Stephen A. Jordan

Investigation of the cylinder separated shear-layer physics by large-eddy simulation

Abstract The transition process to turbulence occurring within the separated shear layers of a circular cylinder is investigated by the large-eddy simulation methodology. The Reynolds number ( Re =8000) is sub-critical, meaning that upstream separation is laminar. In this study, we desire to improve our understanding of the shear-layer properties as well as assess the capability of the solution methodology to accurately resolve the fundamental characteristics. In the computation, the dynamic eddy-viscosity model is implemented to handle the turbulent scales cut off by the grid-filter. However, the grid-scale level within the shear layer resolves the majority of turbulent scales of interest. The governing equations are re-cast into a curvilinear coordinate framework to accommodate a non-orthogonal grid comprised of line clustering near the cylinder periphery and within the shear-layer region. Two fundamental frequencies persist throughout the entire transition process; one identifying von Karman shedding and the other denoting the Bloor “transition wave”. Only two other mixed modes are clearly discernible. Transition begins approximately 1/4 diameters from separation and concludes about one diameter further downstream. All the characteristic trends of the shear layer, in terms of their growth rate and dependence on Re , that were established by M.F. Unal, D. Rockwell [J. Fluid Mech. 190 (1988) 491] have been verified by the present simulation to the higher Re .

Dynamic Subgrid-Scale Modeling for Large-Eddy Simulations in Complex Topologies

The dynamic eddy-viscosity relationship is a suitable choice for modeling the subgridscales (SGS) in a large-eddy simulation (LES) of complex turbulent flows in irregular domains. This algebraic relationship is easy to implement and its dynamic coefficient will give negligible turbulent viscosity contributions in the flow regions that are irrotational or laminar. Its fine-scale turbulence predictions can be qualitatively reasonable if the local grid resolution maintains the SGS field predominantly within the equilibrium range of turbulent energy spectra. This performance is given herein by two curvilinear coordinate forms of the dynamic Smagorinsky model that are formally derived and a-priori tested using the resolved physics of the cylinder wake. The conservative form evaluates the dynamic coefficient in the computational (transformed) space whereas its non-conservative counterpart operates in the physical domain. Although both forms equally captured the real normal SGS stress reasonably well, the real shear stress and dissipation rates were severely under-predicted. Mixing the eddy-viscosity choice with a scale-similarity model can ease this latter deficiency

The spatial resolution properties of composite compact finite differencing

Demand for spectral-like spatial routines to resolve fine-scale physics is easily satisfied by compact finite differencing. Commonly, the lower-order multi-parameter families at (and near) non-periodic boundaries are independently tuned to meet or exceed the high-order resolution character of the field stencil. Unfortunately, that approach quantifies a false influence of the boundary scheme on the resultant interior dispersive and dissipative consequences of the compound template. Knowing that each composite template owns three ingredients that define their numerical character, only their formal accuracy and global stability have been properly treated in a coupled fashion. The present work presents a companion means for quantifying the resultant spatial resolution properties. The procedure particularly focuses on the multi-parameter families used to diminish the dispersive and dissipative errors at the non-periodic boundaries. The process introduces a least-squares technique of the target field stencil to optimize the free parameters of the boundary scheme. Application of the optimized templates to both the linear convection and Burgers equations at a fictitious non-periodic boundary showed major reductions of the predictive error.

A Priori Assessments of Numerical Uncertainty in Large-Eddy Simulations

Current suggestions for estimating the numerical uncertainty in solutions by the Large-Eddy Simulation (LES) methodology require either a posteriori input or reflect global assessments. In most practical applications, this approach is rather costly for the user and especially time consuming due to the CPU effort needed to reach the statistical steady state. Herein, we demonstrate two alternate a priori graphical exercises. An evaluation of the numerical uncertainty uses the turbulent quantities given by the area under the wave number spectra profiles. These profiles are easily constructed along any grid line in the flow domain prior to the collection of the turbulent statistics. One exercise involves a completion of the spectrum profile beyond the cutoff wave number to the inverse of Kolmorgorov’s length scale by a model of isotropic turbulence. The other extends Richardson Extrapolation acting on multiple solutions. Sample test cases of both LES solutions and direct numerical simulations as well as published experimental data show excellent agreement between the integrated matched spectra and the respective turbulent statistics. Thus, the resultant uncertainties themselves provide a useful measure of accumulated statistical error in the resolved turbulent properties.

