Possible individual assignments for 3T380

Numerical integration schemes and shell models

Links between dissipation, intermittency, and helicity in the GOY model revisited.

High-resolution simulations within the GOY shell model are used to study various scaling relations for turbulence. A power-law relation between the second-order intermittency correction and the crossover from the inertial to the dissipation range is confirmed. Evidence is found for the intermediate viscous dissipation range proposed by Frisch and Vergassola. It is emphasized that insufficient dissipation-range resolution systematically drives the energy spectrum towards statistical mechanical equipartition. In fully resolved simulations the inertial-range scaling exponents depend on both model parameters; in particular, there is no evidence that the conservation of a helicity-like quantity leads to universal exponents.

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Bowman, J. C., Doering, C. R., Eckhardt, B., Davoudi, J., Roberts, M., & Schumacher, J. (n.d.). Links between dissipation, intermittency, and helicity in the GOY model revisited. Physica D-Nonlinear Phenomena, 218(1), 1–10.

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**Hamiltonian structure of the Sabra shell model of turbulence: exact calculation of an anomalous scaling exponent.**

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We show that the Sabra shell model of turbulence, which was introduced recently, displays a Hamiltonian structure for given values of the parameters. The requirement of scale independence of the flux of this Hamiltonian allows us to compute exactly a one-parameter family of anomalous scaling exponents associated with 4th-order correlation functions.

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L'vov, V., Podivilov, E., & Procaccia, I. (1999). Hamiltonian structure of the Sabra shell model of turbulence: exact calculation of an anomalous scaling exponent. EPL-Europhysics Letters, 46, 609.

Improved shell model of turbulence.

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We introduce a shell model of turbulence that exhibits improved properties in comparison to the standard and very popular? Gledzer, Ohkitani, and Yamada ?GOYGOYGOY model. The nonlinear coupling is chosen to mini- mize correlations between different shells. In particular, the second-order correlation function is diagonal in the shell index and the third-order correlation exists only between three consecutive shells. Spurious oscillations in the scaling regime, which are an annoying feature of the GOY model, are eliminated by our choice of nonlinear coupling. We demonstrate that the model exhibits multiscaling similar to the GOY model. The scaling expo- nents are shown to be independent of the viscous mechanism as is expected for Navier-Stokes turbulence and other shell models. These properties of the model make it optimal for further attempts to achieve understanding of multiscaling in nonlinear dynamics.

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L'vov, V., Podivilov, E., Pomyalov, A., Procaccia, I., & Vandembroucq, D. (1998). Improved shell model of turbulence. Physical Review E, 58(2), 1811–1822. doi:10.1103/PhysRevE.58.1811

Pseudo-spectral methods

In this paper the problem of the stability of plane Poiseuille flow is considered, using expansions in Chebyshev polynomials to approximate the solutions of the Orr-Sommerfeld equation. Explore the Chebyshev-tau method for this problem. Starting point may the the development of a Chebyshev-tau algorithm for the 1D diffusion equation with Neumann or Dirichlet BCs.

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Orszag, JFM 50, 689-703, 1971.

Spectral methods (Fourier Galerkin, Fourier pseudospectral, Chebyshev Tau, Chebyshev collocation, spectral element) and standard finite differences are applied to solve the Burgers equation. This equation admits a (nonsingular) thin internal layer that must be resolved if accurate numerical solutions are to be obtained.

Investigate this problem with 1D Fourier, 1D Chebyshev-Tau and with a 1D FD method. Start with a simple 1D diffusion problem and increase the complexity of the problem step-by-step.

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Basdevant et al., Comp. Fluids 14, 23-41, 1986.

This work addresses the application of Fourier spectral methods for flows in closed non-periodic domains with the use of volume penalization. Although this paper focusses on a specific flow problem in elliptic domains it serves as a nice starting point to explore the use of volume penalization techniques. A 2D Fourier spectral code is available but the volume penalization needs to be built into the code and needs to be tested with the help of a few test problems.

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Keetels et al., PRE 78, 036301, 2008.

