Turbulence, algebraic stress models

Gatski, T. B., Speziale, C. G. On explicit algebraic stress models for complex turbulent flows. /. Fluid Mech., vol. 154, pp. 59-78, 1993. 

Abid, R., Ramsey, C., Gatski, T. Prediction of nonequilibriura turbulent flows with explicit algebraic stress models. AIAA J., vol. 33, pp. 2026-2031, 1995. 

More advanced models, for example the algebraic stress model (ASM) and the Reynolds stress model (RSM), are not based on the eddy-viscosity concept and can thus account for anisotropic turbulence thereby giving still better predictions of flows. In addition to the transport equations, however, the algebraic equations for the Reynolds stress tensor also have to be solved. These models are therefore computationally far more complex than simple closure models (Kuipers and van Swaaij, 1997). 

Zhang, J., Nieh, S. and Zhou, L. (1992). A New Version of Algebraic Stress Model for Simulating Strongly Swirling Turbulent Flows. Numerical Heat Transfer, PartB Fundamentals, 22,49. Zhou, L. (1993). Theory and Numerical Modeling of Turbulent Gas-Particle Flows and Combustion. Boca Raton, Fla. CRC Press. 

The flow pattern is calculated from conservation equations for mass and mometum, in combination with the Algebraic Stress Model (ASM) for the turbulent Reynolds stresses, using the Fluent V3.03 solver. These equations can be found in numerous textbooks and will not be reiterated here. Once the flow pattern is known, the mixing and transport of chemical species can be calculated from the following model equation ... 

Launder, B.E. (1971), An improved algebraic stress model of turbulence. Imperial College, Mechanical Engineering Department Report, TM/TN/A/11. 

The standard two-equation k-e model has been used for almost all of the simulations referred to in this chapter because it is the most tested and reliable turbulence model available. Although it will not give the amount of information that a mean Reynolds stress or an algebraic stress model will give, it requires an order of magnitude less CPU time and gives predictions of the mean velocities that are of comparable accuracy to the higher order models. 

In the algebraic stress model or ASM , the transport equations for the Reynolds stresses are rewritten as algebraic expressions by assuming that the transport of the stresses around the flow field is proportional to the transport of the turbulent kinetic energy, k. 

Turbulence modeling capability (range of models). Eddy viscosity k-1, k-e, and Reynolds stress. k-e and Algebraic stress. Reynolds stress and renormalization group theory (RNG) V. 4.2 k-e. low Reynolds No.. Algebraic stress. Reynolds stress and Reynolds flux. k- Mixing length (user subroutine) and k-e. 

Currently the widely used turbulence models are standard K-s model, RNG K-e model and the Reynolds stress model (RSM). Standard K-s model is based on isotropic turbulence model, its simulation result error of separator flow field is large (Shan Yongbo, 2005). RNG K-s model has improved with a standard K-s model, but there are still larger defects. To improve the cyclone vortex field strength prediction results a greater extent, algebraic stress turbulence model based... 

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

Computational experience has revealed that the two-equation models, employing transport equations for the velocity and length scales of the fluctuating motion, often offer the best compromise between width of application and computational economy. There are, however, certain types of flows where the k-e model fails, such as complex swirling flows, and in such situations more advanced turbulence models (ASM or RSM) are required that do not involve the eddy-viscosity concept (Launder, 1991). According to the ASM and the RSM the six components of the Reynolds stress tensor are obtained from a complete set of algebraic equations and a complete set of transport equations. These models are conceptually superior with respect to the older turbulence models such as the k-e model but computationally they are also (much) more involved. 

The Smagorinsky Model (cf. Ref. [51]) is an algebraic model in the same spirit as the Prandtl mixing length model discussed in section RANS Turbulence Modeling. In the Smagorinsky model, the SGS stresses are assumed to be proportional to the rate of strain, that is, = VtS, and the kinematic eddy viscosity is determined from the expression... 

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