Effects of Galerkin-truncation on Inviscid Equations of Hydrodynamics. Samriddhi Sankar Ray (U. Frisch, S. Nazarenko, and T.

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1 Effects of Galerkin-truncation on Inviscid Equations of Hydrodynamics Samriddhi Sankar Ray (U. Frisch, S. Nazarenko, and T. Matsumoto) Laboratoire Lagrange, Observatoire de la Côte d Azur, Nice, France. International Centre for Theoretical Sciences Tata Institute of Fundamental Research, Bangalore, India. Indo-French Center for Applied Mathematics Kick-Off Meeting, Laboratoire J. A. Dieudonné Parc Valrose, Nice, France. November Phys. Rev. E, 84, 6 ()

2 The Indo French Collaboration Debarghya Banerjee (Bangalore) Marc Etienne Brachet (Paris) Sagar Chakraborty (Copenhagen/Kanpur) Joris van der Hoeven (Paris) Rahul Pandit (Bangalore) Walter Pauls (Nice/Goettingen)

3 Outline Introduction : Why Worry About Truncated Systems? Galerkin Truncation The Tyger Phenomenon : D Burgers Equation Implications for Tracing δ Conclusions and Perspective

4 Why Worry About Truncated Systems? Inverse energy cascade in D turbulence. (Kraichnan, 967) Physics Statistical Physics Approach Thermalisation D Euler : Temporal evolution of the energy spectra. (Bustamante & Brachet, Phys. Rev. E, in press, ). Mathematics Clue to Finite-time Blow-up E(k,t) = C(t)k n(t) k δ(t)

5 Equilibrium Statistical Mechanics and Turbulence Equilibrium statistical mechanics is concerned with conservative Hamiltonian dynamics, Gibbs states,... Turbulence is about dissipative out-of-equilibrium systems. In 95 Hopf and Lee apply equilibrium statistical mechanics to the D Euler equation and obtain the equipartition energy spectrum which is very different from the Kolmogorov spectrum. In 967 Kraichnan uses equilibrium statistical mechanics as one of the tools to predict the eistence of an inverse energy cascade in D turbulence.

6 Equilibrium Statistical Mechanics and Turbulence In 989 Kraichnan remarks the truncated Euler system can imitate NS fluid: the high-wavenumber degrees of freedom act like a thermal sink into which the energy of low-wave-number modes ecited above equilibrium is dissipated. In the limit where the sink wavenumbers are very large compared with the anomalously ecited wavenumbers, this dynamical damping acts precisely like a molecular viscosity. In 5 Cichowlas, Bonaiti, Debbasch, and Brachet discovered long-lasting, partially thermalized, transients similar to high-reynolds number flow. E(k) k

7 The Galerkin-truncated D Burgers equation The (untruncated) inviscid Burgers equation, written in conservation form, is t u + (u /) = ; u(,) = u (). Let K G be a positive integer, here called the Galerkin truncation wavenumber, such that the action of the projector P KG : P KG u() = k K G e ikû k. The associated Galerkin-truncated (inviscid) Burgers equation t v +P KG (v /) = ; v = P KG u.

8 Time Evolution of the Truncated Equation t =..5 v()

9 Tygers in the Galerkin-truncated D Burgers equation t =. t =.5 t = v() v() v() Growth of a tyger in the solution of the inviscid Burgers equation with initial condition v () = sin( π/). Galerkin truncation at K G = 7. Number of collocation points N = 6,84. Observe that the bulge appears far from the place of birth of the shock.

10 Tygers only at regions of positive strain u () = sin()+sin( +.9)+sin() v() t = v() t = Three-mode initial condition. Tygers appear at the points having the same velocity as the shock and positive strain.

