NOMENCLATURE English B k production of turbulent kinetic energy by buoyancy B ε production of turbulent dissipation by buoyancy C f friction coefficient D k destruction of turbulent kinetic energy D ε destruction of turbulent dissipation k specific turbulent kinetic energy l mfp mean-free path length l mix mixing length p pressure p* modified pressure P k production of turbulent kinetic energy P ε production of turbulent dissipation q i Reynolds flux S source term S ij mean strain rate tensor t time u velocity u + dimensionless velocity (wall variables) u * friction velocity x horizontal displacement (displacement in the streamwise direction) y + dimensionless vertical distance (wall variables) Greek Symbols δ boundary layer thickness δ v * displacement thickness δ ij Kroenecker delta function ε dissipation rate of turbulent kinetic energy φ generic scalar variable φ' fluctuating component of time-averaged variable φ Φ mean component of time-averaged variable φ φ Reynolds time averaged variable φ Γ diffusion coefficient Γ t turbulent diffusivity κ von-Karman constant µ molecular viscosity v ν t eddy viscosity Π ij pressure-strain correlation tensor ρ density σ turbulent Prandtl-Schmidt number σ k Prandtl-Schmidt number for k τ ij Reynolds stress tensor τ w wall shear stress ω dissipation per unit turbulent kinetic energy (specific dissipation) Ω ij mean rotation tensor 1 1.0 INTRODUCTION Theoretical analysis and prediction of turbulence has been, and to this date still is, the fundamental problem of fluid dynamics, particularly of computational fluid dynamics (CFD). The major difficulty arises from the random or chaotic nature of turbulence phenomena. Because of this unpredictability, it has been customary to work with the time averaged forms of the governing equations, which inevitably results in terms involving higher order correlations of fluctuating quantities of flow variables. The semi-empirical mathematical models introduced for calculation of these unknown correlations form the basis for turbulence modeling. It is the focus of the present study to investigate the main principles of turbulence modeling, including examination of the physics of turbulence, closure models, and application to specific flow conditions. Since turbulent flow calculations usually involve CFD, special emphasis is given to this topic throughout this study. There are three key elements involved in CFD: (1) grid generation (2) algorithm development (3) turbulence modeling While for the first two elements precise mathematical theories exist, the concept of turbulence modeling is far less precise due to the complex nature of turbulent …

Introductory Turbulence Modeling

A method has been developed for the application of wall functions to generalized curvilinear coordinate systems with nonorthogonal grids. Two test cases have been computed using this method with the k-e turbulence model: flow over a flat plate at 0-deg angle of attack using a nonorthogonal grid at the wall and flow over a prolate hemispheroid with a hemispherical nose cap at 0-deg angle of attack. All results are compared with experimental data. In addition, the hemispheroid results are compared with computations using the Baldwin-Lomax algebraic turbulence model and the Chien low-Reynolds-number k-e turbulence model.

Application of wall functions to generalized nonorthogonal curvilinear coordinate systems

A near wall turbulence model and its incorporation into a multiple-time-scale turbulence model are presented. In the method, the conservation of mass, momentum, and the turbulent kinetic energy equations are integrated up to the wall; and the energy transfer rate and the dissipation rate inside the near wall layer are obtained from algebraic equations. The algebraic equations for the energy transfer rate and the dissipation rate inside the near wall layer were obtained from a k-equation turbulence model and the near wall analysis. A fully developed turbulent channel flow and fully developed turbulent pipe flows were solved using a finite element method to test the predictive capability of the turbulence model. The computational results compared favorably with experimental data. It is also shown that the present turbulence model could resolve the over shoot phenomena of the turbulent kinetic energy and the dissipation rate in the region very close to the wall.

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Near-wall turbulence model and its application to fully developed turbulent channel and pipe flows

A second-moment Reynolds Stress Transport Model (RSTM) is used in the present work for computing the flow around a two-dimensional airfoil. An incompressible SIMPLEC code is used, employing a non-staggered grid arrangement. A third-order QUICK scheme is used for the momentum equations, and a second-order, bounded MUSCL scheme is used for the turbulent quantities. As the RSTM is valid only for fully turbulent flow, an eddy viscosity, one-equation model is used near the wall. The two models are matched along a preselected grid line in the fully turbulent region. Detailed comparisons between calculations and experiments are presented for an angle of attack of α = 13.3 deg. The RSTM predictions agree well with the experiments, and approaching stall is predicted for α = 17 deg, which agrees well with experimental data. The results obtained with a two-layer k − e model show poor agreement with experimental data; the velocity profiles on the suction side of the airfoil show no tendency of separation, and no tendency of stall is predicted

Prediction of the Flow Around an Airfoil Using a Reynolds Stress Transport Model

A one-equation turbulence model has been developed from available experimental observations on both attached and separated turbulence flows, and numerical results are presented for flows about NACA 0012 and RAE 2822 airfoils. The model is shown to duplicate the accurate results of the Baldwin-Lomax (1978) algebraic model for the case of attached flow. For cases of separated flow, the model is found to predict shock location and strength closer to the experimentally observed values than both the algebraic and the q-omega two-equation models.

V. C. Patel

Papers

Practical near-wall turbulence models for complex flows including separation