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.

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.

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

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.

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

Improved shell model of turbulence.

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 exponents 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.

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

Structure-preserving and exponential discretizations of initial value problems (new 2013!)

A problem discussed in this paper is the computation of a trajectory of a charge particle in an electromagnetic field. Extension to include flow, e.g. in a plasma.

John C. Bowman (Canadian Applied Mathematics Quarterly, vol. 14, no. 3. 2006)

Finite volume methods

Finite-Volume Discretization (new 2013!)

  1. Consider a cell-centered finite-volume discretization of the convection-diffusion equation in which for the discretization of the convection term a limiter is used that is economical with the introduction of numerical diffusion, by exploiting the physical diffusion present in this equation. Derive such a limiter and apply it to a test case to be delivered upon request by Barry Koren. Use forward Euler for the time integration and analyze the positivity requirement. (Bonus in case of success: opportunity to write a scientific publication.)
  2. Consider the convection equation and derive your own multi-dimensional upwind finite-volume discretization, which exploits in principle all six cells surrounding each interior cell face.


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Pseudo-spectral methods

Accurate solution of the Orr–Sommerfeld stability equation

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.

Orszag, JFM 50, 689-703, 1971.

Spectral and finite difference solutions of the Burgers equation

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.

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

Spontaneous angular momentum generation of two-dimensional fluid flow in an elliptic geometry

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 volume penalization needs to be built into the code and needs to be tested with the help of a few test problems.

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

Decaying two-dimensional turbulence with rigid walls

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.

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

Lattice Boltzmann method

Flow Through Randomly Curved Manifolds (new 2014!)

We present a computational study of the transport properties of campylotic (intrinsically curved) media. It is found that the relation between the flow through a campylotic media, consisting of randomly located curvature perturbations, and the average Ricci scalar of the system, exhibits two distinct functional expressions, depending on whether the typical spatial extent of the curvature perturbation lies above or below the critical value maximizing the overall scalar of curvature. Furthermore, the flow through such systems as a function of the number of curvature perturbations is found to present a sublinear behavior for large concentrations, due to the interference between curvature perturbations leading to an overall less curved space. We have also characterized the flux through such media as a function of the local Reynolds number and the scale of interaction between impurities. For the purpose of this study, we have also developed and validated a new lattice Boltzmann model.

Mendoza, M., Succi, S., & Herrmann, H. J. (2013). Flow Through Randomly Curved Manifolds. Scientific Reports, 3. doi:10.1038/srep03106


Finite-volume lattice Boltzmann schemes (new 2013!)

Simple and practical finite-volume schemes for the lattice Boltzmann equation are derived in two and three dimensions through the application of modern finite-volume methods. The schemes use a finite-volume vortex- type formulation based on quadrilateral elements in two dimensions and trilinear hexahedral elements in three dimensions. It is shown that the schemes are applicable to domains with irregular boundaries of arbitrary shape in two and three dimensions.

Xi, H., Peng, G., & Chou, S.-H. (1999). Finite-volume lattice Boltzmann schemes in two and three dimensions. Physical Review E, 60(3), 3380–3388. doi:10.1103/PhysRevE.60.33

Lattice Boltzmann for Shallow Water Flows (new 2013!)

In this assignment the student will implement a 2D lattice Boltzmann code and derive the necessary changes to move from a code solving fluid flows in the limit of the Navier-Stokes equations to the shallow water equations. For the shallow water equations it is assumed that the flow in the direction of the height of a fluid layer is neglectable in comparison to flows in the horizontal plane. This is for example the case for the description of ocean flows or rivers. With this approximation a 3D problem reduces to a 2D problem and thus the computational effort is substantially reduced.

J. G. Zhou, Lattice Boltzmann Methods for Shallow Water Flows, Springer 2004

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.

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.

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.

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

Turbulence drives microscale patches of motile phytoplankton

Patchiness plays a fundamental role in phytoplankton ecology by dictating the rate at which individual cells encounter each other and their predators. The distribution of motile phytoplankton species is often considerably more patchy than that of non-motile species at submetre length scales, yet the mechanism generating this patchiness has remained unknown. Here we show that strong patchiness at small scales occurs when motile phytoplankton are exposed to turbulent flow. We demonstrate experimentally that Heterosigma akashiwo forms striking patches within individual vortices and prove with a mathematical model that this patchiness results from the coupling between motility and shear. When implemented within a direct numerical simulation of turbulence, the model reveals that cell motility can prevail over turbulent dispersion to create strong fractal patchiness, where local phytoplankton concentrations are increased more than 10-fold. This ‘unmixing’ mechanism likely enhances ecological interactions in the plankton and offers mechanistic insights into how turbulence intensity impacts ecosystem productivity.

