Introduction to fundamentals of Turbulence, Large Eddy Simulation ...

Ciclo di seminari Università degli studi di Roma “Tor Vergata” May, 2013

Introduction to fundamentals of Turbulence, Large Eddy Simulation and subgrid-scale modeling Charles Meneveau Johns Hopkins University Baltimore, USA Additional reference: http://www.scholarpedia.org/article/Turbulence:_Subgrid-Scale_Modeling

JHU Mechanical Engineering

Ciclo di seminari Universita degli studi di Roma, Tor Vergata May, 2013

OVERVIEW: Mercoledì: •  •  •  •  • 

Introduction to fundamentals of Turbulence Intro to Large Eddy Simulation (LES) Intro to Subgrid-scale (SGS) modeling The dynamic SGS model Some sample applications from our group

Venerdì: •  Dynamic model for LES over rough multiscale surfaces •  LES studies of large wind farms

JHU Mechanical Engineering

Turbulence = eddies of many sizes

From: Multimedia Fluid Mechanics, Cambridge Univ. Press

Turbulence = eddies of many sizes + large-scale vortices

From: Multimedia Fluid Mechanics, Cambridge Univ. Press

Turbulence in aerospace systems: Jets and eddies during shuttle launch

Large Eddy Simulation of flow in thrust-reversers Blin, Hadjadi & Vervisch (2002) J. of Turbulence.

Turbulence in reacting flows: Premixed flame in I.C. engine, combustion

Numerical simulation of flame propagation in decaying isotropic turbulence

Turbulence in environment and geophysics

Turbulence in renewable energy

From J.N. Sørensen, Annual Rev. Fluid Mech. 2011:"

Turbulence in astrophysics

Turbulence in astrophysics

Simplest turbulence: Isotropic turbulence Corrsin wind tunnel at the Johns Hopkins University Active Grid M = 6 "

Contractions

Test Section

Flow

Decaying turbulence

Johns Hopkins University: Baltimore, Maryland USA JHU: Latrobe Hall (Mechanical Engineering) Downtown Baltimore

Maryland Hall

Corrsin wind tunnel at JHU

Stan Corrsin 1920-1986

Turbulence is: •  multiscale, •  mixing, •  dissipative, •  chaotic, •  vortical •  well-defined statistics, •  important in practice

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 (linear momentum) •  First law of thermodynamics (energy) •  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 =0 ∂x j ∂u j ∂t

+

∂uk u j ∂xk

1 ∂p =− + ν∇ 2u j + g j ρ ∂x j

aj =

Fj m

Traditional approach: Reynolds decomposition

∂u j =0 ∂x j ∂u j ∂t

+

∂uk u j ∂xk

1 ∂p =− + ν∇ 2u j + g j ρ ∂x j

t

Traditional approach: Reynolds decomposition

∂u j =0 ∂x j ∂u j ∂t

+

∂uk u j ∂xk

1 ∂p =− + ν∇ 2u j + g j ρ ∂x j

Reynolds’ equations: ∂u j =0 ∂x j ∂u j ∂t

+

∂u j uk ∂xk

t

(

1 ∂p ∂ 2 =− + ν∇ u j + g j − u j uk − u j uk ρ ∂x j ∂xk

)

u ′j uk′ Reynolds’ stress tensor: Requires closure (“turbulence problem”)

E(k)  k −5 / 3

Rate of energy cascade and dissipation !

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

L

ηK

Turbulence problem: “last unsolved problem in classical physics” (e.g. Feynman 1979) Sample challenges (unsolved) Reynolds’ stress tensor: u ′j u k′ Requires closure (“turbulence problem”) First-principles derivation of Kolmogorov’sk-5/3 spectrum “With turbulence, it's not just " a case of physical theory " being able to handle only " simple cases—we can't do any." We have no good fundamental " theory at all.” (Feynman, 1979," Omni Magazine, Vol. 1, No.8)."

E(k)  k −5 / 3

Direct Numerical Simulation (DNS): N-S equations:

∂u j ∂t

+

∂u k u j ∂xk

∂p =− + ν∇ 2 u j + g j ∂x j

Moderate Re (~ 103), DNS possible

∂u j =0 ∂x j

High Re (~ 107), DNS impossible

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

ˆ u(k,t) = FFT [u(x,t)] ˆ k ⋅ u(k,t) = 0,

ˆ ∂u(k,t) ˆ ˆ = P(k) ⋅ (u × ω )(k) − ν k 2 u(k,t) + P(k) ⋅ f(k,t) ∂t (u × ω )(k) = FFT [u(x,t) × ∇ × u(x,t)]

FFT

x2

k2

ˆ u(k,t)

u(x,t)

x1

k1

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

FFT

x2

k2

ˆ u(k,t)

u(x,t)

x1

k1

Computational state-of-the-art: 1283 - 2563: Routine, can be run on small PC with (2563 x 12 x 4) Bytes = 100 MB of RAM 10243 : needs cluster with O(100) nodes and O(64 GB) RAM 40963 : world record (Earth Simulator, Japan, 2003, 30 - TFlops), 4 TB RAM

