Algebraic stress models

Almost all modern CFD codes have a k - model. Advanced models like algebraic stress models or Reynolds stress model are provided FLUENT, PHOENICS and FLOW3D. Table 10-3 summarizes the capabilities of some widely used commercial CFD codes. Other commercially CFD codes can be readily assessed on the web from hptt//www.cfd-online.com This is largest CFD site on the net that provides various facilities such as a comprehensive link section and discussion forum. 

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). 

A cure against these longer CPU times is the Algebraic Stress Model (ASM) described by, e.g., Rodi (1984) and used and recommended by, e.g., Bakker (1992) and Bakker (1996). Most commercial codes do no longer support an ASM. 

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 ... 

The precise form of the function varies depending on the approximated forms of the dissipative and redistributive terms and how the convective and diffusive transport terms are eliminated. Launder (1971) neglected the latter terms entirely while Rodi (1984) assumed that the convective transport is proportional to the transport of k with an equivalent assumption for the diffusion term. Rodi s algebraic stress model can be written ... 

Wherever the convective and diffusive transport of Reynolds stresses are important (flows that are far from equilibrium), algebraic stress models may prove to be inadequate, and solution of the full transport equations of the Reynolds stresses may become necessary. Extensive efforts have been made to model the terms appearing in the exact transport equations of Reynolds stresses. The redistributive term has been the... 

Algebraic Stress Models (ASM) Accounts for anisotropy Combines generality of approach with the economy of the k-s model Good performance for isothermal and buoyant thin shear layers Restricted to flows where convection and diffusion terms are negligible Performs as poorly as k-e in some flows due to problems with s equation Not widely validated... 

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. 

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