Incompressible MHD simulations
|
|
- Cameron O’Brien’
- 5 years ago
- Views:
Transcription
1 Incompressible MHD simulations Felix Spanier 1 Lehrstuhl für Astronomie Universität Würzburg Simulation methods in astrophysics Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 1 / 20
2 Outline Where do we need incompressible MHD? Theory of HD&MHD Turbulence Numerical methods for HD Numerical methods for MHD Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 2 / 20
3 Heliosphere Closest available space plasma: The solar wind includes shocks and density fluctuation The large scale evolution seems to be compressible... Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 3 / 20
4 Heliosphere... but in the frame of the solar wind, we find a turbulent spectrum Alfvén waves travelling Sounds like incompressible turbulence! Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 3 / 20
5 Interstellar medium Infamous picture this is usually used to suggest turbulence it really shows density fluctuations Density fluctuations? That s compressible turbulence? Really? Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 4 / 20
6 / er g cm s -3-1 Interstellar medium -3-1 / er g cm s Damping rates 1 1e+10 1e-05 Viskos Joule Landau Neutral Viskos Joule Neutral 1 1e-10 1e-05 1e-15 1e-10 1e-15 1e-20 1e-20 1e-25 1e-25 1e-30 1e-17 1e-16 1e-15 1e-14 1e-13 1e-12 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06-1 k / cm Alfvén wave damping 1e-30 1e-17 1e-16 1e-15 1e-14 1e-13 1e-12 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06-1 k / cm Sound wave damping Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 4 / 20
7 Interstellar medium So is this incompressible MHD? Wave damping suggests that for large k only Alfvén waves survive Density fluctuations suggest there is compressible turbulence Two solutions: Alfvén turbulence with enslaved density fluctuations Not waves but shocks govern the ISM Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 4 / 20
8 Euler Equation Force on a volume element of fluid F = p df (1) Divergence theorem F = p dv (2) Newton s law for one volume element m du dt = F (3) Using density instead of total mass ρ du dv = dt p dv (4) ρ du dt = p (5) Lagrange picture Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 5 / 20
9 Euler Equation Changing to material derivative Euler s equation du dt du = u dt + (dr )u (1) t u t = u t Here the velocity is a function of space and time + (u )u (2) + (u )u = 1 p (3) ρ u = u(r, t) (4) This can now be applied to the momentum transport equation t (ρu i) = ρ u i t + u i ρ t (5) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 5 / 20
10 Euler Equation Partial derivative of u is known from Euler Π ik = stress tensor u i t u i = u k 1 p (1) x k ρ x i t (ρu i) = ρu k u i x k p x i u i ρu k x k (2) = p x i = δ ik p x k (ρu i u k ) x k (3) (ρu i u k ) x k (4) ρu i t = Π ik x k (5) Π ik = pδ ik ρu i u k (6) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 5 / 20
11 Continuity equation t ( ρ t + (ρu) ρ t For incompressible fluids (ρ=const) ρ dv = ρu dv (7) ) dv = 0 (8) + (ρu) = 0 (9) u = 0 (10) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 6 / 20
12 Viscid fluid Π ik = pδ ik + ρu i u k σ ik (11) σ ik = shear stress For Newtonian fluids this depends on derivatives of the velocity Coefficients have to fulfill σ ik = a u i x k + b u k x i no viscosity in uniform fluids(u=const) no viscosity in uniform rotating floew It follows a = b ( ui σ ik = η + u k x k x i 2 3 δ ik + c u l x l δ ik (12) ) u l u l + ζδ ik (13) x l x l Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 7 / 20
13 Viscid fluid Stress tensor for incompressible fluids u i = 0 (11) x i ( ui σ ik = η + u ) k (12) x k x i σ ik x k = η 2 u i x 2 k (13) And we find the Navier-Stokes-equation : u t + (u )u + p = ν u (14) ν = η ρ = Viscosity Pressure p = (u )u + ν u (15) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 7 / 20
14 Vorticity Using u = 0 one can take the curl of the Navier-Stokes-equation, resulting in a PDE for ω = u ( u) (u ( u)) t = ν ( ( u)) (16) ω t = (ω )u + ν ω (17) ω is quasi-scalar for 2d-velocities We need the stream function Ψ to derive velocities ω = u (18) u = Ψ (19) Ψ = ω (20) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 8 / 20
15 The MHD equations Compressible MHD Solutions ρ v t ρ t + (ρv) = 0 + ρ(v )v = p 1 4π B B B t = (v B) Different wave modes are solutions to the MHD equations Alfvén modes (incompressible, aligned to the magnetic field) Fast magnetosonic (compressible) slow magnetosonic (compressible) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 9 / 20
16 The MHD equations Incompressible MHD Solutions ρ v t v = 0 + ρ(v )v = p 1 4π B B B t = (v B) Different wave modes are solutions to the MHD equations Alfvén modes (incompressible, aligned to the magnetic field) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 9 / 20
17 The MHD equations Elsasser variables w = v + b v A e z w + = v b + v A e z v = w + + w 2 b = w + + w 2v A e z 2 Solutions Different wave modes are solutions to the MHD equations Alfvén modes (incompressible, aligned to the magnetic field) w ± correspond to forward/backward moving waves Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 9 / 20
18 Short review of Kolmogorov Kolmogorov assumes energy transport which is scale invariant Time-scale depends on eddy-turnover times Detailed analysis of the units reveals 5/3-law Simplified picture but enough to illustrate what we are doing next E(k) Inertial- range k Productionrange Dissipationrange Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 10 / 20