1,0243 DNS: iso-velocity filled contours (data from: JHU public database cluster, Claire Verhulst & Jason Graham Matlab visualization)

Energetic signature of the large, intermediate (and some smaller) turbulent eddies:

u1 (x, y, z0 ,t 0 )

~ 40 grid points 1,024 grid points 1,024 grid points

1,024 grid points

Take 10243 DNS of forced isotropic turbulence (standard pseudo-spectral Navier-Stokes simulation, dealiased):

10244 space-time history 27 Tbytes, Rel~ 430

http://turbulence.pha.jhu.edu

Y. Li, E. Perlman, M. Wan, Y. Yang, R. Burns, C.M., R. Burns, S. Chen, A. Szalay & G. Eyink: “A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence”. Journal of Turbulence 9, No 31, 2008. So far 12 papers published using data from public database

Take 10243 DNS of forced isotropic turbulence (standard pseudo-spectral Navier-Stokes simulation, dealiased): Running Fortran or C

Running Matlab

Manual query On web-browser

10244 space-time history 27 Tbytes, Rel~ 430

http://turbulence.pha.jhu.edu

Y. Li, E. Perlman, M. Wan, Y. Yang, R. Burns, C.M., R. Burns, S. Chen, A. Szalay & G. Eyink: “A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence”. Journal of Turbulence 9, No 31, 2008. So far 12 papers published using data from public database

New paradigm: Client computer (e.g. my laptop) runs the analysis using Fortran, C, Matlab codes, fetching data as needed from databases through a web-service we adapted Fortran, C and Matlab to “surf the web” for data

Y. Li, E. Perlman, M. Wan, Y. Yang, R. Burns, C. Meneveau, R. Burns, S. Chen, A. Szalay & G. Eyink: Journal of Turbulence 9, No 31, 2008.

Coupling GPU’s with databases: material surfaces (24 million particles) advected in turbulent 1,0243 field Real-time flow-viz (download 1 snapshot at fixed t, animated on GPU)

Dr. Kai Buerger (TUM, summer 2011) M. Karweit (MS Thesis, & Album of Fluid Motion, CUP)

iso-vorticity surfaces (JHU database, Dr. Kai Buerger visualization

Vortical signature of the smallest turbulent eddies:

)

(∇ × u)2

ai =

Fi m

Data mining: high-intensity vorticity event associated with vortex reconnections

Colors: surfaces of iso-vorticity magnitude 1024

64

Video generated by Dr. Jonathan P. Graham using VAPOR software (NCAR)

Basic (and applied) research challenge: Coarse-graining - Large-Eddy-Simulation (LES): Coarse-graining for more affordable simulations

u1 (x, y, z0 ,t 0 )

4x109 d.o.f.

u1 (x, y, z0 ,t 0 )

105 d.o.f.

Ongoing Research Questions: •  How do small-scales affect large scale motions (and vice-versa)? •  How can we replace the effects of small scales on large scales (SGS modeling)?

Large-eddy-simulation (LES) and filtering: G(x): Filter

N-S equations: ∂u j ∂uk u j ∂p + =− + ν∇2 u j ∂t ∂x k ∂x j

∂u j =0 ∂x j

Δ u1 (x, y, z0 ,t 0 )

ui (x,t) = GΔ * ui = ∫ GΔ (x − x ')ui (x ')d 3x ' D

Large-eddy-simulation (LES) and filtering: G(x): Filter

N-S equations: ∂u j ∂uk u j ∂p + =− + ν∇2 u j ∂t ∂x k ∂x j

Filtered N-S equations:

∂u j =0 ∂x j

Δ u1 (x, y, z0 ,t 0 )

 ∂u ∂u j  ∂p ku j + =− + ν∇ 2u j ∂t ∂xk ∂x j D

Large-eddy-simulation (LES) and filtering: G(x): Filter

N-S equations: ∂u j ∂uk u j ∂p + =− + ν∇2 u j ∂t ∂x k ∂x j

∂u j =0 ∂x j

Filtered N-S equations:

Δ u1 (x, y, z0 ,t 0 )

 ∂u ∂u j  ∂p ku j + =− + ν∇ 2u j ∂t ∂xk ∂x j ∂u ˜j ∂u ˜j ˜ ∂p ∂ 2 ˜ ˜ + uk =− + ν∇ u j − τ ∂t ∂x k ∂x j ∂x k jk where SGS stress tensor is:

τ ij = uiu j − u˜i u˜ j

D

Most common modeling approach: eddy-viscosity ⎛ ∂ui ∂u j ⎞ τ = −ν sgs ⎜ + = −2ν sgs Sij ⎟ ⎝ ∂x j ∂xi ⎠ d ij

Functional form in analogy to kinetic theory of gases (Chapman-Enskog expansions, etc..) “Eddies ~ molecules” (???)

ν sgs = (c sΔ) | S˜ | 2

cs: “Smagorinsky coefficient”

Detour: some remarks on eddy-viscosity ⎛ ∂ui ∂u j ⎞ τ = −ν sgs ⎜ + = −2ν sgs Sij ⎟ ⎝ ∂x j ∂xi ⎠ d ij

Functional form in analogy to kinetic theory of gases (Chapman-Enskog expansions, etc.. “Eddies ~ molecules” (???)

