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Show More complete descriptions of these approaches can be found in the provided references. Details of Fluent's implementation for each can be found in [8]. Standard k-e model The standard k-e (SKE) model [9] is the most robust and widely used first order turbulence closure model. The Boussinesq hypothesis is used to describe the transport of momentum via turbulence as (•^r -pupj=ju, rr >\ 3w, 3w, - - + l ox, 3x J ' J -|K) with dimensional analysis providing the correlation between turbulence (eddy) viscosity, u,, and the turbulence kinetic energy, k, and dissipation rate, e. ju,=pC - ;CM=0.09 The constants C1£, C2E, ak, aE and Prx were set to 1.44, 1.92, 1.0, 1.3 and .85, respectively. The parameter C3E is variable, with the limits of 1 for buoyant shear layers with the main flow direction aligned with the gravitational vector and 0 for buoyant shear layers perpendicular to the gravity vector. Standard k-e model deficiencies It is well known that the SKE model does not always predict normal stresses consistent with the physics of turbulent flows [10]. By combining the Boussinesq hypothesis and the definition of eddy viscosity, it can be shown that the normal stresses (defined strictly positive) will become negative for mean strain rates large enough to satisfy k du 1 = 3.7 £ dx 3Cp Similarly, it can also be shown that the Schwarz inequality for shear stresses Modeled transport equations for k and e, which include additional semi-empirical constants and . phenomenological considerations, complete the closure and are given by -Dk d P Dt~ dx, -De p~b7 p + - L '*; + Gk+Gb-pe _9_ dx. dx, + Cl f^(Gi+C3 fGj-C2 cpy uaup <u~au~p (no implied summation) can be violated when the mean strain rate is large. Another weakness of the SKE model of particular importance to axisymmetric jet flame modeling is the "round jet anomoly". Predictions of spreading rate for momentum dominated jets using the SKE model have been unexpectedly poor (spreading rate is overpredicted). These errors are considered to be mainly due to the modeled dissipation equation. Realizable k-e model Here Gk represents the generation of turbulent kinetic energy due to mean velocity gradients and is modeled using a Boussinesq approximation as Gk=/i,S2 ;SmfisvSv ;S„ = r9«( duj dx, dx, \ J 'J Gb is the generation of turbulent kinetic energy due to buoyancy G> = Si - M, dp pPr, a*, The realizable k-e (RKE) turbulence model proposed by Shih et al. [10] was intended to address these deficiencies of traditional k-e models. Though quite similar to the SKE formulation (the /c-equation is identical except for model constants), the R K E model differs in two significant ways. Firstly, the RKE model incorporates the eddy-viscosity model of Reynolds [11] which involves a nonconstant C ^ Here CM is a function of the mean strain and rotation rates, the angular velocity of the system rotation and the |
