Introduction to Turbulence, its Numerical Modeling and Downscaling

560.700: Applications of Science-Based Coupling of Models October, 2008 Introduction to Turbulence, its Numerical Modeling and Downscaling Charles Meneveau Department of Mechanical Engineering Center for Environmental and Applied Fluid Mechanics Johns Hopkins University Mechanical Engineering

Contents Give an introduction to turbulent flow characteristics and physics Provide a first, basic understanding of most standard RANS turbulence model used in industrial CFD Provide basic understanding of Direct Numerical (DNS) and Large Eddy Simulation (LES) Smagorinsky model UPSCALING: dynamic approach and illustrations Renormalized Numerical Simulation

Turbulent flows: multiscale, mixing, dissipative, chaotic, vortical well-defined statistics, important in practice From: Multimedia Fluid Mechanics, Cambridge Univ. Press

Turbulent flows: multiscale, mixing, dissipative, chaotic, vortical well-defined statistics, important in practice From: Multimedia Fluid Mechanics, Cambridge Univ. Press

Turbulence in atmospheric boundary layer

Turbulence in reacting flows: Premixed flame in I.C. engine, combustion Numerical simulation of flame propagation in decaying isotropic turbulence

Turbulence in aerospace systems: LES of flow in thrust-reversers Blin, Hadjadi & Vervisch (2002) J. of Turbulence.

Turbulence in thermofluid equipment: From: Multimedia Fluid Mechanics, Cambridge Univ. Press

Simplest turbulence: Isotropic decaying turbulence JHU Corrsin wind tunnel Active Grid M = 6 " Active Grid M = 6 " Test Section u 2 Contractions x 2 u 1 x 1 Flow

Physical quantities describing fluid flow Density field Velocity vector field Pressure field Temperature field (or internal energy, or enthalpy etc..) Physical laws governing fluid flow Conservation of mass Newton s second law First law of thermodynamics Equation of state Some constraints in closure relations from second law of TD Navier Stokes equations for a Newtonian, incompressible fluid

Navier-Stokes equations, incompressible, Newtonian!u j!x j = 0!u j!t +!u u k j = " 1!p + $% 2 u!x k #!x j + g j j a j = F j m

Traditional approach: Reynolds decomposition!u j!x j = 0!u j!t +!u u k j = " 1!p + $% 2 u!x k #!x j + g j j t

Traditional approach: Reynolds decomposition!u j!x j = 0!u j!t +!u u k j = " 1!p + $% 2 u!x k #!x j + g j j Reynolds equations:!u j!x j = 0 t!u j!t +!u u j k = " 1!p + $% 2 u!x k #!x j + g j "! u j!x j u k " u j u k k ( )

Kinematic Reynolds stress (minus):!u j!x j = 0!u j!t +!u u j k = " 1!p + $% 2 u!x k #!x j + g j "! u j!x j u k " u j u k k ( )! jk R = u j u k " u j u k Written as velocity co-variance tensor: 0 0! R jk = (u j + u" j )(u k + u k " ) # u j u k = u j u k + u j u k " + u k u" j + u" j u k " # u j u k = u" j " u k 0

Spectral representation of co-variance tensor: u! j u k! = 1 (2" ) $$$ # 3 jk (k 1, k 2, k 3 ) d 3 k! jk (k 1, k 2, k 3 ): Spectral tensor of turbulence (how much energy there is in each wave vector k) In homogeneous isotropic turbulence (simplest case, with no preferred directions) the spectral tensor function of a vector can be expressed based on a single scalar function of magnitude of wavenumber, E(k):! jk (k) = 1 % # 4"k 2 jk $ k k j k ( & ' k 2 ) * E(k)

Turbulence Physics: the energy cascade (Richardson 1922, Kolmogorov 1941) L From: Multimedia Fluid Mechanics, Cambridge Univ. Press! K

Turbulence Physics: the energy cascade (Richardson 1922, Kolmogorov 1941) L Injection of kinetic energy into turbulence (from mean flow)!! u" 3 L u! " 1 3 u! i u i! u! 2 Time! u! 2 L / u!!! L u 3 Constant Flux of energy across scales: ε!! u "(r 1) 3 r 1!! u "(r 2 )3 r 2! K! = 2" # u$ % i # u i $ + # u$ j #x ' j & #x j #x i Dissipation of kinetic energy into heat (due to molecular friction) ( * )