Limitations of basic eddy-viscosity:

⎛ ∂ui ∂u j ⎞ τ = −ν sgs ⎜ + = −2ν sgs Sij ⎟ ⎝ ∂x j ∂xi ⎠ d ij

Turbulence is not like a “can of sand”

Limitations of basic eddy-viscosity:

⎛ ∂ui ∂u j ⎞ τ = −ν sgs ⎜ + = −2ν sgs Sij ⎟ ⎝ ∂x j ∂xi ⎠ d ij

Turbulence is not like a “can of sand”

but more like a “can of worms”

Visualizations of multi-scale vortices in turbulence

Limitations of basic eddy-viscosity:

⎛ ∂ui ∂u j ⎞ τ = −ν sgs ⎜ + = −2ν sgs Sij ⎟ ⎝ ∂x j ∂xi ⎠ d ij

Turbulence is not like a “can of sand”

but more like a “can of worms”

Still, in LES eddy-viscosity seems to work “better than it should” Also, many models need eddy-viscosity additions in ad-hoc “regularizations” Next slides: an “excuse” for eddy-viscosity via “fluid dynamics” arguments

A “fluid-mechanical” rationale for basic eddy-viscosity:

(

 τ ≡ u i u j − ui u j d ij

) ( ) d

≈ u i′u ′

d

A “fluid-mechanical” rationale for basic eddy-viscosity:

(

 τ ≡ u i u j − ui u j d ij

u′(0)

) ( ) d

≈ u i′u ′

d

∂(ui + ui′) ∂(ui + ui′) + (u k + uk′ ) = forces ∂t ∂xk u′(t)

 A

A “fluid-mechanical” rationale for basic eddy-viscosity:

(

 τ ≡ u i u j − ui u j d ij

u′(0)

) ( ) d

≈ u i′u ′

d

∂(ui + ui′) ∂(ui + ui′) + (u k + uk′ ) = forces ∂t ∂xk u′(t)

 A

dui′ ∂ui =− uk′ + [ forces '− ∇(u 'u ')] dt ∂xk

A “fluid-mechanical” rationale for basic eddy-viscosity:

(

 τ ≡ u i u j − ui u j d ij

u′(0)

) ( ) d

≈ u i′u ′

d

∂(ui + ui′) ∂(ui + ui′) + (u k + uk′ ) = forces ∂t ∂xk u′(t)

 A “Production-only” approximation:

dui′ ∂ui =− uk′ + [ forces '− ∇(u 'u ')] dt ∂xk

du′  u′, ≈ − Ai dt

(stretching and tilting of vel. fluctuation by large-scale velocity gradients - consistent with vortex stretching)

Li, Chevillard, Eyink & CM Phys Rev. E, 2009.

∂ui  Aik = ∂xk

A “fluid-mechanical” rationale for basic eddy-viscosity:

du′  u′ ≈ − Ai dt u′(0)

u′(t)

A

A “fluid-mechanical” rationale for basic eddy-viscosity:

du′  u′ ≈ − Ai dt

 i u′(0) u′(t) = exp(− At)

u′(0)

u′(t)

A

A “fluid-mechanical” rationale for basic eddy-viscosity:

du′  u′ ≈ − Ai dt

 i u′(0) u′(t) = exp(− At)

1  2 ⎛ ⎞  u′(t) ≈ ⎜ I − At + (At) − ...⎟ iu′(0) ⎝ ⎠ 2

u′(0)

u′(t)

A

A “fluid-mechanical” rationale for basic eddy-viscosity:

du′  u′ ≈ − Ai dt u′(0)

1  2 ⎛ ⎞  u′(t) ≈ ⎜ I − At + (At) − ...⎟ iu′(0) ⎝ ⎠ 2

(

)

(

)

 i u′(0) ⎤ ⎡ u′T (0) i I − A  Tt ⎤ u′ ⊗ u′ = u′u′T ≈ ⎡⎣ I − At ⎦⎣ ⎦ u′(t)

A

A “fluid-mechanical” rationale for basic eddy-viscosity:

du′  u′ ≈ − Ai dt u′(0)

1  2 ⎛ ⎞  u′(t) ≈ ⎜ I − At + (At) − ...⎟ iu′(0) ⎝ ⎠ 2

(

)

(

)

(

)

(

)

 i u′(0) ⎤ ⎡ u′T (0) i I − A  Tt ⎤ u′ ⊗ u′ = u′u′T ≈ ⎡⎣ I − At ⎦⎣ ⎦ u′(t)

A

 ≈ I − At  i u ′u ′ T A  u ′u ′ T A

t =0

 Tt i I−A

A “fluid-mechanical” rationale for basic eddy-viscosity:

du′  u′ ≈ − Ai dt u′(0)

1  2 ⎛ ⎞  u′(t) ≈ ⎜ I − At + (At) − ...⎟ iu′(0) ⎝ ⎠ 2

(

)

(

)

(

)

(

)

 i u′(0) ⎤ ⎡ u′T (0) i I − A  Tt ⎤ u′ ⊗ u′ = u′u′T ≈ ⎡⎣ I − At ⎦⎣ ⎦ u′(t)