Durham W.M., Climent E., Barry M., De Lillo F., Boffetta G., Cencini M. & Stocker R., 2013, "Turbulence drives microscale patches of motile phytoplankton", Nature Communications, 4, 2148

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 􏰌-DNA 􏰁used here as a monodisperse model polymer􏰀 with an optical tweezer. The experiments show a drag force that is larger than expected from the Stokes formula and the independently measured viscosity of the DNA solution. We attribute this to the accumulation of DNA in front of the colloid and the reduced DNA density behind the colloid. This hypothesis is corroborated by a simple drift-diffusion model for the DNA molecules, which reproduces the experimental data surprisingly well, as well as by corresponding Brownian dynamics simulations.

Gutsche, C., Kremer, F., Krüger, M., Rauscher, M., Weeber, R., & Harting, J. (2008). Colloids dragged through a polymer solution: Experiment, theory, and simulation. The Journal of Chemical Physics, 129(8), 084902. doi:10.1063/1.2965127


Multi-Particle Collision Dynamics: A Particle-Based Mesoscale Simulation Approach to the Hydrodynamics of Complex Fluids


In this review, we describe and analyze a mesoscale simulation method for fluid flow, which was introduced by Malevanets and Kapral in 1999, and is now called multi-particle collision dynamics (MPC) or stochastic rotation dynamics (SRD). The method consists of alternating streaming and collision steps in an en- semble of point particles. The multi-particle collisions are performed by grouping particles in collision cells, and mass, momentum, and energy are locally conserved. This simulation technique captures both full hydrodynamic interactions and thermal fluctuations. The first part of the review begins with a description of several widely used MPC algorithms and then discusses important features of the origi- nal SRD algorithm and frequently used variations. Two complementary approaches for deriving the hydrodynamic equations and evaluating the transport coefficients are reviewed. It is then shown how MPC algorithms can be generalized to model non-ideal fluids, and binary mixtures with a consolute point. The importance of angular-momentum conservation for systems like phase-separated liquids with dif- ferent viscosities is discussed. The second part of the review describes a number of recent applications of MPC algorithms to study colloid and polymer dynamics, the behavior of vesicles and cells in hydrodynamic flows, and the dynamics of vis- coelastic fluids.

Gompper, G., Ihle, T., Kroll, D. M., & Winkler, R. G. (2009). Advances in Polymer Science. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/12_2008_5

Self-consistent dissipative particle dynamics algorithm   

We propose an implementation of dissipative particle dynamics that is free of the inconsistencies that plagued earlier algorithms. The present algorithm satisfies a form of microscopic reversibility. As a consequence, we recover the correct equilibrium properties. Moreover, we can use much larger time steps than previously. We report a detailed comparison between simulated transport properties and the theoretical predictions. We find that the existing theory is only valid under very special conditions. A more general theory is still lacking.

Pagonabarraga, I., Hagen, M., & Frenkel, D. (1998). Self-consistent dissipative particle dynamics algorithm. EPL-Europhysics Letters, 42, 377. IOP Publishing.

Mesoscale simulations: Lattice Boltzmann and particle algorithms.


I introduce two mesoscale algorithms, lattice Boltzmann and stochastic rotation dynamics, and show how they can be used to investigate the hydrodynamics of complex fluids. For each method I describe the algorithm, show that it solves the Navier–Stokes equations, and then discuss physical problems where it is particularly applicable. For lattice Boltzmann the examples I choose are phase ordering in a binary fluid and drop dynamics on a chemically patterned surface. For stochastic rotation dynamics I consider the hydrodynamics of dilute polymer solutions, concentrating on shear thinning and translocation across a barrier.

Yeomans, J. M. (2006). Mesoscale simulations: Lattice Boltzmann and particle algorithms. Physica A-Statistical Mechanics And Its Applications, 369(1), 159–184. doi:10.1016/j.physa.2006.04.011

Smoothed particle hydrodynamics

In this review the theory and application of Smoothed particle hydrodynamics (SPH) since its inception in 1977 are discussed. Emphasis is placed on the strengths and weaknesses, the analogy with particle dynamics and the numerous areas where SPH has been successfully applied.

J J Monaghan 2005 Rep. Prog. Phys. 68 1703 doi:10.1088/0034-4885/68/8/R01

MHD (with Rony Keppens, year 2012, still possible if interested...)

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.  

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.

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