Introduction to Turbulence, its NumericalModeling and Downscaling

Charles Meneveau

Department of Mechanical Engineering Center for Environmental and Applied Fluid Mechanics

Johns Hopkins University

Mechanical Engineering

560.700: Applications of Science-Based Coupling of Models

October, 2008

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 flamepropagation in decaying

isotropic turbulence

Turbulence in aerospace systems:

LES of flow in thrust-reversersBlin, Hadjadi & Vervisch (2002) J. of Turbulence.

Turbulence in thermofluid equipment:

From: Multimedia Fluid Mechanics, Cambridge Univ. Press

Simplest turbulence: Isotropic decaying turbulence

Contractions

Active GridM = 6 "

1x

2x

Test Section

1u

2u

Flow

Active GridM = 6 "

JHU Corrsin wind tunnel

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

•Density field•Velocity vector field•Pressure field •Temperature field (or internal energy, or enthalpy etc..)

Physical quantities describing fluid flow

Navier Stokes equations for a Newtonian, incompressible fluid

Navier-Stokes equations, incompressible, Newtonian

!uj

!t+

!uku

j

!xk

= "1

#

!p

!xj

+ $%2uj+ g

j

!uj

!x j= 0

aj=Fj

m

Traditional approach: Reynolds decomposition

!uj

!t+

!uku

j

!xk

= "1

#

!p

!xj

+ $%2uj+ g

j

!uj

!x j= 0

t

!uj

!t+

!uku

j

!xk

= "1

#

!p

!xj

+ $%2uj+ g

j

!uj

!x j= 0

!uj

!t+!u

ju

k

!xk

= "1

#

!p

!xj

+ $%2uj+ g

j"

!

!xk

uju

k" u

ju

k( )

!uj

!xj

= 0

Reynolds’ equations:

t

Traditional approach: Reynolds decomposition

!uj

!t+!u

ju

k

!xk

= "1

#

!p

!xj

+ $%2uj+ g

j"

!

!xk

uju

k" u

ju

k( )

!uj

!xj

= 0

! jk

R= ujuk " ujuk

Written as velocity co-variance tensor:

! jk

R= (uj + "uj )(uk + "uk ) # ujuk = ujuk + uj "uk + uk "u j + "uj "uk # ujuk = "u j "uk

0 0

0

• Kinematic Reynolds stress (minus):

• Spectral representation of co-variance tensor:

!u j !uk =1

(2" )3

# jk (k1,k

2,k

3) d

3k$$$

! jk (k1,k2 ,k3): Spectral tensor of turbulence

(how much energy there is in each wave vector k)

In homogeneous isotropic turbulence (simplest case, with no preferreddirections) the spectral tensor function of a vector can be expressed based ona single scalar function of magnitude of wavenumber, E(k):

! jk (k) =1

4"k2# jk $

kjkk

k2

%&'

()*

E(k)

• Turbulence Physics: the energy cascade (Richardson 1922, Kolmogorov 1941)

From: Multimedia Fluid Mechanics, Cambridge Univ. Press

L

!K

• Turbulence Physics: the energy cascade (Richardson 1922, Kolmogorov 1941)

Injection of kinetic energy intoturbulence (from mean flow)

!u2

Time!

!u2

L / !u!

!u3

L

Dissipation of kinetic energy intoheat (due to molecular friction)

! !"u3

L

! !"u (r1)

3

r1

! = 2"# $ui#x

j

# $ui#x

j

+# $uj

#xi

%

&'

(

)*

! !"u (r2 )

3

r2

L

!K

Con

stan

t Flu

x of

ene

rgy

acro

ss s

cale

s: ε

!u "1

3ui!u

i!

• Turbulence Physics: the energy cascade (Richardson 1922, Kolmogorov 1941)

Injection of kinetic energy intoturbulence (from mean flow)

Dissipation of kinetic energy intoheat (due to molecular friction)

!