A

 ≈ I − At  i u ′u ′ T A  u ′u ′ T A isotropy :

(

ce Δ S

t =0

)I 2

 Tt i I−A

A “fluid-mechanical” rationale for basic eddy-viscosity:

du′  u′ ≈ − Ai dt

1  2 ⎛ ⎞  u′(t) ≈ ⎜ I − At + (At) − ...⎟ iu′(0) ⎝ ⎠ 2

(

)

(

)

(

)

(

)

 i u′(0) ⎤ ⎡ u′T (0) i I − A  Tt ⎤ u′ ⊗ u′ = u′u′T ≈ ⎡⎣ I − At ⎦⎣ ⎦

u′(0)

u′(t)

A  u ′u ′ A T

tA

(

≈ ce Δ S

)

2

 ≈ I − At  i u ′u ′ T A  u ′u ′ T A isotropy :

(

 A )i(I − A  T t A ) ⎤ ≈ ce Δ S ⎡⎣(I − At ⎦

(

ce Δ S

t =0

 Tt i I−A

)I 2

) ⎡⎣I − (A + A ) t 2

T

2 S

A

+ O(t 2 ) ⎤⎦

A “fluid-mechanical” rationale for basic eddy-viscosity:

du′  u′ ≈ − Ai dt

1  2 ⎛ ⎞  u′(t) ≈ ⎜ I − At + (At) − ...⎟ iu′(0) ⎝ ⎠ 2

(

)

(

)

u′(0)

u′(t)

 ≈ I − At  i u ′u ′ T A  u ′u ′ T A

A  u ′u ′ A T

tA

(

≈ ce Δ S

τ = u′u′ A d ij

(

)

(

)

 i u′(0) ⎤ ⎡ u′T (0) i I − A  Tt ⎤ u′ ⊗ u′ = u′u′T ≈ ⎡⎣ I − At ⎦⎣ ⎦

T

d tA

)

2

isotropy :

(

 A )i(I − A  T t A ) ⎤ ≈ ce Δ S ⎡⎣(I − At ⎦

(

≈ −2 ce Δ S

)t 2

ν sgs

A

S

(

ce Δ S

t =0

 Tt i I−A

)I 2

) ⎡⎣I − (A + A ) t 2

T

2 S

A

+ O(t 2 ) ⎤⎦

A “fluid-mechanical” rationale for basic eddy-viscosity:

τ = u ′u ′ d ij

T d tA

(

≈ −2 ce Δ S

)t 2

A

S

ν sgs choosing

tA

1 ∝ S

ν sgs = (c sΔ) | S˜ | cs: “Smagorinsky coefficient” 2

Effects of tij upon resolved motions: Energetics (kinetic energy): ∂ 12 u˜ j u˜ j ∂ 12 u˜ j u˜ j ∂ (....) − 2νS˜ jk S˜ jk − −τ jk S˜ jk + u˜ k =− ∂t ∂x k ∂x j

(

)

∂u j ⎞ 1 ⎛ ∂u Sij = ⎜ i + 2 ⎝ ∂x j ∂xi ⎟⎠

Π Δ = − τ jk S˜ jk

u′ 3 ε= L

u1 (x, y, z0 ,t 0 )

Inertial-range flux π Δ

E(k) ΠΔ

LES resolved

Δ SGS

k

Theoretical calibration of cs (D.K. Lilly, 1967): τ ij = −2(cs Δ)2 | S | Sij

Π Δ = ε = − τ ij Sij

ε≈c Δ 2

Sij Sij

2

3/2

Sij Sij

E(k) = cK ε 2 / 3 k −5 / 3 π Δ

E(k)

ε = cs2 Δ 2 2 | S | Sij Sij 2 s

u′ 3 ε= L

3/2

ΠΔ

1 ∂ui ⎛ ∂ui ∂u j ⎞ = + = ⎜ ⎟ 2 ∂x j ⎝ ∂x j ∂xi ⎠

LES resolved

SGS

1 1 k2 2 3 2 E(k) 3 = ∫∫∫ [k j Θii (k) + ki k j Θij (k)]d k = ∫∫∫ [k ( ( δ − )) + 0]d k ii 2 |k|< π / Δ 2 |k|< π / Δ 4π k 2 k2 2/3 1 = cK ε 2

π /Δ

∫ 0

k −5 / 3+ 2

3−1 4π k 2 dk = cK ε 2 / 3 2 4π k

4/3 ⎛ 3 π ⎛ ⎞ ⎞ 2 2 3/2 2/3 ε ≈ cs Δ 2 ⎜ cK ε ⎜ ⎟ 4 ⎝ Δ ⎠ ⎟⎠ ⎝

π /Δ

∫ 0

1/ 3 2/3 3 ⎛ π ⎞ k dk = cK ε ⎜ ⎟ 4 ⎝ Δ⎠

3/2

⎛ 3c ⎞ ⇒ 1 ≈ cs2π 2 ⎜ K ⎟ ⎝ 2 ⎠

3/2

cK = 1.6 ⇒ cs ≈ 0.16

4/3

⎛ 3c ⎞ ⇒ cs = ⎜ K ⎟ ⎝ 2 ⎠

−3/ 4

π −1

k

Theoretical calibration of cs (D.K. Lilly, 1967): τ ij = −2(cs Δ)2 | S | Sij

Π Δ = ε = − τ ij Sij

ε≈c Δ 2

Sij Sij

2

3/2

Sij Sij

E(k) = cK ε 2 / 3 k −5 / 3 π Δ

E(k)

