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This turbulent model is a simple single-equation model that solves the modeled transport equation for vortex viscosity. In general, this model is used for low Reynolds numbers and areas affected by viscosity within the boundary layer. This model is used in aviation, low-separation flows (such as airfoil, wing and aircraft fuselage, and missile), and flows under a pressure gradient.
Two-equation turbulence models allow the determination of both, a turbulent length and time scale by solving two separate transport equations. The standard k-epsilon model in ANSYS Fluent falls within this class of models and has become the workhorse of practical engineering flow calculations in the time since it was proposed by Launder and Spalding. Robustness, economy, and reasonable accuracy for a wide range of turbulent flows explain its popularity in industrial flow and heat transfer simulations. It is a semi-empirical model.
The standard k-epsilon model is a model based on model transport equations for the turbulence kinetic energy (k) and its dissipation rate (e). The model transport equation for k is derived from the exact equation, while the model transport equation for e was obtained using physical reasoning
In the derivation of the k-epsilon model, the assumption is that the flow is fully turbulent, and the effects of molecular viscosity are negligible. The standard k-epsilon model is therefore valid only for fully turbulent flows.
The RNG k-epsilon model was derived using a statistical technique called the renormalization group theory. It is similar in form to the standard K-epsilon model, but includes the following refinements:
- The RNG model has an additional term in its epsilon equation that improves the accuracy of rapidly strained flows.
- The effect of swirl on turbulence is included in the RNG model, enhancing accuracy for swirling flows.
- The RNG theory provides an analytical formula for turbulent Prandtl numbers, while the standard k-epsilon model uses user-specified, constant values.
- While the standard k-epsilon model is a high-Reynolds number model, the RNG theory provides an analytically-derived differential formula for effective viscosity that accounts for low-Reynolds number effects. Effective use of this feature does, however, depend on the appropriate treatment of the near-wall region.
These features make the RNG k-epsilon model more accurate and reliable for a wider class of flows than the standard k-epsilon model.
The RNG-based k-epsilon turbulence model is derived from the instantaneous Navier-Stokes equations, using a mathematical technique called “renormalization group” (RNG) methods. The analytical derivation results in a model with constants different from those in the standard k-epsilon model, and additional terms and functions in the transport equations for k and epsilon.
The k-epsilon RNG turbulence model is used to model transient flows, flows in geometries with high curvatures, and air conditioning problems.
Realizable k-ε Model
The realizable k-epsilon model differs from the standard k-epsilon model in two important ways:
- The realizable k-epsilon model contains an alternative formulation for the turbulent viscosity.
- A modified transport equation for the dissipation rate, epsilon, has been derived from an exact equation for the transport of the mean-square vorticity fluctuation.
The term “realizable” means that the model satisfies certain mathematical constraints on the Reynolds stresses, consistent with the physics of turbulent flows. Neither the standard k-epsilon model nor the RNG k-epsilon model is realizable.
Both the realizable and RNG k-epsilon models have shown substantial improvements over the standard k-epsilon model where the flow features include strong streamline curvature, vortices, and rotation. Since the model is still relatively new, it is not clear in exactly which instances the realizable k-epsilon model consistently outperforms the RNG model. However, initial studies have shown that the realizable model provides the best performance of all the k-epsilon model versions for several validations of separated flows and flows with complex secondary flow features.
One of the weaknesses of the standard k-epsilon model or other traditional k-epsilon models lies with the modeled equation for the dissipation rate (epsilon). The well-known round-jet anomaly (named based on the finding that the spreading rate in planar jets is predicted reasonably well, but prediction of the spreading rate for axisymmetric jets is unexpectedly poor) is considered to be mainly due to the modeled dissipation equation.
The realizable k-epsilon model proposed by Shih et al. was intended to address these deficiencies of traditional k-epsilon models by adopting the following:
- A new eddy-viscosity formula involving a variable Cm originally proposed by Reynolds.
- A new model equation for dissipation (epsilon) based on the dynamic equation of the mean-square vorticity fluctuation.
One limitation of the realizable k-epsilon model is that it produces non-physical turbulent viscosities in situations when the computational domain contains both rotating and stationary fluid zones (for example, multiple reference frames, rotating sliding meshes). This is due to the fact that the realizable k-epsilon model includes the effects of mean rotation in the definition of the turbulent viscosity. This extra rotation effect has been tested on single moving reference frame systems and showed superior behavior over the standard k-epsilon model. However, due to the nature of this modification, its application to multiple reference frame systems should be taken with some caution.
Standard Wall Function
If the k-epsilon turbulence model is used for the simulation, it is not possible to simulate flow vortex near the walls, a wall function must be defined to investigate the fluid behavior near the wall. These wall functions are near-wall analytical flow profiles obtained by explicitly solving the near-wall flow equations; hence, they are more accurate than numerical methods (in simpler models). Also, because of the need for no accurate mesh near the wall, they greatly reduce the computational time. It should be noted that when using the k-epsilon model, due to the lack of accurate mesh near the walls, it is necessary to check the wall functions using Y-Plus (Y+). The suitable Y+ should be between 30 and 300 for the standard model.
Enhanced Wall Function
When the k-epsilon turbulence model is used and it is not feasible to simulate flow vortices near the walls, a function of the wall must be defined to investigate the fluid behavior near the wall. These wall functions are near-wall analytical flow profiles obtained by explicitly solving the near-wall flow equations; hence, they are better than numerical methods, and also due to the lack of the need for accurate mesh near the wall, they greatly reduce the computational cost. It should be noted that when using the k-epsilon model, due to the lack of meshing near the walls, it is necessary to check the wall functions using Y+ so that in enhanced mode the appropriate value for Y+ must be about 1. The enhanced wall function is mainly suitable for low Reynolds number and finer grid models.
The k-epsilon and Reynolds stress turbulence models do not have the ability to simulate vortexes near walls, and therefore, to model vortexes and solve flows near walls we should use the wall function. While the k-omega and Sparat-Allmaras turbulence models, with suitable meshing, have the ability to simulate and solve the flow directly near the walls. The k-omega model has different types including standard, GEKO, BSL, and SST. The standard type has applications such as better performance in low-velocity and reverse flows due to reverse pressure gradient, used incompressible flows, free and transient shear, suitable for mixing layers, plate and radial jets, suitable for wall enclosures and also suitable for aerodynamic and turbomachinery problems. While the SST (shear stress transport) model acts as a combination of the k-Ԑ model in the areas near the wall and the k-ω model in the open space. Capabilities of this type of turbulent flow are suitable in cases such as airfoils, transient shock waves, and flows containing reverse pressure gradients, and of course, have disadvantages such as the high probability of instability and weak convergences due to a transfer from one turbulence model to another. The k-epsilon model also has the ability to calculate shear flow corrections and low-Re corrections. In fact, this model can use shear flow corrections to increase accuracy in predicting fluid shear flow behavior.
The simulations that are related to the external flow, apply the K-Omega SST model. This model of k-omega operates as a hybrid function, which results in a gradual transfer of flow from the k-omega model for near-wall regions to the k-epsilon model in areas beyond the boundary layer. This model is used for reverse pressure gradient flows and in airfoil simulations. Since the wall function does not define in the k-omega model, finer grids should be used in areas close to the airfoil walls. However, in this turbulence model, the probability of divergence increases due to the transition from one model to another.
The k-omega standard turbulence model for modeling currents including velocity deceleration and separation due to reverse pressure gradient, mixing layer flows and plate and radial jets, high rotational flows, and issues related to aerodynamic forces and turbochargers are applicable.
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