Real-space Manifestations of Bottlenecks in Turbulence Spectra - - PowerPoint PPT Presentation

Real-space Manifestations of Bottlenecks in Turbulence Spectra

Rahul Pandit

Centre for Condensed Matter Theory, Department of Physics Indian Institute of Science, Bangalore, India. 28 May 2012 Wolfgang Pauli Institute, Vienna, Austria.

Work done with:

◮ Uriel Frisch, Laboratoire Lagrange, OCA, UNS, CNRS, BP

4229, 06304 Nice Cedex 4, France;

◮ Samriddhi Sankar Ray, Laboratoire Lagrange, OCA, UNS,

CNRS, BP 4229, 06304 Nice Cedex 4, France;

◮ Ganapati Sahoo, Max Planck Institute for Dynamics and

Self-Organization, Am Fassberg 17, 37077 G¨

• ttingen,

Germany;

◮ Debarghya Banerjee, Centre for Condensed Matter Theory,

Department of Physics, Indian Institute of Science, Bangalore, India.

◮ Support: OTARIE (France), CSIR, DST, UGC (India) and

SERC (IISc)

Outline

• 1. Energy-spectra bottlenecks: Direct numerical simulations

(DNS), experiments, and earlier studies.

• 2. Hyperviscous hydrodynamical equations:

◮ 1D deterministic hyperviscous Burgers (DHB) equation; ◮ 1D stochastic hyperviscous Burgers (SHB) equation; ◮ 3D deterministic hyperviscous Navier-Stokes (3DHNS)

equation.

• 3. Real-space manifestations of bottlenecks:

◮ DHB: boundary-layer theory and (DNS); ◮ SHB: DNS; ◮ 3DHNS: DNS.

• 4. Conclusions

Bottlenecks : DNS

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Bottleneck effect in three-dimensional turbulence simulations, W. Dobler,

• N. Erland, L. Haugen,T. A. Yousef, and A. Brandenburg, Phys. Rev. E,

68, 026304 (2003).

Bottlenecks : DNS

0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.001 0.01 0.1 1

a

257 471 732 1131 Rλ = 167 257 471 732 1131 Rλ = 167

kη k5/3E(k)/ ε 2/3 ∏(k)/ ε

Study of HighReynolds Number Isotropic Turbulence by Direct Numerical Simulation, Takashi Ishihara, Toshiyuki Gotoh, and Yukio Kaneda, Annu.

• Rev. Fluid Mech., 41, 165 (2009).

Bottlenecks : DNS

100 10–1 10–3 10–2 10–1 10–2 10–1 100 1.6 1.8 2.0 2.2

Ψ (kη) kη

11/3 38 90 140 240 400 650 1000 Rλ 1.0 1.5 2.0 2.5 −3 −2 −1

Ψ (kη) d[log E(k)] d[log k] kη (a) (b)

10–3 10–2 10–1 10–3 10–2 10–1

The bottleneck effect and the Kolmogorov constant in isotropic turbulence, D. A. Donzis and K. R. Sreenivasan, J. Fluid Mech., 657, 171 (2010).

Bottlenecks : Experiments

On the universal form of energy spectra in fully developed turbulence, Z-S. She and E. Jackson, Phys. Fluids A, 5, 1526 (1993).

Bottlenecks : Experiments

Local isotropy in turbulent boundary layers at high Reynolds number, S.

• G. Saddoughi and S. V. Veeravalli, J. Fluid Mech., 268, 333 (1994).

Hyperviscous Hydrodynamical Models

• 1. Deterministic hyperviscous Burgers equation (DHB).
• 2. Stochastic hyperviscous Burgers equation (SHB).
• 3. 3D hyperviscous Navier–Stokes equation (3DHNS).

DHB equation

∂tu + u∂xu = −ναk−2α

r

(−∂2

x)αu + f (x, t) ◮ u - velocity field ◮ να - coefficient of hyperviscosity ◮ kr - a reference wavevector ◮ α - dissipativity ◮ We use f (x, t) = sin(x)

SHB equation

∂tu + u∂xu = −ναk−2α

r

(−∂2

x)αu + f (x, t);

f (x, t) is a zero-mean, space-periodic Gaussian random force with ^ f (k1, t1)^ f (k2, t2) = 2D0|k|−1δ(t1 − t2)δ(k1 + k2).

3DHNS equation

∂u ∂t + u · ∇u(x, t) = −∇p − να

• −∇2αu(x, t) + f(x, t);

∇ · u = 0. p - pressure ; ρ = 1. The force f(x, t) is specified in terms of its spatial Fourier transform, as ^ f(k, t) = PΘ(kf − k) 2E(kf , t) ^ u(k, t), where Θ(kf − k) is 1 if k ≤ kf and 0 otherwise, P is the power input, and E(kf , t) =

k≤kf E(k, t); we choose kf = 1.

