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Spectral analysis of mixing in chaotic flows via the mapping matrix formalism: Inclusion of molecular diffusion and quantitative eigenvalue estimate in the purely convective limit
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10.1063/1.4738598
/content/aip/journal/pof2/24/7/10.1063/1.4738598
http://aip.metastore.ingenta.com/content/aip/journal/pof2/24/7/10.1063/1.4738598

Figures

Image of FIG. 1.
FIG. 1.

Illustration of the computation of the entries M ij of the mapping matrix M. The cell Ω j at t = t 0 is covered with a number of markers that are tracked during flow in Δt to arrive at the final cross section t = t 0 + Δt. The ratio of the number of markers received by the recipient cell Ω i to the initial number of markers in Ω j is determined. In this case M ij = 3/16.

Image of FIG. 2.
FIG. 2.

Panel (a): ⟨Λmp⟩ vs the number of particles per unit interval N p for the TPSF at Pe = 103 and N c = 50: symbols (•) refers to T p = 0.8, (□) to T p = 1.6. The horizontal lines represent the corresponding values of Λmp obtained by means of Fourier analysis. Panel (b): vs N p for the TPSF at T p = 0.8, Pe = 103. The arrow indicates increasing values of N c : (□) N c = 15, (■) N c = 30, (○) N c = 50, (•) N c = 65. Lines (A) and (B) represent the scaling . Panel (c): vs N p for several TPSF protocols: (□) T p = 0.8, Pe = 102 N c = 30, (■) T p = 0.8, Pe = 103 N c = 30, (○) T p = 1.18, Pe = 103, N c = 30, (•) T p = 1.6, Pe = 103, N c = 50. Panel (d): Prefactor of the normalized variance vs N c for the TPSF at T p = 0.8, Pe = 103. The solid line represents the scaling .

Image of FIG. 3.
FIG. 3.

Invariant rescaling of vs N c × N p for several TSPC protocols at a fixed value of Pe = 103: (□) T p = 0.8, N c = 15; (■) T p = 0.8, N c = 30; (○) T p = 0.8, N c = 50; (•) T p = 0.8, N c = 65; (△) T p = 1.6, N c = 30; (▲) T p = 1.6, N c = 50; (▽) T p = 1.18, N c = 30. The solid line represents the scaling with a = 5.

Image of FIG. 4.
FIG. 4.

(line (a) and symbols (•)), and (line (b) and symbols (□)) vs Pe for the TPSF at T p = 0.8, N c = 30. Line (c) represents the scaling .

Image of FIG. 5.
FIG. 5.

Rescaled eigenvalue distribution vs for the TPSF at T p = 0.8, Pe = 103. Symbols (□) refers to N c = 50, N p = 5 (dominant eigenvalue), (■) to N c = 50, N p = 20 (dominant eigenvalue), (○) to N c = 50, N p = 20 (third eigenvalue), (•) to N c = 30, N p = 100. The solid line represents the normalized Gaussian distribution .

Image of FIG. 6.
FIG. 6.

Dominant decay exponent of the purely convective mapping matrix as a function of the specific particle number N p . Symbols (□) and line (a) refer to T p = 0.8, N c = 60, (■) and line (b) to T p = 0.8 and N c = 100, (○) and line (c) to T p = 1.6 and N c = 60, (•) and line (d) to T p = 1.6 and N c = 100. lines (e) to (f) and symbols (▽, ▲, △) to T p = 2 at N c = 60, 80, 100, respectively.

Image of FIG. 7.
FIG. 7.

Panel (a): ⟨Λmp⟩ vs N p for the TPSF at T p = 0.8, Pe = 103. Line (A) and (□) dominant (second), line (B) and (○) third, line (C) and (•) fourth decay exponent. The solid horizontal lines correspond to the values obtained by Fourier analysis. Panel (b): vs N p for the second third and fourth decay exponent (symbols as in panel (a)). Lines (A) and (B) correspond to the scaling .

