•Michal Macek michal.macek@isibrno.cz •Group of Cryogenics and Superconductivity •The Czech Academy of Sciences Institute of Scientific Instruments • • •Turbulent convection PřF MUNI: F3250 - Moderní témata fyziky kondenzované fáze :: 30.11.2023 Institute of Scientific Instruments of the CAS ALISI research departments In 1967 In 2023 www.isibrno.cz • Cryogenics at ISI CAS Brno © Jiri Frolec © Tomas Danek © Tomas Danek I:\Prezentace tymu\Workshop_UPT_2014\Dílčí prezentace\p1010034_U2.jpg nf_celky_pc_12_u EWA apparatus (near-field regime) •Heat transport by Convection PRL 109, 224302 (2012), PRB 99, 024511 (2019) Emister apparatus (far-field regime) •Heat transport by Radiation … discussed in detail today Heat Transport at Low Temperatures (~5K) ConEV apparatus (Rayleigh-Bénard convection) Outline of Lecture 1.Basic aspects 2. I. I. I. 2. Experiments in Brno: Classical turbulent Rayleigh-Benard convection I. I. I. I. 3. New research directions [1 a] Heat Transport [1 b] Turbulent Flows [1 c] Helium 4He [2 a] Nu(Ra) - Heat transport efficiency [2 b] Re(Ra) - Dynamics of coherent structures (wind) [3 a] Attractors in RBC [3 b] Modulated Convection and Rotation [3 c] Visualization of 4He flows – metastable molecular excimers [3 d] Classical and Quantum heat transfers - analogies There’s never too much good art… Basic Heat transport mechanisms: Recall… •1. ? • • •2. ?? • • •3. ??? • Basic Heat transport mechanisms: Recall… Leonardo da Vinci (1452-1519) Katsushika Hokusai (1760-1849) Vincent van Gogh (1853-1890) Turbulence ‘Mount Fuji viewed from the sea,’ from One Hundred Views of Mount Fuji, ca. 1834. British Museum ‘Starry Night’ 1889. MoMA, New York ‘Drawing of Water Vortex,’ cca. 1510 - 1513. Royal Collection Windsor, UK 1.1 3/40 There’s never too much good art… Leonardo da Vinci (1452-1519) ‘Drawing of Water Vortex,’ cca. 1510 - 1513. Royal Collection Windsor, UK Richardson cascade Adapted from: Uriel Frisch. 1995. Turbulence: The Legacy of A. N. Kolmogorov. •Starting at forcing scale •Ending at viscous dissipation scale “Big whirls have little whirls, That feed on their velocity, And little whirls have lesser whirls, And so on to viscosity.” Lewis Fry Richardson (1881 – 1953) [a take/parody on Jonathan Swift’s "On Poetry: a Rhapsody" (1733)] space Turbulence: Scales And Cascades There’s never too much good art… Turbulence: Scales And Cascades Richardson – Kolmogorov cascade Adapted from: Uriel Frisch. 1995. Turbulence: The Legacy of A. N. Kolmogorov. •Starting at forcing scale •Ending at viscous dissipation scale Andrey Nikolaevich Kolmogorov (1903-1987) space log(k) log E(k) E(k) ~ k -5/3 Kolmogorov scale: η = (ν3/ε)1/4 At r < η viscosity dominates, flow is laminar For L0 > r > η, inertial cascade of vortices. viscosity negligible flow is turbulent “Big whirls have little whirls, That feed on their velocity, And little whirls have lesser whirls, And so on to viscosity.” Lewis Fry Richardson (1881 – 1953) [a take/parody on Jonathan Swift’s "On Poetry: a Rhapsody" (1733)] •Starting at forcing scale •Ending at viscous dissipation scale •Broad and smooth kinetic energy spectrum Andrey Nikolaevich Kolmogorov (1903-1987) There’s never too much good art… Characteristic scales of turbulence in Nature Quantum vortex in superfluid He II M100 galaxy as seen through Hubble telescope Maelstrom, Saltstraumen, Norway Turbulence: Major open scientific problem Clear Ideas by Rene Magritte, 1958 Clay Mathematics Institute $1 000 000 prize: Navier–Stokes existence and smoothness problem: Even basic properties of the solutions to Navier–Stokes have never been proven. For the three-dimensional system of equations, and given some initial