The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

A hybrid RANS-LES approach with delayed-DES and wall-modelled LES capabilities

Strategies for turbulence modelling and simulations

The numerical simulation of high Reynolds number flows is hampered by model accuracy if the Reynolds-averaged Navier–Stokes (RANS) equations are used, and by computational cost if direct or large-eddy simulations (LES) that resolve the near-wall layer are employed. The cost of a calculation scales like the Reynolds number to the power 3 for direct numerical simulations, or 2.4 for LES, making the resolution of the wall layer at high Reynolds number infeasible even with the most advanced computers. In LES, an attractive alternative to compute high-Re flows is the use of wall-layer models, in which only the outer layer is resolved, while the near-wall region is modeled. Three broad classes of approaches are presently used: bypassing this region altogether using wall functions, solving a separate set of equations in the near-wall region, weakly coupled to the outer flow, or simulating the near-wall region in a global, Reynolds-averaged, sense. These approaches are discussed and their ranges of applicability are highlighted. Various unresolved issues in wall-layer modeling are presented.

Wall-layer models for large-eddy simulations

The Structure of Turbulent Shear Flow By Dr. A. A. Townsend. Pp. xii + 315. 8¾ in. × 5½ in. (Cambridge: At the University Press.) 40s.

/pdf/the-structure-of-turbulent-shear-flow-2yf3szhaqb.pdf

The Structure of Turbulent Shear Flow

Calculations of unsteady 2D flow around a square cylinder at incidence (α=0°−45°) are presented. The Reynolds numbers are low (Re=45–200) so that the flow is presumably laminar. A von Karman vortex sheet is predicted behind the cylinders with a periodicity which agrees well with experiments. An incompressible SIMPLEC code is used with a non-staggered grid arrangement. A third-order QUICK scheme is used for the convective terms. The time discretization is implicit and a second-order Crank–Nicolson scheme is employed. At the outlet of the computational domain a convective Sommerfeld boundary condition is compared with a traditional Neumann condition. The convective boundary condition is shown to be more effective in reducing the CPU time, reducing the upstream influence of the outlet and thus reducing the necessary downstream extent of the domain. A study of the effects of spatial resolution and blockage is also provided. The onset of vortex shedding is investigated by using the Stuart–Landau equation at various angles of incidence and for a solid blockage of 5%. A number of quantities such as Strouhal number and drag, lift and moment coefficients are calculated. © 1998 John Wiley & Sons, Ltd.

Low-Reynolds-number flow around a square cylinder at incidence: study of blockage, onset of vortex shedding and outlet boundary condition

Direct numerical simulations of two-dimensional (2D) and 3-D unsteady flow around a square cylinder for moderate Reynolds numbers (Re=150–500) are performed, employing an implicit fractional step method finite-volume code with second-order accuracy in space and time. The simulations, which are carried out with a blockage ratio of 5.6%, indicate a transition from 2-D to 3-D shedding flow between Re=150 and Re=200. Both spanwise instability modes, A and B, are present in the wake transitional process, similar to the flow around a circular cylinder. However, seemingly in contrast to a circular cylinder, the transitional flow around a square cylinder exhibits a phenomenon of distinct low-frequency force pulsations (Re=200–300). For 3-D simulations, the Strouhal number and the mean drag coefficient are in general agreement with existing experiments. Between Re=300 and 500, an increase in the spanwise coupling of fluctuating forces is indicated. The influence of the spanwise aspect ratio using periodic boundary conditions, a finer grid, and a finer time step is also investigated.

Simulation of three-dimensional flow around a square cylinder at moderate Reynolds numbers

Flows over two tandem cylinders were analysed using the newly developed collocated unstructured computational fluid dynamics (CUCFD) code, which is capable of handling complex geometries. A Reynolds number of 100, based on cylinder diameter, was used to ensure that the flow remained laminar. The validity of the code was tested through comparisons with benchmark solutions for flow in a lid-friven cavity and flow around a single cylinder. For the tandem cylinder flow, also mesh convergence was demonstrated, to within a couple of percent for the RMS lift coefficient. The mean and fluctuating lift and drag coefficients were recorded for centre-to-centre cylinder spacings between 2 and 10 diameters. A critical cylinder spacing was found between 3.75 and 4 diameters. The fluctuating forces jumped appreciably at the critical spacing. It was found that there exists only one reattachment and one separation point on the downstream cylinder for spacings greater than the critical spacing. The mean and the fluctuating surface pressure distributions were compared as a function of the cylinder spacing. The mean and the fluctuating pressures were significantly different between the upstream and the downstream cylinders. These pressures also differed with the cylinder spacing.

Numerical predictions of low Reynolds number flows over two tandem circular cylinders

http://www.tfd.chalmers.se/~lada/postscript_files/paper-resolution-IJHFF.pdf

Large Eddy Simulations: How to evaluate resolution

Large eddy simulation of flow past a rigid prism of a square cross section with one side facing the oncoming flow at Re=2.2×10 4 is performed. An incompressible code is used employing an implicit fractional step method finite volume with second-order accuracy in space and time. Three different subgrid scale models: the Smagorinsky, the standard dynamic, and a dynamic one-equation model, are applied. The influence of finer grid, shorter time step, and larger computational spanwise dimension is investigated. Some global quantities, such as the Strouhal number and the mean and rms values of lift and drag, are computed. A scheme for correcting the global results for blockage effects is presented

Lars Davidson

Papers

Low-Reynolds-number flow around a square cylinder at incidence: study of blockage, onset of vortex shedding and outlet boundary condition

Simulation of three-dimensional flow around a square cylinder at moderate Reynolds numbers

Numerical predictions of low Reynolds number flows over two tandem circular cylinders

Large Eddy Simulations: How to evaluate resolution

Large Eddy Simulation of Flow Past a Square Cylinder: Comparison of Different Subgrid Scale Models