ε = cs2 Δ 2 2 | S | Sij Sij 2 s

u′ 3 ε= L

3/2

ΠΔ

1 ∂ui ⎛ ∂ui ∂u j ⎞ = + = ⎜ ⎟ 2 ∂x j ⎝ ∂x j ∂xi ⎠

LES resolved

SGS

1 1 k2 2 3 2 E(k) 3 = ∫∫∫ [k j Θii (k) + ki k j Θij (k)]d k = ∫∫∫ [k ( ( δ − )) + 0]d k ii 2 |k|< π / Δ 2 |k|< π / Δ 4π k 2 k2 2/3 1 = cK ε 2

π /Δ

∫ 0

k −5 / 3+ 2

3−1 4π k 2 dk = cK ε 2 / 3 2 4π k

4/3 ⎛ 3 π ⎛ ⎞ ⎞ 2 2 3/2 2/3 ε ≈ cs Δ 2 ⎜ cK ε ⎜ ⎟ 4 ⎝ Δ ⎠ ⎟⎠ ⎝

π /Δ

∫ 0

1/ 3 2/3 3 ⎛ π ⎞ k dk = cK ε ⎜ ⎟ 4 ⎝ Δ⎠

3/2

⎛ 3c ⎞ ⇒ 1 ≈ cs2π 2 ⎜ K ⎟ ⎝ 2 ⎠

3/2

cK = 1.6 ⇒ cs ≈ 0.16

4/3

⎛ 3c ⎞ ⇒ cs = ⎜ K ⎟ ⎝ 2 ⎠

−3/ 4

π −1

k

Theoretical calibration of cs (D.K. Lilly, 1967): τ ij = −2(cs Δ)2 | S | Sij

Π Δ = ε = − τ ij Sij

ε≈c Δ 2

Sij Sij

2

3/2

Sij Sij

E(k) = cK ε 2 / 3 k −5 / 3 π Δ

E(k)

ε = cs2 Δ 2 2 | S | Sij Sij 2 s

u′ 3 ε= L

3/2

ΠΔ

1 ∂ui ⎛ ∂ui ∂u j ⎞ = + = ⎜ ⎟ 2 ∂x j ⎝ ∂x j ∂xi ⎠

LES resolved

SGS

1 1 k2 2 3 2 E(k) 3 = ∫∫∫ [k j Θii (k) + ki k j Θij (k)]d k = ∫∫∫ [k ( ( δ − )) + 0]d k ii 2 |k|< π / Δ 2 |k|< π / Δ 4π k 2 k2 2/3 1 = cK ε 2

π /Δ

∫ 0

k −5 / 3+ 2

3−1 4π k 2 dk = cK ε 2 / 3 2 4π k

4/3 ⎛ 3 π ⎛ ⎞ ⎞ 2 2 3/2 2/3 ε ≈ cs Δ 2 ⎜ cK ε ⎜ ⎟ 4 ⎝ Δ ⎠ ⎟⎠ ⎝

π /Δ

∫ 0

1/ 3 2/3 3 ⎛ π ⎞ k dk = cK ε ⎜ ⎟ 4 ⎝ Δ⎠

3/2

⎛ 3c ⎞ ⇒ 1 ≈ cs2π 2 ⎜ K ⎟ ⎝ 2 ⎠

3/2

cK = 1.6 ⇒ cs ≈ 0.16

4/3

⎛ 3c ⎞ ⇒ cs = ⎜ K ⎟ ⎝ 2 ⎠

−3/ 4

π −1

k

Theoretical calibration of cs (D.K. Lilly, 1967): τ ij = −2(cs Δ)2 | S | Sij

Π Δ = ε = − τ ij Sij

ε≈c Δ 2

Sij Sij

2

3/2

Sij Sij

E(k) = cK ε 2 / 3 k −5 / 3 π Δ

E(k)

ε = cs2 Δ 2 2 | S | Sij Sij 2 s

u′ 3 ε= L

3/2

ΠΔ

1 ∂ui ⎛ ∂ui ∂u j ⎞ = + = ⎜ ⎟ 2 ∂x j ⎝ ∂x j ∂xi ⎠

LES resolved

SGS

1 1 k2 2 3 2 E(k) 3 = ∫∫∫ [k j Θii (k) + ki k j Θij (k)]d k = ∫∫∫ [k ( ( δ − )) + 0]d k ii 2 |k|< π / Δ 2 |k|< π / Δ 4π k 2 k2 2/3 1 = cK ε 2