conditions, mathematicians have not yet proved that smooth solutions always exist, or that if they do exist, they have bounded energy per unit mass. Turbulence onset and development: Reynolds number inertial effects (characteristic scales of velocity U and length L) Dimensionless “flow-similarity” number Osborne Reynolds (1842-1912) Turbulent flows – inertia dominates 2.1 22/40 Laminar flows – viscosity dominates (1883) Turbulent mixing: very efficient heat transport • Temperature is a scalar quantity (of course), • like e.g. concentration of cream in coffee • – it is governed by similar (advection-diffusion) equations. • Velocity of the fluid flow is a vector field (of course). •Surface of the drop deforms and grows in area rapidly, due to turbulent flow fluctuations. • •Compared to transport by molecular diffusion, mixing by turbulent flows is thus able to enhance scalar transport by many orders of magnitude. • •Example: In a 1km thick atmospheric boundary layer, turbulent convection transfers heat at least 105 more efficiently(!) Helium •“Classical flows” • in liquid (He-I) and gaseous (GHe) phases • Helium 4He: Three fluid phases 4He Critical point (CP): Pc =227 kPa, Tc = 5.20 K •“Quantum flows” in superfluid (He-II) phase A Liquid That Pours Itself! The Self-Siphoning Fluid: Polyethylene Glycol : https://www.youtube.com/watch?v=t3neqUhoDRA Material properties obtained from HEPAK database State and transport in 4He Material property databases •REFPROP •HEPAK •XHEPAK Plaster model of internal energy U(S,V) of a “water-like substance” handmade by J. C. Maxwell (!) for J. W. Gibbs (found at Yale Sloane Physics Lab.) Fluid properties Group of Cryogenics and Superconductivity Long-time tradition in development and construction of cryogenic instruments for Basic Research: Cryogenics and Superconductivity group at ISI Brno •Heat transport by Convection ConEV apparatus (Convection Experimental Vessel) discussed in this talk PRL 107 014302 (2011), PRL 109 154301 (2012), PRL 110 199402 (2013), RSI 81 085103 (2010), PNAS 110 8036 (2013), JFM 785 270 (2015), JFM 832 721 (2017), PRE 99 (R) 011101 (2019), … https://lh3.googleusercontent.com/fxgH52ZEhtL3v_-NS7yBQd4g2RILyeRje6y4hVdqE2RX_Xg_aHJLSFFFzqdF9QP7N MzRHy4LZrhBGlnEmVJT8Euip_qjp3TEEdeTdp2Er6A-LRt9JyryNH0hWdwIl5kx8B6tk1X5r_ajPY9lTPm4u4a3nSs0E3kJpnhQ 9WbGd3ubH6MxU_m18X0D_pwvGNlgVl1vgfZgmoeXGTIqG-C6IjOp2eETfPovYeryngb2orZnTDX4WOcR5n5HYkTR1JDEYnDI7tN OjeQ5ZpPfF8YSgxeFbm3aX_Tkh5ETHgmuI9OWCGcDlyjWJdKzjMY2CsSxJwelJWy63qR6DqhP4QXJywXhkpydrHPyqDY4dCCf5H wRqweHao7kXE8PUJ_cwjb52M7wOn_e4PYwvFoTuReLJQ8uf4ux1dV8nu41d3qgIe6aTvylCj2tNJL6ClN8bS5l258FZ0D_It5QT ib9BUFYPhk5IJhHULVKuCsK-cnYLO8Z_KqWV9kMZCLcyQhv7OaF2vgENUAxrDJaOJhyRnPyJ8b6zlt6Vu9vlQrkwJ6wK8NczWPQ NUAJzG3Ap2ua1G2Eo672uuR-dcNzB8TExCwQpYhiF-HDzxrNdS95GNJyjawwyGr7uVNLcJ19a3UVq-N3FGJlwOHfeX5kJQGCeVf xi2htekI5gWqdP9X7DRypLmE9vnUeCsIezecB5p5Iy0o=w930-h697-no?authuser=0 Rayleigh Bénard convection (RBC) cell Size L = H = 30 cm RBC cell inside the cryostat Rayleigh Bénard convection (RBC) cell Size L = H = 30 cm RBC cell inside the cryostat Rayleigh-Bénard convection – Rayleigh-Bénard model of convection Control parameters for RBC (adjustable): Order parameters for RBC (response of the system): Ra, Pr, Nu, Re: Dimensionless numbers related to intensity of turbulence Rayleigh number Prandtl number simulation from Shishkina et al. PRL 114 (2015) Nusselt number Reynolds number Finite Cell: Diameter D Aspect ratio Ra > Rac = 1708 ~ ∞ ∞ Tb = Tt + ΔT heat Oberbeck-Boussinesq (OB) fluid: •constant fluid properties within DT: • • •density r is assumed to linearly depend on temperature T a - thermal expansion coefficient λ - fluid thermal conductivity