19 Short review of Kolmogorov Kolmogorov assumes energy transport which is scale invariant Time-scale depends on eddy-turnover times Detailed analysis of the units reveals 5/3-law Simplified picture but enough to illustrate what we are doing next Dimensional Argument 5/3-law can be derived by assuming scale independent energy transport and dimensional arguments. This gives a unique solution. Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 10 / 20
20 Kraichnan-Iroshnikov General idea: Turbulence is governed by Alfvén waves Collision of Alfvén waves is mechanism of choice Eddy-turnover time is then replaced by Alfvén time scale Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 11 / 20
21 Kraichnan-Iroshnikov Energy E = (V 2 + B 2 )dx is conserved Cascading to smaller energies Dimensional arguments cannot be used to derive τ λ = λ/v λ Dimensionless factor v λ /v A can enter Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 11 / 20
22 Wave-packet interaction Kraichnan-Iroshnikov Solutions: w ± = f (r v A t) w = v + b v A e z w + = v b + v A e z Figure: Packets interact, energy is conserved, shape is not Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 11 / 20
23 Kraichnan-Iroshnikov Packet interaction Amplitudes δw + λ δw λ δv λ δb λ during one collision δv λ (δv 2 λ/λ)(λ/v A ) Number of collision required to change wave packet N (δv λ / δv λ ) 2 (v A /δv λ ) 2 τ IK Nλ/v A λ/δv λ (v A /δv λ ) E IK = δv 2 k k 2 k 3/2 Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 11 / 20
24 Goldreich-Sridhar 3/2 spectrum not that often observe Let s get a new theory We come back to Kraichnan-Iroshnikov: Alfvén waves govern the spectrum Turbulent energy is transported through multi-wave interaction Goldreich and Sridhar made up a number of articles dealing with this problem They distinguish between strong and weak turbulence Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 12 / 20
25 Goldreich-Sridhar Weak Turbulence What is weak turbulence? Weak turbulence describes systems, where waves propagate if one neglects non-linearities. If one includes non-linearities wave amplitude will change slowly over many wave periods. The non-linearity is derived pertubatevily from the interaction of several waves Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 12 / 20
26 Goldreich-Sridhar Weak Turbulence Goldreich-Sridhar claim that Kraichnan-Iroshnikov can be seen as three-wave interaction of Alfvén waves They also believe, this is a forbidden process, since resonance condition reads k z1 + k z2 = k z k z1 + k z2 = k z 4-wave coupling is now their choice! Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 12 / 20
27 Goldreich-Sridhar Weak Turbulence k 1 + k 2 = k 3 + k 4 ω 1 + ω 2 = ω 3 + ω 4 k 1z + k 2z = k 3z + k 4z k 1z k 2z = k 3z k 4z Parallel component unaltered 4-wave interaction just changes k of quasi-particles Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 12 / 20
28 Goldreich-Sridhar Weak Turbulence d 2 v λ dt 2 δv λ d 2 v λ dt 2 (k zv A ) 2 d dt (k v 2 λ) k v λ dv λ dt since (v ) v λ k for Alfvén waves k 2 v 3 λ ( ) δv λ 2 ( ) 4 v λ k v k v N k z v λ k z v λ v 2 λ = E(k z, k )d 3 k constant rate of cascade E(k z, k ) ɛ(k z )v a k 10/3 This spectrum now hold, when the velocity change is small Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 12 / 20
29 Goldreich-Sridhar Strong Turbulence Assume perturbation v λ on scales λ kz 1 and λ k 1 ζ λ k v λ k z v λ Anisotropy parameter N ζ 4 λ isotropic excitationk k z L 1 ( va v L ) 4 (k L) 4/3 Everything is fine as long as ζ λ 1 GS assume frequency renormalization when this is not given Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 12 / 20
30 Goldreich-Sridhar Strong Turbulence When energy is injected isotropically ζ L 1 we have a critical balance k z v A k v λ Alfvén timescale and cascading time scale match For the critical balance and scale-independent cascade we find E(k, k z ) k z k 2/3 L 1/3 v v A (k L) 1/3 ( ) v 2 A k f 10/3 L1/3 k z L 1/3 k 2/3 Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 12 / 20
31 Goldreich-Sridhar Strong Turbulence Consequences A cutoff scale exists Eddies are elongated Figure: From Cho, Lazarian, Vishniac (2003) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 12 / 20
32 Turbulence Conclusion There is a number of turbulence models out there It is not yet clear which is correct All analytical turbulence models are incompressible The only thing we know for sure is that there is power law in the spectral energy distribution Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 13 / 20
33 Numerical methods Methods for the Navier-Stokes-equation Projection solve compressible equations, project on incompressible part Vorticity use vector potential to always fulfill divergence-free condition Spectral use Fourier space to do the same as above Methods are presented for the Navier-Stokes-equation but also apply to MHD Spectral methods will be presented for Elsässer variables in detail Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 14 / 20
34 Projection method Different possible schemes Here a scheme proposed by [1]. Step 1: Intermediate Solution We are looking for a solution u, which is not yet divergence free u u n t = ((u )u) n+1/2 p n 1/2 + ν u n (21) This requires some standard scheme to solve the Navier-Stokes-equation (cf. Pekka s talk) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 15 / 20
35 Projection method Step 2: Dissipation Use Crank-Nicholson for stability in the dissipation step yields ν u n 1 2 ν (un + u ) (21) u = u n t ((u )u + p) (22) ( 1 ν t ) ( 2 u = 1 + ν t ) 2 u (23) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 15 / 20
