Shearless turbulence mixing: numerical experiments onthe intermediate asymptotics

M. Iovieno, D. Tordella

Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di TorinoCorso Duca degli Abruzzi 24, 10129 Torino, Italia

– D.Tordella, M.Iovieno 2005 “Numerical experiments on the intermediate asymptotics of shear-free turbulentdiffusion”, Journal of Fluid Mechanics, to appear.

– D.Tordella, M.Iovieno “The dependance on the energy ratio of the shear-free interaction between two isotropic

turbulence” Direct and Large Eddy Simulation 6 - ERCOFTAC Workshop, Poitiers, Sept 12-14, 2005.

– D.Tordella, M.Iovieno “Self-similarity of the turbulence mixing with a constant in time macroscale gradient” 22nd

IFIP TC 7 Conference on System Modeling and Optimization, Torino, July 18-22, 2005.

– M.Iovieno, D.Tordella 2002 “The angular momentum for a finite element of a fluid: A new representation andapplication to turbulent modeling”, Physics of Fluids, 14(8), 2673–2682.

– M.Iovieno, C.Cavazzoni, D.Tordella 2001 “A new technique for a parallel dealiased pseudospectral Navier-Stokescode.” Computer Physics Communications, 141, 365–374.

– M.Iovieno, D.Tordella 1999 “Shearless turbulence mixings by means of the angular momentum large eddy model”,

American Physical Society - 52thDFD Annual Meeting.

Shearless turbulence mixing.

E(x)

x

Mixing Layer

turbulence turbulence

Region 2:Low−EnergyHigh−Energy

Region 1:

• no mean shear⇒ no turbulence production

• the mixing layer is generated by theturbulence inhomogeneity, i.e.:

� by the gradient of turbulent energyand� by the gradient of integral scale

Previous investigations:

Esperiments with grid turbulence:

– Gilbert B. J. Fluid Mech. 100, 349–365 (1980).

– Veeravalli S., Warhaft Z. J. Fluid Mech. 207,191–229 (1989).

Numerical simulations (DNS):

– Briggs D.A., Ferziger J.H., Koseff J.R., Monismith S.G. J. Fluid Mech. 310, 215–

241 (1996).

– Knaepen B., Debliquy O., Carati D. J. Fluid Mech. 414, 153–172 (2004).

• in (passive) grid turbulence the higher energy is always associated tolarger integral scales, so the two parameters are not independent ⇒guess about no intermittency in the absence of scale gradient andturbulence production.

• numerical simulations reproduced the 3,3:1 laboratory experiment byVeeravalli and Warhaft.

New decay properties

• the two parameters, the turbulent kinetic energy ratio E and theintegral scale ratio L, has been independently varied

• the persistency of intermittency in the limit of no scale gradient (L →1) and absence of turbulence production has been investigated.

In particular we present:

•Part 1: results from numerical simulations (DNS and LES, 2005 JFM,to appear)

•Part 2: intermediate asymptotics analysis (L → 1, 2005 IFIP TC7and DLES6; L �= 1, in preparation)

Part 1: numerical experiments

Numerical simulations (DNS and LES) have been carried out with

• Fixed energy ratio E ∼ 6.7 and varying scale ratio 0.38 ≤ L ≤ 2.7

•No scale gradient (L = 1) and variable energy ratio 1 ≤ E ≤ 58.3

•Reynolds number: Reλ ≈ 45 (DNS, LES) and Reλ ≈ 450 (LES only,IAM model, Iovieno & Tordella Phys.Fluids 2002)

•Numerical method: Fourier-Galerkin pseudospectral on a 2π3 cu-be and a 2π × 2π × 4π parallelepiped (Iovieno-Cavazzoni-TordellaComp.Phys.Comm. 2001)Resolution: DNS = 1282 × 256, LES = 322 × 64

• Initial conditions: two turbulent fields coming from simulations ofdecaying homogeneous isotropic turbulence.

