[PPT] - The Mean Impulse Response of Homogeneous Isotropic Turbulence: the PowerPoint Presentation

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The Mean Impulse Response of Homogeneous Isotropic Turbulence: the first (DNS based) measurement

Marco Carini and Maurizio Quadrio

Dipartimento di Ingegneria Aerospaziale Politecnico di Milano

XX AIDAA Congress Milano, 30 June 2009

M. Carini & M. Quadrio (DIA-PoliMI)

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Outline

Introduction The impulse response and its measurement Results Conclusions

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Introduction

Numerical simulation of turbulence

Wu and Moin JFM 2009

◮ Turbulence dominates most

engineering flows;

◮ Available strategies: RANS, LES; ◮ Modeling (Reynolds stress, Subgrid

scale stress) always required;

◮ Closure theories useful for modeling.

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Introduction Linear impulse response and closure theories

A statement of the closure problem

Homogeneous equations in Fourier space (κ, p, q wave vectors) using symbolic notation:

◮ First-order eq. (momentum)

„ ∂ ∂t + νκ2 « b u(κ, t) = b ub u ,

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Introduction Linear impulse response and closure theories

A statement of the closure problem

Homogeneous equations in Fourier space (κ, p, q wave vectors) using symbolic notation:

◮ First-order eq. (momentum)

„ ∂ ∂t + νκ2 « b u(κ, t) = b ub u ,

◮ Second-order moment eq.

„ ∂ ∂t + ν(κ2 + q2) « b u(κ, t)b u(q, t) = b ub ub u ,

M. Carini & M. Quadrio (DIA-PoliMI)

HIT Impulse Response XX AIDAA Congress, 30 June 2009 4 / 28

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Introduction Linear impulse response and closure theories

A statement of the closure problem

Homogeneous equations in Fourier space (κ, p, q wave vectors) using symbolic notation:

◮ First-order eq. (momentum)

„ ∂ ∂t + νκ2 « b u(κ, t) = b ub u ,

◮ Second-order moment eq.

„ ∂ ∂t + ν(κ2 + q2) « b u(κ, t)b u(q, t) = b ub ub u ,

◮ Third-order moment eq.

„ ∂ ∂t + ν(κ2 + q2 + p2) « b u(κ, t)b u(q, t)b u(p, t) = b ub ub ub u ,

M. Carini & M. Quadrio (DIA-PoliMI)

HIT Impulse Response XX AIDAA Congress, 30 June 2009 4 / 28

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Introduction Linear impulse response and closure theories

A statement of the closure problem

Homogeneous equations in Fourier space (κ, p, q wave vectors) using symbolic notation:

◮ First-order eq. (momentum)

„ ∂ ∂t + νκ2 « b u(κ, t) = b ub u ,

◮ Second-order moment eq.

„ ∂ ∂t + ν(κ2 + q2) « b u(κ, t)b u(q, t) = b ub ub u ,

◮ Third-order moment eq.

„ ∂ ∂t + ν(κ2 + q2 + p2) « b u(κ, t)b u(q, t)b u(p, t) = b ub ub ub u ,

◮ And so on.

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Introduction Linear impulse response and closure theories

An overview of closure theories

Turbulence in fluids, Lesieur, 2008

Navier-Stokes Statistical moment hierarchy Stochastic models Q.N. E.D.Q.N. E.D.Q.N.M. Q.N.M. R.C.M. M.R.C.M. T.F.M. D.I.A. R.N.G. L.H.D.I.A. S.B.L.H.D.I.A. L.E.T. L.D.I.A. L.R.A.

RENORMALIZATION THEORIES Lagrangian theories Eulerian theories

diagrammatic expansions functional power reversions Markovianization Markovianization Markovianization

Φκ,p,q

α,β,σ(t)

Φα,β,σ(t) Φα,β,σ ψκ,p,q = ψ0 µκ = 0 ˜ µ = νκ2 + µκ

u

u u u = 0

u

u u u = −µκ u u u

G.R.I. G.R.I.

