Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов

D Turbulent kinetic energy k + = ∇ ⋅ (μ_{t, k} ∇ k) + S _k E Turbulence dissipation ε + = ∇ ⋅ (μ_{t, ɛ} ∇ ε) + S _ε F Electric charge/potential E + 0 = ∇ ⋅ (k_e ∇ E) + 0 G Species A mf _A + = + S _A

      In Table 2, the rows D and E show the components of turbulence quantities for turbulent kinetic energy, k, and turbulence dissipation rate, ε. Glass flows are never turbulent, thanks to the stabilizing influence of glass extreme viscosity, in contrast to flows of air, gas, oxygen, combustion fumes, and other fluids that are part of glass processes.

      3.3 Turbulence

      Turbulence represents an unstable flow condition that is naturally unsteady. Nevertheless, turbulent flows are commonly treated as steady because the velocity field can be well described with time‐averaged values, and similarly for other field variables such as temperature. That is, for any given location, a field variable can be decomposed into a time‐invariant average, plus a fluctuating component. The time and length scales associated with turbulent fluctuations are small compared with the important features of a glass process, but the effects of these fluctuating quantities are profound and must accounted for in simulation models.

      There are many model formulations to account for turbulence, most having variations tuned for specific flow conditions. It is beyond the scope of this chapter to review even a few of them. The so‐called k–ε model is the most prolific, which is why its two equations are mentioned in Table 2. The source terms S_k and S_ε are related to the rate of strain of the average flow.

      (10)

      and

      (11)

      where C_1,ε and C_2,ε are model constants, and S represents the modulus of the strain rate tensor determined from the time‐averaged velocity field. (The rate of strain tensor is defined as .) Sometimes, these sources are embellished with additional terms to account for effects such as buoyancy or compressibility.

      The purpose of a turbulence model is to account for transport occurring through the turbulent eddies that are not explicitly resolved in the simulation. Such transport occurs through advection as represented by the second term on the left side of the generic transport Eq. (9), averaged over time for steady‐state problems. As mentioned above, these eddies are smaller than typical features of interest, but they are much larger than the molecular scale associated with diffusion through the first term on the right side of Eq. (9). The k–ε model accounts for turbulent transport by treating it as a diffusive phenomenon characterized by a “turbulent viscosity,” μ_t,

      (12)

where C_μ is another model constant, which adds up to the Newtonian viscosity to yield an effective viscosity, μ_eff,

      (13)

      It is usually the case that μ_t ≫ μ and μ_eff ≈ μ_t. Likewise, similar substitutions are made for the diffusion coefficients of other transport equations.

      3.4 Radiative Heat Transfer

      Heat transfer by conduction and convection is accounted for by Eq. (8) (or C in Table 2), which is also compatible with radiative heat transfer applied as a boundary condition on an opaque surface. However, glass and other media (e.g. combustion gases) are commonly semitransparent in glass processes, that is, they emit, absorb, and scatter IR radiation volumetrically.

      Radiative heat transfer significantly differs from transport by advection and diffusion so that it cannot be mathematically described by an equation of the form of Eq. (9). It is instead governed by an integrodifferential equation, known as the radiative transfer equation (RTE) [3],

(14)

where represents the radiation intensity at a position and in the beam direction ;