Decay exponents

•The two homogeneous fields decay algebrically in time, according totheoretical (and experimental) results (see Karman and Howarth 1938,Sedov 1944, Batchelor 1953, Speziale 1995)

E = A(t + t0)−n

•Decay rates n1, n2 are higher than the limit, n = 1, for high Reynoldsnumber, but still close to this value (n1 ≈ n2 ≈ 1.2 − 1.4), so thatthe energy and scale ratios remain nearly constant (up to 10%) duringthe decay

L(t)

L(0)=

⎛⎜⎜⎜⎝1 +

t

t01

⎞⎟⎟⎟⎠1− n1

2⎛⎜⎜⎜⎝1 +

t

t02

⎞⎟⎟⎟⎠−1+

n2

2

E(t)E(0)

=

⎛⎜⎜⎜⎝1 +

t

t02

⎞⎟⎟⎟⎠n2

⎛⎜⎜⎜⎝1 +

t

t01

⎞⎟⎟⎟⎠−n1

•All mixings have an intermediate self-similar stage of decay

Energy similarity profiles

−4 −3 −2 −1 0 1 2 3 4(x-xc)/Δ

0.0

0.2

0.4

0.6

0.8

1.0

( E

- E

2 )

/ ( E

1- E

2 )

A

B

C1

C2

(a)

Region 1

Region 2

(High Energy)

(Low Energy)

100 101 102

κ

10−5

10−4

10−3

10−2

10−1

E(κ

)

A

B

C1−c

C2

C1−s

(b)

Δ(t) = mixing layer thickness, �(t) = 13

∑i

∫∞0 Rii(r,t)dr

Rii(0,t), where Rii is the

longitudinal velocity correlation (see e.g. Batchelor, 1953).

Higher order moments: skewness and kurtosis profiles

S =u3

u232

K =u4

u22 ⇒ S ≈ 0, K ≈ 3 in homogeneous isotropic turb.

Case A: E = 6.7, L = 1, the two fields have the same integral scale.

−4 −3 −2 −1 0 1 2 3 4(x-xc)/Δ

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

S

3.74.55.25.6LES − IAM modelDNS , Briggs et al.

t/τ1

(a)

0.42

0.8

−4 −3 −2 −1 0 1 2 3 4(x-xc)/Δ

2.5

3.0

3.5

4.0

4.5

K

3.74.55.25.6LES − IAM modelDNS , Briggs et al.

t/τ1

(b)4.15

0.61

Case C: E = 6.5, L = 1.5 : the gradients of energy and scales havethe same sign: larger scale turbulence has more energy

−3 −2 −1 0 1 2 3(x-xc)/Δ

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

S

(C1−c) t/τ1 = 5.2(C1−s) t/τ1 = 5.2V&W−b, x/M = 164V&W−p, x/M = 110

(a)

−3 −2 −1 0 1 2 3(x-xc)/Δ

3.0

4.0

5.0

6.0

K

(C1−c) t/τ1 = 5.2(C1−s) t/τ1 = 5.2V&W−b, x/M = 164V&W−p, x/M = 110

(b)

Penetration - position of the maximum of skewness/kurtosis

0 10 20 30 40 50 60E1 /E2

0.0

0.5

1.0

1.5

((x s+

x k)/2

-xc)

/Δ

(a)

DNSLES, IAM model, Reλ=45LES, IAM model, Reλ=450Knaepen et al. (2004)Pope & Haworth (1987)

<1 =1 >1l1/l2

0.0 0.2 0.4 0.6 0.8 1.0 1.2((xs+xk)/2-xc)/Δ

0.0

0.1

0.2

0.3

0.4

0.5

* (E

/E1

)

Δ

LES, IAM model, Reλ=45LES, IAM model, Reλ=450DNS, Reλ=45

(b) Penetration with L = 1

Scaling law (energy ratio):

