Partially Resolved Numerical Simulation CRTI RD Project Review Meeting Canadian Meteorological Centre August 22-23, 2006

Introduction Urban Environment in City Scale (~1 km range) Contains regions of massive separation, recirculation, and turbulent wakes, etc. Conventional RANS approach is unable to correctly simulate above application Hybrid RANS/LES approach is needed

Hybrid RANS/LES approaches Conventional approach spatially filtered equations are solved in the LES region time averaged equations are solved in the RANS region Incompatibility of flow properties between two regions occurs Partially Resolved Numerical Simulation (PRNS) temporal filtered equations are solved in both the LES and RANS regions A unified simulation approach that spans the spectrum from RANS towards LES and DNS

Partially Resolved Numerical Simulation Basic Idea RANS and LES have the same form of filtered transport equation A unified model can be developed to determine the degree of modeling required to represent the unresolved turbulent stresses This can be achieved in principle through the rescaling of RANS models

PRNS model Modeling Approach A PRNS model is obtained by multiplying a dimensionless resolution control function (F R ) to a RANS model: Depends on the physical resolution requirement, the model can serve as: a RANS model (F R  1) where all turbulence scales are modeled, or provides no modeling (F R  0) where all scales are resolved In between these two limits, the model behaves like a LES-type subscale model where only the unresolved scales are modeled The fidelity of PRNS depends on the value of F R as well as on the specific formulation of the model where

Formulation of resolution control function (F R ) Speziale (1998): Batten et al. (2000): Liu and Shih (2006):

Our Formulation of F R Hsieh, Lien and Yee (2006): Assume energy spectrum E(  )   2/3  -5/3 for  i   c   k cc KK ii log  logE(  ) F R =1 F R =0

Implementation of F R Method 1: F R is only applied in momentum equations to reduce the magnitude of turbulent stresses from RANS-like calculation k and  are the large scale turbulent energy and dissipation rate Adopted by Speziale (1998) and Hsieh, Lien and Yee (2006)

Implementation of F R Method 2: F R is applied in momentum and turbulence transport equations to reduce the amplitude of eddy viscosity k and  are the subscale scale turbulent energy and dissipation rate Adopted by Batten et al. (2000) and Liu and Shih (2006)

Implementation of F R Summary k,  t =C  k 2 /  Method 1: Method 2:

Reynolds stresses calculation in PRNS Reynolds stresses has contributions from both the resolved and unresolved scales Reconstruction of modeled Reynolds stresses To use PRNS calculation on a RANS-type coarse grid, a new formulation which require the reconstruction of is proposed An ad hoc modeled Reynolds stresses is introduced, where the non-linear effect of subscale stresses are absorbed into F R The optimal value of the exponent n needs to be determined over a range of flow conditions, and currently n=0.3 is used

Test case Case 6.2 Fully developed channel flow over a matrix of 250 wall-mounted cubes (Meinders and Hanjalic, 1999) Benchmark problem for the 8th ERCOFTAC workshop (1999)

Results Numerical results are obtained based on: 45x45x45 nodes (streamwise by spanwise by vertical) Standard k-  turbulence model with wall functions are used for URANS and PRNS calculations

Streamwise mean velocity

Reynolds stress (u’u’)

Reynolds stress (w’w’)

Resolution control function distribution

Time History at (x,y,z) = (0.5, 1.3, 0)

Energy Spectrum at (x,y,z) = (0.5, 1.3, 0)

Conclusions PRNS provides a unified simulation strategy for high Reynolds number complex turbulent flows Implementation of PRNS to any CFD code that runs URANS simulation is straightforward and requires very minimum modifications PRNS has been demonstrated in improving the prediction of turbulent mean quantities

References Batten, P., Goldberg, U., and Chakravarthy, S. (2000), “Sub-grid turbulence modeling for unsteady flow with acoustic resonance”, AIAA paper Hellsten, A., Rautaheimo, P. (1999). “Workshop on refined turbulence modelling”, Proceedings of the 8th ERCOFTAC/IAHR/COST Workshop, 17–18 June, 1999, Helsinki, Finland. Helsinki University of Technology. Meinders E.R. and Hanjalic K. (1999), “Vortex structure and heat transfer in turbulent flow over a wall-mounted matrix of cubes”, Int. J. Heat Fluid Flow, Vol. 20, pp Liu N.S. and Shih T.H. (2006), “Turbulence modeling for very large-eddy simulation”, AIAA Journal, Vol. 44, pp Speziale, C.G. (1998), “Turbulent modeling for time-dependent RANS and VLES: a review”, AIAA Journal, Vol. 36, pp