Image of FIG. 8.
FIG. 8.

Panels (a)–(c): Eigenvalue spectrum of the diffusive mapping matrix (N c = 50) of the sine flow at T p = 0.8 for several values of the Péclet number: (a) Pe = 104, (b) Pe = 106, (c) Pe = 108. Panel (d) depicts the eigenvalue spectrum of the purely convective mapping matrix.

Image of FIG. 9.
FIG. 9.

Dominant decay exponent of the mapping matrix Λmp (dots •) vs Pe at T p = 0.8, N c = 50. Line (a) represents the scaling of the dominant decay exponent Λ of the advection-diffusion equation obtained by Cerbelli et al.,29 dotted horizontal line (b) is the limit value of Λmp in the purely convective case.

Image of FIG. 10.
FIG. 10.

Dominant scaling exponent of the purely convective mapping matrix (horizontal lines), and its intersection with the corresponding exponent Λ(Pe) of the advection-diffusion operator (solid lines (colored red) and symbols (□)). The arrows indicate increasing values of N c = 30, 50, 80, 150, 200. Symbols (•) correspond to the intersection points between Λ(Pe) and , defining the effective Péclet number Pe eff. Panel (a): T p = 0.56. Panel (b): T p = 0.8. Panel (c): T p = 1.18.

Image of FIG. 11.
FIG. 11.

Effective Péclet number Pe eff vs the square of linear lattice size N c . Symbols (□) refer to T p = 0.56, (○) to T p = 0.8, (•) to T p = 1.18. The solid lines correspond to the scaling .

Image of FIG. 12.
FIG. 12.

Dominant scaling exponent of the purely convective mapping matrix vs the lattice Péclet number for different periods of the sine flow. Line (a) and (□) T p = 0.56, line (b) and (■) T p = 0.8, line (c) and (○) T p = 1.18, lines (d) and (e) and (•) T p = 1.6.

Image of FIG. 13.
FIG. 13.

Dominant decay exponent of the mapping matrix with diffusion as a function of the Péclet number. Data refer to N c = 50. Line (a) and (□) refer T p = 0.56, line (b) and (○) to T p = 0.8, line (c) and (•) T p = 1.18. The solid lines are the results of Eq. (39).

Image of FIG. 14.
FIG. 14.

Dominant decay exponent of the mapping matrix with diffusion Λmp (symbols) for the TPSF at a function of the Péclet number: (□) N c = 50, (○) N c = 100, (•) N c = 200. The solid lines represent the function equation (39). Panel (a): T p = 0.56. The dotted line represents the scaling Λmp(Pe) ∼ Pe −β, with β = 0.87. Panel (b): T p = 0.8. The dotted line represents the scaling Λmp(Pe) ∼ Pe −β, with β = 0.745. Panel (c): T p = 1.18. The dotted line represents the scaling Λmp(Pe) ∼ Pe −β, with β = 0.6.

Image of FIG. 15.
FIG. 15.

Comparison of Λmp (symbols (•)) vs Pe for T p = 1.18, N c = 50, and Λ(Pe) obtained by Fourier analysis (symbols (□)). The solid line is the scaling Λ(Pe) ∼ Pe −β where β = 0.55.

Image of FIG. 16.
FIG. 16.

Dominant decay exponent of the mapping matrix with diffusion Λmp (symbols) for the TPSF at T p = 1.6 as a function of the Péclet number: (□) N c = 30, (○) N c = 100, (•) N c = 200. The arrow indicates increasing values of N c .

Image of FIG. 17.
FIG. 17.

Prefactor p 0 entering Eq. (31) vs the flow period T p . For the value at T p = 2, see the discussion in Sec. VI in connection with Figure 22.

Image of FIG. 18.
FIG. 18.

Comparison of the prediction of dominant decay exponent by using the purely convective mapping matrix with respect to the results obtained by the spectral analysis of the advection-diffusion operator. Symbols (•) refer to , (○) to Λ. Line (a) corresponds to T p = 0.56, (b) to T p = 0.8, (c) to T p = 1.18.