**This paper on decaying 2D Navier-Stokes turbulence addresses flow on a circular domain with the help of a Galerkin expansion of the flow variables in Bessel functions. This represents a non-trivial approach and this assignment focusses on exploration of this particular method. What are the pros and cons with respect to the "classical" approaches such as Fourier-Chebishev spectral method? As a first step it is advised to consider the 2D diffusion problem on a circular domain with either Dirichlet or Neumann BCs.**

Li and Montgomery, Phys. Lett. A 218, 281-291, 1996.

Turbulence in More than Two and Less than Three Dimensions.

We investigate the behavior of turbulent systems in geometries with one compactified dimension. A novel phenomenological scenario dominated by the splitting of the turbulent cascade emerges both from the theoretical analysis of passive scalar turbulence and from direct numerical simulations of Navier- Stokes turbulence.

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Celani, A., Musacchio, S., & Vincenzi, D. (2010). Turbulence in More than Two and Less than Three Dimensions. Physical Review Letters, 104(18).

**MHD**

In this assignment, we revisit results originally obtained in a seminal paper by Nigel Weiss (Proc. Roy. Soc. A 293, 310-328, 1966). The paper discusses how an initially uniform magnetic field becomes distorted and amplified in total energy content, when embedded in a time-independent, prescribed (incompressible) flow field consisting of several convective eddies. The problem is purely 2D, and in fact linear in the unknowns.

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Nigel Weiss (Proc. Roy. Soc. A 293, 310-328, 1966)

In this assignment, the aim is to use/develop a modern incompressible MHD code to revisit the original results by Steven Orszag and Cha-Mei Tang (J. Fluid Mech., 90, 129-143, 1979). We wish to solve a purely 2D incompressible MHD evolution, on a double-periodic [0,2π]2 domain.

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Steven Orszag and Cha-Mei Tang (J. Fluid Mech., 90, 129-143, 1979)

Lattice Boltzmann method

Fluctuating lattice Boltzmann.

The lattice Boltzmann algorithm efficiently simulates the Navier-Stokes equation of isothermal fluid flow, but ignores thermal fluctuations of the fluid, important in mesoscopic flows. We show how to adapt the algorithm to include noise, satisfying a fluctuation-dissipation theorem (FDT) directly at lattice level: this gives correct fluctuations for mass and momentum densities, and for stresses, at all wave vectors k. Unlike previous work, which recovers FDT only as k → 0, our algorithm offers full statistical mechanical consistency in mesoscale simulations of, e.g., fluctuating colloidal hydrodynamics.

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Adhikari, R., Stratford, K., Cates, M., & Wagner, A. (2005). Fluctuating lattice Boltzmann. Europhysics Letters (EPL), 71(3), 473–479. doi:10.1209/epl/i2004-10542-5

**Lattice Boltzmann method with self-consistent thermo-hydrodynamic equilibria**

Lattice kinetic equations incorporating the effects of external/internal force fields via a shift of the local fields in the local equilibria are placed within the framework of continuum kinetic theory. The mathematical treatment reveals that in order to be consistent with the correct thermo-hydrodynamical description, temperature must also be shifted, besides momentum. New perspectives for the formulation of thermo- hydrodynamic lattice kinetic models of non-ideal fluids are then envisaged. It is also shown that on the lattice, the definition of the macroscopic temperature requires the inclusion of new terms directly related to discrete effects. The theoretical treatment is tested against a controlled case with a non-ideal equation of state.

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Sbragaglia, M., Benzi, R., Biferale, L., Chen, H., Shan, X., & Succi, S. (2009). Lattice Boltzmann method with self-consistent thermo-hydrodynamic equilibria. Journal Of Fluid Mechanics, 628(-1), 299–309. Cambridge University Press. doi:10.1017/S002211200900665X

Lattice Boltzmann method at finite Knudsen numbers.

A modified lattice Boltzmann model with a stochastic relaxation mechanism mimicking “virtual” collisions between free-streaming particles and solid walls is introduced. This modified scheme permits to compute plane channel flows in satisfactory agreement with analytical results over a broad spectrum of Knudsen numbers, ranging from the hydrodynamic regime, all the way to quasi-free flow regimes up to Kn ∼ 30.

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Toschi, F., & Succi, S. (2005). Lattice Boltzmann method at finite Knudsen numbers. Europhysics Letters (EPL).

Particle based methods

**Colloids dragged through a polymer solution: Experiment, theory, and simulation.**

We present microrheological measurements of the drag force on colloids pulled through a solution of