Turbulence turbulent viscosity

In the k-e model of turbulence, turbulent viscosity is related to k and e by the following equation ... 

Turbulent kinetic energy dissipation rate density (m s ) = Turbulent viscosity (kg m sec )... 

The production of turbulence is maximum close to walls, where both shear rate and turbulent viscosity, ut, are high. In pipe flow, the maximum is close to y+ = 12. A proper design of a chemical reactor for efficient mixing at low Re should allow the generated turbulence to be transported with the mean flow from the region where it is produced to the bulk of the fluid where it should dissipate. 

In another class of models, pioneered by Elghobashi and Abou-Arab (1983) and Chen (1985), a particle turbulent viscosity, derived by extending the concept of turbulence from the gas phase to the solid phase, has been used. This is the so-called k—s model, where the k corresponds to the granular temperature and s is a dissipation parameter for which another conservation law is required. By coupling with the gas phase k—s turbulence model, Zhou and Huang (1990) developed a k—s model for turbulent gas-particle flows. The k—s models do not... 

It is then assumed that due to this separation in scales, the so-called subgrid scale (SGS) modeling is largely geometry independent because of the universal behavior of turbulence at the small scales. The SGS eddies are therefore more close to the ideal concept of isotropy (according to which the intensity of the fluctuations and their length scale are independent of direction) and, hence, are more susceptible to the application of Boussinesq s concept of turbulent viscosity (see page 163). 

Usually, however, the stresses are modeled with the help of a single turbulent viscosity coefficient that presumes isotropic turbulent transport. In the RANS-approach, a turbulent or eddy viscosity coefficient, vt, covers the momentum transport by the full spectrum of turbulent scales (eddies). Frisch (1995) recollects that as early as 1870 Boussinesq stressed turbulence greatly increases viscosity and proposed an expression for the eddy viscosity. The eventual set of equations runs as... 

In its turn, the turbulent viscosity may be position dependent and generally may be modeled in terms of a model, very usually a k-e model ... 

Note that the Eqs. (1), (2), and (8) are really and essentially different due to the absence or presence of different turbulent transport terms. Only by incorporating dedicated formulations for the SGS eddy viscosity can one attain that LES yield the same flow field as DNS. RANS-based simulations with their turbulent viscosity coefficient, however, essentially deliver steady flow fields and as such are never capable of delivering the same velocity fields as the inherently transient LES or DNS, irrespectively of the refinement of the computational grid ... 

With a view to any simulation, a few important items have to be addressed. First of all, it has to be decided whether the flow to be simulated is 2-D, 21/2-D, or 3-D. When the flow is, e.g., axis-symmetrical and steady, a 2-D simulation may suffice. For a flow field in which all variables, including the azimuthal velocity component, may not depend on the azimuthal coordinate, a 21/2-D simulation may be most appropriate. Most other cases may require a full 3-D simulation. It is tempting to reduce the computational job by casting the 3-D flow field into a 2-D mode. The experience, however, is that in 2-D simulations the turbulent viscosity tends to be overestimated in this way, the flow... 

The left-hand sides of Eqs. (25)-(29) have the same form as Eq. (5) and represent accumulation and convection. The terms on the right-hand side can be divided into spatial transport due to diffusion and source terms. The diffusion terms have a molecular component (i.e., /i and D), and turbulent components. We should note here that the turbulence models used in Eqs. (26) and (27) do not contain corrections for low Reynolds numbers and, hence, the molecular-diffusion components will be negligible when the model is applied to high-Reynolds-number flows. The turbulent viscosity is defined using a closure such as... 

The k-s turbulence model was developed and described by Launder and Spalding (1972). The turbulent viscosity, pt, is defined in terms of the turbulent kinetic energy, k, and its rate of dissipation, z. 

In a system with both heat and mass transfer, an extra turbulent factor, kx, is included which is derived from an adapted energy equation, as were e and k. The turbulent heat transfer is dictated by turbulent viscosity, pt, and the turbulent Prandtl number, Prt. Other effects that can be included in the turbulent model are buoyancy and compressibility. 