n - kinematic viscosity k - thermal diffusivity Rayleigh-Bénard convection – a simple model system Oberbeck-Boussinesq (OB) fluid: •constant fluid properties within DT • • •density r is assumed to linearly depend on temperature a - thermal expansion coefficient n - kinematic viscosity k - thermal diffusivity λ - fluid thermal conductivity a - thermal expansion coefficient n - kinematic viscosity k - thermal diffusivity λ - fluid thermal conductivity •Navier-Stokes equation = Newton equation for continuum - a viscous fluid with pressure and upward buoyancy forcing: • •Heat advection • •Incompressibility condition: • •Heat conduction: (Fourier’s law) a - thermal expansion coefficient n - kinematic viscosity k - thermal diffusivity λ - fluid thermal conductivity ρ0 - mean density g - gravity acceleration Equations and Scale-Similarity of solutions u - velocity field θ - temperature field p - pressure field •Fluid properties: (considered constant!) • “Boussinesq equations” of RBC equation: •Dynamical variables: • Rayleigh-Bénard convection – Equations and Scale-Similarity of solutions “Non-dimensionalize” using spatial, temporal, velocity, temperature and pressure scales: a ,n, k, ρ0 (and g) Ra, Pr Character of RBC solutions depends only on 2 essential parameters – Ra & Pr - instead of 5! Zastav sa a diskutuj Ra!!!! Atmosphere: Ra ≈ 1017 Sun: Ra ≈ 1030 Ocean: Ra ≈ 1020 Natural convection often occurs on large scale distances L and thus is characterized by very high values of Ra number. Large-scale Convection in Nature working fluid properties system size Laboratory experiments: high Ra at low L. Examples Ra Atmosphere ≈ 1017 Ocean ≈ 1020 Laboratory ≈ 1017 Computer ≈ 1011 Fluid Temperature a/nk Air 20°C 0.122 Water 20°C 14.4 Helium 4He (gas) 5.5 K 1.41×108 Helium I (liquid) 2.25 K 3.25×10-5 Ra values attainable with different fluids 4He Critical point (CP): Pc =227 kPa, Tc = 5.20 K HIGHEST Ra near the critical point Liquid Vapor Gas Other interesting regions besides CP: 1.Saturated Vapor Curve (SVC) 2.Critical Isochore (CI) Rayleigh-Bénard Convection in 4He Brno cell L = 30 cm, Ra from 106 up to 1015 SVC CI CP Coherent structures / patterns observed in a fluid-filled pot heated from below History detour #1: …it is not as simple … Shadowgraph seen from above •Formulated eqations for system with ∞ plates •Predicted critical value of the control parameter (now “Rayleigh number”) for conduction - convection transition Rac = 1708 Experimentally confirmed with high accuracy History detour #2: {\color{red}\rho_0 Edward Lorenz: computer-ready model for weather prediction (PhD Thesis, 1948, MIT) strange attractor History detour #3: “butterfly wing effect” – sensitivity to initial conditions: deterministic chaos Simple 3D dynamical system derived from Boussinesq equations of RBC Lorenz system as simplified Rayleigh–Bénard convection Grenoble 4He cryogenic experiment X. Chavanne, F. Chilla, B. Castaing, B. Hébral, B. Chabaud and J. Chaussy, Phys. Rev. Lett. 79, 3648 (1997) •Transition in the vicinity of Ra ~ 1011 - interpreted as regime with Nu ~ Ra1/2. •Transition observed also in the cell geometry G = 1.14 and 0.23. Critical point #1: Onset of (laminar) Convection “Critical point #2”: Transition to Turbulence Conduction Laminar convection Regimes of RBC: Heat Transport Nusselt number Nu (dimensionless heat transfer coefficient): Power law dependences in turbulent regime(s) are good approximation Ra Turbulent convection Grenoble 4He cryogenic experiment X. Chavanne, F. Chilla, B. Castaing, B. Hébral, B. Chabaud and J. Chaussy, Phys. Rev. Lett. 79, 3648 (1997) •Transition in the vicinity of Ra ~ 1011 - interpreted as regime