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Introduction Linear impulse response and closure theories

Renormalization approach

Navier-Stokes D.I.A. R.N.G. L.H.D.I.A. S.B.L.H.D.I.A. L.E.T. L.D.I.A. L.R.A.

RENORMALIZATION THEORIES Lagrangian theories Eulerian theories

diagrammatic expansions functional power reversions G.R.I. G.R.I.

Second-order closure obtained by:

◮ introducing the mean impulse response

tensor Gij;

◮ resorting to complicated mathematical

tools (from quantum mechanics);

◮ deriving an integro-differential closed set

f equations in the unknowns:

◮ Qij(κ, τ) = b

ui(κ, t)b uj(−κ, t − τ);

◮ Gij(κ, τ).

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Introduction Linear impulse response and closure theories

The Direct Interaction Approximation theory

Kraichnan JFM 1959

Robert H. Kraichnan (Philadelfia 1928 - Santa Fe 2008)

◮ The first theory introducing the

concept of impulse response tensor;

◮ At the root of all triadic closures; ◮ Avoids unphysical behaviors; ◮ No empirical parameters; ◮ Deviation from Kolmogorov -5/3

law.

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Introduction Linear impulse response and closure theories

Why measuring Gij in homogeneous isotropic turbulence ?

MOTIVATIONS

◮ The related closure theories are first developed there; ◮ simplest turbulent flow; ◮ a measure of Gij is missing; ◮ Gij measure might sort out controversial issues.

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The impulse response and its measurement Analytical tools

Navier-Stokes equations in wave-number space

Each space direction assumed statistically homogeneous:        κi ui(κ, t) = 0, ∂ ∂t + νκ2

ui(κ, t) = Mijm(κ)
uj(p, t)

um(κ − p, t)dp + Pij(κ) fj(κ, t), with:

◮ Pij(κ) projection tensor in Fourier space, Pij(κ) = δij − κ−2κiκj; ◮ Mijm(κ) ≡ −i/2(κmPij(κ) + κjPim(κ)); ◮

fj(κ, t) volume stirring force.

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The impulse response and its measurement Analytical tools

The linear impulse response definition

Non-linear system: linear response respect to infinitesimal variations ∆(·).

∆ui(x, t) = Gij(x, x′, t, t′)∆fj(x′, t′)dx′ dt′

∆u(x, t) ∆f(x′, t′) T = t′ T = t > t′ volume force turbulent fluctuations

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The impulse response and its measurement Analytical tools

Impulse response properties

◮ The mean stationary response in Fourier space:

Gij(κ, κ′, τ)
= Gij(κ, τ)δ(κ − κ′);
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The impulse response and its measurement Analytical tools

Impulse response properties

◮ The mean stationary response in Fourier space:

Gij(κ, κ′, τ)
= Gij(κ, τ)δ(κ − κ′);

◮ Statistical isotropy:

Gij(κ, τ) = Pij(κ)G(κ, τ), where G(κ, τ) is the mean impulse response function.

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The impulse response and its measurement Analytical tools

Impulse response properties

◮ The mean stationary response in Fourier space:

Gij(κ, κ′, τ)
= Gij(κ, τ)δ(κ − κ′);

◮ Statistical isotropy:

Gij(κ, τ) = Pij(κ)G(κ, τ), where G(κ, τ) is the mean impulse response function.

◮ Real and bounded:

|G(κ, τ)| ≤ G(κ, 0+) = 1, ∀ τ > 0 and ∀ κ.

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The impulse response and its measurement Kraichnan’s heritage

The Stokes or viscous response function

◮ Dropping non-linear terms in the NS momentum eq., Stokes momentum eq.

is obtained: ∂ ∂t + νκ2

ui(κ, t) = Mijm(κ)
uj(p, t)

um(κ − p, t)dp + Pij(κ) fj(κ, t).

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The impulse response and its measurement Kraichnan’s heritage

The Stokes or viscous response function

◮ Dropping non-linear terms in the NS momentum eq., Stokes momentum eq.

is obtained: ∂ ∂t + νκ2

ui(κ, t) = Mijm(κ)
uj(p, t)

um(κ − p, t)dp + Pij(κ) fj(κ, t).

