Formation and propagation of shock-generated vortex rings Martin Brouillette and Christian H´ ebert Laboratoire d’ondes de choc Universit´ e de Sherbrooke Sherbrooke (Qu´ ebec) CANADA 1
Outline 1. Introduction 2. Experimental considerations 3. Vortex propagation 4. Vortex formation — Circulation standpoint 5. Other features — Shock formation by vortex 6. Summary 2
Introduction Explore compressible turbulence via the standpoint of compressible vorticity and its building blocks. Compressible vorticity is also important from both fundamental and practical standpoints, in: • Blade-vortex interaction, including sound generation, for rotary wing aircraft applications. • Shock-vortex interaction, including sound generation, for jet noise ap- plications. 3
Objectives Experimental study of isolated vortices as building blocks of compressible turbulence. For example, in the Richtmyer-Meshkov instability: Spike Bubble In particular, — What exactly are compressible vortices? — How are they different from incompressible vortical structures? 4
Specific experimental objectives • Characterize the effects of the generator on the production and propa- gation of compressible vortices. • Examine the effects of compressibility and scale on these properties. • Compare with incompressible results. 5
Experimental considerations Experiments are performed with a modified shock tube: Diaphragm 45-500mm 1.84m Adjustable� � end wall Open end Driver Driven Section � Punch Pressure� � � transducers Open driven end, with 3 different exit nozzle diameters: 6.4, 12.7 and 25.4 mm Mounting�flanges Driven Nozzle 300mm 51mm Exit 6
Features of this setup: • Produce shear-driven vortices (Kelvin-Helmoltz instability) as opposed to baroclinically-driven vortices (Rayleigh-Taylor or Richtmyer-Meshkov instabilities) • High vorticity production rates. • Fluid piston analogous to high speed spike in RMI. 7
Effect of driver length on fluid ejection history Standard (= long) driver Adjustable� Expansion waves� end wall Reflected� from shock diffraction� expansion waves� at open end from end wall� � � Diaphragm� � Position� � � � � Constant� � ejection� velocity Tube� Shock wave exit Analogous to RMI followed by RTI. 8
Tuned driver Velocity � ejection� program of � Shock and reflected� shortest duration expansions arrive at� same time Analogous to “almost only” RMI. 9
Flow diagnostics • Piezoelectric pressure transducers. • Flow visualization (shadowgraph, schlieren, holographic interferome- try). 10
Vortex propagation Three regimes of propagation Low shock Mach number → regime 1 11
Regime 1 — Development of circonferential instabilities (oblique view) Oblique spark shadowgraph, M s = 1 . 32, D p = 38 mm: t = 1 . 42 ms, x/D p = 3 . 30 12
As M s is increased: Regime 2 — Appearance of shocks Chocs 13
As M s is further increased: Regime 3 — Secondary vorticity generation Anneau secondaire Chocs Anneau principal 14
Vortex propagation (37 mm orifice) Standard driver Tuned driver Regime 1 1 < M s < 1 . 34 1 < M s < 1 . 44 Regime 2 1 . 34 < M s < 1 . 45 1 . 44 < M s < 1 . 60 Regime 3 M s > 1 . 45 M s > 1 . 60 We know that for the same shock Mach number, impulse is larger for standard driver. Regimes appear at lower Mach numbers for the standard case. 15
Vortex propagation — Position vs time Results normalized with orifice diameter D p and maximum fluid velocity U p as: x ∗ = x t ∗ = tU p D p D p 18.00 16.00 Ms=1.65 tuned Tuned 14.00 12.00 Standard 10.00 t * Ms=1.20 Dp=0.5" standard 8.00 Ms=1.30 Dp=0.5" standard Ms=1.10 Dp=1.0" standard 6.00 Ms=1.20 Dp=1.0" standard Ms=1.30 Dp=1.0" standard 4.00 Ms=1.30 Dp=1.5" tuned Ms=1.51 Dp=1.5" tuned 2.00 Ms=1.65 Dp=1.5" tuned 0.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 x * 16