36 Projection method Step 3: Projection step To project u on the divergence-free part, an auxiliary field φ is calculated, which fulfills (u t φ) = 0 (21) u φ = t (22) u n+1 = u t φ n+1 (23) Here a Poisson equation has to be solved. This is the tricky part and is the most time consuming. Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 15 / 20
37 Projection method Step 4: Pressure gradient Finally the pressure is updated. This can be calculated using the divergence of the Navier-Stokes-equation p n+1/2 = p n 1/2 + φ n+1 ν t 2 φn+1 (21) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 15 / 20
38 Projection method Total scheme u u n t + p n+1/2 = [(u )u] n+1/2 + ν 2 2 (u n + u ) (21) t 2 φ n+1 = u (22) u n+1 = u t φ n+1 (23) p n+1/2 = p n 1/2 + φ n+1 ν t 2 2 φ n+1 (24) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 15 / 20
39 Vorticity-Streamline The natural formulation for divergence-free flows is the stream function u = Ψ Step 1: Calculate velocity u n = Ψ n (25) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 16 / 20
40 Vorticity-Streamline Step 2: Solve the Navier-Stokes-equation t u = (u ) u + ν u (25) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 16 / 20
41 Vorticity-Streamline Step 3: Calculate Vorticity t ω = ( t u) (25) ω n+1 = ω n + dt ω (26) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 16 / 20
42 Vorticity-Streamline Step 4: Calculate Streamfunction Solve Poisson equation Ψ n+1 = ω n+1 (25) Problem: This would result in a mesh-drift instability. Use staggered grid! Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 16 / 20
43 Vorticity-Streamline Complete scheme u = Ψ (25) (u x ) i,j+1/2 = Ψ i,j+1 Ψi, j (26) dy (u y ) i+1/2,j = Ψ i+1,j Ψi, j (27) ( dx ( ) ) 1/2 (ux ) u i+1/2,j+1/2 = i+1,j+1/2 + (u x ) i,j+1/2 1/2 ( ) (28) (u y ) i+1/2,j+1 + (u y ) i+1/2,j Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 16 / 20
44 Spectral methods Basic idea: Use Fourier transform to transform PDE into ODE Problem: This is complicated for the nonlinear terms. Spectral Navier-Stokes-equation ũ α t k α ũ α = 0 ( = ik γ δ αβ k ) αk β ) (ũβ k 2 u γ νk 2ũ α (29) Quantities with a tilde are fouriertransformed. What s so special about the red and green term? Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 17 / 20
45 Spectral methods The green term Start with the Navier-Stokes-equation we can identify u t + (u )u + p = 0 (29) ik γ δ αβ ũ β u γ = (u )u (30) and taking the divergence of the Navier-Stokes-equation p = (u )u (31) so the second term is the gradient of the pressure ( ) kα k β ) ik γ (ũβ k 2 u γ = p (32) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 17 / 20
46 Spectral methods Step 1: Fouriertransform Starting with ũ α the velocity has to be transformed into u α Then u α u β can be calculated (only 6 tensor elements are needed) This is transformed back to Fourier space ũ α u β Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 17 / 20
47 Spectral methods Step 2: Anti-Aliasing In the high wavenumber regime of u α we will find flawed data. Due to the fact, that not all data is correctly attributed for by the Fourier transform, aliased data has to be removed For k > 1/2k max set ũ α u β (k) = 0 Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 17 / 20
48 Spectral methods Step 3: Update velocity ũ α = ũ n α t(ik γ ( δ αβ k αk β k 2 ) (ũβ u γ ) νk 2ũ α ) (29) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 17 / 20
49 Spectral methods Step 4: Projection Project u α on its divergence free part u n+1 = u k(k u ) k 2 (29) Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 17 / 20
50 Spectral methods Why to use Spectral Methods? Pro Easy to implement Relies on fast FFT-algorithms Easy to parallelize Contra high aliasing-loss Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 17 / 20
51 Fourier Transforms Fourier transforms are available in large numbers on the market FFTW 2 ubiquitious library, allows parallel usage FFTW 3 the improved version, no parallel support Intel MKL includes DFT, much faster than FFTW, simple parallelized version, limited to Intel-like systems P3DFFT UC San Diego s Fortran based FFT, highly parallelizable, requires FFTW-3 Sandia FFT Sandia Lab s C based FFT, highly parallelizable, requires FFTW-2 Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 18 / 20
52 Parallel Computing Usually a parallel fluid simulation requires exchange of field borders between processors. Spectral methods are slightly different: The calculation of u α u β and the update step are completely local. They have no derivatives and do not need neighboring points Every non-local interaction is done in the FFT So the only thing we have to do, is using local coordinates and a parallel FFT Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 19 / 20
53 Extension to MHD Elsässer notation ( t v A k z ) wα = i k α k β k ( γ 2 k 2 w + β w γ + w β w + γ ik β wα w + β ν 2 k 2n wα ( t + v A k z ) w α + = i k α k β k ( ) γ 2 k 2 w + β w γ + w β w γ + ) ik β w + α w β ν 2 k 2n w + α The equations are treated similarly to the Navier-Stokes-equation. Two major changes: Two coupled equations Magnetic field introduced via v A k z term Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 20 / 20
54 [1] D. L. Brown, R. Cortez, and M. L. Minion. Accurate Projection Methods for the Incompressible Navier-Stokes Equations. Journal of Computational Physics, 168: , Apr Felix Spanier (Uni Würzburg) Simulation methods in astrophysics 20 / 20
Fundamentals of Turbulence
Fundamentals of Turbulence Stanislav Boldyrev (University of Wisconsin - Madison) Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas What is turbulence? No exact definition.
