2.2
Turbulence Modelling
Work presented in this thesis, used Unsteady Reynolds-Averaged Navier-Stokes (URANS)[13], DES[45] and SAS computational methods. DES computations utilised the S-A turbulence model by Spalart and Allmaras[46]while URANS and SAS compu- tations utilised thek−ω SST turbulence model, a hybridisation of thek−ω [47] and
k−ε model, by Menter[48].
2.2.1
Unsteady Reynolds-Averaged Navier-Stokes Simulation
The least expensive method in terms of computational cost is the RANS method. RANS splits the flow variables into a mean flow component with a fluctuating component superimposed on it. However, such an approach means that the mean terms are known but the fluctuating terms are not. Early studies of cavity flows using CFD focused on the use of Unsteady Reynolds-Averaged Navier-Stokes, which uses a time- varying mean component. To account for the effect of the fluctuating part of the flow, a turbulence closure is needed, which models all of the turbulent scales.
URANS models tend to predict well the larger scales associated with the lower frequency discrete acoustic tones but fail to provide the same accuracy in capturing the smaller, higher frequency and more intermittent time scales. The broadband noise is not captured by these models either. The presence of these multiple acoustic tones and of a large number of turbulent scales may mean that achieving a good level of accuracy and consistency with turbulence modelling is difficult for cavity flows. In order to attain reasonable behaviour from URANS, some studies modify the turbulence models or limit the level of eddy viscosity produced by the model. Therefore the shortfalls of URANS computations justify the use of more advanced methods.
2.2.2
Large Eddy Simulation
With Direct Numerical Simulation (DNS) being too expensive and URANS unable to predict the unsteadiness within the cavity correctly, studies using other methods also appeared in literature. The philosophy behind LES is to resolve the larger, energy- containing eddies on the grid, while a sub-grid scale (SGS) model is used to model the smaller, more isotropic scales. Since the smallest eddies are modelled, the grids can be less refined and the time-steps can be larger then those required for DNS.
2.2. TURBULENCE MODELLING 25
Despite the potential of LES, there are problems in resolving the near-wall turbulent stresses since the required resources approach those of DNS. As a general rule of thumb, pure LES allows the use of 10 times higher Reynolds numbers than DNS for current computational resources. This means that it is of limited use for complex flows. One solution is to use wall functions at the solid boundaries and therefore very coarse grid spacings can be used. Another solution was to make a hybrid RANS-LES model.
2.2.3
Detached-Eddy Simulation
Even though studies using LES appear in literature, the computer resources required to accurately resolve all the flow features are still high. Therefore, for flows with a large range of turbulent scales (such as wall-bounded flows), proposals for hybrid RANS- LES models were put forward. The hybrid schemes use RANS for simulating near- wall areas and LES for the separated flow regions. The most popular was proposed by Spalartet al.[49] and is termed Detached-Eddy Simulation. Spalart argued that when computing solutions of 3D real-life geometries using ‘Navier-Stokes and a Turbulence Model’ equations (i.e. RANS or URANS), the accuracy of the converged solution can only be as good as that of the turbulence model. When applied to cavities, this new method was found to give good agreement with experimental data and at a reduced cost compared to LES simulations. Its underlying principle involved using RANS for the boundary layers and LES everywhere outside. Therefore the word ‘detached’ was used to imply that the ‘attached’ eddies internal to the boundary layer will be modelled and the detached ones resolved.
Spalartet al.[49] modified the one equation S-A turbulence model to achieve a DES equivalent. The destruction term for the eddy viscosity is proportional to(ν˜/d)2, where d is the distance to the nearest wall. In the Smagorinsky model,[50] the sub- grid scale (SGS) eddy viscosity is scaled with the grid spacing, ∆. Therefore for the DES formulation, the d is replaced by a length proportional to ∆. In practice, the modified distance is expressed as a comparison between the actual distance to the wall and that calculated byCDES ∆, whereCDES is a constant and ∆computes the size of
the maximum cell length: ˜
d=min(d,CDES∆) (2.14)
2.2. TURBULENCE MODELLING 26
When CDES ∆ ≪ d, LES is triggered and the S-A model acts as an SGS model.
RANS is activated when the converse occurs. The boundary between LES and RANS is therefore completely dependent on the geometry and on the density of the computational domain. In the CFD flow solver,CDES has the value 0.65. Also note
that other metric relations are possible for∆.
A concern is the ‘grey area’ in whichCDES ∆is on the same order asd. In the
separated regions, the detached boundary layer should grow ‘free-shear-flow’ eddies; however, it is not seeded with eddies from the boundary layer since they are suppressed by the RANS model.
2.2.4
Scale-Adaptive Simulation
While DES is capable of accurately predicting these flows, it still takes a considerable amount of time on a large number of processors. Since its introduction by Menter
et al.[51–53]in 2003, the SAS approach gained popularity due to its LES-like behaviour in highly separated flow regions and found place in several studies. A detailed explanation of the theory and description of the model was given by Menter and Egorov
[54] following which Egorov et al.[55] presented the application of the SAS model,
implemented in ANSYS-FLUENT and ANSYS-CFX, for a range of complex flows. The governing equations of the SST-SAS model differ from those of the SST- RANS model[48]by the additional SAS source term,QSASin the transport equation for
the turbulence eddy frequency, given by:
QSAS=max [ ρζ2κS2 ( L LvK )2 −C2ρk σΦ max ( 1 ω ∂ω∂xj ∂ω ∂xj ,1 k ∂k ∂xj ∂k ∂xj ) ,0 ] (2.16)
wherel is the length scale andLvK is the von Karman length scale in classic boundary
layer definition, L= √ k c1µ/4ω , LvK =κ U′(y) U′′(y) (2.17)
and the constantsζ2= 3.51,σΦ= 2/3 andC= 2.
SAS is an improved URANS model that can produce spectral content for un- steady flows by adjusting the turbulence length scale to the local flow inhomogeneities and balancing the contributions of modelled and resolved parts of the turbulent stresses.