Turbulence Physics: the energy cascade (Richardson 1922, Kolmogorov 1941) Injection of kinetic energy into turbulence (from mean flow) Constant Flux of energy across scales: ε! k L = 1 L k k k! = 1! K E = f (k,!) Dimensional Analysis (Pi-theorem: 3-2=1): E k 5/3! 2/3 = const Dissipation of kinetic energy into heat (due to molecular friction) E(k) = c K! 2/3 k "5/3

Solid line is equivalent of a 3D radial spectrum equal to Solid experimental E(k) = c K! 2/3 k "5/3, c K ~ 1.6 support for K-41: ε Supports (approximately) the notion that ε is the only relevant physical scale in the inertial range, + L at large scales + ν at small scales

N-S equations: Direct Numerical Simulation:!u j!t +!u k u j!x k = "!p!x j + #$ 2 u j + g j!u j!x j = 0 Moderate Re (~ 10 3 ), DNS possible High Re (~ 10 7 ), DNS impossible From: Multimedia Fluid Mechanics, Cambridge Univ. Press

DNS - pseudo-spectral calculation method (Orszag 1971: for isotropic turbulence - triply periodic boundary conditions) û(k,t) = FFT[u(x,t)] k! û(k,t) = 0, "û(k,t) "t = P(k)!(u!# $ )(k) % &k2û(k,t) + P(k)! ˆf(k,t) (u! " )(k) = FFT[u(x,t)! #! u(x,t)] u(x,t) x 2 FFT k 2 û(k,t) x 1 k 1

DNS - pseudo-spectral calculation method (Orszag 1971: for isotropic turbulence - triply periodic boundary conditions) u(x,t) x 2 FFT k 2 û(k,t) x 1 k 1 Computational state-of-the-art: 128 3-256 3 : Routine, can be run on small PC with (256 3 x 12 x 4) Bytes = 100 MB of RAM 1024 3 : needs cluster with O(100) nodes and O(64 GB) RAM 4096 3 : world record (Earth Simulator, Japan, 2003, 30 - TFlops), 4 TB RAM

Time evolution of velocity in isotropic turbulence On JHU Cluster 1,024 3 grid points shown: u 2 on a 256 2 planar subset Source: JHU Turbulence Database Cluster Li, Perlman, Yang, Wan, Burns, Chen, Szalay, Eyink, Meneveau, J. of Turbulence 9, No 31, 2008

JHU turbulence database cluster with Prof. Alex Szalay (Physics & Astronomy), Randal Burns (Computer Sci.), Shiyi Chen (Mech. Eng. & Greg Eyink (Applied Math & Stat). Homework assignment using database See handout Read out velocity along a fixed line, and animate as function of time Filter using a box filter, and animate Compute the energy spectra of unfiltered and filtered velocity

Regions of large vorticity in isotropic turbulence World-record DNS (Nagoya group in Japan): On Earth Simulator (2003) 4,096 3 grid points Source: Kaneda & Ishihara, J. of Turbulence, 2006

Large-eddy-simulation (LES) and filtering:! u j!t! u j + u k = "! p + #$ 2 u!x k!x j "! % j!x jk k Δ G(x): Filter

Large-eddy-simulation (LES) and filtering: N-S equations: G(x): Filter!u j!t +!u ku j!x k = "!p!x j + #$ 2 u j Filtered N-S equations:!u j!x j = 0 Δ! u j!t! u j!t +!u ku j = "! p + #$ 2!x k!x j u j! u j + u k = "! p + #$ 2 u!x k!x j "! % j!x jk k Δ where SGS stress tensor is:! ij = u i u j " u i u j

% $!u! d ij = "# i sgs + $!u j ' & $x j $x i ( * ) = "2#! sgs S ij Length-scale: ~ Δ (instead of L), Velocity-scale ~ Δ S! sgs ~ " 2! S! sgs = ( c s ") 2 S c s : Smagorinsky constant

c s =0.16 works well for isotropic, high Reynolds number turbulence But in practice (complex flows) c s = c s (x,t) Ad-hoc tuning? Examples: Transitional pipe flow: from 0 to 0.16 Near wall damping for wall boundary layers (Piomelli et al 1989) y c s = 0.16 c s = c s (y)