More informationThe Effect of Magnetic Turbulence Energy Spectra and Pickup Ions on the Heating of the Solar Wind
The Effect of Magnetic Turbulence Energy Spectra and Pickup Ions on the Heating of the Solar Wind C. S. Ng Geophysical Institute, University of Alaska Fairbanks A. Bhattacharjee, P. A. Isenberg, D. Munsi,
More informationNONLINEAR MHD WAVES THE INTERESTING INFLUENCE OF FIREHOSE AND MIRROR IN ASTROPHYSICAL PLASMAS. Jono Squire (Caltech) UCLA April 2017
NONLINEAR MHD WAVES THE INTERESTING INFLUENCE OF FIREHOSE AND MIRROR IN ASTROPHYSICAL PLASMAS Jono Squire (Caltech) UCLA April 2017 Along with: E. Quataert, A. Schekochihin, M. Kunz, S. Bale, C. Chen,
More informationNatalia Tronko S.V.Nazarenko S. Galtier
IPP Garching, ESF Exploratory Workshop Natalia Tronko University of York, York Plasma Institute In collaboration with S.V.Nazarenko University of Warwick S. Galtier University of Paris XI Outline Motivations:
More informationTURBULENCE IN FLUIDS AND SPACE PLASMAS. Amitava Bhattacharjee Princeton Plasma Physics Laboratory, Princeton University
TURBULENCE IN FLUIDS AND SPACE PLASMAS Amitava Bhattacharjee Princeton Plasma Physics Laboratory, Princeton University What is Turbulence? Webster s 1913 Dictionary: The quality or state of being turbulent;
More informationIntroduction to Magnetohydrodynamics (MHD)
Introduction to Magnetohydrodynamics (MHD) Tony Arber University of Warwick 4th SOLARNET Summer School on Solar MHD and Reconnection Aim Derivation of MHD equations from conservation laws Quasi-neutrality
More informationEuler equation and Navier-Stokes equation
Euler equation and Navier-Stokes equation WeiHan Hsiao a a Department of Physics, The University of Chicago E-mail: weihanhsiao@uchicago.edu ABSTRACT: This is the note prepared for the Kadanoff center
More informationMacroscopic plasma description
Macroscopic plasma description Macroscopic plasma theories are fluid theories at different levels single fluid (magnetohydrodynamics MHD) two-fluid (multifluid, separate equations for electron and ion
More informationMHD turbulence in the solar corona and solar wind
MHD turbulence in the solar corona and solar wind Pablo Dmitruk Departamento de Física, FCEN, Universidad de Buenos Aires Motivations The role of MHD turbulence in several phenomena in space and solar
More informationMagnetohydrodynamic Waves
Magnetohydrodynamic Waves Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 17, 2016 These slides are largely based off of 4.5 and 4.8 of The Physics of
More informationDissipation Scales & Small Scale Structure
Dissipation Scales & Small Scale Structure Ellen Zweibel zweibel@astro.wisc.edu Departments of Astronomy & Physics University of Wisconsin, Madison and Center for Magnetic Self-Organization in Laboratory
More informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationEnergy spectrum of isotropic magnetohydrodynamic turbulence in the Lagrangian renormalized approximation
START: 19/Jul/2006, Warwick Turbulence Symposium Energy spectrum of isotropic magnetohydrodynamic turbulence in the Lagrangian renormalized approximation Kyo Yoshida (Univ. Tsukuba) In collaboration with:
More informationQuick Recapitulation of Fluid Mechanics
Quick Recapitulation of Fluid Mechanics Amey Joshi 07-Feb-018 1 Equations of ideal fluids onsider a volume element of a fluid of density ρ. If there are no sources or sinks in, the mass in it will change
More informationTutorial School on Fluid Dynamics: Aspects of Turbulence Session I: Refresher Material Instructor: James Wallace
Tutorial School on Fluid Dynamics: Aspects of Turbulence Session I: Refresher Material Instructor: James Wallace Adapted from Publisher: John S. Wiley & Sons 2002 Center for Scientific Computation and
More informationChapter 7. Basic Turbulence
Chapter 7 Basic Turbulence The universe is a highly turbulent place, and we must understand turbulence if we want to understand a lot of what s going on. Interstellar turbulence causes the twinkling of
More informationMagnetohydrodynamic Turbulence: solar wind and numerical simulations
Magnetohydrodynamic Turbulence: solar wind and numerical simulations Stanislav Boldyrev (UW-Madison) Jean Carlos Perez (U. New Hampshire) Fausto Cattaneo (U. Chicago) Joanne Mason (U. Exeter, UK) Vladimir
More informationComputational Fluid Dynamics 2
Seite 1 Introduction Computational Fluid Dynamics 11.07.2016 Computational Fluid Dynamics 2 Turbulence effects and Particle transport Martin Pietsch Computational Biomechanics Summer Term 2016 Seite 2
More informationMagnetohydrodynamic Turbulence
Magnetohydrodynamic Turbulence Stanislav Boldyrev (UW-Madison) Jean Carlos Perez (U. New Hampshire), Fausto Cattaneo (U. Chicago), Joanne Mason (U. Exeter, UK) Vladimir Zhdankin (UW-Madison) Konstantinos
More informationA Three-Fluid Approach to Model Coupling of Solar Wind-Magnetosphere-Ionosphere- Thermosphere
A Three-Fluid Approach to Model Coupling of Solar Wind-Magnetosphere-Ionosphere- Thermosphere P. Song Center for Atmospheric Research University of Massachusetts Lowell V. M. Vasyliūnas Max-Planck-Institut
More informationMultiscale Computation of Isotropic Homogeneous Turbulent Flow
Multiscale Computation of Isotropic Homogeneous Turbulent Flow Tom Hou, Danping Yang, and Hongyu Ran Abstract. In this article we perform a systematic multi-scale analysis and computation for incompressible