Con

stan

t Flu

x of

ene

rgy

acro

ss s

cale

s: ε

k

kL=1

L

k! =1

!K

k

E = f (k,!)

Dimensional Analysis (Pi-theorem: 3-2=1):

E k5 /3

!2 /3

= const

E(k) = cK!2 /3k"5 /3

Solid experimentalsupport for K-41:

Supports (approximately)the notion that ε is the onlyrelevant physical scale inthe inertial range,+ L at large scales+ ν at small scales

ε E(k) = c

K!

2/3k"5/3, c

K~ 1.6

Solid line is equivalent of a 3D radial spectrum equal to

Direct Numerical Simulation:

N-S equations:!u j

!t+!uku j

!xk= "

!p

!x j+ #$

2uj + gj

!uj

!x j= 0

Moderate Re(~ 103),DNS possible

High Re(~ 107),DNS impossible

From: Multimedia Fluid Mechanics, Cambridge Univ. Press

DNS - pseudo-spectral calculation method(Orszag 1971: for isotropic turbulence - triply periodic boundary conditions)

k ! u(k, t) = 0, "u(k, t)

"t= P(k) ! (u # $!)(k) %&k

2u(k, t) + P(k) ! f(k, t)

u(k, t) = FFT [u(x,t)]

(u ! "!)(k) = FFT [u(x,t) ! # ! u(x,t)]

FFT

x1

x2

k2

k1

u(k, t)

u(x,t)

FFT

x1

x2

k2

k1

u(k, t)

u(x,t)

Computational state-of-the-art:

1283 - 2563: Routine, can be run on smallPC with (2563 x 12 x 4) Bytes = 100 MB of RAM

10243 : needs cluster with O(100) nodes and O(64 GB) RAM40963 : world record (Earth Simulator, Japan, 2003, 30 - TFlops), 4 TB RAM

DNS - pseudo-spectral calculation method(Orszag 1971: for isotropic turbulence - triply periodic boundary conditions)

Time evolution of velocity in isotropic turbulence

Source: JHU Turbulence Database ClusterLi, Perlman, Yang, Wan, Burns, Chen, Szalay, Eyink, Meneveau, J. of Turbulence 9, No 31, 2008

On JHU Cluster 1,0243 grid points shown: u2 on a 2562 planar subset

JHU turbulence database clusterwith 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 (Nagoyagroup in Japan):

Source: Kaneda & Ishihara,J. of Turbulence, 2006

On Earth Simulator (2003)

4,0963 grid points

Δ

G(x): Filter

Large-eddy-simulation (LES) and filtering:

!˜ u j

!t+ ˜ u k

!˜ u j

!xk

= "!˜ p

!x j

+ #$2˜ u j "

!

!xk

% jk

Δ

Δ

G(x): Filter

Large-eddy-simulation (LES) and filtering:

N-S equations:

!˜ u j

!t+ ˜ u k

!˜ u j

!xk

= "!˜ p

!x j

+ #$2˜ u j "

!

!xk

% jk

Filtered N-S equations:

!uj

!t+!ukuj

!xk= "

!p

!x j

+ #$2

u j

!˜ u j

!t+!ukuj

!xk

= "!˜ p

!x j

+ #$2

˜ u j

where SGS stress tensor is:

! ij = uiu j " ˜ u i ˜ u j

!uj

!x j= 0

! ijd= "# sgs

$ !ui$x j

+$ !uj

$xi

%

&'

(

)* = "2# sgs

!Sij

!sgs = cs"( )2

| ˜ S |

Length-scale: ~ Δ (instead of L), Velocity-scale ~ Δ |S|

! sgs ~ "

2| !S |

cs: “Smagorinsky constant”

But in practice (complex flows)

cs

= cs(x, t)

Ad-hoc tuning?