Measure empirical Smagorinsky coefficient for atmospheric surface layer as function of height and stability (thermal forcing or damping): HATS - 2000 (with NCAR researchers: Horst, Sullivan) Kettleman City (Central Valley, CA) $ c s = & & %! " jk S jk 2# 2 S S ij S ij ' ) ) ( 1/ 2 Example result: effect of atmospheric stability on coefficient from sonic anemometer measurements in atmospheric surface layer (Kleissl et al., J. Atmos. Sci. 2003) c s = c s (x,t) Neutral stratification Stable stratification

How to avoid tuning and case-by-case adjustments of model coefficient in LES? The Dynamic Model (DOWNSCALING) (Germano et al. Physics of Fluids, 1991)

Germano identity and dynamic model (Germano et al. 1991): Exact ( rare in turbulence): u i u j! u i u j = u i u j! u i u j E(k) τ k LES resolved SGS

Germano identity and dynamic model (Germano et al. 1991): Exact ( rare in turbulence): u i u j! u i u j = u i u j - u i u j + u i u j! u i u j E(k) τ k LES resolved SGS

Germano identity and dynamic model (Germano et al. 1991): Exact ( rare in turbulence): u i u j! u i u j = u i u j - u i u j + u i u j! u i u j E(k) T ij = L ij! (T ij! " ij ) = 0! ij + L ij L T τ k LES resolved SGS

Germano identity and dynamic model (Germano et al. 1991): Exact ( rare in turbulence): u i u j! u i u j = u i u j - u i u j + u i u j! u i u j E(k) T ij =! ij + L ij L ij! (T ij! " ij ) = 0!2( c s 2" ) 2 S S ij!2( c s ") 2 S S ij L T τ k Assumes scale-invariance: L ij! c s 2 M ij = 0 where M ij = 2! 2 ( S S " 4 S S ) ij ij LES resolved SGS

Germano identity and dynamic model (Germano et al. 1991): L ij! c s 2 M ij = 0 Over-determined system: solve in some average sense (minimize error, Lilly 1992):! = ( L ij " c 2 s M ij ) 2 Minimized when: c s 2 = L ij M ij M ij M ij Averaging over regions of statistical homogeneity or fluid trajectories Lagrangian dynamic model (CM, Tom Lund & Bill Cabot, JFM 1996): Average in time, following fluid particles for Galilean invariance: A = t #!" A(t') (t-t') 1 T e! T dt

Examples: Diurnal cycle: start stably stratified, then heating. Resulting dynamic coefficient (averaged): Consistent with HATS field measurements:

Examples: LES of flow in thrust-reversers Blin, Hadjadi & Vervisch (2002) J. of Turbulence.

Examples: LES of flow in turbomachinery Zou, Wang, Moin, Mittal. (2007) Journal of Fluid Mechanics.

Downtown Baltimore: URBAN CONTAMINATION AND TRANSPORT wind Momentum and scalar transport equations solved using LES and Lagrangian dynamic subgrid model. Buildings are simulated using immersed boundary method. Yu-Heng Tseng, C. Meneveau & M. Parlange, 2006 (Env. Sci & Tech. 40, 2653-2662)

Modeling turbulent flow over fractal trees with renormalized numerical simulation (RNS) Charles Meneveau PhD thesis of Dr. Stuart Chester Co-investigator: M.B. Parlange Funding: National Science Foundation NSF Divisions of Atmospheric Sciences (CMG)

academic fractal trees: sample LES 5 gen branches

Even simpler fractal trees: in-plane fractal (D<2), perpendicular to flow

Fractal Tree Description: Iterated Function Systems Self-similar fractals: scale ratio r < 1 Iterated Function System (IFS) description : Affine mapping maps trunk to i th sub-branch # sub-branches per branch r 2 l 0 rl 0 Prefractal: Finite # branch generations g Inner cutoff scale l 0 g to get limiting fractal Barnsley & Demko, Proc. R. Soc. Lond. A (1985) N B = 3 r = 1/2 g = 2

Branch-Resolved Simulations (BRS-LES) Flow u = 0 τ wall g = 2 u = 0 u = 0 d 8h Mesh spacing h Fluid: Large-Eddy Simulation (LES), Smagorinsky model and/or scale-dependent Lagrangian dynamic model (Bou-Zeid et al. Phys Fluids 2005) Filtered-coarse-grained Navier-Stokes equations:!u j!t + u k!u j!x k = "!p!x j + #$ 2 u j "!!x k % jk Smagorinsky closure:! ij dev = "2c s 2 # 2 S S ij

