrid-scale (SGS) model of Germano et al. [Phys. Fluids A 3, 1760 (1991)] is generalized for the large eddy simulation (LES) of compressible flows and transport of a scalar. The model was applied to the LES of decaying isotropic turbulence, and the results are in excellent agreement with experimental data and direct numerical simulations. The expression for the SGS turbulent Prandtl number was evaluated using direct numerical simulation (DNS) data in isotropic turbulence, homogeneous shear flow, and turbulent channel flow. The qualitative behavior of the model for turbulent Prandtl number and its dependence on molecular Prandtl number, direction of scalar gradient, and distance from the wall are in accordance with the total turbulent Prandtl number from the DNS data.

A series of tests were performed to help extend the use of a subgrid-scale model to compressible and wall-bounded flows. A priori tests were done in the case of the incompressible turbulent channel flow. They showed that a 1-D formulation of the structure-function model is more appropriate, leading to a satisfactory behavior of the model at the walls without requiring any damping function. This model is consistent with the original formulation of Metais & Lesieur (1990). In large-eddy simulations of compressible isotropic turbulence, both models performed well up to an initial rms Mach number of 0.6.

Differentially rotating disks of gases and solids occur in several astrophysical systems, in particular in the inner parts of protostellar nebulae, of which our own solar system is thought to be a relic. The objectives of this paper are to: (1) study localized turbulence in circumstances approximating those found in accretion disks using previously existing expertise in performing direct numerical simulation of turbulent, incompressible channel flows with low Reynolds numbers; (2) determine the limitations of such calculations; and (3) extend the type of numerical simulation (e.g., to include density and stratification and compressibility effects and to accommodate higher Reynolds numbers with sub-grid scale modeling) so that the relevant physical effects are realistically captured.

Abstract Turbulent convection may have played a major role in determining the structure and evolution of the primordial solar nebula, but current, incomplete models of convection and turbulence give very different results and remain largely untested in the absence of detailed astronomical observations. Numerical simulations provide an “experimental” database for comparison with these models, and, to this end, direct numerical simulations of turbulent convection were performed with modifications intended to mimic some of the unique physical features of thin accretion disks, such as the primordial solar nebula: internal heating, a gravitational acceleration that is linearly proportional to the distance from midplane of the nebula, and rapid rotation. Peclet numbers in the simulations are comparable to those in solar nebula models; Rossby numbers in the simulations are an order of magnitude larger than those in solar nebula models because of the unrealistically high Prandtl and low Reynolds numbers required ...