More informationViscous Fluids. Amanda Meier. December 14th, 2011
Viscous Fluids Amanda Meier December 14th, 2011 Abstract Fluids are represented by continuous media described by mass density, velocity and pressure. An Eulerian description of uids focuses on the transport
More informationPHYS 643 Week 4: Compressible fluids Sound waves and shocks
PHYS 643 Week 4: Compressible fluids Sound waves and shocks Sound waves Compressions in a gas propagate as sound waves. The simplest case to consider is a gas at uniform density and at rest. Small perturbations
More informationMax Planck Institut für Plasmaphysik
ASDEX Upgrade Max Planck Institut für Plasmaphysik 2D Fluid Turbulence Florian Merz Seminar on Turbulence, 08.09.05 2D turbulence? strictly speaking, there are no two-dimensional flows in nature approximately
More informationScaling relations in MHD and EMHD Turbulence
Scaling relations in MHD and EMHD Turbulence Jungyeon Cho Chungnam National University, Korea Outline MHD Non-MHD E(k) MHD turb. small-scale turb. ~1/r i k Topic 1. Strong MHD Turbulence Alfven wave Suppose
More informationHydrodynamics. Stefan Flörchinger (Heidelberg) Heidelberg, 3 May 2010
Hydrodynamics Stefan Flörchinger (Heidelberg) Heidelberg, 3 May 2010 What is Hydrodynamics? Describes the evolution of physical systems (classical or quantum particles, fluids or fields) close to thermal
More informationAlfvén wave turbulence: new results with applications to astrophysics. Sébastien GALTIER Université Paris-Sud & Institut Universitaire de France
Alfvén wave turbulence: new results with applications to astrophysics Sébastien GALTIER Université Paris-Sud & Institut Universitaire de France 1 Recent co-workers : - Barbara Bigot (France/USA) - Ben
More informationGreg Hammett Imperial College, London & Princeton Plasma Physics Lab With major contributions from:
Greg Hammett Imperial College, London & Princeton Plasma Physics Lab With major contributions from: Steve Cowley (Imperial College) Bill Dorland (Imperial College) Eliot Quataert (Berkeley) LMS Durham
More informationTurbulence and transport
and transport ` is the most important unsolved problem of classical physics.' - Richard Feynman - Da Vinci 14521519 Image: General Atomics Reynolds, Richardson, Kolmogorov Dr Ben Dudson, University of
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More informationChapter 5. The Differential Forms of the Fundamental Laws
Chapter 5 The Differential Forms of the Fundamental Laws 1 5.1 Introduction Two primary methods in deriving the differential forms of fundamental laws: Gauss s Theorem: Allows area integrals of the equations
More informationWaves in plasma. Denis Gialis
Waves in plasma Denis Gialis This is a short introduction on waves in a non-relativistic plasma. We will consider a plasma of electrons and protons which is fully ionized, nonrelativistic and homogeneous.
More informationIsotropic homogeneous 3D turbulence
Chapter 6 Isotropic homogeneous 3D turbulence Turbulence was recognized as a distinct fluid behavior by Leonardo da Vinci more than 500 years ago. It is Leonardo who termed such motions turbolenze, and
More informationReview of fluid dynamics
Chapter 2 Review of fluid dynamics 2.1 Preliminaries ome basic concepts: A fluid is a substance that deforms continuously under stress. A Material olume is a tagged region that moves with the fluid. Hence
More informationModelling of turbulent flows: RANS and LES
Modelling of turbulent flows: RANS and LES Turbulenzmodelle in der Strömungsmechanik: RANS und LES Markus Uhlmann Institut für Hydromechanik Karlsruher Institut für Technologie www.ifh.kit.edu SS 2012
More informationHeating of Test Particles in Numerical Simulations of MHD Turbulence and the Solar Wind
Heating of Test Particles in Numerical Simulations of MHD Turbulence and the Solar Wind Ian Parrish UC Berkeley Collaborators: Rémi Lehe (ENS), Eliot Quataert (UCB) Einstein Fellows Symposium October 27,
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationDedalus: An Open-Source Spectral Magnetohydrodynamics Code
Dedalus: An Open-Source Spectral Magnetohydrodynamics Code Keaton Burns University of California Berkeley Jeff Oishi SLAC National Accelerator Laboratory Received ; accepted ABSTRACT We developed the magnetohydrodynamic
More informationAstrophysical Fluid Dynamics (2)
Astrophysical Fluid Dynamics (2) (Draine Chap 35;;Clarke & Carswell Chaps 1-4) Last class: fluid dynamics equations apply in astrophysical situations fluid approximation holds Maxwellian velocity distribution
More informationAn Introduction to Theories of Turbulence. James Glimm Stony Brook University
An Introduction to Theories of Turbulence James Glimm Stony Brook University Topics not included (recent papers/theses, open for discussion during this visit) 1. Turbulent combustion 2. Turbulent mixing
More information2. Conservation Equations for Turbulent Flows
2. Conservation Equations for Turbulent Flows Coverage of this section: Review of Tensor Notation Review of Navier-Stokes Equations for Incompressible and Compressible Flows Reynolds & Favre Averaging
More informationUNCONDITIONAL STABILITY OF A PARTITIONED IMEX METHOD FOR MAGNETOHYDRODYNAMIC FLOWS
UNCONDITIONAL STABILITY OF A PARTITIONED IMEX METHOD FOR MAGNETOHYDRODYNAMIC FLOWS CATALIN TRENCHEA Key words. magnetohydrodynamics, partitioned methods, IMEX methods, stability, Elsässer variables. Abstract.
