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- 1. WIND ENERGY METEOROLOGY UNIT 7 ATMOSPHERIC FLOW MODELING III: LARGE-EDDY SIMULATION Detlev Heinemann ENERGY METEOROLOGY GROUP INSTITUTE OF PHYSICS OLDENBURG UNIVERSITY FORWIND – CENTER FOR WIND ENERGY RESEARCHMittwoch, 15. Juni 2011
- 2. ATMOSPHERIC FLOW MODELING III: LES SIMULATION OF TURBULENT ATMOSPHERIC FLOW We know: ‣ Turbulence consists of three-dimensional, chaotic, or random motion that spans a range of scales that increases rapidly with Reynolds number ‣ Complete numerical integration of the exact equations governing the turbulent velocity field (Navier–Stokes equations) is known as direct numerical simulation (DNS). ‣ Because of limited computing power, DNS is restricted to low- Reynolds-number turbulence, which exists in laboratory flows, e.g., in wind tunnels. 2Mittwoch, 15. Juni 2011
- 3. ATMOSPHERIC FLOW MODELING III: LES SIMULATION OF TURBULENT ATMOSPHERIC FLOW TURBULENCE MODELING HIERARCHY Direct Numerical Simulation (DNS) • Solution of the Navier-Stokes equations without use of an explicit turbulence – limited to low Reynolds numbers • Powerful research tool Large Eddy Simulation (LES) • Direct resolution of the large, energy-containing scales of the turbulent flow, model only the small eddies • High computational cost in boundary layers Reynolds-average Navier-Stokes (RANS) • Model the entire spectrum of turbulent motions • Uneven performance in flows outside of the calibration range of the modelsMittwoch, 15. Juni 2011
- 4. ATMOSPHERIC FLOW MODELING III: LES SIMULATION OF TURBULENT ATMOSPHERIC FLOW TURBULENCE MODELING HIERARCHY Direct Numerical Simulation (DNS) • Solution of the Navier-Stokes equations without use of increase in cost an explicit turbulence – limited to low Reynolds numbers • Powerful research tool Large Eddy Simulation (LES) • Direct resolution of the large, energy-containing scales of the turbulent flow, model only the small eddies • High computational cost in boundary layers Reynolds-average Navier-Stokes (RANS) • Model the entire spectrum of turbulent motions increase in • Uneven performance in flows outside of the calibration empiricism range of the modelsMittwoch, 15. Juni 2011
- 5. ATMOSPHERIC FLOW MODELING III: LES SIMULATION OF TURBULENT ATMOSPHERIC FLOW TURBULENCE MODELING HIERARCHY Direct Numerical Simulation (DNS) • Solution of the Navier-Stokes equations without use of increase in cost an explicit turbulence – limited to low Reynolds numbers • Powerful research tool Large Eddy Simulation (LES) • Direct resolution of the large, energy-containing scales “hybrid” methods combine RANS of the turbulent flow, model only the small eddies and LES (e.g., Detached-Eddy • High computational cost in boundary layers Simulation) Reynolds-average Navier-Stokes (RANS) • Model the entire spectrum of turbulent motions increase in • Uneven performance in flows outside of the calibration empiricism range of the modelsMittwoch, 15. Juni 2011
- 6. ATMOSPHERIC FLOW MODELING III: LES SIMULATION OF TURBULENT ATMOSPHERIC FLOW TURBULENCE MODELING HIERARCHY l η = l/Re3/4 Direct numerical simulation (DNS) Large eddy simulation (LES) Reynolds averaged Navier-Stokes equations (RANS) 4Mittwoch, 15. Juni 2011
- 7. ATMOSPHERIC FLOW MODELING III: LES SIMULATION OF TURBULENT ATMOSPHERIC FLOW spatial and temporal resolution of scales in “inertial subrange” 5Mittwoch, 15. Juni 2011
- 8. ATMOSPHERIC FLOW MODELING III: LES ENERGY SPECTRUM OF TURBULENCE 6Mittwoch, 15. Juni 2011
- 9. ATMOSPHERIC FLOW MODELING III: LES ENERGY SPECTRUM OF TURBULENCE 7Mittwoch, 15. Juni 2011
- 10. ATMOSPHERIC FLOW MODELING III: LES SIMULATION OF TURBULENT ATMOSPHERIC FLOW Example: ‣ PBL: largest turbulent eddies are on the order of kilometers and the smallest on the order of millimeters --> spectrum of turbulent motion spans more than six orders of magnitude ‣ To numerically integrate the Navier–Stokes equations for this turbulent flow would require at least 1018 numerical gridpoints (today 1010 is possible...) 8Mittwoch, 15. Juni 2011
- 11. ATMOSPHERIC FLOW MODELING III: LES SIMULATION OF TURBULENT ATMOSPHERIC FLOW Consequence: ‣ Only a portion of the scale range can be explicitly resolved, --> larger eddies or most important scales of the flow ‣ Remaining scales must be roughly represented or parameterized in terms of the resolved portion ‣ philosophy behind large-eddy simulation (LES) ‣ PBL turbulence: Large eddies contain most of the turbulent kinetic energy (TKE) --> energy-containing eddies they are responsible for most of the turbulent transport 9Mittwoch, 15. Juni 2011
