Introduction to DNS DNS of wall-bounded flow Modelling of turbulent flows: RANS and LES Turbulenzmodelle in der Str¨ omungsmechanik: RANS und LES Markus Uhlmann Institut f¨ ur Hydromechanik Karlsruher Institut f¨ ur Technologie www.ifh.kit.edu SS 2012 Lecture 2 1 / 36
Introduction to DNS DNS of wall-bounded flow LECTURE 2: DNS as numerical experiments 2 / 36
Introduction to DNS DNS of wall-bounded flow Questions to be answered in the present lecture What are the possibilities & limitations of numerical simulations of the full Navier-Stokes equations? ◮ what are the goals of DNS? Part I ◮ what is the history of DNS? ◮ what are the computational requirements? ◮ how to treat the boundary conditions? ◮ DNS results for coherent structure dynamics in Part II wall-bounded flows 3 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements Definition of “direct numerical simulation” (DNS) Solve the Navier-Stokes equations for turbulent flow, resolving all relevant temporal and spatial scales. ◮ for incompressible fluid solve: ∂ t u + ( u · ∇ ) u + 1 ν ∇ 2 u ρ ∇ p = ∇ · u = 0 with suitable initial & boundary conditions. 4 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements Spectral view: DNS versus LES energy spectrum dissipation spectrum 1 0.8 DNS 0.6 κE ( κ ) κD ( κ ) 0.4 LES 0.2 0 −4 −3 −2 −1 0 10 10 10 10 10 κη ◮ DNS resolves spatial scales down to Kolmogorov scale η 5 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements Physical space view: DNS versus RANS Example: channel flow instantaneous DNS data ( u ′ ) ⇒ DNS statistics 1 0.8 � � u ′ u ′ � /U 0 0.6 0.4 � u � /U 0 0.2 0 0 0.2 0.4 0.6 0.8 1 → flow direction ◮ DNS needs to be integrated in time to obtain statistics ◮ � u i � , � u ′ i u ′ j � are variables in RANS computation 6 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements Objectives of DNS studies (Today) DNS is a research method, not an engineering tool. ◮ computational effort: → today not feasible to perform DNS for practical application ◮ main purpose of DNS: → development of turbulence theory ⇒ improvement of simplified models 7 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements 1. DNS as “precise experiment” or “perfect measurement” If we can simulate the flow with high-fidelity: ◮ full 3D, time-dependent flow field is available ◮ virtually any desired quantity can be computed (e.g. pressure fluctuations, pressure-deformation tensor) ◮ there are no limitations by measurement sensitivity (e.g. size of probes near a wall) � analysis only limited by mind of researcher (it is important to ask the right questions) ⇒ DNS complements existing laboratory experiments 8 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements 2. DNS as “virtual experiment” When experiments are too costly/impossible to realize: ◮ numerical simulations provide great flexibility ◮ idealizations can be realized with ease: ◮ e.g. homogeneous-isotropic flow conditions ◮ periodicity ◮ absence of gravitational force ◮ . . . ⇒ DNS replaces laboratory experiments 9 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements 3. DNS as “non-natural experiment” When non-physical configurations need to be simulated: ◮ we have the possibility to modify the equations ◮ we can apply arbitrary constraints ◮ examples from the past are: ◮ filtering (damping) turbulence in some part of the domain ◮ suppress individual terms in the equations ◮ applying artificial boundary conditions ◮ . . . ⇒ DNS directly serves turbulence theory 10 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements “The question of simulating turbulent flows is largely one of economics, clever programming, and access to a big machine.” Fox and Lilly (1972) Reviews of Geophysics and Space Physics 11 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements Historical development of DNS 1972 first ever DNS of hom.