cs=0.16 works well for isotropic,high Reynolds number turbulence

Examples: Transitional pipe flow: from 0 to 0.16

Near wall damping for wall boundary layers (Piomelli et al 1989)

cs= c

s(y)

ycs= 0.16

Measure “empirical” Smagorinsky coefficient for atmospheric surface layeras function of height and stability (thermal forcing or damping):

cs =

! " jk˜ S jk

2#2| ˜ S | ˜ S ij

˜ S ij

$

%

& &

'

(

) )

1/ 2

Example result: effect of atmosphericstability on coefficient from sonicanemometer measurements inatmospheric surface layer(Kleissl et al., J. Atmos. Sci. 2003)

HATS - 2000(with NCARresearchers:

Horst, Sullivan)Kettleman City

(Central Valley, CA)

Stable stratification

Neu

tral

str

atifi

catio

n

cs

= cs(x, t)

How to avoid “tuning” and case-by-caseadjustments of model coefficient in LES?

The Dynamic Model (DOWNSCALING)(Germano et al. Physics of Fluids, 1991)

E(k)

k

LES resolved SGS

τ

Germano identity and dynamic model(Germano et al. 1991):

Exact (“rare” in turbulence):

uiuj ! ˜ u i ˜ u j = uiu j ! ˜ u i ˜ u j

E(k)

k

LES resolved SGS

τ

Germano identity and dynamic model(Germano et al. 1991):

Exact (“rare” in turbulence):

uiuj ! ˜ u i ˜ u j = uiu j - ˜ u i ˜ u j + ˜ u i ˜ u j ! ˜ u i ˜ u j

E(k)

k

LES resolved SGS

L τT

Germano identity and dynamic model(Germano et al. 1991):

Exact (“rare” in turbulence):

Tij = ! ij + Lij

uiuj ! ˜ u i ˜ u j = uiu j - ˜ u i ˜ u j + ˜ u i ˜ u j ! ˜ u i ˜ u j

Lij ! (Tij ! " ij ) = 0

E(k)

k

LES resolved SGS

L τT

Germano identity and dynamic model(Germano et al. 1991):

Exact (“rare” in turbulence):

Tij = ! ij + Lij

uiuj ! ˜ u i ˜ u j = uiu j - ˜ u i ˜ u j + ˜ u i ˜ u j ! ˜ u i ˜ u j

!2 cs 2"( )2

| ˜ S | ˜ S ij

!2 cs"( )2

| ˜ S | ˜ S ij

Assumes scale-invariance:

Lij ! (Tij ! " ij ) = 0

where

Lij ! cs2

Mij = 0

Mij = 2!2

| ˜ S | ˜ S ij " 4 | ˜ S | ˜ S ij( )

Germano identity and dynamic model(Germano et al. 1991):

Minimized when:Averaging over regions ofstatistical homogeneityor fluid trajectories

Lagrangian dynamic model (CM, Tom Lund & Bill Cabot, JFM 1996):Average in time, following fluid particles for Galilean invariance:

A = A(t ') !"

t

#1

Te!

(t - t' )

T dt

Lij ! cs2

Mij = 0

Over-determined system:solve in “some average sense”(minimize error, Lilly 1992):

! = Lij " cs2

Mij( )2

cs2

=Lij Mij

Mij Mij

• Diurnal cycle: start stably stratified, then heating….

Examples:

Resulting dynamic coefficient (averaged):

Consistent with HATS field measurements:

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.

Examples:

Downtown Baltimore:

URBAN CONTAMINATIONAND TRANSPORT

Yu-Heng Tseng, C. Meneveau & M. Parlange, 2006 (Env. Sci & Tech. 40, 2653-2662)

Momentum and scalar transport equations solved using LES andLagrangian dynamic subgrid model. Buildings are simulated usingimmersed boundary method.

wind

Modeling turbulent flow over fractal treeswith renormalized numerical

simulation (RNS)

Charles Meneveau

PhD thesis of Dr. Stuart ChesterCo-investigator: M.B. Parlange

Funding: National Science FoundationNSF 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