More informationMath 575-Lecture Viscous Newtonian fluid and the Navier-Stokes equations
Math 575-Lecture 13 In 1845, tokes extended Newton s original idea to find a constitutive law which relates the Cauchy stress tensor to the velocity gradient, and then derived a system of equations. The
More informationChapter 1. Introduction to Nonlinear Space Plasma Physics
Chapter 1. Introduction to Nonlinear Space Plasma Physics The goal of this course, Nonlinear Space Plasma Physics, is to explore the formation, evolution, propagation, and characteristics of the large
More informationGetting started: CFD notation
PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =
More informationMulti-D MHD and B = 0
CapSel DivB - 01 Multi-D MHD and B = 0 keppens@rijnh.nl multi-d MHD and MHD wave anisotropies dimensionality > 1 non-trivial B = 0 constraint even if satisfied exactly t = 0: can numerically generate B
More informationLecture 14. Turbulent Combustion. We know what a turbulent flow is, when we see it! it is characterized by disorder, vorticity and mixing.
Lecture 14 Turbulent Combustion 1 We know what a turbulent flow is, when we see it! it is characterized by disorder, vorticity and mixing. In a fluid flow, turbulence is characterized by fluctuations of
More informationNonequilibrium Dynamics in Astrophysics and Material Science YITP, Kyoto
Nonequilibrium Dynamics in Astrophysics and Material Science 2011-11-02 @ YITP, Kyoto Multi-scale coherent structures and their role in the Richardson cascade of turbulence Susumu Goto (Okayama Univ.)
More informationStatistical studies of turbulent flows: self-similarity, intermittency, and structure visualization
Statistical studies of turbulent flows: self-similarity, intermittency, and structure visualization P.D. Mininni Departamento de Física, FCEyN, UBA and CONICET, Argentina and National Center for Atmospheric
More information13. ASTROPHYSICAL GAS DYNAMICS AND MHD Hydrodynamics
1 13. ASTROPHYSICAL GAS DYNAMICS AND MHD 13.1. Hydrodynamics Astrophysical fluids are complex, with a number of different components: neutral atoms and molecules, ions, dust grains (often charged), and
More informationSeveral forms of the equations of motion
Chapter 6 Several forms of the equations of motion 6.1 The Navier-Stokes equations Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationDiffusive Transport Enhanced by Thermal Velocity Fluctuations
Diffusive Transport Enhanced by Thermal Velocity Fluctuations Aleksandar Donev 1 Courant Institute, New York University & Alejandro L. Garcia, San Jose State University John B. Bell, Lawrence Berkeley
More informationTheoretical Foundation of 3D Alfvén Resonances: Time Dependent Solutions
Theoretical Foundation of 3D Alfvén Resonances: Time Dependent Solutions Tom Elsden 1 Andrew Wright 1 1 Dept Maths & Stats, University of St Andrews DAMTP Seminar - 8th May 2017 Outline Introduction Coordinates
More informationfluid mechanics as a prominent discipline of application for numerical
1. fluid mechanics as a prominent discipline of application for numerical simulations: experimental fluid mechanics: wind tunnel studies, laser Doppler anemometry, hot wire techniques,... theoretical fluid
More informationTurbulence - Theory and Modelling GROUP-STUDIES:
Lund Institute of Technology Department of Energy Sciences Division of Fluid Mechanics Robert Szasz, tel 046-0480 Johan Revstedt, tel 046-43 0 Turbulence - Theory and Modelling GROUP-STUDIES: Turbulence
More information7 The Navier-Stokes Equations
18.354/12.27 Spring 214 7 The Navier-Stokes Equations In the previous section, we have seen how one can deduce the general structure of hydrodynamic equations from purely macroscopic considerations and
More informationReduced MHD. Nick Murphy. Harvard-Smithsonian Center for Astrophysics. Astronomy 253: Plasma Astrophysics. February 19, 2014
Reduced MHD Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 19, 2014 These lecture notes are largely based on Lectures in Magnetohydrodynamics by Dalton
More informationStability of Shear Flow
Stability of Shear Flow notes by Zhan Wang and Sam Potter Revised by FW WHOI GFD Lecture 3 June, 011 A look at energy stability, valid for all amplitudes, and linear stability for shear flows. 1 Nonlinear
More informationLES of Turbulent Flows: Lecture 3
LES of Turbulent Flows: Lecture 3 Dr. Jeremy A. Gibbs Department of Mechanical Engineering University of Utah Fall 2016 1 / 53 Overview 1 Website for those auditing 2 Turbulence Scales 3 Fourier transforms
More informationEngineering. Spring Department of Fluid Mechanics, Budapest University of Technology and Economics. Large-Eddy Simulation in Mechanical
Outline Geurts Book Department of Fluid Mechanics, Budapest University of Technology and Economics Spring 2013 Outline Outline Geurts Book 1 Geurts Book Origin This lecture is strongly based on the book:
More information36. TURBULENCE. Patriotism is the last refuge of a scoundrel. - Samuel Johnson
36. TURBULENCE Patriotism is the last refuge of a scoundrel. - Samuel Johnson Suppose you set up an experiment in which you can control all the mean parameters. An example might be steady flow through
More informationBasic concepts in viscous flow
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic Adapted from Chapter 1 of Cambridge Texts in Applied Mathematics 1 The fluid dynamic equations Navier-Stokes equations Dimensionless
More informationTurbulent spectra generated by singularities
Turbulent spectra generated by singularities p. Turbulent spectra generated by singularities E.A. Kuznetsov Landau Institute for Theoretical Physics, Moscow Russia In collaboration with: V. Naulin A.H.
