One-Dimensional Turbulence  ·  Turbulent Round Jet (Ma = 0.9)

Kerstein, JFM 392 (1999)  ·  Sharma, Klein & Schmidt, Phys. Fluids 34, 085134 (2022)
Jet evolution
Radial profile
Triplet map
Energy spectrum
x/D = 1.0 NOZZLE EXIT · Potential core — coherent shear layer Eddy events: 0  |  t* = 0.000
JET EVOLUTION (x-r spacetime)
r/D horizontal · x/D vertical
colour = u/U_j   gold lines = b½

Parameters

Eddies fire automatically every frame via Poisson process. Button triggers one immediately.

Live statistics

x/D
1.0
U₃ / Uⱼ
1.000
b½ / D
0.094
urms / U₃
0.000
Eddy events
0
η₃ / D
—
Jet nozzle exit. Potential core — coherent shear layer, turbulence developing at lip line.

ODT model equations

① Diffusion — deterministic, runs between every eddy event:
∂uᵢ/∂t = ν ∂²uᵢ/∂y²
ν = 1/ReD. Smooths gradients; competes against eddy stirring. Sets Kolmogorov scale η below which eddies are suppressed by Z.
② Triplet map — stochastic eddy event on [y₀, y₀+l]:
f(y) = y₀ + { 3(y−y₀),  y−y₀ ∈ [0, l/3]
          2l−3(y−y₀), [l/3, 2l/3]
          3(y−y₀)−2l, [2l/3, l]
Compresses to ⅓, pastes 3 copies, flips centre. Measure-preserving ⇒ scale-local energy cascade (Kerstein JFM 1999 §2).
③ Eddy rate λ(l, y₀) — Poisson process:
1/τ = √[ (1/l⁶)·uⁿ² − Zν²/l⁴ ]
uⁿ = ∫ u(f(y))·K(y) dy is the kernel-weighted velocity (dim: U·L²). K(y) = y − f(y) is the triplet-map displacement kernel — piecewise linear, zero at endpoints, integrates to zero. Only eddies with real τ⁻¹ are accepted. Z suppresses sub-Kolmogorov eddies.
Full rate: λ = C/l² · 1/τ (Kerstein JFM 1999 Eq.2.14; Klein et al. 2019 Eq.9)
④ Large-eddy suppression (LS) [Sharma PoF 2022 §II.D]:
Reject if l > β‑LS · x  (β‑LS = 0.5)
Prevents unphysically large eddies in the free jet. Controls jet spreading rate. Essential for free-shear ODT; without it spreading is over-predicted.
⑤ Advection — ODT line moves downstream:
dx/dt = Ubulk(x)  ⟹  x/D grows
U₃/Uⱼ = 6D/x  ·  b½/D = 0.094·x/D
Self-similar scalings (Panchapakesan & Lumley 1993). Profile shape and eddy statistics evolve continuously as the line advects downstream.

PoF 2022 context

Ma=0.9 unheated round jet, finite co-flow. ODT results:
  • ✦ U₃ ∝ (x/D)⁻¹ reproduced without closure
  • ✦ urms/U₃ → 0.18–0.22 on centreline
  • ✦ E(k) ∼ k⁻⁵⁄³ inertial subrange
  • ✦ ReD-independence of self-similar statistics
Phys. Fluids 34, 085134 (2022)
Sharma, Klein & Schmidt
doi:10.1063/5.0101270 ↗

Legend

Instantaneous u(r, x)
Running mean Ū(r)
Eddy event / b½ envelope
Gaussian self-similar ref.
Velocity (low → high)