π /Δ

∫ 0

k −5 / 3+ 2

3−1 4π k 2 dk = cK ε 2 / 3 2 4π k

4/3 ⎛ 3 π ⎛ ⎞ ⎞ 2 2 3/2 2/3 ε ≈ cs Δ 2 ⎜ cK ε ⎜ ⎟ 4 ⎝ Δ ⎠ ⎟⎠ ⎝

π /Δ

∫ 0

1/ 3 2/3 3 ⎛ π ⎞ k dk = cK ε ⎜ ⎟ 4 ⎝ Δ⎠

3/2

⎛ 3c ⎞ ⇒ 1 ≈ cs2π 2 ⎜ K ⎟ ⎝ 2 ⎠

3/2

cK = 1.6 ⇒ cs ≈ 0.16

4/3

⎛ 3c ⎞ ⇒ cs = ⎜ K ⎟ ⎝ 2 ⎠

−3/ 4

π −1

k

cs=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 cs = 0.16

cs = cs (y)

How does cs vary under realistic conditions? Interrogate data: Measure: Measure:

Π Δ = − τ jk S˜ jk Π Smag 2 Δ = 2Δ | S˜ | S˜ij S˜ij 2 cs

Obtain“empirical” Smagorinsky coefficient

⎛ − τ S˜ jk jk ⎜ cs = ⎜ 2 ˜ ˜ ˜ ⎝ 2Δ | S | Sij Sij

= f(x,conditions…):

⎞ ⎟ ⎟ ⎠

1/ 2

An example result from atmospheric turbulence…:

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)

⎛ − τ S˜ jk jk ⎜ cs = ⎜ 2Δ2 | S˜ | S˜ S˜ ij ij ⎝

Kettleman City (Central Valley, CA)

c s = c s (x,t)

Stable stratification Neutral stratification

Example result: effect of atmospheric stability on coefficient from sonic anemometer measurements in atmospheric surface layer (Kleissl et al., J. Atmos. Sci. 2003)

⎞1/ 2 ⎟ ⎟ ⎠

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

Germano identity and dynamic model (Germano et al. 1991): Exact (“rare” in turbulence):

ui u j − u˜i u˜ j = uiu j

− u˜ iu˜ j

E(k)

!

LES resolved

SGS

k

Germano identity and dynamic model (Germano et al. 1991): Exact (“rare” in turbulence):

ui u j − u˜i u˜ j = uiu j

- u˜ iu˜ j + u˜ iu˜ j

− u˜ iu˜ j

E(k)

!

LES resolved

SGS

k

Germano identity and dynamic model (Germano et al. 1991): Exact (“rare” in turbulence):

ui u j − u˜i u˜ j = uiu j

Tij =

- u˜ iu˜ j + u˜ iu˜ j

τ ij

− u˜ iu˜ j

E(k)

+ Lij

Lij − (Tij − τ ij ) = 0 L

! T

LES resolved

SGS

k

Germano identity and dynamic model (Germano et al. 1991): Exact (“rare” in turbulence):

ui u j − u˜i u˜ j = uiu j

- u˜ iu˜ j + u˜ iu˜ j

τ ij

Tij =

− u˜ iu˜ j

E(k)

+ Lij

Lij − (Tij − τ ij ) = 0 −2 (c s 2Δ ) | S˜ | S˜ij 2

2 −2 (c sΔ ) | S˜ | S˜ij

L T

Assumes scale-invariance:

Lij − c Mij = 0 2 s

(

2 where M ij = 2Δ | S˜ | S˜ij − 4 | S˜ | S˜ij

!

LES resolved

)

SGS

k

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

Lij − c s Mij = 0 2

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

Ε = (Lij − c Mij ) 2 s

Minimized when:

c = 2 s

Lij Mij Mij Mij

2

Averaging over regions of statistical homogeneity or fluid trajectories

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

Lij − c s Mij = 0 2

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

Ε = (Lij − c Mij ) 2 s

Minimized when:

c = 2 s

Lij Mij

2

Averaging over regions of statistical homogeneity or fluid trajectories

Mij Mij

Homework: (a) Prove the above equation, and

  qi = u i T − ui T

(b) repeat entire formulation for the SGS heat flux vector modeled using an SGS diffusivity (find Cscalar)

∂T  qi = Cscalar Δ | S | ∂xi 2

Similarity, tensor eddy-viscosity, and mixed models τ ijmnl

~ ∂u~ ∂ u ~~ j = Cnl Δ2 i − 2(CS Δ) 2 S Sij ∂xk ∂xk Two-parameter dynamic mixed model

Lij ≡ Tij − τ ij = u~iu~j − u~i u~j

~ ~ ~~ 2 ∂ui ∂u j Tij = −2(CS 2Δ) S Sij + Cnl (2Δ ) ∂xk ∂xk 2

•  Mixed tensor Eddy Viscosity Model:

•  Taylor-series expansion of similarity (Bardina 1980) model (Clark 1980, Liu, Katz & Meneveau (1994), …) •  Deconvolution: (Leonard 1997, Geurts et al, Stolz & Adams, Winckelmans etc..) •  Significant direct empirical evidence, experiments: •  Liu et al. (JFM 1999, 2-D PIV) •  Tao, Katz & CM (J. Fluid Mech. 2002): tensor alignments from 3-D HPIV data •  Higgins, Parlange & CM (Bound Layer Met. 2003): tensor alignments from ABL data •  From DNS: Horiuti 2002, Vreman et al (LES), etc…

−τ dij

S˜ij

Reality check: Do simulations with these closures produce realistic statistics of ui(x,t)?

