Shearless turbulence mixing: numerical experiments on the ... · Shearless turbulence mixing:...
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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.
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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
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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.
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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)
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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.
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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
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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).
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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
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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)
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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
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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
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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
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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.
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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.
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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
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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
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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.
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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)
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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).
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⇒ 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
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⇒ 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
⎤⎥⎥⎥⎦
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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
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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)
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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
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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.
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. . . 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)
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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)
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. . . 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).
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. . . 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)
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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
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•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 →
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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