Image of FIG. 19.
FIG. 19.

Comparison of the dominant decay exponent of the purely convective mapping matrix (•) and the dominant decay exponent of the corresponding advection-diffusion operator (○, and □) at T p = 1.6.

Image of FIG. 20.
FIG. 20.

Modulus of the second eigenfunction of the purely convective mapping matrix at T p = 1.6. Panel (a): N c = 100. Panel (b): N c = 200.

Image of FIG. 21.
FIG. 21.

Zoom-in of the Poincaré map of the TPSF at T p = 2 near the periodic point (0.25, 0.25).

Image of FIG. 22.
FIG. 22.

Dominant decay exponent of the purely convective mapping matrix vs the lattice Péclet number for T p = 2. The solid line (a) represents the scaling with β = 0.38, solid lines (b) and (c) the scaling with β = 0.45.

Image of FIG. 23.
FIG. 23.

Modulus (multiplied by 10−3) of the second eigenfunction of the purely convective mapping matrix at T p = 2. Panel (a): N c = 50. Panel (b): N c = 52.

Image of FIG. 24.
FIG. 24.

Comparison of the dominant decay exponent of the purely convective mapping matrix (symbols • and □) and the dominant decay exponent of the corresponding advection-diffusion operator (○) for T p = 2. Symbols (•) refer to the lower branch of for N c = 2(2n + 1), (□) to the upper branch for N c = 4n, with n integer. Solid lines (a) and (b) represent the scalings with β = 0.37 and β = 0.45, respectively.

Image of FIG. 25.
FIG. 25.

L 2-norm decay of a scalar field ϕ vs time t. Symbols refer to the coarse-grained evolution via the purely convective mapping matrix, solid lines (a)–(c) the results obtained by solving the advection-diffusion equation for a value of Pe number equal to the effective Péclet number pertaining to each discretization. Symbols (□) refer to N c = 30, (○) to N c = 50, (•) to N c = 80. Panel (a): T p = 0.8. Panel (b): T p = 1.18.

Image of FIG. 26.
FIG. 26.

Poincaré sections of the PPM. Panel (a): βppm = 1, (b) βppm = 8, (c) βppm = 10, (d) βppm = 20.

Image of FIG. 27.
FIG. 27.

Dominant scaling exponent of the purely convective mapping matrix vs the lattice Péclet number Pe c for the PPM at different values of βppm Line (a) and (□) βppm = 1, line (b) and (■) βppm = 8, line (c) and (○) βppm = 10, lines (d) and (e) and (•) βppm = 20. Dashed lines represent the scalings , for the values of the exponent β reported in Table II.

Image of FIG. 28.
FIG. 28.

Behavior of the dominant decay exponent vs Pe for very high values of Pe eff obtained using a power-method. Line (a) refers to , line (b) to the scaling , where β = 0.5.

Image of FIG. 29.
FIG. 29.

Zoom-in of the Poincaré map of the TPSF at T p = 1.6 showing the occurrence of a small quasi-periodic island.

Tables

Generic image for table
Table I.

Scaling exponent β characterizing the behavior of the dominant decay exponent with Pe for different values of the period of the TPSF. The second column, indicated with (mp) refers to the exponent derived from the data of the purely convective mapping matrix, the third column refers to the values obtained in Cerbelli et al. 29 by considering the continuum advection-diffusion operator.

Generic image for table
Table II.

Scaling exponents β for the PPM for the different values of βppm considered in the paper.

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/content/aip/journal/pof2/24/7/10.1063/1.4738598
2012-07-31
2014-04-25
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Spectral analysis of mixing in chaotic flows via the mapping matrix formalism: Inclusion of molecular diffusion and quantitative eigenvalue estimate in the purely convective limit
http://aip.metastore.ingenta.com/content/aip/journal/pof2/24/7/10.1063/1.4738598
10.1063/1.4738598
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