•  Need good data •  Need good simulations •  Next: Summary of results from Kang et al. (JFM 2003) • Smagorinsky model, • Dynamic Smagorinsky model, • Dynamic 2-parameter mixed model

Remake of Comte-Bellot & Corrsin (1967) decaying isotropic turbulence experiment at high Reynolds number Active Grid M=6" Contractions

u2 x2

Test Section

u1 x1

X-wire probe array D1 = 0.01m D2 = 0.02m D3 = 0.04m D4 = 0.08m

Flow

20M h1 = 0.005m D1 = 0.01m

30M

h3 = 0.02m D3 = 0.04m

0.91m

40M D

48M

u2 x2

u1 Flow x1

20 M

30 M

40 M

48M

Initial condition for LES

Results

(m3s-2)

10

0

10-1 x /M=20 10

1

-2

x /M=48

10-3

1

E(κ)

10-4 10-5 10-6 10-7 100

101

102

κ

103

104

(m-1)

LES of Temporally Decaying Turbulence

Experiment LES Taylor Hypothesis Spatially Decaying Grid Turbulence Temporally Decaying Turbulence x1 = u1 t 2-D Box Filter 3-D Filter  

Pseudo-spectral code: 1283 nodes, carefully dealiased (3/2N)

 

All parameters are equivalent to those of experiments.

 

Initial energy distribution: 3-D energy spectrum at x1/M = 20

 

LES Models:

standard Smagorinsky-Lilly model, dynamic Smagorinsky and dynamic mixed tensor eddy-visc. model

Results: Dynamic Model Coefficients   Dynamic Smagorinsky

τ

dyn −Smag ij

= −2

Lij M ij M ij M ij

  Dynamic Mixed tensor eddy visc:

Δ S˜ S˜ij 2

τ

mnnl ij

~ ∂u~i ∂u j ~~ = Cnl Δ − 2(CS Δ) 2 S Sij ∂xk ∂xk 2

0.2

0.2

0.15

0.15

C

C

C

C

s

nl

nl

C

s

s

0.1

C

0.1

0.05

C

C

nl

s

s

1/12 from the first order approximation of Lij

0.05

0 0

10

20

x /M 1

30

40

50

0 0

10

20

x /M 1

30

40

50

3-D Energy Spectra (LES vs experiment)   Dynamic Smagorinsky 10

10

0

-1

x /M=20 1

10

-2

Exp.

x /M=48 1

E(κ) 10

-3

10

-4

10

-5

10

•  cutoff grid filter (D = 0.04 m) •  cutoff test filter at 2D 0

10

1

10

κ

2

10

3

3-D Energy Spectra (LES vs experiment)   Dynamic Mixed 10

10

tensor eddy-visc. model

0

-1

x /M=20 1

10

-2

x /M=48 1

E(κ) 10

-3

10

-4

10

-5

10

•  cutoff grid filter (D = 0.04 m) •  cutoff test filter at 2D 0

10

1

10

κ

2

10

3

PDF of SGS Stress (LES vs experiment) ~ ~ τ 12 ≡ u1u2 − u1u2

 SGS Stress 25

x /M=48 1

20

Smagorisnky

15

Dynamic Smagorisnky

PDF 10

Dynamic mixed tensor eddy-visc model predicts PDF of the SGS stress accurately.

Dynamic Mixed Nonlinear

5

Exp. 0 -0.4

-0.3

-0.2

-0.1

0

0.1

τ 12 / u~12rms

0.2

0.3

0.4

Germano identity and dynamic model (Germano et al. 1991): Exact (“rare” in turbulence):

ui u j − u˜i u˜ j = uiu j

- u˜ iu˜ j + u˜ iu˜ j

τ ij

Tij =

− u˜ iu˜ j

E(k)

+ Lij

Lij − (Tij − τ ij ) = 0 −2 (c s 2Δ ) | S˜ | S˜ij 2

2 −2 (c sΔ ) | S˜ | S˜ij

L T

Assumes scale-invariance:

Lij − c Mij = 0 2 s

(

2 where M ij = 2Δ | S˜ | S˜ij − 4 | S˜ | S˜ij

!

LES resolved

)

SGS

k

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

Lij − c s Mij = 0 2

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

Ε = (Lij − c Mij ) 2 s

Minimized when:

c = 2 s

Lij Mij

2

Averaging over regions of statistical homogeneity or fluid trajectories

Mij Mij Problem: what to do for non-homogeneous flows without directions over which to average (“learn”, or “assimilate” largerscale statistics?)