◮ The Stokes response function G(0)(κ, τ) can be derived analytically:

G(0)(κ, τ) = exp(−νκ2τ).

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The impulse response and its measurement Kraichnan’s heritage

The DIA approximate solution

After manipulating DIA eqs. in their homogeneous isotropic form Kraichnan derived (JFM 1959): G(κ, τ) = exp(−νκ2τ)J1(2u0κτ) u0κτ , where:

◮ J1 is the Bessel’s function of the first kind; ◮ u0 is the root mean squared of the velocity field.

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The impulse response and its measurement Kraichnan’s heritage

The DIA approximate solution

After manipulating DIA eqs. in their homogeneous isotropic form Kraichnan derived (JFM 1959): G(κ, τ) = exp(−νκ2τ)J1(2u0κτ) u0κτ , where:

◮ J1 is the Bessel’s function of the first kind; ◮ u0 is the root mean squared of the velocity field.

Only local energy-containing range time scale (u0κ)−1 appears in the inviscid part of DIA solution:

J1(2u0κτ) u0κτ .

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The impulse response and its measurement Kraichnan’s heritage

Kraichnan’s picture of random convection

Kraichnan PoF 1964

1 2 3 4 5 6 7 8 9 10 −0.2 0.2 0.4 0.6 0.8 1 1.2

u0κτ

(—) J1(2u0κτ) u0κτ and (r) exp(−1/2u2

0κ2τ 2).

◮ The idealized random Gaussian

convection problem: G(κ, τ) = exp(−1/2v2

0κ2τ 2).

where v0 is the r.m.s of the random uniform convection velocity.

◮ Sweeping of small eddies by big ones

dominates Eulerian two points two time statistics;

◮ Spurious sweeping or convective

time-scale (u0κ)−1 regulates response time decay.

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The impulse response and its measurement The measurement technique

Two (unpractical) strategies for measuring G

Problem

A turbulent flow has a large noise, while forcing amplitude must be small: S/N ratio is small !

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The impulse response and its measurement The measurement technique

Two (unpractical) strategies for measuring G

Problem

A turbulent flow has a large noise, while forcing amplitude must be small: S/N ratio is small !

Response to impulsive forcing

— Small amplitude for linearity since forcing is concentrated. + All frequencies obtained at once.

Response to sinusoidal forcing

+ Large amplitude for linearity since forcing power is distributed. — One single frequency obtained at a time.

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The impulse response and its measurement The measurement technique

The right way

Resorting to input-output correlations:

Rin,in Rout,in H

Rin,out(s) =

H(s − s′)Rin,in(s′)ds′,

◮ when the input is white noise Rin,in(s′) = δ(s′), then Rin,out(s) = H(s).

Response to white noise forcing

+ Large amplitude for linearity since forcing power is uniformly distributed; + All frequency obtained at once.

Details

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The impulse response and its measurement The measurement technique

Computational Tools

◮ DNS pseudo-spectral code

developed on purpose.

◮ SMP parallel computing. ◮ Simulations performed on

Supercomputing system located at Universit´ a di Salerno.

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The impulse response and its measurement The measurement technique

Vorticity isosurface ω = 2.5 ωr.m.s.

Reλ = 46, N = 643 Reλ = 55, N = 1283

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The impulse response and its measurement The measurement technique

Vorticity isosurface ω = 2.5 ωr.m.s.

Reλ = 77, N = 1923 Reλ = 94, N = 2563

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Results

A test case: the Stokes response N = 323

A reference solution known analytically

G(0)(κ, τ) = exp(−νκ2τ).