The Reynolds-averaged approach is widely used for engineering calculations, and typically includes models such as Spalart-Allmaras, k-e and its variants, k-co, and the Reynolds stress model (RSM). The Boussinesq hypothesis, which assumes pt to be an isotropic scalar quantity, is used in the Spalart-Allmaras model, the k-s models, and the k-co models. The advantage of this approach is the relatively low computational cost associated with the computation of the turbulent viscosity, fit. For the Spalart-Allmaras model, one additional transport equation representing turbulent viscosity is solved. In the case of the k-e and k-co models, two additional transport equations for the turbulence kinetic energy, k, and either the turbulence dissipation rate, s, or the specific dissipation rate, co, are solved, and pt is computed as a function of k and either e or co. Alternatively, in the RSM approach, transport equations can be solved for each of the terms in the Reynolds stress tensor. An additional scale-determining equation (usually for s) is also required. This means that seven additional transport equations must be solved in 3D flows. 

Using the definition for the turbulent viscosity (jit — /An /xmoi), which gives a result similar to the standard k-s model with only a small difference in the modeling constant, the effective viscosity is now defined as a function of k and s in Eq. (16) in algebraic form. 

The RNG model provides its own energy balance, which is based on the energy balance of the standard k-e model with similar changes as for the k and e balances. The RNG k-e model energy balance is defined as a transport equation for enthalpy. There are four contributions to the total change in enthalpy the temperature gradient, the total pressure differential, the internal stress, and the source term, including contributions from reaction, etc. In the traditional turbulent heat transfer model, the Prandtl number is fixed and user-defined the RNG model treats it as a variable dependent on the turbulent viscosity. It was found experimentally that the turbulent Prandtl number is indeed a function of the molecular Prandtl number and the viscosity (Kays, 1994). 

RANS turbulence models are the workhorse of CFD applications for complex flow geometries. Moreover, due to the relatively high cost of LES, this situation is not expected to change in the near future. For turbulent reacting flows, the additional cost of dealing with complex chemistry will extend the life of RANS models even further. For this reason, the chemical-source-term closures discussed in Chapter 5 have all been formulated with RANS turbulence models in mind. The focus of this section will thus be on RANS turbulence models based on the turbulent viscosity hypothesis and on second-order models for the Reynolds stresses. 

In order to use this equation for CFD simulations, the unclosed term involving the Reynolds stresses ((u, u j)) must be modeled. Turbulent-viscosity-based models rely on the following... 

The next level of turbulence models introduces a transport equation to describe the variation of the turbulent viscosity throughout the flow domain. The simplest models in this category are the so-called one-equation models wherein the turbulent viscosity is modeled by... 

The last term on the right-hand side is unclosed and represents scalar transport due to velocity fluctuations. The turbulent scalar flux ( , varies on length scales on the order of the turbulence integral scales Lu, and hence is independent of molecular properties (i.e., v and T).17 In a CFD calculation, this implies that the grid size needed to resolve (4.70) must be proportional to the integral scale, and not the Batchelor scale as required in DNS. In this section, we look at two types of models for the scalar flux. The first is an extension of turbulent-viscosity-based models to describe the scalar field, while the second is a second-order model that is used in conjunction with Reynolds-stress models. 

The turbulent diffusivity defined by (4.74) is proportional to the turbulent viscosity defined by (4.46). Turbulent-diffusivity-based models for the scalar flux extend this idea to arbitrary mean scalar gradients. The standard gradient-diffusion model has the form... 

Other terms which can be defined quantitatively are introduced in the following sections. Some other terms, such as turbulence, viscosity, and diffusivity are used without definition. For a full explanation of these terms, we refer the reader to standard texts in fluid meehanies, heat transfer, and mass transfer. 

Prandtl s mixing length hypothesis (Prandtl, 1925) was developed for momentum transport, instead of mass transport. The end result was a turbulent viscosity, instead of a turbulent diffusivity. However, because both turbulent viscosity and turbulent diffusion coefficient are properties of the flow field, they are related. Turbulent viscosity describes the transport of momentum by turbulence, and turbulent diffusivity describes the transport of mass by the same turbulence. Thus, turbulent viscosity is often related to turbulent diffusivity as... 

The Reynolds stress model requires the solution of transport equations for each of the Reynolds stress components as well as for dissipation transport without the necessity to calculate an isotropic turbulent viscosity field. The Reynolds stress turbulence model yield an accurate prediction on swirl flow pattern, axial velocity, tangential velocity and pressure drop on cyclone simulation [7,6,13,10],... 

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