11 From tygers to thermalization t =. 4 4 t =.4 t =.5 v(), u() v(), u() v(), u() v(), u() t =.8 v(), u() t =. v(), u() t = t = t = 4.5 v(), u() v(), u()

12 From tygers to thermalization.. t = u () t =..4 t =.9.5 u ().5. u () t =.7.5 t =.5.8 t =..4. u ().5. u () u () t = t = u () u ()

13 Phenomenological Eplanation : D Burgers A localized strong nonlinearity, such as is present at a preshock or a shock, acts as a source of a truncation wave. Away from the source this truncation wave is mostly a plane wave with wavenumber close to K G. The radiation of truncation waves begins only at or close to the time of formation of a preshock. Resonant interactions are confined to particles such that τ v τ v v s λ G. If τ is small the region of resonance will be confined to a small neighborhood of widths K G / around the point of resonance. In a region of negative strain a wave of wavenumber close to K G will be squeezed and thus disappearing beyond the truncation horizon which acts as a kind of black hole.

14 Scaling properties of the early tygers log w K G / log K G log a K G / log K G width K G / amplitude K G / (using phase miing arguments) (using energy conservation arguments)

15 Scaling properties of the early tygers Scaling of the tyger widths : By the time t, truncation is significant only for a lapse of time O(K / G ). The phase miing argument tells us that the coherent build up of a tyger will affect only those locations whose velocity differs from that at resonance by an amount π v K / K / G K G. G Since at such times, the velocity v of the truncated solution is epected to stay close to the velocity u of the untruncated solution and the latter varies linearly with near the resonance point, the width of the t tyger is itself proportional to K / G. Scaling of the tyger amplitudes : The Galerkin-truncated Burgers equation conserves energy. The apparent energy loss due to truncation λ G / d K 5/ G. Conservation demands that this energy-loss is transferred to the tygers which gives the tyger-amplitude scaling as K / G. The above argument is appealing but not rigorous.

16 Weak solutions?.5 t =.4.5 t =.5.5 t =.6 Filtered v(),u().5.5 Filtered v(),u().5.5 Filtered v(),u() Plots of solution of the Galerkin-truncated Burgers equation, with K G = 5,46 (green) and K G =,845 (black), low-pass filtered at wavenumber K =, at various times. Initial condition v () = sin()+sin(.74). The untruncated solution is shown in red.

17 Birth of tygers : Systematic theory Define discrepancy ũ v u to obtain ( ) t ũ +P KG uũ + ũ u = (I P KG ), ũ() =. Decompose u = u < +u >, where u < P KG u and u > (I P KG )u. Similarly the perturbation u P KG ũ. Finally we obtain : t u +P KG ( u < u + (u ) ) = P KG (u < u > + (u> ) ).

18 f The beating input Im ˆf k 4 5 6

19 Birth of tygers : Approimations t u +P KG ( u < u + (u ) ) = P KG (u < u > + (u> ) ). Strategy :. The term (u ) is discarded;. The perturbation u is set to zero at time t G ;. The untruncated solution is frozen to its t value. With the three approimations the temporal dynamics of the perturbation near t is KG dτû d k = A kk û k +ˆf k, û k () =, k = K G A kk ik û <,k k, ˆf k ik (û < pû> q + û> pû> q ). p+q=k

20 Are the approimations justified?..5 ak G / log K G Log-linear plot of the compensated amplitude of the tyger K / G a(k G), calculated (i) from ũ, (ii) from the linearised approimation for u, and (iii) from the freezing plus reinitialization approimation, all versus K G.

21 Fourier space solution of the perturbation Im û K G = k Im û 4 K G = 5, k 5 Im û K G =, k / 4 The boundary layer in Fourier space near K G. Shown are the imaginary parts of û (t ) for three values of K G. The origin is at the preshock. The even-odd oscillations indicate that most of the activity is at the tyger, a distance π away.

22 Scaling function for the boundary layer.5..5 F (KG k)/ K / G The envelopes of the various boundary layers shown in earlier (with preshock contributions subtracted out), collapsed into a single curve after rescaling. Red circles: K G =,, blue circles: K G = 5,, red squares: K G =,, blue squares: K G = 5,. The thick black line is the eponential fit.

23 Tygers can be reduced to a problem in linear algebra We can solve KG dτû d k = at time τ = t t G : k = K G A kk û k +ˆf k, û k () = u (τ ) = A ( ) e τ A I f = ( n= ) A n τ n+ f. (n+)! From this it becomes clear that much will be controlled by the spectral properties of the operator A. We are thus led to consider the associated eigenvalue/eigenvector equation Aψ = λψ.