- 12. ATMOSPHERIC FLOW MODELING III: LES SIMULATION OF TURBULENT ATMOSPHERIC FLOW ‣ Explicite calculation of these large eddies and approximate representation of the effects of smaller ones ‣ Accuracy of LES increases as the grid resolution becomes finer ‣ LES is a compromise between DNS, in which all turbulent fluctuations are resolved, and the traditional Reynolds-averaging approach in which all fluctuations are parameterized and only ensemble-averaged statistics are calculated ‣ With increasing computer power a much broader application of LES to more complicated geophysical turbulence problems is anticipated 10Mittwoch, 15. Juni 2011
- 13. ATMOSPHERIC FLOW MODELING III: LES THE LES TECHNIQUE Basis for an LES of the PBL: Navier–Stokes equations for an incompressible fluid where ui satisfy the continuity equation: ui: flow velocities in the three spatial Xi: ith-component of body forces directions p: pressure fluctuation ρ: air density t time ν: kinematic viscosity of the fluid xi spatial coordinates 11Mittwoch, 15. Juni 2011
- 14. ATMOSPHERIC FLOW MODELING III: LES THE LES TECHNIQUE ‣ PBL applications: ‣ major body forces are gravity and Coriolis forces ‣ Xi can be approximated as ‣ where the gravitational acceleration gi is nonzero only in the x3 (or z) direction, θ is the virtual potential temperature, T0 is the temperature of some reference state, and f is the Coriolis parameter. ‣ An additional transport equation is required for θ if buoyancy is considered. ‣ The numerical integration of these equations is DNS ‣ for LES they need to be spatially filtered 12Mittwoch, 15. Juni 2011
- 15. ATMOSPHERIC FLOW MODELING III: LES THE LES TECHNIQUE Deriving the volume-filtered Navier-Stokes equations: first decomposing all dependent variables, e.g., ui, into a volume ~ average, ui, and a subgrid-scale (SGS) (or subfilter) component, ui‘‘: Here the volume-averaged or resolved-scale variable is defined as where G is a three-dimensional (low-pass) filter function, e.g., Gaussian, top-hat or sharp wave cutoff filter. 13Mittwoch, 15. Juni 2011
- 16. ATMOSPHERIC FLOW MODELING III: LES THE LES TECHNIQUE Filtering process Example: One-dimensional random signal Solid curve: Total signal (fluctuating in x). Dashed curve: Smoother field after applying the filter operator G (so-called filtered field or resolved-scale motion). Difference between total and resolved signals representing the SGS fluctuations. (Partitioning between resolved and SGS components depends on the filter; i.e., cutoff scale and sharpness) 14Mittwoch, 15. Juni 2011
- 17. ATMOSPHERIC FLOW MODELING III: LES THE LES TECHNIQUE Filtering process ‣ Filtering is a local spatial averaging over the filter width Δ ‣ Increasing Δ ‣ removes more scales from the velocity field and ‣ increases the contribution of τij --> filter width should be part of the expressions for the models of τij 15Mittwoch, 15. Juni 2011
- 18. ATMOSPHERIC FLOW MODELING III: LES THE LES TECHNIQUE Filters ‣ Sharp Fourier cutoff filter in wave space ‣ Gaussian ‣ Tophat filter in physical space 16Mittwoch, 15. Juni 2011
- 19. ATMOSPHERIC FLOW MODELING III: LES THE LES TECHNIQUE Effect of Filters Unfiltered and filtered velocity spectra 17Mittwoch, 15. Juni 2011
- 20. ATMOSPHERIC FLOW MODELING III: LES THE LES TECHNIQUE Applying the filtering procedure, term-by-term, to the Navier- Stokes equation leads to equations that govern large (resolved- scale) eddies: ~ - first term on the right-hand side: advection of ui by the resolved-scale ~ motion uj - second term: SGS contribution - remaining terms: identical to their counterparts in NS, except that they depend on filtered (resolved-scale) fields 18Mittwoch, 15. Juni 2011
- 21. ATMOSPHERIC FLOW MODELING III: LES THE LES TECHNIQUE An alternative version can be derived using the identity and expressing the SGS stress (or flux) tensor as 19Mittwoch, 15. Juni 2011