Branch-Resolved Simulations (BRS-LES) Flow u = 0 τ wall g = 2 u = 0 u = 0 d 8h Mesh spacing h Fluid: Large-Eddy Simulation (LES), Smagorinsky model and/or scale-dependent Lagrangian dynamic model (Bou-Zeid et al. Phys Fluids 2005),! dev ij = "2c 2 s # 2 S S ij Spectral in horizontal / 2nd-order finite difference in vertical Tree configuration: single spanwise row Inflow from precursor simulation of turbulent channel flow Resolved branches: u = 0 in branches, Immersed Boundary (IB) method (Mohd-Yusof, 1997) τ wall from rough wall log-layer velocity profile Min. branch diameter ~8 grid pts for reasonable drag (Y.H. Tseng et al. Env. Sci. & Tech.) Simulations with cutoff branch generation g = 0, 1, 2, 3 512 x 256 x 256 resolution, 64 CPUs

L g = 0 Drag Calculations Using Branch-Resolved Simulations L g = 1 Superposition g BRS g = 2 g = 3

Upper bound on C T Augment tree with resolved plates (to replace unresolved branches) Approach fractal limit from above g = 0 Plates ΔC T Reduce uncertainty? Do it affordably? g = 1 g = 2 Model forces on unresolved branches

Proposed Downscaling Strategy: RNS (Chester et al., J. Comp. Phys. 2007) Use scale invariance: Exploit resolved scales for parameterization of unresolved scales (similar to dynamic model - Germano et al 1991) Resolved branches: scale-model of unresolved branches Measurements on scale-model Rescale & apply to model full-scale prototype = unresolved branches Fluid problem: use LES Resolved Unresolved RNS: downscale forces

Unresolved Branch Force Force field β = branch at generation g+1 and up Resolved branch force: IB method generations 0,, g Unresolved branch force: f β (x) = momentum sink: β & descendants generations g+1 and up Filtered branch indicator function Representation of β & descendants on mesh Support of f β (x) gen. g+1 Total force on β and descendants β Drag model neglect lift, etc.: T gen. g Drag coeff. (unknown) Mean velocity near β Frontal area β & descendants g = 1

c D (g + 1) = f (Re, geometry) Re : eliminate Re-dependence RNS: Determination of c D Identical for all branches at a given generation Self-similar geometry: assume scale-invariance of c D Measure c D (g + 1) from resolved branches Force model for resolved branch b at generation g: Minimize total square error gen. g b β gen. g+1 (Iterate in time) resolved unresolved

Results: Resolved Geometry Only trunk resolved: g = 0 Coarse 128 64 64 resolution Resistive effect of RNS force along tree shape ~79 % of total tree drag is modeled drag modeling effects of unresolved branches is important Procedure is stable! Instantaneous Mean Time history of c D

Animation, case g=1, periodic in x

Overall tree drag predictions RNS drag estimate consistent with BRS results, intermediate between bounds. BRS + Plates RNS g = 0RNS RNS drag estimate differs from superposition of cylinders estimate by 12.8 % BRS Relatively insensitive to mesh and tree (RNS g level) resolution g = 0 g = 1 g = 2

RNS Predictions: Effect of Fractal Dimension Two geometries: +, N B = 3 y, N B = 2 Scale ratio: r = 0.20, 0.25,, 0.50 [, 0.53] Similarity fractal dimension D = log N B / log r Two flow configurations: single row (SR), lattice (L) Drag increases with fractal dimension: y-config: Instantaneous SR + SR Y L + L Y Mean

Angled branches: 100 0 80 0

3D application: Tripod Tree c n c a

More applications: More realistic trees Corals (data & RANS: see Chang, Iaccarino, Elkins, Eaton & Monismith CTR Annu Res Briefs)

Other Possible RNS Applications (?) Flow in porous media (Sai Rapaka, Meneveau & Chen) fractal distributions of permeability Flow in blood vessels / lungs

Summary Examples of downscaling: - Dynamic model: to determine Smagorinsky coefficient from resolved fluctuations (eddies) - RNS: determine drag coefficient of small-scale branchings from resolved branches - General theme: using the dynamical simulation for the scales that are being resolved teach us as much as possible about the physics (often better compared to ad-hoc models)