-iso. turbulence by Orszag & Patterson 1981 homogeneous shear flow by Rogallo 1987 plane channel flow by Kim, Moin & Moser 1986-88 flat-plate boundary layer by Spalart 1990-95 homogeneous compressible flow (Erlebacher/Blaisdell/Sarkar) 1997 solid particle transport in channel flow (Pan & Banerjee) 2005 deformable bubbles in channel flow (Lu et al.) ◮ # of publications in Phys. Fluids: 1990 – 14, 2008 – 76 ◮ # of grid points: 10 4 − → 10 11 12 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements Numerical requirements for DNS Homogeneous turbulence ◮ uniform grid with N × N × N points: periodic box ∆ x = ∆ y = ∆ z = L N ◮ assume a periodic field → use Fourier series with wavenumbers: κ ( α ) = 2 πi L , where: − N/ 2 ≤ i ≤ N/ 2 i L ⇒ largest wavenumber: κ max = πN L Fourier modes: exp( I κ x ) ◮ operation count per time step: using fast κ = ( κ (1) , κ (2) , κ (3) ) O ( N 3 log N ) Fourier transform 13 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements Homogeneous turbulence – spatial resolution Large scale resolution ◮ largest flow scales need to be much smaller than box size � otherwise: artifacts of periodicity! ◮ rule of thumb: (box) L ≥ 8 L 11 (integral scale) ◮ recall: lowest non-zero wavenumber in DNS is κ 0 = 2 π L ⇒ κ 0 L 11 = π (largest scale) 4 found to be adequate by comparison with experiments 14 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements Homogeneous turbulence – large scale resolution (2) 0.25 0.20 E( κ ) Energy-containing range 0.15 kL 11 0.10 ◮ smallest wavenumber: 0.05 setting κ 0 L 11 = π 4 0.00 0 1 2 3 4 5 6 7 8 9 10 κ L 11 ⇒ ≈ 95% of energy resolved grid turbulence, Comte-Bellot & Corrsin 1971 ◦ Re λ = 60 . . . 70 15 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements Homogeneous turbulence – small scale resolution Small scale resolution ◮ need to resolve the dissipation range � otherwise: there is no sink for kinetic energy → “pile-up” ◮ rule of thumb: κ max η ≥ 1 . 5 or ∆ x ≤ πη 1 . 5 16 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements Homogeneous turbulence – small scale resolution (2) Dissipation range ◮ representing up to: κ max η = 1 . 5 ⇒ most dissipation resolved Pope’s model spectrum, Re λ = 600 17 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements Homogeneous turbulence – number of grid points Combined small/large scale requirements ◮ N = L ∆ x = 12 L 11 πη ◮ how does the scale ratio L 11 /η evolve with Re ? ◮ from the model spectrum: L 11 /L ≈ 0 . 43 for large Re (recall L ≡ k 3 / 2 /ε ) ◮ defining Re L ≡ k 1 / 2 L η = Re 3 / 4 we obtain: L ν L i.e. N 3 ≈ 4 . 4 Re 9 / 4 ⇒ finally: N ≈ 1 . 6 Re 3 / 4 L L � steep rise with Reynolds! 18 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements Homogeneous turbulence – temporal resolution Resolving the small-scale motion ◮ typically need: (time step) ∆ t = 0 . 1 τ η (Kolmogorov scale) Sampling sufficient large-scale events ◮ each simulation needs to be run for a time T given by: T ≈ 4 k ( k/ε is characteristic of large scales) ε ⇒ obtain for the number of time steps M : M = T ∆ t = 4 0 . 1 Re 1 / 2 ◮ L 19 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements Homogeneous turbulence – total operation count Total number of operations per DNS, using spectral method: ◮ N tot = N op · M ∼ N 3 log( N ) · M ∼ Re 11 / 4 log( Re L ) L Simulation parameters for “landmark” studies: N Re L computer speed # processors 32 180 10 Mflop/s 1 Orszag & Patterson 1972 512 4335 46 Gflop/s 512 Jimenez et al. 1993 4096 216000 16 Tflop/s 4096 Kaneda et al. 2003 20 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements Result of high-Reynolds DNS of hom.-iso. turbulence Kolmogorov scaling of data by Kaneda et al. (2003) E ( κ ) κ 5 / 3 /ε 2 / 3 εL u 3 0 κη Re λ ◮ Kolmogorov scaling largely confirmed 21 / 36
Purpose of DNS Introduction to DNS History of DNS DNS of wall-bounded flow Numerical requirements Evolution of computer speed single-processor CPU speed performance of multi-processor systems 15 10 flop/s 12 10 9 10 1990 1995 2000 2005 2010 year (from Hirsch 2007) (data from top500.org ) ◮ large CPU speed increase ◮ massively-parallel machines maintain exp-growth ◮ limitation: power & heat ⇒ peak performance doubles every 18 months 22 / 36
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