Affine mapping maps trunk to ith sub-branch

# sub-branchesper branch

NB = 3r = 1/2

l0

rl0

g = 2

r2l0

Fractal Tree Description: Iterated Function Systems• Self-similar fractals: scale ratio r < 1• Iterated Function System (IFS) description∗:

• Prefractal:–Finite # branch generations g–Inner cutoff scale

– g → ∞ to get limiting fractal

•∗ Barnsley & Demko, Proc. R. Soc. Lond. A (1985)

g = 2

Flow

u = 0

τwall

u = 0

u = 0

Mesh spacing h

d ≥ 8h

Branch-Resolved Simulations (BRS-LES)

• Fluid: Large-Eddy Simulation (LES), Smagorinsky model and/or scale-dependent Lagrangian dynamic model (Bou-Zeid et al. Phys Fluids 2005)

!uj

!t+ uk

!uj

!xk= "

!p

!x j+ #$

2uj "

!

!xk% jk

Filtered-coarse-grained Navier-Stokes equations:

Smagorinsky closure: ! ijdev

= "2cs2#

2| S | Sij

g = 2

Flow

u = 0

τwall

u = 0

u = 0

Mesh spacing h

d ≥ 8h

Branch-Resolved Simulations (BRS-LES)

• Fluid: Large-Eddy Simulation (LES), Smagorinsky model and/or scale-dependent Lagrangian dynamic model (Bou-Zeid et al. Phys Fluids 2005),

• 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)

– τwall 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

! ijdev

= "2cs2#

2| S | Sij

BRS

Superposition g →∞L

Lg = 0

g = 1

g = 3g = 2

Drag Calculations Using Branch-Resolved Simulations

• Reduce uncertainty?• Do it affordably?

Model forceson unresolvedbranches

ΔCTg = 0

g = 1 g = 2

Plates

Upper bound on CT• Augment tree with resolved plates (to replace unresolved branches)• Approach fractal limit “from above”

Resolved UnresolvedRNS:downscale forces

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 branch force:IB methodgenerations 0, …, g

Unresolved branch force:fβ(x) = momentum sink: β & descendantsgenerations g+1 and up

β = branch at generation g+1 and up

β

T

g = 1Mean velocity“near” β

Frontal areaβ & descendants

Drag coeff.(unknown)

gen. g+1

gen. g

Unresolved Branch Force• Force field

• Filtered branch indicator function–Representation of β & descendants on mesh–Support of fβ(x)

• Total force on β and descendants

• Drag model—neglect lift, etc.:

β b gen. g+1

gen. g

(Iterate in time) resolvedunresolved

RNS: Determination of cD• cD(g + 1) = f (Re, geometry)

– Re → ∞: eliminate Re-dependence– Identical for all branches at a given generation – Self-similar geometry: assume scale-invariance of cD

⇒ Measure cD(g + 1) from resolved branches

• Force model for resolved branch b at generation g:

• Minimize total square error

Resolved Geometry

MeanInstantaneous Time history of cD

Results:

• 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 !

Animation, case g=1, periodic in x

g = 0 g = 1 g = 2

RNSg = 0

Overall tree drag predictions

• RNS drag estimate consistent with BRS results, intermediate between bounds.

• RNS drag estimate differs from superposition of cylinders estimate by 12.8 %

• Relatively insensitive to mesh and tree (RNS g level) resolution

BRS

BRS + Plates RNS

Mean

y-config: Instantaneous

SR +SR YL +L Y

RNS Predictions: Effect of Fractal Dimension• Two geometries:

–“+”, NB = 3–“y”, NB = 2

• Scale ratio: r = 0.20, 0.25, …, 0.50 [, 0.53]• Similarity fractal dimension D = −log NB / log r• Two flow configurations: single row (SR), lattice (L)• Drag increases with fractal dimension:

Angled branches:

1000 800

cn

ca

3D application: “Tripod” Tree

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)