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

E =

∫ (L

ij

−∞

− C M ij 2 s

)

2

1 −(t T-t') e dt' T

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

E =

∫ (L

ij

− C M ij 2 s

−∞

)

2

1 −(t T-t') e dt' T t

t

1 e dt' ∫ T 2 Cs = −∞ t (t -t' ) 1 − T ∫−∞ Mij Mij T e dt' Lij M ij

δ E =0 ⇒

(t -t' ) − T

ℑLM

1 − (tT-t') = ∫ Lij M ij e dt' T −∞ t

ℑMM

) 1 −(t -t' = ∫ Mij Mij e T dt' T −∞

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

E =

∫ (L

ij

− C M ij 2 s

−∞

)

2

1 −(t T-t') e dt' T t

t

1 e dt' ∫ T 2 Cs = −∞ t (t -t' ) 1 − T ∫−∞ Mij Mij T e dt' Lij M ij

δ E =0 ⇒

(t -t' ) − T

ℑLM

1 − (tT-t') = ∫ Lij M ij e dt' T −∞ t

ℑMM

) 1 −(t -t' = ∫ Mij Mij e T dt' T −∞

With exponential weight-function, equivalent to relaxation forward equations: ∂ℑLM ∂ℑ 1 + u˜k LM = (Lij M ij − ℑLM ) ∂t ∂xk T ∂ℑMM ∂ℑ 1 + u˜k MM = (Mij Mij − ℑMM ) ∂t ∂x k T

ℑLM (x,t) c = ℑMM (x,t) 2 s

Lagrangian dynamic model has allowed applying the Germanoidentity to a number of complex-geometry engineering problems LES of flows in internal combustion engines: Haworth & Jansen (2000) Computers & Fluids 29.

Examples: LES of flow over wavy walls Armenio & Piomelli (2000) Flow, Turb. & Combustion.

LES of structure of impinging jets: Tsubokura et al. (2003) Int Heat Fluid Flow 24.

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.

Examples: Human swimming flow structures (Rajat Mittal, 2008)

Michael Phelps local Baltimore boy

Ciclo di seminari Universita degli studi di Roma, Tor Vergata May, 2013

OVERVIEW: Mercoledì: •  •  •  •  • 

Introduction to fundamentals of Turbulence Intro to Large Eddy Simulation (LES) Intro to Subgrid-scale (SGS) modeling The dynamic SGS model Some sample applications from our group

Venerdì: •  Dynamic model for LES over rough multiscale surfaces •  LES studies of large wind farms

JHU Mechanical Engineering

Examples: LES of convective atmospheric boundary layer: Kumar, M. & Parlange (Water Resources Research 42, 2006) •  Transport equation for temperature •  Boussinesq approximation •  Coriolis forcing •  Lagrangian dynamic model with assumed b=Cs(2 D)/Cs(D) •  Constant (non-dynamic) SGS Prandtl number Prsgs=0.4 •  Imposed surface flux of sensible heat on ground •  Diurnal cycle: start stably stratified, then heating…. Imposed ground heat flux during day:

Examples: •  Diurnal cycle: start stably stratified, then heating…. Resulting dynamic coefficient (averaged):

Consistent with HATS field measurements:

Large-eddy-Simulation of atmospheric flow over fractal trees: Chester, M & Parlange (J. Comp. Phys. 225, 2007; J. Env. Fluid Mech. 7, 2007)

Urban contamination and Transport Tseng, M. & Parlange, Env. Sci & Tech. 40, 2006

Downtown Baltimore: wind

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

Agriculture: What is the isolation distance to avoid cross-polination? Collaboration with Marcelo Chamecki (Penn State U)

Deposition flux?

Liso

LES: Eulerian approach C(x,y,z,t)

((

2 ∂C −1  + ( u − ws e 3 ) i ∇C = ∇ i Cs−dyn Δ Scsgs | S | C ∂t

Vertical settling velocity ws

)

Scale-dep dynamic eddy-viscosity eddy-diffusivity SGS Log-law type boundary condition For C, log-law, corrected by settling velocity

Low ws emitter field

High ws

Deposition flux? Liso

)

LES results: Downstream evolution of concentration profiles

Downstream evolution of deposition flux

1-D “reduction” (back to early 1900s) - vertical profile

Similarity solution for any eta (Rouse #)

Deposition downstream of field:

Sc ws γ = κ u*

deltaL is proper length-scale for deposition flux, not L

Comparison with LES:

γ =

Sc ws = 0.125 κ u*

Over field

Downstream

Scaling of deposition flux: field edge delta instead of L

γ =

γ =

Sc ws = 0.313 κ u*

Traditional approach (current rules)

Sc ws =0 κ u*

deltaL

Different L L

Sc ws γ = = 0.313 κ u*

γ =

Sc ws =0 κ u*

Collapse for different L

Scaling with deltaL

Isolation distance as function of flux threshold:

E.g., ws=0.06 m/s (d=40 micro-m), u*=0.5 m/s, Sc=1, k=0.4,

L=500m -> deltaL=30m,

ID = 4.5km !!

Sc ws γ = = 0.3 κ u*

E.g., Phi=10-3 -> ID/deltaL ~150

Useful references on LES and SGS modeling: •  P. Sagaut: “Large Eddy Simulation of Incompressible Flow” (Springer, 3rd ed., 2006) •  U. Piomelli, Progr. Aerospace Sci., 1999 •  C. Meneveau & J. Katz, Annu Rev. Fluid Mech., 2000