Principal Results

◮ We develop a quantitative, analytical understanding of

bottlenecks in the hyperviscous Burgers equation.

◮ For this it is crucial to examine the solution in real space,

where we can use boundary-layer-type analysis, in the vicinities

• f shocks, to uncover oscillations in the velocity profile.

◮ We validate our DHB solutions with a pseudospectral DNS.

Principal Results

◮ The key feature of real-space oscillations carries over to

• scillations in velocity correlation functions in hyperviscous

hydrodynamical equations that display genuine turbulence.

◮ We show this in the second part of our study by using DNS. ◮ This association of bottlenecks and oscillations in velocity

correlation functions, similar the association of peaks in the static structure factor S(k), of a liquid in equilibrium, with damped oscillations in the pair correlation function, has not been made so far.

S(k) and g(r) : liquid-state of Ar at 85 K

Structure Factor and Radial Distribution Function for Liquid Argon at 85K, J. L. Yarnell et al., Phys. Rev. A, 7, 2130 (1973).

Earlier Theoretical Models

◮ Inhibited cascade model (G. Falkovich, 1994). ◮ Local scaling exponents (D. Lohse and A. M¨

uller-Groeling, 1995).

◮ Aborted Thermalisation (Frisch et al., 2008).

Inhibited cascade model (Falkovich)

◮ Evolution equation of pair correlation functions. ◮ Lowest non-linearity expressed via a triple correlation function. ◮ Pair-correlator expressed as a power function. ◮ The non-linear interaction in the inertial range and in the

dissipation range gives a term, which, if greater than the viscous dissipation, yields a bottleneck.

Bottleneck phenomenon in developed turbulence, G. Falkovich, Phys. Fluids, 6, 1411 (1994).

Local scaling exponents (Lohse and M¨ uller-Groeling)

◮ Analysis of experimental data to obtain the energy spectrum

and the second-order structure function.

◮ The local scaling exponent of the second-order structure

function decreases monotonically.

◮ The local scaling exponent of the spectrum has a minimum

and a maximum.

◮ Energy pile-up using Falkovich’s argument.

Bottleneck effects in turbulence: Scaling phenomenon in r versus p space,

• D. Lohse and A. M¨

uller-Groeling, Phys. Rev. Lett., 74, 1747 (1995).

Hyperviscous Burgers and Navier-Stokes Equations

∂tv + v∂xv = −µK −2α

G

(−∂2

x)αv

∂tv + v · ∇v = −∇p − µK −2α

G

(−∇2)αv, ∇ · v = 0

◮ µ > 0, KG > 0, and α is the dissipativity. ◮ Dissipation rate µ(k/KG)2α tends to zero for all k < KG and

to infinity for k > KG, when α → ∞.

◮ For fixed µ and KG, the solution of the hyperviscous equations

tend to the Galerkin-truncated equations as α tends to ∞.

◮ True for Navier-Stokes, Burgers, MHD, DIA and EDQNM. ◮ False for MRCM and resonant wave interaction theory.

◮ Galerkin-truncation leads to thermalization (Lee, 1952; Hopf,

1952; Kraichnan, 1958).

EDQNM: Hyperviscosity

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Hyperviscosity, Galerkin truncation and bottlenecks in turbulence, U. Frisch, S. Kurien, R. Pandit, W. Pauls, S. S. Ray, A. Wirth, and J-Z Zhu,

• Phys. Rev. Lett., 101, 144501 (2008).

Hyperviscous Burgers Equation : Large α

0.5 1 1.5 2 2.5 3 −7 −6 −5 −4 −3 −2 −1 log(k) log(E(k)) α = 1000 t = 1 t = 2 t = 4 t = 10 t = 20 t = 30 k−2 scaling

Hyperviscosity, Galerkin truncation and bottlenecks in turbulence, U. Frisch, S. Kurien, R. Pandit, W. Pauls, S. S. Ray, A. Wirth, and J-Z Zhu,

• Phys. Rev. Lett., 101, 144501 (2008).

Thermalization and Bottlenecks

◮ Large α produces a big bottleneck. ◮ Thermalization is accompanied by Gaussianization. ◮ Spurious effects expected: depletion of intermittency.