ηs + ηk

2∼ a

⎛⎜⎜⎜⎝E1

E2− 1

⎞⎟⎟⎟⎠b

a 0.36, b 0.298

Scaling law (energy gradient):

∇∗(E/E1) (1− E−1)/2

ηs + ηk

2∼ a

⎛⎜⎜⎜⎜⎝

2∇∗(E/E1)

1− 2∇∗(E/E1)

⎞⎟⎟⎟⎟⎠b

Penetration - position of maximum of skewness/kurtosis, E = 6.7

0.5 1.0 1.5 2.0l1/l2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

(xs-

x c)/Δ

, (x

k-x c)

/Δxs, DNSxk, DNS

Part 2: similarity analysis

Properties of the numerical solutions:

•A self-similar decay is always reached

• It is characterized by a strong intermittent penetration, which dependson the two mixing parameters:

– the turbulent energy gradient

– the integral scale gradient

This behaviour must be contained in the turbulent motion equations:

• the two-point correlation equation which allows to consider both themacroscale and energy gradient parameters(Bij(x, r, t) = ui(x, t)uj(x + r, t));

• the one-point correlation equation, the limit r → 0, which allows toconsider the effect of the energy gradient only.

Single-point second order correlation equations

To carry out the similarity analysis for L = 1, we consider the secondorder moment equations for single-point velocity correlations

∂tu2 + ∂xu3 = −2ρ−1∂xpu + 2ρ−1p∂xu− 2εu + ν∂2xu

2 (1)

∂tv21 + ∂xv2

1u = 2ρ−1p∂y1v1 − 2εv1 + ν∂2xv

21 (2)

∂tv22 + ∂xv2

2u = 2ρ−1p∂y2v2 − 2εv2 + ν∂2xv

22 (3)

where:u is the velocity fluctuation in the inhomogeneous direction x,v1, v2 are the velocity fluctuations in the plane (y1, y2) normal to x,ε is the dissipation.

boundary conditions:

outside the mixing, turbulence is homogeneous and isotropic:• For x→ −∞ (high-energy turbulence):

u2 = v21 = v2

2 =2

3E1(t)

pu = u3 = v21u = v2

2u = 0

• For x→ +∞ (low-energy turbulence):

u2 = v21 = v2

2 =2

3E2(t)

pu = u3 = v21u = v2

2u = 0

initial conditions:

u2 = v21 = v2

2 =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

23E1(0) if x < 023E2(0) if x ≥ 0

pu = 0

Hypothesis and semplifications

•The two homogenous turbulences decay in the same way, thus

E1(t) = A1(t + t0)−n1, E2(t) = A2(t + t0)

−n2

the exponents n1, n2 are close each other (numerical experiments,Tordella & Iovieno, 2005). Here, we suppose n1 = n2 = n = 1, avalue which corresponds to Rλ� 1 (Batchelor & Townsend, 1948).

• In the absence of energy production, the pressure-velocity correlationhas been shown to be approximately proportional to the convectivefluctuation transport (Yoshizawa, 1982, 2002)

−ρ−1pu = au3 + 2v2

1u

2, a ≈ 0.10,

• Single-point second order moments are almost isotropic through themixing:

u2 v2i

These semplifications imply that the pressure-velocity correlations canbe represented as:

−ρ−1pu = αu3, α =3a

1 + 2a≈ 0.25.

Thus the problem is reduced to

∂tu2 + (1− 2α)∂xu3 = −2εu + ν∂2xu

2

with the boundary and initial conditions previously described.

Similarity hypothesis

The moment distributions are determined by

• the coordinates x, t

• the energies E1(t), E2(t)

• the scales �1(t), �2(t).