1 2 3 4 5 6 7 0.2 0.4 0.6 0.8 1

τνκ2 G(0)(κ, τ)

κ/κ0 = 4 κ/κ0 = 5 κ/κ0 = 6 κ/κ0 = 7 κ/κ0 = 8 κ/κ0 = 9 κ/κ0 = 10 Stokes exact 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

τνκ2 G(0)(κ, τ) κ/κ0 = 8

∆τνκ2 =0.0059 ∆τνκ2 =0.029 ∆τνκ2 =0.059 ∆τνκ2 =0.15 Stokes exact

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Results

The measured response: Reλ = 94, N = 2563

Kolmogorov scale κ = κd

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0.5 1 1.5 2 2.5 3 3.5 4 4.5 9.741e−01 9.741e−01 9.562e−01 9.599e−01 9.184e−01 9.226e−01 8.835e−01 8.817e−01 8.431e−01 8.495e−01 7.799e−01 7.751e−01 7.520e−01 7.606e−01 6.570e−01 6.520e−01 6.494e−01 6.620e−01 5.287e−01 5.249e−01 5.452e−01 5.602e−01 4.035e−01 4.044e−01 4.422e−01 4.610e−01 2.906e−01 2.982e−01 3.446e−01 3.687e−01 1.955e−01 2.104e−01 2.586e−01 2.868e−01 1.209e−01 1.420e−01 1.066e−01 1.149e−01 4.292e−02 5.263e−02 4.800e−02 6.416e−02 1.149e−02 2.380e−02 9.670e−01 9.588e−01 8.910e−01 8.827e−01 7.891e−01 7.804e−01 6.701e−01 6.626e−01 6.009e−01 5.711e−01 5.085e−01 4.811e−01 4.092e−01 3.961e−01 3.334e−01 3.188e−01 5.445e−01 5.402e−01 4.641e−01 4.325e−01 3.620e−01 3.346e−01 2.569e−01 2.502e−01 1.878e−01 1.808e−01 4.224e−01 4.229e−01 3.389e−01 3.100e−01 2.403e−01 2.166e−01 1.428e−01 1.443e−01 9.007e−02 9.160e−02 3.131e−01 3.180e−01 2.338e−01 2.102e−01 1.485e−01 1.304e−01 6.786e−02 7.592e−02 3.417e−02 4.147e−02 2.177e−01 2.296e−01 1.525e−01 1.349e−01 8.594e−02 7.308e−02 2.580e−02 3.648e−02 7.736e−03 1.678e−02 1.401e−01 1.592e−01 9.451e−02 8.196e−02 4.693e−02 3.811e−02 6.700e−03 1.600e−02 −1.545e−03 6.065e−03 9.690e−01 9.655e−01 9.699e−01 9.552e−01 9.350e−01 9.436e−01 9.363e−01 9.308e−01 9.251e−01 9.169e−01 9.119e−01 9.019e−01 8.959e−01 8.859e−01 8.812e−01 8.689e−01 9.138e−01 9.065e−01 8.992e−01 8.797e−01 8.494e−01 8.502e−01 8.311e−01 8.185e−01 8.006e−01 7.847e−01 7.698e−01 7.494e−01 7.338e−01 7.128e−01 6.990e−01 6.753e−01 8.397e−01 8.277e−01 8.063e−01 7.811e−01 7.403e−01 7.315e−01 7.009e−01 6.798e−01 6.506e−01 6.268e−01 6.028e−01 5.736e−01 5.505e−01 5.208e−01 5.004e−01 4.693e−01 7.527e−01 7.349e−01 7.006e−01 6.688e−01 6.186e−01 6.010e−01 5.608e−01 5.333e−01 4.956e−01 4.674e−01 4.373e−01 4.044e−01 3.772e−01 3.456e−01 3.226e−01 2.917e−01 3.534e−01 3.252e−01 2.935e−01 2.627e−01 2.353e−01 2.083e−01 1.866e−01 1.621e−01 2.353e−01 2.112e−01 1.819e−01 1.572e−01 1.331e−01 1.140e−01 9.690e−02 8.058e−02 1.460e−01 1.281e−01 1.041e−01 8.665e−02 6.722e−02 5.667e−02 4.482e−02 3.582e−02 8.349e−02 7.246e−02 5.470e−02 4.400e−02 2.984e−02 2.558e−02 1.850e−02 1.424e−02 4.391e−02 3.827e−02 2.667e−02 2.058e−02 1.125e−02 1.049e−02 6.870e−03 5.063e−03