Methods

◮ DHB : Boundary–layer theory and DNS ◮ SHB : DNS ◮ 3DHNS : DNS

DHB : Boundary–layer Theory

◮ The velocity eventually goes to a steady state, which is a

solution of the ordinary differential equation (ODE) that is

• btained by dropping the time-derivative term.

◮ When α = 1, this nonlinear ODE is not integrable, but its

limit as να → 0 is the same as for ordinary dissipation, namely, it has a shock at x = π, where the solution jumps from u− = +2 to u+ = −2.

DHB: Boundary–layer Theory

◮ For small but finite να, the shock is broadened and its

structure can be analyzed by a boundary-layer technique using the stretched spatial variable X ≡ (x − π)/νβ, with β =

1 2α−1,

and expanding the boundary-layer velocity in powers of να.

◮ To leading order

d dX u2 2

• = (−1)α+1 d2α

dX 2αu0, u0(±∞) = ∓2.

DHB : Boundary–layer Theory

◮ For large X the equation can be linearized because u0 is close

to its asymptotic constant value.

◮ For example, for large negative X, we set u0 = 2 + w, discard

the quadratic term in w, and obtain, after integrating once, (−1)α+1d2α−1w/dX 2α−1 = 2w.

DHB: Boundary–layer Theory

◮ This constant-coefficient ODE has solutions of the form

µ exp(καX), where µ is arbitrary and the “eigenvalue” κα is any of the (2α − 1)th roots of (−1)α+12.

◮ Only the eigenvalues that have a positive real part are

acceptable, because w should vanish at −∞.

◮ If all the modes with such eigenvalues are actually present

then, for X → −∞, the solution tends to +2 in an oscillatory fashion and it is dominated by the mode n⋆ (and its complex conjugate), which has the smallest positive real part.

DHB : Boundary–layer Theory

◮ In terms of the unstretched coordinates, this means that, in

the neighborhood of the shock, the solution for even α displays damped oscillations with wavelength λth

α = 2πνβ α

• 2βsin[(2n⋆ + 1)βπ)]

−1 and with an e-folding rate K th

α = 2βν−β α cos[(2n⋆ + 1)βπ)].

DHB: Boundary–layer Theory

◮ Such damped oscillations imply the presence of a pair of

complex k poles in wave-number space, whose signature, for real k, is a Lorentzian which can be a bump or a trough, near wave number 2π/λα, with width ∼ Kα and amplitude ∼ K −1

α . ◮ A semi-numerical analysis shows that the solution of the DHB

yields a bottleneck (bump).

DHB : Boundary–layer Theory

◮ By using a numerical, shooting method, we obtain evidence

that, for α = 2, there is a unique solution that has u ′

0(X)|X=0 = −2.121530817618 . . . and u ′′ 0 (X)|X=0 = 0. ◮ We can also obtain the value of this first derivative at the

• rigin with ≃ 10% accuracy by assuming that the solution has

singularities on the imaginary axis at X = Z⋆ = ±i ∆ (a Painlev´ e-type argument indicates that, near such a singularity, to the leading order, u0(X) ≈ 120/(X − Z⋆)).

DHB: Boundary–layer Theory

◮ The vanishing of the second derivative implies that this

unique solution is odd in the X variable.

◮ Direct numerical integration of the boundary-layer equation is

a greater challenge than the full DHB equation.

DHB : Boundary–layer Theory

◮ By using the value of u ′ 0(0), obtained by the shooting method,

and u ′′

0 (0) = 0, we solve the third-order, boundary-layer

equation for α = 2 numerically.

◮ We find XC ≃ 1.15, the value of X at which u0(X) first

crosses the −2 asymptote. Next, we calculate u ′

0(XC) by

using the Taylor expansion u0(X) ∼ u ′

0(0)X + u ′′′ 0 (0)X 3/3! + u ′′′′′

(0)X 5/5! + ... along with the known values of u ′

0, and u ′′′ 0 (0) = 2 and

u ′′′′′ (0) = −(u ′

0(0))2.

DHB : Boundary–layer Theory

◮ The linear theory suggests

u0(X) = −2 + Ae−K2(X−XC ) sin 2π(X − XC)/λ2 for X ≥ XC; thence we obtain u ′

0(XC) in terms of A and λ2. ◮ By using the values of u ′ 0(XC) (from the Taylor expansion

above) and λ2, we obtain A ≃ −0.983, which is within 1.7%

• f the value of A (≃ −0.966) that we get from the solution of

the boundary-layer equation.

DHB : Boundary–layer Theory

◮ We now address the question of whether the Fourier-space

manifestation of these oscillations is a bump or a trough.

◮ The Fourier transform γ(k) of the real and even function

−u ′(X) is real and even; and γ(k) is the square root of the compensated energy spectrum.