More informationNonlinear processes associated with Alfvén waves in a laboratory plasma
Nonlinear processes associated with Alfvén waves in a laboratory plasma Troy Carter Dept. Physics and Astronomy and Center for Multiscale Plasma Dynamics, UCLA acknowledgements: Brian Brugman, David Auerbach,
More informationEffects of Forcing Scheme on the Flow and the Relative Motion of Inertial Particles in DNS of Isotropic Turbulence
Effects of Forcing Scheme on the Flow and the Relative Motion of Inertial Particles in DNS of Isotropic Turbulence Rohit Dhariwal PI: Sarma L. Rani Department of Mechanical and Aerospace Engineering The
More informationA unifying model for fluid flow and elastic solid deformation: a novel approach for fluid-structure interaction and wave propagation
A unifying model for fluid flow and elastic solid deformation: a novel approach for fluid-structure interaction and wave propagation S. Bordère a and J.-P. Caltagirone b a. CNRS, Univ. Bordeaux, ICMCB,
More informationIntroduction to Turbulence and Turbulence Modeling
Introduction to Turbulence and Turbulence Modeling Part I Venkat Raman The University of Texas at Austin Lecture notes based on the book Turbulent Flows by S. B. Pope Turbulent Flows Turbulent flows Commonly
More informationMath background. Physics. Simulation. Related phenomena. Frontiers in graphics. Rigid fluids
Fluid dynamics Math background Physics Simulation Related phenomena Frontiers in graphics Rigid fluids Fields Domain Ω R2 Scalar field f :Ω R Vector field f : Ω R2 Types of derivatives Derivatives measure
More informationJet Stability: A computational survey
Jet Stability Galway 2008-1 Jet Stability: A computational survey Rony Keppens Centre for Plasma-Astrophysics, K.U.Leuven (Belgium) & FOM-Institute for Plasma Physics Rijnhuizen & Astronomical Institute,
More informationFormation and Long Term Evolution of an Externally Driven Magnetic Island in Rotating Plasmas )
Formation and Long Term Evolution of an Externally Driven Magnetic Island in Rotating Plasmas ) Yasutomo ISHII and Andrei SMOLYAKOV 1) Japan Atomic Energy Agency, Ibaraki 311-0102, Japan 1) University
More information20. Alfven waves. ([3], p ; [1], p ; Chen, Sec.4.18, p ) We have considered two types of waves in plasma:
Phys780: Plasma Physics Lecture 20. Alfven Waves. 1 20. Alfven waves ([3], p.233-239; [1], p.202-237; Chen, Sec.4.18, p.136-144) We have considered two types of waves in plasma: 1. electrostatic Langmuir
More informationThe Janus Face of Turbulent Pressure
CRC 963 Astrophysical Turbulence and Flow Instabilities with thanks to Christoph Federrath, Monash University Patrick Hennebelle, CEA/Saclay Alexei Kritsuk, UCSD yt-project.org Seminar über Astrophysik,
More information( ) Notes. Fluid mechanics. Inviscid Euler model. Lagrangian viewpoint. " = " x,t,#, #
Notes Assignment 4 due today (when I check email tomorrow morning) Don t be afraid to make assumptions, approximate quantities, In particular, method for computing time step bound (look at max eigenvalue
More informationIdeal Magnetohydrodynamics (MHD)
Ideal Magnetohydrodynamics (MHD) Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 1, 2016 These lecture notes are largely based on Lectures in Magnetohydrodynamics
More informationFluctuation dynamo amplified by intermittent shear bursts
by intermittent Thanks to my collaborators: A. Busse (U. Glasgow), W.-C. Müller (TU Berlin) Dynamics Days Europe 8-12 September 2014 Mini-symposium on Nonlinear Problems in Plasma Astrophysics Introduction
More informationAA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 1 / 29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS Hierarchy of Mathematical Models 1 / 29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 2 / 29
More informationPlasma Physics for Astrophysics
- ' ' * ' Plasma Physics for Astrophysics RUSSELL M. KULSRUD PRINCETON UNIVERSITY E;RESS '. ' PRINCETON AND OXFORD,, ', V. List of Figures Foreword by John N. Bahcall Preface Chapter 1. Introduction 1
More informationUNIVERSITY of LIMERICK
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4607 SEMESTER: Autumn 2012-13 MODULE TITLE: Introduction to Fluids DURATION OF
More informationComputational Astrophysics
16 th Chris Engelbrecht Summer School, January 2005 3: 1 Computational Astrophysics Lecture 3: Magnetic fields Paul Ricker University of Illinois at Urbana-Champaign National Center for Supercomputing
More informationCosmic-ray Acceleration and Current-Driven Instabilities
Cosmic-ray Acceleration and Current-Driven Instabilities B. Reville Max-Planck-Institut für Kernphysik, Heidelberg Sep 17 2009, KITP J.G. Kirk, P. Duffy, S.O Sullivan, Y. Ohira, F. Takahara Outline Analysis
More informationAA210A Fundamentals of Compressible Flow. Chapter 1 - Introduction to fluid flow
AA210A Fundamentals of Compressible Flow Chapter 1 - Introduction to fluid flow 1 1.2 Conservation of mass Mass flux in the x-direction [ ρu ] = M L 3 L T = M L 2 T Momentum per unit volume Mass per unit
More informationContinuum Mechanics Lecture 5 Ideal fluids
Continuum Mechanics Lecture 5 Ideal fluids Prof. http://www.itp.uzh.ch/~teyssier Outline - Helmholtz decomposition - Divergence and curl theorem - Kelvin s circulation theorem - The vorticity equation
More informationStructure Formation and Particle Mixing in a Shear Flow Boundary Layer
Structure Formation and Particle Mixing in a Shear Flow Boundary Layer Matthew Palotti palotti@astro.wisc.edu University of Wisconsin Center for Magnetic Self Organization Ellen Zweibel University of Wisconsin
More informationLecture 1: Introduction to Linear and Non-Linear Waves
Lecture 1: Introduction to Linear and Non-Linear Waves Lecturer: Harvey Segur. Write-up: Michael Bates June 15, 2009 1 Introduction to Water Waves 1.1 Motivation and Basic Properties There are many types
More informationBefore we consider two canonical turbulent flows we need a general description of turbulence.
