CENTRO HISTÓRICO
3.2.2. Intervenciones regulares
For turbulence closure several turbulence models exist in CFD which can be employed to perform this task, some of which will be discussed in this section of the report. The process of Reynolds averaging to turbulence modelling requires that the Reynolds stresses −𝜌〈𝑢𝑖𝑢𝑗〉 "which is a second order tensor that represents a second-order
moment of the velocity components at a single point in space” are appropriately modelled to obtain the desired solution (Andersson et al., 2011). The individual Reynolds stresses in the stress tensor has 9 components but since the Reynolds stress tensor is symmetric there are 3 normal stresses and 3 shear tensor stresses which means that the Reynolds tensors contains 6 unknowns that must be modelled. An ideal way to close the Reynolds Averaged Navier Equation would be to derive a transport equation for each of the Reynolds stresses, but this would result in third-order moments of velocity components requiring the derivation of equations for the third order moments would result in a fourth order moments; this goes on indefinitely and is regarded as the closure problem.
One of the common methods use to achieve closure is to use the Boussinesq hypothesis based on the assumption that the components of the Reynolds stress tensor −𝜌〈𝑢𝑖𝑢𝑗〉 are proportional to the gradient of the mean velocity
(Andersson et al., 2011).The Boussinesq hypothesis proposes that the transport of momentum by turbulence is a diffusive process and that the Reynolds stresses can be modelled by using a turbulence eddy viscosity; this is known as the eddy viscosity model (Fluent, 2012).
𝜏𝑖𝑗= −𝜌〈𝑢𝑖𝑢𝑗〉 = 𝑣𝑇( 𝜕〈𝑈𝑖〉 𝜕𝑥𝑗 +𝜕〈𝑈𝑗〉 𝜕𝑥𝑖 ) −2 3𝑘𝛿𝑖𝑗 (18)
Where 𝑘 is the turbulence kinetic energy per unit mass and it is defined as the half trace of the Reynolds stress tensor; 𝑘 =1
2〈𝑢𝑖𝑢𝑖〉; 𝑣𝑇 is the eddy viscosity, which together with 𝑘 need to achieve closure. This is achieved via
turbulence models such as the Mixing length models, Spalart-Allmaras, 𝑘 − 𝜀 and 𝑘 − 𝜔 and other eddy viscosity models. This approach has the advantage of reducing the computational cost which is related to determining the
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eddy viscosity 𝑣𝑇 via the turbulence models. The turbulent-viscosity models are based on characteristics velocity,𝑢, and length scales, 𝑙 describing the local turbulent viscosity 𝑣𝑇. An equation to show the relationship between
the characteristic velocity, length scales and local turbulent viscosity is given below (Kinetic theory of gas):
𝑣𝑇 = 𝐶𝑣𝑢𝑙 (19)
Where 𝐶𝑣 𝑖𝑠 a constant of proportionality; this equation makes sense since these scales are responsible for most
of the turbulent transport. Therefore, all turbulence models based on the Eddy viscosity concept contain sets of additional equations required to determine the velocity and length scales to describe the eddy viscosity (local turbulence). For instance, in One equation turbulence model only one additional transport equation (Partial Differential Equation) is solved to determines the characteristic velocity, 𝑢, and, the length scale, most then be specified algebraically in exception of Spalart Allmaras model where the partial differential equation is solved to determine the turbulence eddy viscosity. For the 𝑘 ω and 𝑘 𝜀 models two additional transport equations are solved, one for the turbulence kinetic energy 𝑘, and the other for the turbulence dissipation rate 𝜀 or the specific dissipation rate known as ω are solved and the turbulent viscosity is solved for, as a function of 𝑘 𝑎𝑛𝑑 ω or as 𝑘 𝑎𝑛𝑑 𝜀. These are related to the turbulent viscosity mathematically by (Andersson et al., 2011):
𝑣𝑇= 𝐶𝑣𝑢𝑙 = 𝐶𝑣 𝑘2 𝜀 Since 𝜀 = 𝑘 3 2 𝑙 for the 𝑘 𝜀 𝑚𝑜𝑑𝑒𝑙 (20) and 𝑣𝑇 = 𝐶𝑣𝑢𝑙 = 𝐶𝑣( 𝑘 𝜔) 1/2
For the 𝑘 ω model. (21)
Another approach which could be used to model the stress tensor and used to achieve closure to the RANS equations is the Reynolds Stress Models also known as the second moment closure models (Andersson et al., 2011). These models account for anisotropy due to swirling motion, streamline curves, rapid change in strain rate etc. but are computational expensive. The Reynolds Stress Models solve the transport equation for all components of the specific Reynolds Stress tensor. In addition to the Reynolds Stress Model an extra equation is needed for the turbulence dissipation rate Ɛ, which means that seven additional equations must be solved for 3-D problems, this is because for 3-dimensions the Reynolds stress tensor is symmetric so that only six of the nine components