◮ The rising of the compensated energy spectrum, in the

intermediate regime, between the flat region near k = 0 and the exponential decay at large k, is equivalent to γ′′(k) being positive; and γ′′(k)|k↓0 = 1/2π ∞

0 dXX 2u ′(X). ◮ To solve for γ′′(k), we use u ′(X) either from a numerical

solution or from the linear theory above; we then perform a numerical integration over X; we obtain good agreement (≃ 9%) between the results of both these methods; and, indeed, we find that γ′′(k) is positive, so the spectrum has a bottleneck.

DHB : Numerical Results : Boundary–layer Theory

2 4 6 8 10 12 14 X 2.4 2.3 2.2 2.1 2.0 u X

The blue curve gives the solution of the boundary layer equation for α = 2 and X > XC. u(X) clearly show oscillations. X is the stretched co-ordinates.

DHB : DNS

◮ The reference wavenumber kr = 100; the number of

collocation points N = 214, the time step δt = 10−4, and the hyperviscosity coefficients are ν2 = 5 × 10−3, ν4 = 5 × 10−8, ν8 = 5 × 10−14, and ν16 = 10−20.

◮ In a thin boundary layer around the shock at x = π, there are

conspicuous oscillations.

◮ The characteristic wavelength of these oscillations is λα;

λ16 = 0.012.

DHB : DNS

◮ From plots of the compensated spectra we obtain k16 b = 518,

consistent with our value for λ16 because 2π/k16

b ≃ 0.0121. ◮ The theoretical prediction for the wavelength of these

• scillations yields λth

16 = 0.0120. ◮ We obtain, from our DNS, an e-folding rate K16 ≃ 26.61,

whereas our theoretical prediction yields K th

16 ≃ 26.54. ◮ We find excellent agreement between our theoretical

predictions and our numerical results for both the wavelength

• f the oscillations and the e-folding rate.

DHB : DNS

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5

t E(t)

(a) α = 2 α = 4 α = 8 α = 16

The total energy versus time for α = 2, 4, 8, 16. In the steady state the total energy is in agreement with our analytical calculations.

DHB : DNS

1 2 3 4 5 −6 −5 −4 −3 −2 −1 1

log10 k log10 k2E(k)

(b) α = 2 α = 4 α = 8 α = 16

A log-log plot of the compensated energy spectrum E c(k) versus k.

DHB : DNS

2.8 3 3.2 3.4 −5 −4 −3 −2 −1

log10 k log10 k2E(k)

α = 2 α = 4 α = 8 α = 16

A zoomed in log-log plot of the compensated energy spectrum E c(k) versus k.

DHB : DNS

1 2 3 4 5 6 −2 −1 1 2

x u(x)

(c) α = 2 α = 4 α = 8 α = 16

The steady state DNS solution u(x) versus x for different values of α.

DHB : DNS

3.12 3.13 3.14 3.15 3.16 −2 −1 1 2

x u(x)

α = 2 α = 4 α = 8 α = 16

The steady state DNS solution u(x) versus x for different values of α close to x = π.

DHB : DNS

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.1 0.2 0.3 0.4 0.5 0.6

π − x A8

(e)

A plot of Aα versus the distance from the shock π − x for α = 8.

DHB : DNS

0.02 0.04 0.06 0.08 −12 −10 −8 −6 −4 −2

π − x ln A8

(f)

A lin-log plot of Aα (red *) versus the distance from the shock π − x for α = 8. The black line is a linear fit.

DHB : DNS

0.05 0.1 0.15 0.05 0.1 0.15 0.2 0.25 0.3 0.35

π − x A16

(g)

A plot of Aα versus the distance from the shock π − x for α = 16.

DHB : DNS

0.05 0.1 0.15 0.2 −6 −5 −4 −3 −2 −1

π − x ln A16

(h)

A lin-log plot of Aα (red *) versus the distance from the shock π − x for α = 16. The black line is a linear fit.

SHB : DNS

◮ Our results carry over to the stochastically forced

hyperviscous Burgers equation (SHB).

◮ Let us examine bottlenecks in the SHB equation, with a

white-in-time, Gaussian random force with zero mean, an ultraviolet cutoff at N/8, and a spectrum ∼ k−1.

◮ The velocity field for the SHB shows shocks at various length

scales; and the resulting energy spectrum shows an inertial-range scaling E(k) ∼ k−5/3.

◮ The compensated energy spectrum k5/3E(k), for α = 8,

shows such an inertial range followed by a prominent bottleneck that peaks at a wavenumber k8

b ≃ 890.