Thus, through dimensional analysis,

u2 = E1ϕuu(η,R�1, ϑ1, E ,L)

u3 = E321ϕuuu(η,R�1

, ϑ1, E ,L)

εu = E321�−1

1 ϕεu(η,R�1, ϑ1, E ,L),

where:

η = x/Δ(t), Δ(t) is the mixing layer thickness, R�1= E

121 (t)�1(t)/ν,

ϑ1 = tE121 (t)/�1(t), E = E1(t)/E2(t), L = �1(t)/�2(t)

The high Reynolds number algebraic decay (n = 1) implies:

E = const =E1(0)

E2(0)

L = const =�1(0)

�2(0)

ϑ1 = const =n

f (Rλ1)

R�1∝ t1−n = const

where f (Rλ) =ε�

E3/2is constant during decay (see Batchelor (1953),

Speziale (1995), Sreenivasan (1998)).

⇒ η is the only similarity variable, η = η(x, t).

⇒ similarity conditions:

By introducing the similarity relations in the equation and by imposingthat all the coefficients must be independent from x, t, it is obtained

Δ(t) ∝ �1(t)

⇒ similarity equation:

−1

2η∂ϕuu

∂η+

1

f (Rλ1)(1− 2α)

∂ϕuuu

∂η+

ν

A1f (Rλ1)2

∂2ϕuu

∂η2 =

= ϕuu − 2

f (Rλ1)ϕεu

with boundary conditions

limη→−∞ϕuu(η) =

2

3, lim

η→+∞ϕuu(η) =2

3E−1, lim

η→±∞ϕuuu(η) = 0

⇒ the third-order moment, ϕuuu, can be represented as a function ofthe second order moment, which yields

ϕuuu =1

(1− 2α)

⎡⎢⎢⎢⎣f

2

∫ η−∞ η∂ϕuu

∂ηdη +

ν

A1f

∂ϕuu

∂η

⎤⎥⎥⎥⎦

S =ϕ−3

2uu

(1− 2α)

⎡⎢⎢⎢⎣f

2

∫ η−∞ η∂ϕuu

∂ηdη +

ν

A1f

∂ϕuu

∂η

⎤⎥⎥⎥⎦

With regard to the second-order moments, the numerical experimentssuggest the fit (see also Veeravalli & Wahrhaft, JFM 1989)

3

2ϕuu =

1 + E−1

2− 1− E−1

2erf(η),

This allows to compute the velocity skewness by analitical integration

S =1− E−1√

πe−η2

⎡⎢⎢⎢⎣

f

4(1− 2α)

⎛⎜⎜⎜⎝1− 4ν

A1f2

⎞⎟⎟⎟⎠

⎤⎥⎥⎥⎦×

⎡⎢⎢⎢⎢⎢⎣1 + E−1

2− 1− E−1

2erf(η)

⎤⎥⎥⎥⎥⎥⎦

−32

Normalized energy and skewness distributions; E = 6.7 and L = 1.

−3 −2 −1 0 1 2 3(x-xm)/Δ

0.0

0.2

0.4

0.6

0.8

1.0

φ uu

3.74.55.25.6Similarity profile

t/τ1(a)

−3 −2 −1 0 1 2 3(x-xc)/Δ

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

S

3.74.55.25.6LES − IAM modelDNS , Briggs et al.Similarity solution

t/τ1

(b)

Position of the skewness maximum

0 10 20 30 40 50 60E1 /E2

0.0

0.5

1.0

1.5(x

s-x c

)/Δ

DNSLES, IAM model

<1 =1 >1l1/l2

Similarity solution

Conclusions (. . . up to now)

The intermediate asymptotics of the turbulence diffusion in the absenceof production of turbulent kinetic energy is considered.

•An intermediate similarity stage of decay always exists.

•When the energy ratio E is far from unity, the mixing is very intermittent .

•when L = 1, the intermittency increases with the energy ratio Ewith a scaling exponent that is almost equal to 0.29 .

• intermittency smoothly varies when passing through L = 1:it increases when L > 1 (concordant gradient of energy and scale),it is reduced when L < 1 (opposite gradient of energy and scale)

• the self-similar decay of the shearless mixing is consistent withthe similarity solution of the single-point correlation equation.