with Nu ~ Ra1/2. •Transition observed also in the cell geometry G = 1.14 and 0.23. Turbulent convection “standard RBC turbulence” regimes Nu ~ Ra1/3 Nusselt number Nu (dimensionless heat transfer coefficient): Ra “ultimate RBC turbulence” regime Nu ~ Ra2/7 transitions? Nu ~ Ra1/2 Regimes of RBC: Heat Transport Interesting dynamical transitions: In the seemingly trivial laminar regime, many transitions connected to bifurcations. On the other hand, in the seemingly random turbulent regime, appearance of coherent structures Malkus, Priestley, Spiegel (1954) Shraiman and Siggia (1990) Ra* ≈ 1021 – 1024 •fully turbulent boundary layers •ballistic heat transfer independent of k an n http://graphics8.nytimes.com/images/2008/03/08/us/08kraichna.190.jpg “Standard turbulent” RBC Ultimate / asymptotic turbulent RBC laminar boundary layers (BL) + weak or no dependence on Pr weak dependence on Pr Ultimate regime of heat transport in RBC Extrapolations to large-scale flows? exponents g = 1/3 and g = 1/2 Confirmation of existence of the ultimate regime is great challenge in this field of study! RBC_ultimate_v2 g = 1/2 g = 1/3 Motivation for this talk Clanky ktore tvrdia ze vidia~! PRL (Grenoble) PRL (Goettingen) …. Numerical simulation in 2D (Twente, Rome, Goettingen) Room-temperature high-pressure SF6 experiment (Goettingen) Cryogenic 4He experiment (Grenoble) OBSERVED…? NO…? … Cryogenic 4He experiment (Oregon) Cryogenic 4He experiment (Brno) Cryogenic 4He experiment (Brno) Fig04_Upper_RNu_RNuKorSw_uprav P. Urban, P. Hanzelka, T. Kralik, V. Musilova, A. Srnka and L. Skrbek, Phys. Rev. Lett. 109, 154301 (2012). P. Urban, P. Hanzelka, T. Králík, MM, V. Musilová, L. Skrbek Phys. Rev. E 99, 011101 (R) (2019). Transitions observed at ISI Brno: 1. Transition at Ra >~ 1014: Ultimate regime transition or NOB effects ? 2. Transition at Ra~ 1010-1011: ? G = 1 NOB effects near the Critical Point of 4He NOB: Tm ≠ Tc: BL asymmetry J.Drahotský, P. Hanzelka, V. Musilová, MM, R. du Puits and P. Urban EPJ Conf S. 180, 02020 (2018) Brno – 4He Goettingen – SF6 NOB effects near the Saturation Curves of 4He and SF6 Results: Global heat transfer – Nu(Ra) Transition: Nu ~ Ra2/7 - Ra1/3 Fig04_Upper_RNu_RNuKorSw_uprav V. Musilová, T. Králík, M. La Mantia, MM., P. Urban, L. Skrbek. J. Fluid. Mech. 832, 721 – 744 (2017). Change in shape of the coherent flow structure - the mean wind Transitions observed at ISI Brno: 1. Transition at Ra >~ 1014: Ultimate regime transition or NOB effects ? 2. Transition at Ra~ 1010-1011: ? What is the possible nature of this transition will be discussed now… We measure temperature fluctuations in the bulk, at four positions near the midplane… Grenoble 4He cryogenic experiment X. Chavanne, F. Chilla, B. Castaing, B. Hébral, B. Chabaud and J. Chaussy, Phys. Rev. Lett. 79, 3648 (1997) Onset of (laminar) Convection: -Different forms of coherent structures seen (Pattern formation) Transition to Turbulence: -Coherent structures persist! (on average - mean flows) Conduction Laminar convection Turbulent convection Regimes near onset of convection: Studied in cells with Γ >> 1. Regimes of RBC: Coherent structures Adapted from Bodenschats et al., Annu.Re.Fluid.Mech (2000) Shadowgraph images seen from top Increasing Ra Low Ra - instant. snapshot (spiral defect chaos - SDC) Moderate Ra - instant. snapshot (disorder, strong fluctuations) Moderate Ra - averaged image (SDC recovered) Emran, Schumacher, JFM 776, 96 (2015) MM, Schumacher, in preparation D = 10 cm ~ High Ra regimes: Studied in tall cells with Γ < 1. Zwirner & Shishkina, JFM 850, 984 (2018) Reynolds measurements at ISI Brno: Temperature fluctuations in the turbulent bulk Temperature