Resolving Turbulent Wakes

Resolving the turbulent statistics of bluff-body wakes is a challenging task. Frequently, the streamwise grid point spacing approaching the vortex exit boundary is sacrificed to gain near full resolution of the turbulent scales neighboring the body surface. This choice favors the solution strategies of direct numerical and large-eddy simulations (DNS and LES) that house spectral-like resolving characteristics with inherent dissipation. Herein, two differencing stencils are tested for approximating four forms of the convective derivative in the DNS and LES formulations for incompressible flows. The wake spectral characteristics and conventional parameters are computed for Reynolds numbers Re=200 (laminar wake) and Re=3900

A Large-Eddy Simulation of the Shear-Driven Cavity Flow Using Dynamic Modeling

A large-eddy simulation of the three-dimensional shear-driven cavity flow was carried out at a high Reynolds number where the subgrid scale turbulence was represented by a dynamic model. Lillys expression was tested for determining Smagorinskys coefficient in the model without ad hoc measures such as ensemble-averaging or filtering. However, zero cut-off of negative total viscosity was necessary to maintain stable solutions. A discretized filter function is derived for the test filter. Good qualitative and quantitative agreement with published experimental data was obtained indicating that the dynamic model performed quite well. The cavity flow at Reynolds number of 104 is transitional. The highest turbulent production levels not only occurred within the downstream eddy region of the cavity, but also along the upper half of the downstream wall.

Axisymmetric turbulent statistics of long slender circular cylinders

The experimental evidence leads us to believe that long slender circular cylinders have similar axisymmetric turbulent statistics along most of their axial length. The respective boundary layer reaches a maximum thickness (δ) with no further downstream net growth. Despite their small radius (a), these long cylinders still own high radius-based Reynolds numbers (Rea) as well as transverse curvatures (δ/a). The influence of these flow conditions (and others) on the turbulent statistics is still chiefly unknown. The present effort begins an investigation that targets axial similarity (or homogeneity) of the long thin cylinder statistics. The database is a collection of previous experimental measurements and observations as well as the present computational results by the large-eddy simulation methodology. Interestingly, this investigation shows that reaching axial homogeneity is reliant essentially on Rea with lesser influence by the transverse curvature. But the Rea value depends on the turbulent statistic ...

An iterative scheme for numerical solution of steady incompressible viscous flows

Abstract An iterative solution scheme is proposed for application to steady incompressible viscous flows in simple and complex geometries. The iterative scheme solves the vorticity-stream function form of the Navier-Stokes equations in generalized curvilinear coordinates. The flow system of equations are cast into a Newtons iterative form which are solved using the modified strongly implicit procedure. The solution scheme is benchmarked using two test cases, namely: a shear-driven steady laminar flow in a square cavity; and a simple laminar flow in a complex expanding channel. The iterative process to steady-state convergence in both test cases is highly stable and the convergence rate is without spurious oscillations. At convergence, the flow solutions are second-order accurate.

A skin friction model for axisymmetric turbulent boundary layers along long thin circular cylinders

Only a few engineering design models are presently available that adequately depict the axisymmetric skin friction (Cf) maturity along long thin turbulent cylinders. This deficit rests essentially on the experimental and numerical difficulties of measuring (or computing) the spatial evolution of the thin cylinder turbulence. Consequently, the present axisymmetric Cf models have questionable accuracy. Herein, we attempt to formulate a more robust Cf model that owns acceptable error. The formulation is founded on triple integration of the governing equation system that represents a thin cylinder turbulent boundary layer (TBL) at statistical steady-state in appropriate dimensionless units. The final model requires only the radius-based Reynolds number (Rea) and transverse curvature (δ/a) as input parameters. We tuned the accompanying coefficients empirically via an expanded statistical database (over 60 data points) that house new Cf values from large-eddy simulations (LES). The LES computations employed a t...

A simple model of axisymmetric turbulent boundary layers along long thin circular cylinders

Useful empirical and semi-empirical models of the turbulent boundary layer (TBL) and skin friction evolution along planar geometries are not applicable for axisymmetric thin cylinder flows. Their dissimilarity is readily detectable once the TBL thickness exceeds the cylinder radius (a). Although several recent empirically based axisymmetric models recognize this fact, their acceptable fidelity is either restrictive or deficient for general applicability. Herein, we correct this deficit by building a simple model for the specific canonical class of axisymmetric turbulent flows along long thin cylinders with a zero streamwise pressure gradient. Streamwise growth of the TBL thickness (δ/a), integral scales [displacement (δ*/a) and momentum thicknesses (θ/a)] and skin friction coefficient (Cf) can be estimated along the cylinder length via the respective axial mean velocity profile in wall units. This profile is given by Spaldings formula with algebraic expressions for the two input parameters (κ, κβ) that c...