Lagrangian compressible dynamics in a self-similar incompressible jet
Thomas Basset
1† , Bianca Viggiano
2, Thomas Barois
3, Mathieu Gibert
4, Nicolas Mordant
5, Raúl Bayoán Cal
2, Romain Volk
1, Mickaël Bourgoin
11
Laboratoire de Physique, École Normale Supérieure de Lyon, France.
2Department of Mechanical and Materials Engineering, Portland State University, USA.
3Laboratoire Ondes et Matière d’Aquitaine, Université de Bordeaux, France.
4Institut Néel,
Université Grenoble Alpes, France.
5Laboratoire des Écoulements Géophysiques et Industriels, Université Grenoble Alpes, France.
A turbulent water jet seeded with tracers
40cm
) A turbulent round free jet...
- round nozzle with a diameter D = 4 mm
- nozzle exit velocity UJ ' 7 m/s
⇒ ReD = UJD/ν ' 2.8 × 104 ) ... seeded with tracers...
- neutrally buoyant spherical polystyrene tracers with a diameter dp = 250 µm
- specific seeding only through the nozzle
⇒ nozzle seeding ) ... tracked by cameras.
- three high-speed cameras oriented orthogonal to the brown faces (icosahedral geometry) in a back light configuration
- measurement volume spanning 100 mm ≤ z ≤ 180 mm (jet axis z and z = 0 the nozzle exit position)
- 50 movies of 8000 frames recorded at 6000 fps (inlet valve open some seconds before each recording)
Particle tracking velocimetry
0 200 400 600 800 1000 1200
800
600
400
200
30 40 50 60 70 80
à
Particle detection → Stereoscopic reconstruction [1, 2] → Tracking - Cylindrical coordinates (z, r, θ)
- Tracks convolved with a first-order derivative Gaussian kernel ⇒ velocity
⇒ ' 108 particle positions and velocities
References
[1] Machicoane et al. 2019, Rev. Sci. Instrum. 90 (3), 035112.
[2] Bourgoin & Huisman 2020, Rev. Sci. Instrum. 91 (8), 085105.
[3] Pope 2000, Turbulent Flows, Cambridge University Press.
Mean axial velocity hU (z, r )i
Mean velocity field well-known through Eulerian measurements [3]
- Centerline velocity: U0(z) = hU(z, r = 0)i - Half-width: hU(z, r = r1/2(z))i = 12U0(z)
In the self-similar region (for z & 15D = 60 mm)
U0(z)
UJ = B
(z − z0)/D r1/2(z) = S(z − z0)
⇒ Experimental results: z0 ' 4D, B = 5.3 and S = 0.105
Self-similar radial profile: f(η) = hU(z, r)i/U0(z) with η = r/(z − z0)
f(η) = e−Aη2
with A = log(2)/S2
⇒ Same axial velocity field despite the nozzle seeding
Mean radial velocity hV (z, r )i (1/2)
Self-similar radial profile: g(η) = hV (z, r)i/U0(z)
) Continuity equation for the flow velocity field: ∇· U = 0
→b η(ηf (η))0 = (ηg(η))0 [3]
⇒b g(η) = ηf (η) − 1 η
Z η
0 xf (x) dx (1)
Incorrect function (1): flow velocity field U 6= Uϕ tracer velocity field
⇒ Entrained flow not tagged by the tracers due to the nozzle seeding (higher measured radial velocity): influence of entrainment up to the core of the jet
Mean radial velocity hV (z, r )i (2/2)
) Mean tracer density: ϕ(z, r) with ϕ0(z) = ϕ(z, r = 0) Self-similar radial profile: Φ(η) = ϕ(z, r)/ϕ0(z)
φ0(z) ∝ 1 z − z0
⇒ Same behavior for Φ(η) and f(η) (except wider Φ(η))
) Continuity equation for the tracer velocity field: ∇·(ϕ Uϕ) = 0
⇒
Φ b gϕ(η) = ηfϕ(η) (2) with fϕ(η) ' f(η) from experiments
Correct function (2): smaller maximum and negative part for η & 0.2 due to some entrained tracers remaining in the tank
⇒ A new model coherent with experimental results
⇒ g(η) = gϕ(η) − 1 η
Z η
0 xf(x) dx
| {z }
entrainment term
(3)
A diffusive model
Advection-diffusion equation: ∇·(ϕ U − KT∇ϕ) = 0
with KT the turbulent diffusion coefficient, KcT = KT/(U0r1/2) normalized To be coherent with (3) ⇒
b
KcT(η) = − 1 S
Φ(η) Φ0(η)
1 η
Z η
0 xf(x) dx
⇒ Link KT to the entrainment term and the “compressibility” Φ0(η)/Φ(η) KT linked to the known turbulent viscosity νT through the turbulent Prandtl number σT = νT/KT ' 0.6
⇒ Coherent with scalar transport values [3]
Conclusion
Lagrangian tracer velocity field ∇· Uϕ 6= 0 due to the nozzle seeding but
∇·(ϕ Uϕ) = 0 and ∇·(ϕ U −KT∇ϕ) = 0 with the new turbulent diffusion coefficient KT linked with entrainment