Chapter 2 Canonical Turbulent Flows Before we consider two canonical turbulent flows we need a general description of turbulence. 2.1 A Brief Introduction to Turbulence One way of looking at turbulent
More informationSolar Wind Turbulent Heating by Interstellar Pickup Protons: 2-Component Model
Solar Wind Turbulent Heating by Interstellar Pickup Protons: 2-Component Model Philip A. Isenberg a, Sean Oughton b, Charles W. Smith a and William H. Matthaeus c a Inst. for Study of Earth, Oceans and
More informationComputational Astrophysics
Computational Astrophysics Lecture 1: Introduction to numerical methods Lecture 2:The SPH formulation Lecture 3: Construction of SPH smoothing functions Lecture 4: SPH for general dynamic flow Lecture
More informationLEAST-SQUARES FINITE ELEMENT MODELS
LEAST-SQUARES FINITE ELEMENT MODELS General idea of the least-squares formulation applied to an abstract boundary-value problem Works of our group Application to Poisson s equation Application to flows
More informationPlasmas as fluids. S.M.Lea. January 2007
Plasmas as fluids S.M.Lea January 2007 So far we have considered a plasma as a set of non intereacting particles, each following its own path in the electric and magnetic fields. Now we want to consider
More informationDispersive Media, Lecture 7 - Thomas Johnson 1. Waves in plasmas. T. Johnson
2017-02-14 Dispersive Media, Lecture 7 - Thomas Johnson 1 Waves in plasmas T. Johnson Introduction to plasmas as a coupled system Magneto-Hydro Dynamics, MHD Plasmas without magnetic fields Cold plasmas
More informationAccretion Disks I. High Energy Astrophysics: Accretion Disks I 1/60
Accretion Disks I References: Accretion Power in Astrophysics, J. Frank, A. King and D. Raine. High Energy Astrophysics, Vol. 2, M.S. Longair, Cambridge University Press Active Galactic Nuclei, J.H. Krolik,
More informationOn the Global Attractor of 2D Incompressible Turbulence with Random Forcing
On the Global Attractor of 2D Incompressible Turbulence with Random Forcing John C. Bowman and Pedram Emami Department of Mathematical and Statistical Sciences University of Alberta October 25, 2017 www.math.ualberta.ca/
More informationTopics in Fluid Dynamics: Classical physics and recent mathematics
Topics in Fluid Dynamics: Classical physics and recent mathematics Toan T. Nguyen 1,2 Penn State University Graduate Student Seminar @ PSU Jan 18th, 2018 1 Homepage: http://math.psu.edu/nguyen 2 Math blog:
More informationρ Du i Dt = p x i together with the continuity equation = 0, x i
1 DIMENSIONAL ANALYSIS AND SCALING Observation 1: Consider the flow past a sphere: U a y x ρ, µ Figure 1: Flow past a sphere. Far away from the sphere of radius a, the fluid has a uniform velocity, u =
More information6. Renormalized Perturbation Theory
6. Renormalized Perturbation Theory 6.1. Perturbation expansion of the Navier-Stokes equation Kraichnan (1959): direct-interaction approximation Wyld (1961): partial summation for a simplified scalar model
More informationOn the (multi)scale nature of fluid turbulence
On the (multi)scale nature of fluid turbulence Kolmogorov axiomatics Laurent Chevillard Laboratoire de Physique, ENS Lyon, CNRS, France Laurent Chevillard, Laboratoire de Physique, ENS Lyon, CNRS, France
More informationPLASMA ASTROPHYSICS. ElisaBete M. de Gouveia Dal Pino IAG-USP. NOTES: (references therein)
PLASMA ASTROPHYSICS ElisaBete M. de Gouveia Dal Pino IAG-USP NOTES:http://www.astro.iag.usp.br/~dalpino (references therein) ICTP-SAIFR, October 7-18, 2013 Contents What is plasma? Why plasmas in astrophysics?
More informationNumerical methods for the Navier- Stokes equations
Numerical methods for the Navier- Stokes equations Hans Petter Langtangen 1,2 1 Center for Biomedical Computing, Simula Research Laboratory 2 Department of Informatics, University of Oslo Dec 6, 2012 Note:
More informationChapter 2: Fluid Dynamics Review
7 Chapter 2: Fluid Dynamics Review This chapter serves as a short review of basic fluid mechanics. We derive the relevant transport equations (or conservation equations), state Newton s viscosity law leading
More informationSolar Physics & Space Plasma Research Center (SP 2 RC) MHD Waves
MHD Waves Robertus vfs Robertus@sheffield.ac.uk SP RC, School of Mathematics & Statistics, The (UK) What are MHD waves? How do we communicate in MHD? MHD is kind! MHD waves are propagating perturbations
More information