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) HIT Impulse Response XX AIDAA Congress, 30 June 2009 1 / 28
Outline Introduction The impulse response and its measurement Results Conclusions M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 2 / 28
Introduction Numerical simulation of turbulence ◮ Turbulence dominates most engineering flows; ◮ Available strategies: RANS, LES; ◮ Modeling (Reynolds stress, Subgrid scale stress) always required; ◮ Closure theories useful for modeling. Wu and Moin JFM 2009 M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 3 / 28
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 + νκ 2 � b u ( κ , t ) � = � b u b u � , M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 4 / 28
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 + νκ 2 � b u ( κ , t ) � = � b u b u � , ◮ Second-order moment eq. „ ∂ « ∂t + ν ( κ 2 + q 2 ) � b u ( κ , t ) b u ( q , t ) � = � b u b u b u � , M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 4 / 28
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 + νκ 2 � b u ( κ , t ) � = � b u b u � , ◮ Second-order moment eq. „ ∂ « ∂t + ν ( κ 2 + q 2 ) � b u ( κ , t ) b u ( q , t ) � = � b u b u b u � , ◮ Third-order moment eq. „ ∂ « ∂t + ν ( κ 2 + q 2 + p 2 ) � b u ( κ , t ) b u ( q , t ) b u ( p , t ) � = � b u b u b u b u � , M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 4 / 28
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 + νκ 2 � b u ( κ , t ) � = � b u b u � , ◮ Second-order moment eq. „ ∂ « ∂t + ν ( κ 2 + q 2 ) � b u ( κ , t ) b u ( q , t ) � = � b u b u b u � , ◮ Third-order moment eq. „ ∂ « ∂t + ν ( κ 2 + q 2 + p 2 ) � b u ( κ , t ) b u ( q , t ) b u ( p , t ) � = � b u b u b u b u � , ◮ And so on. M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 4 / 28
Introduction Linear impulse response and closure theories An overview of closure theories Turbulence in fluids , Lesieur, 2008 Navier-Stokes Stochastic models Φ α,β,σ R ENORMALIZATION THEORIES diagrammatic expansions R.C.M. functional power Statistical moment hierarchy Markovianization Φ α,β,σ ( t ) R.N.G. L.E.T. D.I.A. reversions Eulerian theories G.R.I. G.R.I. � � u � u � u � u � = 0 � � u � u � u � u � = − µ κ � � u � u � u � L.D.I.A. L.H.D.I.A. L.R.A. M.R.C.M. Φ κ,p,q α,β,σ ( t ) S.B.L.H.D.I.A. ψ κ,p,q = ψ 0 Lagrangian theories µ = νκ 2 + µ κ ˜ Markovianization Q.N. E.D.Q.N. E.D.Q.N.M. T.F.M. µ κ = 0 Markovianization Q.N.M. M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 5 / 28
Introduction Linear impulse response and closure theories Renormalization approach Navier-Stokes Second-order closure obtained by: R ENORMALIZATION THEORIES ◮ introducing the mean impulse response diagrammatic expansions tensor G ij ; functional ◮ resorting to complicated mathematical power L.E.T. R.N.G. D.I.A. reversions tools (from quantum mechanics ); Eulerian theories ◮ deriving an integro-differential closed set G.R.I. G.R.I. of equations in the unknowns: L.D.I.A. L.H.D.I.A. ◮ Q ij ( κ , τ ) = � b L.R.A. u i ( κ , t ) b u j ( − κ , t − τ ) � ; ◮ G ij ( κ , τ ) . S.B.L.H.D.I.A. Lagrangian theories M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 6 / 28
Introduction Linear impulse response and closure theories The Direct Interaction Approximation theory Kraichnan JFM 1959 ◮ 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. Robert H. Kraichnan (Philadelfia 1928 - Santa Fe 2008) M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 7 / 28
Introduction Linear impulse response and closure theories Why measuring G ij in homogeneous isotropic turbulence ? MOTIVATIONS ◮ The related closure theories are first developed there; ◮ simplest turbulent flow; ◮ a measure of G ij is missing; ◮ G ij measure might sort out controversial issues. M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 8 / 28
The impulse response and its measurement Analytical tools Navier-Stokes equations in wave-number space Each space direction assumed statistically homogeneous: κ i � u i ( κ , t ) = 0 , � ∂ � � ∂t + νκ 2 u m ( κ − p , t ) d p + P ij ( κ ) � u i ( κ , t ) = M ijm ( κ ) � u j ( p , t ) � � f j ( κ , t ) , with: ◮ P ij ( κ ) projection tensor in Fourier space, P ij ( κ ) = δ ij − κ − 2 κ i κ j ; ◮ M ijm ( κ ) ≡ − i/ 2( κ m P ij ( κ ) + κ j P im ( κ )) ; ◮ � f j ( κ , t ) volume stirring force. M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 9 / 28
The impulse response and its measurement Analytical tools The linear impulse response definition Non-linear system: linear response respect to infinitesimal variations ∆ ( · ) . T = t > t ′ T = t ′ volume force ∆ u ( x , t ) turbulent fluctuations ∆ f ( x ′ , t ′ ) � � G ij ( x , x ′ , t, t ′ ) ∆f j ( x ′ , t ′ ) d x ′ dt ′ ∆u i ( x , t ) = M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 10 / 28
The impulse response and its measurement Analytical tools Impulse response properties ◮ The mean stationary response in Fourier space: � � � G ij ( κ , κ ′ , τ ) = G ij ( κ , τ ) δ ( κ − κ ′ ); M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 11 / 28
The impulse response and its measurement Analytical tools Impulse response properties ◮ The mean stationary response in Fourier space: � � � G ij ( κ , κ ′ , τ ) = G ij ( κ , τ ) δ ( κ − κ ′ ); ◮ Statistical isotropy: G ij ( κ , τ ) = P ij ( κ ) G ( κ, τ ) , where G ( κ, τ ) is the mean impulse response function. M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 11 / 28
The impulse response and its measurement Analytical tools Impulse response properties ◮ The mean stationary response in Fourier space: � � � G ij ( κ , κ ′ , τ ) = G ij ( κ , τ ) δ ( κ − κ ′ ); ◮ Statistical isotropy: G ij ( κ , τ ) = P ij ( 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. M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 11 / 28
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 + νκ 2 u m ( κ − p , t ) d p + P ij ( κ ) � u i ( κ , t ) = M ijm ( κ ) u j ( p , t ) � f j ( κ , t ) . � � M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 12 / 28
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 + νκ 2 u m ( κ − p , t ) d p + P ij ( κ ) � u i ( κ , t ) = M ijm ( κ ) u j ( p , t ) � f j ( κ , t ) . � � ◮ The Stokes response function G (0) ( κ, τ ) can be derived analytically: G (0) ( κ, τ ) = exp( − νκ 2 τ ) . M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 12 / 28
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( − νκ 2 τ ) J 1 (2 u 0 κτ ) , u 0 κτ where: ◮ J 1 is the Bessel’s function of the first kind; ◮ u 0 is the root mean squared of the velocity field. M. Carini & M. Quadrio (DIA-PoliMI) HIT Impulse Response XX AIDAA Congress, 30 June 2009 13 / 28
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