. . . work in progress

•Consider both scale and energy gradient pareameters by means of thetwo-point correlation equation

• Small scales. . .→ velocity derivative skewness and structure functions

•Reynolds number effect

•Computational accuracy: influence of the domain dimensions

(. . . the end)

Appendix: Numerical discretization. . .

Incompressible Navier-Stokes equations:

∂tui + ∂j(uiuj) = −∂ip +1

Re∇ui

−∇2p = ∂i∂j(uiuj)

Cubic domain (2π × 2π × 2π) with periodic boundary conditions:

ui(x + 2πe(j)) = ui(x) ∀i, jFourier-Galerkin approximation (see Canuto et al., 1988):

uNi (x, t) =

N/2∑k1,k2,k3=−N/2

uNi,k(t)e

ik·x, pN(x, t) =N/2∑

k1,k2,k3=−N/2pNk (t)eik·x

Semi-discrete equations:

∂tuNi = −ikj

(uiuj)− ikip

N − k2

ReuN

i

−k2pN = −kikj

(uiuj), k2 =| k |2Convective terms

(uiuj) are evaluated with pseudo-spectral method.

Time integration with low storage Runge-Kutta 4 (Jameson at al. AIAA J. 1981)

. . . Parallel code

The code uses real-to-real FFT and stores Fourier coefficients in hermitian form (seeIovieno-Cavazzoni-Tordella,Comp.Phys.Comm. 2001)Most operations are local in the wavenumber space with the exception of the pseudo-spectral computation of convective terms

(uiuj):

Basic method (aliasing error)

• inverse FFT of uNi and uN

j

• product in the physical space

• FFT of the product

Scheme for parallel FFT/inverse FFT

1

2

3

perform 2Dinverse FFT transpose (2,3)

1

2

3

1

2

3

P3

P2

P1

P0

P3

P2

P1

P0

P3

P2

P1

P0

Bij

BijT

⇒ To remove the aliasing error data must be expansed from N to M = 32N points in

alla directions (see Canuto et al., 1988).

. . . dealiased pseudo-spectral computation of products

1

3

2

1

2

3

1

3

3

2

1

expand 1 and 2

perform 2Dinverse FFT

and expand 3transpose

2

P3

P2

P1

P0

P3

P2

P1

P0

P3

P2

P1

P0

P3

P2

P1

P0

Bij

BTij

(Scheme for parallel inverse FFT)

Data transposition (inverse FFT)

•Preliminary cicle: each processor transposes and collocates the “diagonal block”:

g(j1, j2 + I(Iirank), j3)← f(j1, j3 + irankMloc, j2)

where

I(i) =

⎧⎪⎪⎨⎪⎪⎩

Nproci ifi < Nproc

I(i) = M −Nproci otherwise

3

2

3

2

P3

P2

P1

P0

P3

P2

P1

P0

N , M = 3N/2 are the number of points,Nproc is the number of processors Nloc = N/Nproc, Mloc = M/Nproc

•Main loop:

for j from 0 to Nproc − 1,– each processor defines the destination and source for communication:

idest = irank + j, isource = irank − j– each processor creates and transposes the block to be sent:

BT (j1, j2, j3)← f(j1, j3 + idestNloc, j2)– Communication occurs by means of a call to MPI send recv replace

– Each processor allocates the received block in the new position:g(j1, j2 + I(Isource), j3)← BT (j1, j2, j3).

end of the loop.

... example with 4 processors in next slide →

j = 1

j = 2

j = 3

Note: Red blocks are transferred during

each step of the loop

3

2

3

2

P3

P2

P1

P0

P3

P2

P1

P0

3

2

3

2

P3

P2

P1

P0

P3

P2

P1

P0

3

2

3

2

P3

P2

P1

P0

P3

P2

P1

P0