Low Reynolds number turbulence model

There are two main approaches to modeling the near-wall region. In one approach, the so-called wall function approach, the viscosity-affected inner regions (viscous and buffer layers) are not modeled. Instead, semi-empirical formulae (wall functions) are used to bridge the viscosity-affected region between the wall and the fully turbulent region. In another approach, special, low Reynolds number turbulence models are developed to simulate the near-wall region flow. These two approaches are shown schematically in Fig. 3.5(b) and 3.5(c). 

In order to support the experimental work, COPO experiments were simulated numerically with the PHOENICS code [8]. Since the main interest was in simulating the flow and heat transfer on the boundaries, a low-Reynolds number turbulence model by Lam and Bremhorst [9], instead of the standard k-e model, was chosen as the turbulence model to be primarily used. In this model, some of the constants in the k- and... 

We are essentially assuming that the small scales are in dynamic equilibrium with the large scales. This may also hold in low-Reynolds-number turbulent flows. However, for low-Reynolds-number flows, one may need to account also for dissipation rate anisotropy by modeling all components in the dissipation-rate tensor s j. 

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... 

Despite the explicit dependence on Reynolds number, in its present form the model does not describe low-Reynolds-number effects on the steady-state mechanical-to-scalar time-scale ratio (R defined by (3.72), p. 76). In order to include such effects, they would need to be incorporated in the scalar spectral energy transfer rates. In the original model, the spectral energy transfer rates were chosen such that R(t) —> AV, = 2 for Sc = 1 and V

model parameter. DNS data for 90 < R-,. suggest that Re, is nearly constant. However, for lower... 

In these equations summations over repeated indices are implied. The values for the empirical constants Cu = 1.44, C2e = 1.92, Gi = 1.0, and ce = 1.3 are widely accepted (Launder and Spaulding, The Numerical Computation of Turbulent Flows, Imperial Coll. Sci. Tech. London, NTIS N74-12066 [1973]). The k-e. model has proved reasonably accurate for many flows without highly curved streamlines or significant swirl. It usually underestimates flow separation and overestimates turbulence production by normal straining. The k-e model is suitable for high Reynolds number flows. See Virendra, Patel, Rodi, and Scheuerer (A1AA J., 23, 1308-1319 [1984]) for a review of low Reynolds number k-e. models. 

This form allows extension to low Reynolds number and near wall flows, unlike the standard k-e model, which is valid only for fully turbulent flows. Despite such... 

In most high Reynolds number flows, the wall function approach gives reasonable results without excessive demands on computational resources. It is especially useful for modeling turbulent flows in complex industrial reactors. This approach is, however, inadequate in situations where low Reynolds number effects are pervasive and the hypotheses underlying the wall functions are not valid. Such situations require the application of a low Reynolds number model to resolve near-wall flows. For the low Reynolds number version of k-s models, the following boundary conditions are used at the walls ... 

The experimental effective diffusivity factor should normally approach the value of 2.25 calculated by Kronig and Brink (K5) at low Reynolds numbers. For drops with turbulent circulation described by Handles and Baron s model, R is given (J2) by... 

Although the low Reynolds number characteristic of most of these flows eliminates the challenges of nonlinearity in the convective term and the associated difficulty in modeling turbulent flows (which is actually not true in some gas-liquid devices), we are instead forced to face the nonlinearity of the source term in the Poisson-Boltzmann equation, the nonlinearity of the coupling of electrodynamics with fluid flow, aud the uncertainty in predicting electroosmotic boundary conditions. 

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