24 Implications for Tracing δ D Euler Taylor Green flows (Bustamante & Brachet, Phys. Rev. E, in press, ). (Left) Energy spectra at different times. (Right) Comparison of fit to E(k,t) = C(t)k n(t) k δ(t) (solid black line) and spectrum at resolution 496 (red markers).

25 (a) Constant C; (b) prefactor n; (c) decrement δ (horizontal lines : δk ma =, dashed black line: eponential); (d) d(ln(δ(t))/dt. 5 (brown triangles), 4 (blue squares), 48 (green diamonds) and 496 (red circles). Implications for Tracing δ D Euler Taylor Green flows (Bustamante & Brachet, Phys. Rev. E, in press, ).

26 A Better Way to Estimate δ From Simulations? Take into account truncation effects. Improve the accuracy in estimating δ(t) by using a better processing of the numerical output. Increase numerical precision in Direct Numerical Simulations.

27 A Better Way to Estimate δ From Simulations? Take into account truncation effects. By using theoretical knowledge of the beating input to negate truncation effects. Selectively eliminating the boundary layer at high k to improve estimation of δ at longer times (purging). Improve the accuracy in estimating δ(t) by using a better processing of the numerical output. Asymptotic Etrapolation : Perform on the data a succession of transformations which successively strip off the leading and subleading terms in the asymptotic epansion (here, for large k ). Increase numerical precision in Direct Numerical Simulations. Useful to go from double precision to higher precision.

28 Digression : Method of Asymptotic Etrapolation Suppose the values of a function G n for integers n up to some high value N are known with a high precision. Goal: to obtain as many terms as possible in the asymptotic epansion of G n for high n. Trying to fit the function by a guessed leading-order asymptotic form with a few free parameters gives a poor accuracy for these parameters. The asymptotic etrapolation method achieves the goal by applying to the data a sequence of suitably chosen transformations that strip off successively the leading and subleading terms. At certain stages in the sequence of transformations, the processed data may allow simple etrapolation. W. Pauls and U. Frisch, J. Stat. Phys. 7, 95 (7). J. van der Hoeven, J. Symb. Comput. 44, (9). S. Chakraborty, U. Frisch, W. Pauls, and S. S. Ray, Phys. Rev. E (R) 85, 5 ().

29 Digression : Method of Asymptotic Etrapolation By undoing the transformations, one obtains the asymptotic epansion of the original data, up to some order. Typical transformations : I (inverse) = /G n ; R (ratio) = G n /G n ; SR (second ratio) = (G n G n )/G n ; D (difference) = G n G n. Relative Error 4 Quadruple Double 5 R ma Relative error of Nelkin eponents χ 4 and χ 6 obtained by asymptotic etrapolation from pseudo-spectral calculations up to a maimum Reynolds number R ma. W. Pauls and U. Frisch, J. Stat. Phys. 7, 95 (7). J. van der Hoeven, J. Symb. Comput. 44, (9). S. Chakraborty, U. Frisch, W. Pauls, and S. S. Ray, Phys. Rev. E (Rapid Comm.) 85, 5 ().

30 Testing For Burgers High Precision Purging u().5.5 log E(k) log k

31 Testing For Burgers High Precision Purging Asymptotic Etrapolation log δ(t * t) δ th δ num /δ th 4 log (t * t) t * t C. Sulem, P.-L. Sulem, and H. Frisch, J. Comp. Phys. 5, 8-6 (98).

32 Conclusions and Perspective Tygers provide a clue as to the onset of thermalization. We do not have a complete understanding of the phenomenon. Tygers do not modify shock dynamics but modify the flow elsewhere because the tygers induce Reynolds stresses on scales much larger than the Galerkin wavelength; hence the weak limit of the Galerkin-truncated solution as K G is NOT the inviscid limit of the untruncated solution. There is good evidence that the key phenomena associated to tygers are also present in the two-dimensional incompressible Euler equation (and also perhaps in three dimensions). It is clear that comple-space singularities approaching the real domain within one Galerkin wavelength are the triggering factor in both the D Euler and the D Burgers case. Can we purge tygers away and thereby obtain a subgrid-scale method which describes the inviscid-limit solution right down to the Galerkin wavelength?