SHB : DNS

◮ We measure the correlation function u(x)u(x + l), which

show oscillations; these are the real-space manifestations of this bottleneck.

◮ The wavelength of these oscillations is ≃ 0.00706; and the

corresponding wavenumber is ≃ 889.97, in agreement with the wavenumber at which the bottleneck shows a peak.

SHB : DNS

1 2 3 4 5 6 −3 −2 −1 1 2

x u(x)

(a)

A snapshot of the velocity field for the SHB equation, with α = 8, in the statistically steady state.

SHB : DNS

10 10

1

10

2

10

3

10

4

10

−3

10

−2

10

−1

10 10

1

k k5/3E(k)

(b)

A log-log plot of the compensated energy spectrum k5/3E(k) versus k.

SHB : DNS

4.16 4.17 4.18 4.19 4.2 −0.327 −0.326 −0.325 −0.324 −0.323 −0.322 −0.321

l <u(x)u(x+l)>

(c)

A plot showing the oscillations in the correlation function u(x)u(x + l), with a wavelength λSHB

α

, which corresponds to the wavenumber at which the bottleneck is seen.

3DHNS : DNS

◮ We integrate it by a pseudospectral method with a 2/3

dealiasing rule, an Adams-Bashforth scheme for time marching, 5123 collocation points, α = 4, and ν4 = 10−14.

◮ We force the 3D HNS equation to a statistically steady state

by using the constant-energy-injection method.

◮ The compensated energy spectrum E c(k) ≡ k5/3E(k) shows

a bottleneck between the inertial and dissipation ranges.

3DHNS : DNS

◮ The correlation function D(l) = u(x) · u(x + l), averaged

• ver five configurations, separated in time by about 6

integral-scale eddy turnover time, shows gentle oscillations, which are the real-space manifestations of this bottleneck.

◮ These oscillations can be seen clearly in Do(l), which is

• btained by subtracting the linear, decaying trend from D(l).

◮ The wavelength of these oscillations is ≃ 0.1665 and the

corresponding wavenumber is ≃ 37.7, in agreement with the wavenumber at which the bottleneck shows a peak.

3DHNS : DNS

1 10 1 10 100 k5/3E(k) k

(d)

The compensated energy spectrum k5/3E(k) for the 3D HNS in the statistically steady state. The bottleneck peaks at wavenumber K HNS

b,α

= 40.

3DHNS : DNS

6.6 6.8 7 7.2 7.4 7.6 7.8 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 D(l)=<u(x+l)u(x)> l

(e)

A plot of the correlation function D(l) = u(x)u(x + l) versus l for the 3D HNS.

3DHNS : DNS

• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02 0.025 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Do(l) l

(f)

A plot of the function Do(l), obtained by subtracting the linear part from the correlation function D(l). The wave length associated with such oscillations is in agreement with the wavelength associated to K HNS

b,α .

Summary of Results : DHB

1 2 3 4 −6 −4 −2 2 log10 k log10 k2E(k) (a) 1 2 3 4 5 6 −2 −1 1 2 x u(x) (b)

3.12 3.14 3.16 −1 1 2 3 x ud(x)

Summary of Results : SHB

0.5 1 1.5 2 2.5 3 −2 −1.5 −1 −0.5 0.5 log10 k log10 k5/3E(k) (a)

4.16 4.18 4.2 −0.326 −0.324 −0.322 l < u(x)u(x + l) >

Summary of Results : 3DHNS

0.5 1 1.5 2 2.5 −1.5 −1 −0.5 0.5 1 log10 k log10 k5/3E(k) (b) 0.2 0.25 0.3 0.35 0.4 0.45 0.5 6.6 6.8 7 7.2 7.4 7.6 l D(l) (c)

0.2 0.3 0.4 0.5 −0.02 0.02 l Do(l)

Conclusions

◮ We have provided a theoretical explanation for energy-spectra

bottlenecks in the DHB equation by combining analytical and numerical studies.

◮ These bottlenecks appear as a natural consequence of

• scillations in the velocity profiles in the vicinity of a shock.

◮ We have shown that energy-spectra bottlenecks in the SHB

and the 3D HNS equations, which exhibit turbulence, are associated with damped oscillations in real-space velocity correlation functions.

◮ Our work confirms that the larger the dissipativity α, the

more pronounced is the bottleneck.

◮ Energy spectra for homogeneous isotropic turbulence in the

3D NS equation (α = 1) show a mild bottleneck; we expect, therefore, that there should be weak oscillations in real-space velocity correlation functions.

◮ The detection of such weak oscillations is an important

challenge for experiments and DNS.