flops/sec & bytes

UNRAVELING TURBULENCE

A.E.P. Veldman and R.W.C.P. Verstappen

University of Groningen

Panta Rhei

Heraclitus (535-475 BC)

Navier-Stokes equations

Claude Navier (1822) George Stokes (1845)

• Conservation of momentum (incompressible)

∂u

∂t

| {z }

evolution

+ (u · grad)u

| {z }

convection non-linear

creator of turbulence

= −1

ρ grad p

| {z }

pressure

+ div(ν grad u)

| {z }

diffusion ν very small

• ratio convection

diffusion ∼ U L

ν = Re(ynolds number)

Osborne Reynolds (1894)

Generation of turbulence

∂u

∂t + u∂u

∂x + · · · = · · · + ν∂²u

∂x² u ∼ sin ωx ⇒ u∂u

∂x ∼ ω sin ωx cos ωx

⇒ ∂u

∂t ∼ −¹₂ω sin 2ωx − ν ω² sin ωx

Energy cascade:

Frequency doubling until diffusion comes into action.

Big whorls have little whorls, Which feed on their velocity,

And little whorls have lesser whorls, And so on to viscosity.

(L.F. Richardson, 1881–1953)

Scaling and complexity

• ^Kolmogorov (1941) scaling

dissipation of energy ǫ = de/dt [L²T⁻³] kinematic viscosity ν [L²T⁻¹]

ℓ ∼ (ν³/ǫ)^1/4; τ ∼ (ν/ǫ)^1/2

size small eddies ℓ ∼ Re^−3/4 × size large eddies L

time scale small eddies τ ∼ Re^−1/2 × time scale large eddies T

• Complexity of flow

(Re^3/4)³ × Re^1/2 = Re^11/4

• Re 10× larger ⇒ complexity 1000× larger!

Leonardo’s turbulence

Eddies in water (≈ 1507)

Laminar vs. turbulent

laminar flow (forced) turbulent flow Turbulent flow is better able to follow curved walls.

Reduces size of wake, and herewith drag!

Exploiting turbulence

Drag reduction by delaying separation

golf ball

speed skater

head

& legs torso

Modelling turbulence

RaNS Reynolds-averaged Navier-Stokes:

model all turbulence (steady)

LES Large-eddy simulation (unsteady):

resolve large eddies;

model smaller eddies

DNS direct numerical simulation:

resolve all flow structures

mean

flow behind a square cylinder

snapshot

Modelling turbulence (2)

Prediction Spalart (2000):

airplane Re = 10⁷⁻⁸

model RaNS LES DNS

grid points 10⁷ 10^11.5 10¹⁶

memory (bytes) 10¹⁰ 10¹⁴.5 10¹⁸.5

time steps 10³ 10^6.7 10^7.7

flops 10¹⁴ 10²² 10²⁷

speed (flop/s) 10⁹ 10¹⁷ 10²²

year 1990 2045 2080

But: Algorithmic improvements bring future sooner!

Square cylinder at Re=22.000

Testcase RaNS and LES - 1997

Attempts by RaNS and LES

Models were tuned to known answer

DNS is pure Navier–Stokes – and similar grid ...

Challenges

• Ten years ago Re = 10⁴ was not really possible.

So how about the following applications?

medium speed Reynolds cyclist (tourist) air 20 km/h 1 · 10⁵ golfball (pro) air 250 km/h 2 · 10⁵ speed skater (pro) air 45 km/h 5 · 10⁵ swimmer (pro) water 5 km/h 3 · 10⁶ car (Dutch) air 80 km/h 5 · 10⁶

shark water 20 km/h 2 · 10⁷

airplane air 900 km/h 3 · 10⁷

ship water 20 km/h upto 10⁹

Note: air is 15× more viscous than water

• A ship (+screw) is even 10¹⁵ more expensive!

Boosting performance

• Computer designers

computer performance: 30× per decade

• Numerical mathematicians

algorithm performance: 30× per decade

⇒ Together: factor 1000 per decade!

i.e. factor 10 in Reynolds number

NB: Similar progress is found in other computational disciplines.

Numerical progress

Discretization of dφ/dx

000000000000000 000000000000000 111111111111111 111111111111111 000000000000000

000000000000000 111111111111111 111111111111111

Lagrangian interpolation Symmetry preservation

h h

- - +

-

h

x_- x x

φ

φ

-

0

+

-

0

φ

+

0 +

h

φ₊ φ

φ

x x

x _{0 +}

dφ

dx = h²₋φ+ + (h²₊ − h²₋)φ0 − h²₊φ₋ h+h−(h+ + h−)

Coefficient of φ⁰ can make sys- tem singular!

dφ

dx = φ+ − φ₋ h+ + h₋

Skew-symmetric expression:

– system never singular!

– no artifical diffusion!

Balance at smallest scales

• Turbulence is subtle balance between production by (non-linear) convection and destruction by diffusion

• Numerical diffusion must not interfere with this balance !!

• Skew-symmetric discretization has no numerical diffusion

• Similar story for skew-symmetric turbulence modelling

Evolution of energy

dφ_h

dt + L_hφ_h = 0

• ‘Energy’ ||φ_h||²_h = φ^∗_hHφ_h evolves in time as d

dt||φ_h||²_h = −(L_hφ_h)^∗Hφ_h − φ^∗_hHL_hφ_h

= −φ^∗_h(HL_h + (HL_h)^∗)φ_h (H represents local grid size)

⇒ – energy is conserved iff HL_h is skew-symmetric – energy dissipates iff symmetric part of H L_h is

positive definite

Channel turbulence (1)

1 2 5 10 20 50 100 200

0 4 8 12 16 20

u+ = y +

u+ = 2.5 ln y + + 5 u+

y+

DNS 4th−order 64x64x32 Kim et al (1987)

Kuroda et al (1995) Gilbert & Kleiser (1991)

Calculated mean flow

Channel turbulence (2)

Grid coarsening: 128 → 16 across channel

Turbulent statistics u^′u^′

0 1 2 3

0 10 20 30 40

DNS 4th-order 64x96x32 DNS 4th-order 64x64x32 DNS 4th-order 64x32x32 DNS 4th-order 64x16x32 DNS 2nd-order 64x64x32

Experiment Kreplin & Eckelman (1979) DNS Kim et al. (1987)

u

y rms

+

Kim et al. (1987) 128 points; now reduced to 32 or even 16!

Algorithmic gain 1000, i.e. 20 years!

Current ‘price’ for DNS

golf ball car ship skater airplane

Reynolds 10⁵ 10⁷ 10⁹

grid points 10¹⁰ 10^14.5 10¹⁹ memory (bytes) 10¹³ 10^17.5 10²²

time steps 10⁵ 10⁶ 10⁷

flops 10¹⁸ 10^23.5 10²⁹

performance (flop/s)^∗ 10¹² 10^17.5 10²³

∗ 2-week run

Enough room left for algorithmic and modelling improvement ... ;-)

Price tag for DNS

2-wk run without further numerics progress

10⁴ 10⁵ 10⁶ 10⁷ 10⁸ 10⁹

Giga Tera Peta

Exa Zeta

Reynolds number

flops/sec & bytes

golf ball

swimmer car

airplane

ship

2007

speed memory

Epilogue

Sir Horace Lamb in 1932 (then aged 83) stated:

I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum- electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic.

Computer simulation may shed some light on the latter.