fluctuations time series (δT ~ 1mK) Four small cubic Ge thermistors (T1 – T4) near sidewalls with respect to LSC direction: T1, T3 – “leading sensors” T2, T4 – “trailing sensors” Statistical analysis (see next slide) •Temperature fluctuations PDFs (in turbulence: non-Gaussian, heavy tails, rare events) • •Power spectra (in turbulence: broad due to “Richardson cascade” transferring energy over wide range of scales) • •Auto- and Cross-correlations (Fourier transform of spectra) • •Reynolds numbers (different types) In the same experiments simultaneously.. Four sensors near sidewalls… Properties of LSC … 1 – PDF: •Histogram •PDF in MATLAB •Moments •Compare with Gaussian PDF 2 – Spectrum: •Fourier transform in MATLAB •Detrend, demean (f=0) •Frequency band filtering • 3 – Correlation functions: •Convolution •Trend effects •Global and side extremes •Zeroes Temperature fluctuations in the turbulent bulk Brno data fitted by a two-fold power law where above / below the transition point Ra = Rac One-point measurements: •Frequency Reynolds number V. Musilová, T. Králík, M. La Mantia, MM., P. Urban, L. Skrbek. J. Fluid. Mech. 832, 721 – 744 (2017). Reynolds numbers (one-point measurements) Oscilace + stochastics of LSC: sideward sloshing seen here (okrem toho twisting, ktory v strede nevidime…atd… Brown, Ahlers, Weiss…V zavislosti na polohe snimacov, su merania citlive na rozne vlastnosti velkych struktur (LCS, plumy…) Two-point measurements: •“Elliptic approximation Reynolds numbers” G.W. He & J.B.Zhang Phys.Rev. E 73 055303 (2006) •Brno data compared with SF6 data by X. He et al. New J. Phys 17 063028 (2015) V. Musilová, T. Králík, M. La Mantia, MM., P. Urban, L. Skrbek. J. Fluid. Mech. 832, 721 – 744 (2017). Reynolds numbers (two-point measurements) Oscilace + stochastics of LSC: sideward sloshing seen here (okrem toho twisting, ktory v strede nevidime…atd… Brown, Ahlers, Weiss…V zavislosti na polohe snimacov, su merania citlive na rozne vlastnosti velkych struktur (LCS, plumy…). Na rozdiel od predchadzajuceho slidu – Re_p - nemame porovnanie s kryo, len s SF_6… “leading sensors” “trailing sensors” M4 (Flatness) M4 (Flatness) Trailing sensors outside main LSC roll for : Tilted elliptical LSC All sensors inside the main LSC roll for : Squarish LSC shape See also Niemela, Sreenivasan Europhys. Lett. (2003) 62, 859 V. Musilová, T. Králík, M. La Mantia, MM., P. Urban, L. Skrbek. J. Fluid. Mech. 832, 721 – 744 (2017). Fluctuation PDFs: Evidence for LSC shape change Co tu vidime… V GL teorii prechod nie je bud vobec (novy fit Stevens), alebo velmi slabo/nevyrazne. Podla GL by prechod v Nu(Ra) mal suvisiet s chovanim medznych vrstiev (stary fit predpoveda v danom mieste prechod v BL typu IV_u – IV_l), Odpoved moze naznacit chovanie LSC, Bulk temperature fluctuations – measurement #f078: Ra = 8.86*1012 , Pr = 1.34, = 1280 T1 T2 T3 T4 Wind reversals X-corr delay X-corr delay PDF skewness PDF skewness sliding window analysis Outlook 1: Attractors in RBC and Data-based Mathematical models Jakub Kašný, FSI VUT BP 2022, to be submitted Outlook 1: Attractors in RBC and Data-based Mathematical models Jakub Kašný, FSI VUT BP 2022, to be submitted Outlook 1: Attractors in RBC and Data-based Mathematical models Jakub Kašný, FSI VUT BP 2022, to be submitted • Outlook 2: Temperature modulation and rotation in Classical RBC and Quantum Counterflow Prof. Ladislav Skrbek (Prague) Outlook 2: Rotating platforms for Classical RBC and Quantum Counterflow 3.2 38/40 Rotating RBC experiment under preparation at ISI Brno and Charles University Prague Outlook 2: Rotating platforms for Classical RBC and Quantum Counterflow Rotating RBC simulation from