- 22. ATMOSPHERIC FLOW MODELING III: LES THE LES TECHNIQUE ‣ Both equations equally describes the evolution of the LE field. They differ in their forms of the resolved advection and SGS terms. ‣ First eq.: SGS term consists of two kinds of influences: cross- products of resolved-SGS components (i.e., ) and a nonlinear product of SGS–SGS components (i.e., ) ‣ Second eq.: SGS term includes all of these influences plus a resolved scale contribution: ‣ --> in principle different SGS models should be used. ‣ For geophysical turbulence, the molecular viscosity term is negligibly small compared with the advection terms and can be neglected. 20Mittwoch, 15. Juni 2011
- 23. ATMOSPHERIC FLOW MODELING III: LES THE LES TECHNIQUE ‣ So far in deriving the LE equations, no approximations have been made. ‣ Because of the spatial filtering procedure, the LE equations contain SGS terms that are unknown and must be modeled in terms of the resolved fields. ‣ Because the magnitudes of SGS terms depend on the filter, its modeling in principle should depend on the filter size and shape. --> To solve the equations, the SGS terms need to be parameterized. 21Mittwoch, 15. Juni 2011
- 24. ATMOSPHERIC FLOW MODELING III: LES SUBGRID-SCALE PARAMETERIZATION ‣ Parameterization introduces uncertainty in LES, particularly in regions where small eddies dominate, i.e., near the surface or behind an obstacle. ‣ In regions where energy-containing eddies are well resolved, LES flow fields are rather insensitive to SGS models (In the interior of PBL, the SGS motions serve mainly as net energy sinks that drain energy from the resolved motions) ‣ Most widely used SGS closure scheme: Smagorinsky–Lilly (S–L) model (Most PBL–LESs adopt a similar scheme) 22Mittwoch, 15. Juni 2011
- 25. ATMOSPHERIC FLOW MODELING III: LES SUBGRID-SCALE PARAMETERIZATION: SMAGORINSKY–LILLY (S–L) MODEL ‣ Relating SGS stresses to resolved-scale strain tensors by with the strain tensor ‣ SGS heat fluxes are similarly related to local gradients in the resolved temperature field by ‣ The SGS eddy viscosity KM and diffusivity KH are expressed as 23Mittwoch, 15. Juni 2011
- 26. ATMOSPHERIC FLOW MODELING III: LES SUBGRID-SCALE PARAMETERIZATION: SMAGORINSKY–LILLY (S–L) MODEL ‣ the Smagorinsky constant cS remains to be determined ‣ Δs is a filtered length scale often taken to be proportional to the grid size ‣ the magnitude of the strain tensor, S, is (2SijSij)1/2 ‣ Pr (~1/3) is the SGS Prandtl number ‣ Important: the SGS fluxes are nonlinear functions of the resolved strain rate (different from the viscous (molecular) stress–strain relationship) 24Mittwoch, 15. Juni 2011
- 27. ATMOSPHERIC FLOW MODELING III: LES SUBGRID-SCALE PARAMETERIZATION: SMAGORINSKY–LILLY (S–L) MODEL Extension to include local buoyancy effects: ‣ KM is modified to depend on local Richardson number Ri (the ratio of buoyancy to shear production terms of TKE budget): where Ric is the critical Richardson number often set between 0.2–0.4, and n = 1/2 is often used ‣ When Ri reaches Ric, turbulence within that grid cell vanishes and the eddy viscosity is shut off. 25Mittwoch, 15. Juni 2011
- 28. ATMOSPHERIC FLOW MODELING III: LES SUBGRID-SCALE PARAMETERIZATION: SMAGORINSKY–LILLY (S–L) MODEL Extension: Explicit calculation of the SGS–TKE e ‣ Relating KM and KH to e via where - cK is a diffusion coefficient to be determined - ℓ is another SGS length scale, which is often taken as the minimum of two length scales (assuming a direct effect of local stability on the local SGS length scale) 26Mittwoch, 15. Juni 2011
- 29. ATMOSPHERIC FLOW MODELING III: LES SUBGRID-SCALE PARAMETERIZATION: SMAGORINSKY–LILLY (S–L) MODEL The SGS TKE e evolves from the following equation: ‣ Terms on the right-hand side: - advection of e by the resolved-scale motion - turbulent and pressure transports - local shear production (nonlinear scrambling) - local buoyancy production - molecular dissipation ‣ are approximated by (s. slide 15): transport terms: molecular dissipation rate with cε: dissipation coefficient 27Mittwoch, 15. Juni 2011
- 30. ATMOSPHERIC FLOW MODELING III: LES SUBGRID-SCALE PARAMETERIZATION: SMAGORINSKY–LILLY (S–L) MODEL SGS model parameters: cS, cK, and cε are usually chosen to be consistent with Kolmogorov inertial-subrange theory, i.e., assuming that the SGS motions are isotropic with a k-5/3 spectral slope. Commonly used values are: cS ~ 0.18, cK ~ 0.10 cε ~ 0.19 + 0.74 ℓ/Δs. With these model parameters, LESs are in a way forced – in an ensemble-mean sense – to drain energy at a rate sufficient to produce a k-5/3 spectral slope near the filter cutoff scale. 28Mittwoch, 15. Juni 2011