κ/κd τκdu0 G(κ, τ)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

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Results

Assessment convective scaling

1 2 3 4 5 6 7 0.2 0.4 0.6 0.8 1

τκu0 G(κ, τ) Reλ = 94

κ/κd = 0.36 κ/κd = 0.46 κ/κd = 0.57 κ/κd = 0.68 κ/κd = 0.79 κ/κd = 0.89 κ/κd = 1

◮ G(κ, τ) for several κ in the

dissipative range;

◮ convective scaled time separation

τκu0;

◮ Data collapse: convective scaling is

effective

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Results

Comparison with analytical solutions

1 2 3 4 5 6 7 0.2 0.4 0.6 0.8 1

τκu0 G(κ, τ) κ/κ0 =56 κ/κd =1

∆τ u0κ =0.061 ∆τ u0κ =0.038 Viscous Gaussian−convective DIA

◮ Viscous Gaussian-Convective (GC)

response: G(κ, τ) = exp(−νκ2τ−1/2u2

0κ2τ 2); ◮ Both DIA and GC fit well for

τκu0 ≪ 1;

◮ GC provides a good fitting for the

whole response.

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Conclusions

◮ The measurement technique has proved to be succesful; ◮ Kraichnan’s theoretical predictions about convective scaling of the response

are confirmed;

◮ Surprisingly the viscous Gaussian-convective solution provides a good

approximation to measured data;

◮ Our results (and conclusions) are limited to low Re;

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Conclusions

The future

◮ Assessment of the response behavior in presence of a well developed inertial

range of scales, i.e. at higher Reλ;

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Conclusions

The future

◮ Assessment of the response behavior in presence of a well developed inertial

range of scales, i.e. at higher Reλ;

Open issue

McComb et al. (JFM 1989) recovered Kolmogorov scaling solving numerically DIA and LET eqs. at Reλ ≈ 1000, while at low Reλ, Reλ < 40, convective scaling was found to be effective.

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HIT Impulse Response XX AIDAA Congress, 30 June 2009 25 / 28

SLIDE 35

Conclusions

The future

◮ Assessment of the response behavior in presence of a well developed inertial

range of scales, i.e. at higher Reλ;

Open issue

McComb et al. (JFM 1989) recovered Kolmogorov scaling solving numerically DIA and LET eqs. at Reλ ≈ 1000, while at low Reλ, Reλ < 40, convective scaling was found to be effective.

Towards modeling

If Komolgorov scaling is restored:

◮ G will provide information about turbulence dynamics. ◮ Fully characterization of the measured response will pave the way for a new

class of models.

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HIT Impulse Response XX AIDAA Congress, 30 June 2009 25 / 28

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Conclusions

THANK YOU FOR YOUR ATTENTION !

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Measuring G using white noise forcing

◮ White noise

wi(t) volume forcing perturbation: ∆fi(κ, t) = ǫwi(κ, t) with ǫ ≪ 1.

∆

ui(κ, t)∆ fj(−κ, t − τ)

=

= +∞

−∞

Gim(κ, t − t′)∆(κ′ − κ)

∆

fm(κ′, t′)∆ fj(−κ′, t − τ)

dt′dκ′.

◮ Sampling property of white noise delta-correlation:

∆

fn(κ′, t′)∆ fj(−κ′, t − τ)

= δnjδ(t′ − t + τ).

◮ Output correlation leads to scaled response:

∆

ui(κ, t)∆ fj(−κ, t − τ)

= ǫ2Gij(κ, τ).
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HIT Impulse Response XX AIDAA Congress, 30 June 2009 27 / 28

SLIDE 38

Only the perturbed velocity field u(κ, t) is observable!

◮ Linear response VS turbulent fluctuations decomposition:

u(κ, t) =

uǫ(κ, t) + ∆ u(κ, t).

◮ Expanding the input-output correlation:

ui(t)∆

fj(t − τ)

ǫ2

= 1 ǫ2

uǫi(t)∆

fj(t − τ)

+
∆

ui(t)∆ fj(t − τ)

.

◮ Red term is averaged out since fully non-linear turbulent fluctuations and

white noise perturbation are independently generated random processes. Then it follows:

ui(κ, t)∆

fj(−κ, t − τ)

ǫ2

= Gij(κ, τ).

Back

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