Richard Stevens https://stevensrjam.github.io/Website/research_rrb.html] Increasing rotation rate Many visualization methods exist: Smoke, Ink or Dye… Particle Tracking Velocimetry, Particle Image Velocimetry, Laser Doppler Velocimetry Shadograph, Schlieren, etc… Outlook 3: Flow visualization at high Ra Tracer “Clouds” (may be more cvonvenient for RBC BL) Tracer Line Ernst Mach &Peter Salcher, cca. 1890 None allow to reach “ultimate” Ra! (enabling direct velocity information in RBC) A promissing visualization method was developed McKinsey et al. Phys. Rev. Lett. 95, 111101 (2005) W. Guo et al. Phys. Rev. Lett., 105, 045301 (2010) J. Gao et al. Rev. Sci. Instrum. 86, 093904 (2015) Long-living (> 10s) molecular triplet excimer He2* McKinsey et al. Phys. Rev. A 59, 200 (1999) - Excimers form after He ionization (by fs-lasers, radioactivity, intense electric field ionization, etc…) -Can be visualized by molecular tagging via laser-induced fluorescence (LIF) Quantum counter-flow (CF) of superfluid He-II: -Thermally induced relative flow of the normal and superfluid components -Analogous heat transfer scaling laws experimentally observed for CF and classical RBC [Skrbek, Gordeev, Soukup, PRE (2003)] -Analogous roles of gravitational potential in RBC and chemical potential in CF; -Detailed theory being developed (MM) … W. Guo et al. PNAS (2013) Outlook 3: Flow visualization at high Ra Many visualization methods exist: PTV, PIV, shadograph… None has so far been realized in fluids allowing to reach “ultimate” Rayleigh numbers (ultimate turbulent heat transport). Besides (and more importantly): At high Ra, only temperature information is experimentally known; Visualization technique would enable measuring velocity field too! Thanks to: Jan Ježek, Minh Tuan Pham, Jakub Grim Laser ionization of Helium gas by the Brilliant nanosecond laser succesfully accomplished R&D for turbulent flow visualization in helium gas at ISI Brno figure.lp.vsp-em._full._visible._spectrum._range.001.png Big Thanks to My ISI Cryo - Colleagues! and colleagues in Prague, Ilmenau, Florida… Prague group of prof. Ladislav Skrbek Ilmenau group of prof. Joerg Schumacher and prof. Ronald du Puits Summary: •Turbulence is an open theoretical and experimental problem •One of major open questions: Existence of the ultimate regime, relevant e.g. at extremely large spatial scales in Nature •Can be studied in Lab in cryogenic Helium •Information on velocity field missing at high Ra - Need for high Ra visualization experiments, possibly with He2* excimers •Analogous transitions laws in classical and quantum heat transfer not well understood… Thank You! Open BSc. and MSc. Thesis Topics: •[Experiment] •Rotating Rayleigh-Benard Convection (with Pavel Urban) • •[Theory] •Classical-Quantum Analogy for heat transport laws in Rayleigh-Benard Convection in GHe • and Counter-Flow in He-II (with Michal Macek) • •[Data Analysis] •Attractors in turbulent Rayleigh Benard Convection (with Michal Macek) •Ultimate Regime of Convection or NOB Convection? (with Michal Macek) • Analyze and compare classical Rayleigh-Benard Convection data from: •Cryogenic GHe experiments at ISI Brno •Dry air experiments at Barrell of Ilmenau by group of prof. De Puits (DE) •Highly parallel direct numerical simulations by group of prof. Schumacher (TU Ilmenau, DE) • Content of the lectures 1.What is turbulent convection and why do we study it at ISI CAS Brno 2. 2. Statistics of turbulence: Stochastic models + signal analysis with practical examples in MATLAB 3. 3. Deterministic models for turbulence – How to derive Lorenz system from Hydrodynamic equations … 4. 4. Data-driven models for turbulent convection