- 31. ATMOSPHERIC FLOW MODELING III: LES SUBGRID-SCALE PARAMETERIZATION Problem: Above SGS models are based on ensemble average concepts but are used inside LES on an instantaneous basis, i.e., to represent SGS effects at every gridpoint and time step. However, small-scale turbulent motion is anisotropic and intermittent, and locally the energy transfer can either be forwardscatter (from large to small scales) or backscatter (from small to large scales), which causes deviations from the equilibrium k-5/3 law. Eddy viscosity SGS models also assume that SGS stresses and strains are perfectly aligned, and hence the local dissipation rate ε = - τij Sij is always positive, thus preventing backscatter of energy 29Mittwoch, 15. Juni 2011
- 32. ATMOSPHERIC FLOW MODELING III: LES SUBGRID-SCALE PARAMETERIZATION These deficiencies of eddy viscosity models have motivated continued development of new SGS models, including ‣ stochastic models where a random field is imposed at the SGS level, thus permitting a backscatter of energy, ‣ dynamic models where the Smagorinsky coefficient is dynamically predicted using a resolved field filtered at two different scales ‣ velocity estimation models that attempt to model the SGS velocity fluctuations ui‘‘ instead of SGS stresses τij. 30Mittwoch, 15. Juni 2011
- 33. ATMOSPHERIC FLOW MODELING III: LES NUMERICAL SETUP ‣ Choice of LES grid and domain sizes depend on the physical flow of interest and the computer capability ‣ Grid-scale motion in LES is nearly isotropic -> Requiring a grid mesh close to isotropic ‣ From the chosen gridpoints, say 100x100x100, an LES domain is chosen to resolve several largest (dominant) turbulent eddies and at the same time resolve eddies as small as possible into the inertial-subrange scales. Example: For a convective PBL with 1 km depth, a 5 km x 5 km x 2 km domain of LES with 100x100x100 gridpoints would cover 3 to 5 large dominant eddies in each horizontal direction and at the same time resolve small eddies down to about 100 mx100 mx40 m in size, assuming model resolution is twice the grid size. For the stable PBL where dominant eddies are smaller, a smaller domain (and consequently a finer grid) is preferred. 31Mittwoch, 15. Juni 2011
- 34. ATMOSPHERIC FLOW MODELING III: LES BOUNDARY CONDITIONS Surface boundary conditions ‣ LES cannot resolve the viscous layer close to the surface; its lowest grid level lies in the surface layer ‣ M–O similarity theory is used as a surface boundary condition to relate surface fluxes to resolved-scale fields at each grid point just above the surface ‣ The primary empirical input parameter to these formulas is the surface roughness ‣ Different from the smooth-wall condition in engineering flows ‣ M–O theory describes ensemble-mean flux–gradient relationships in the surface layer and may not apply well at the local LES grid scale. (Especially when LES horizontal grid size is comparable to or smaller than the height of the first grid level.) 32Mittwoch, 15. Juni 2011
- 35. ATMOSPHERIC FLOW MODELING III: LES BOUNDARY CONDITIONS Upper boundary conditions ‣ The upper boundary of a typical LES domain is usually set to be well above the PBL top, in order to avoid influences on simulated PBL flows from artificial upper boundary conditions. At the top of the domain, turbulence is negligible and a no- stress condition is applicable. Because turbulent motions in the PBL may excite gravity waves in the stably stratified inversion layer, a means of handling gravity waves is often applied. Typically, a radiation condition, which allows for an upward escape of gravity waves, or a wave- absorbing sponge layer is used at the top of the simulation domain. 33Mittwoch, 15. Juni 2011
- 36. ATMOSPHERIC FLOW MODELING III: LES BOUNDARY CONDITIONS Lateral boundary conditions ‣ Most PBL LESs use periodic boundary conditions. (inflow at each gridpoint on a sidewall is equal to the outflow on opposite sidewall) ‣ appropriate for PBLs with homogeneous terrain ‣ no explicit statement of the sidewall boundary (turbulence) conditions --> computational convenience ‣ no simulation of realistic meteorological flows with inhomogeneous surface! 34Mittwoch, 15. Juni 2011

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