Observations: • Speed of vortex rings increases with shock strength. • Rings produced with the tuned driver propagate slower U ∗ ≈ 0 . 34 than with the standard driver U ∗ ≈ 0 . 42. • Within experimental error, not possible to detect compressibility ef- fects. 17
Vortex formation In incompressible experiments, typically use a piston to eject a slug of fluid (liquid). • Ejection Mach number near zero. • Normalized ejected slug length relatively much smaller than in the present study. • Vortex propagation mostly free from the effects of the generating jet. Examine vortex formation in terms of circulation deposition his- tory: Use a normalized circulation Γ ∗ = U ∗ d ∗ 18
Normalized circulation vs normalized time 1.00 Gharib�et�al�(1998) 0.90 0.80 0.70 0.60 � � 0.50 Ms=1.20 Dp=0.5'' standard Ms=1.30 Dp=0.5'' standard 0.40 Ms=1.10 Dp=1.0'' standard 0.30 Ms=1.20 Dp=1.0'' standard Ms=1.30 Dp=1.0'' standard 0.20 Ms=1.30 Dp=1.5'' tuned 0.10 Ms=1.51 Dp=1.5'' tuned 0.00 0.00 2.00 4.00 6.00 8.00 10.00 12.00 t * 19
Observations: • Vortex ring is formed when a vorticity saturation threshold is reached. • Concept of vortex formation number (Gharib et al. 1998). • Formation number higher for compressible rings. • Maximum circulation similar between incompressible results and stan- dard driver results. • Lower circulation with tuned driver. • Non-zero “initial” circulation (purely impulsive ejection history). 20
Other features — Shock formation by vortex Onset of appearance of shock wave within recirculating region: Chocs For the standard driver, this shock appears at M s = 1 . 34 ( U p = 339 m/s). For the tuned driver, this shock appears at M s = 1 . 44 ( U p = 425 m/s). 21
This threshold is reached when flow velocities within ring recirculating region become sonic u/c = 1. But since u ∼ Γ /d this threshold occurs when: Γ d c = 1 With Γ ∼ Ud , then Γ = Γ ∗ U p D p . and this threshold can then be expressed as: Γ ∗ U p D p = 1 d c If this criterion is satisfied for both tuned and standard cases, then: Γ ∗ tuned U p tuned D p tuned = Γ ∗ std U p std D p std d tuned c tuned d std c std For identical test gases c tuned = c std , for identical orifices D p tuned = D p std and we observe that d tuned = d std . Therefore Γ ∗ = U p std tuned Γ ∗ U p tuned std is satisfied if postulate is correct! 22
Experimental data: Standard driver: Γ ∗ std = 0 . 76 Tuned driver : Γ ∗ tuned = 0 . 61 Γ ∗ tuned = 0 . 80 Γ ∗ std Standard driver: transition at M s = 1 . 34 U p std = 339 m / s Tuned driver: transition at M s = 1 . 44 U p tuned = 425 m / s. U p std = 0 . 80 U p tuned at the onset of appearance of the shock within the recirculating region. Postulate appears satisfied! • For a given size, shock appears at a given ring circulation. • The estimation of ring circulation rests on solid ground. 23
Consequences: In a compressible turbulent flow, shocklets would appear if sufficient vorticity is locally present. For a purely impulsive (delta function) fluid ejection history, since the max- imum vorticity deposition Γ ∗ is small, a shock would appear at a very large ejection velocity. Our limited experiments at M s = 2 support this. 24
Conclusions • The behavior of compressible vortices is somewhat similar to that of incompressible vortices, but they attain circulation saturation slower. • Vortex rings can only absorb a maximum amount of circulation. • The most sustained and higher vorticity production rate lead to faster normalized formation and higher circulation. • Can use this point of view to explain he appearance of shocks within vortical structures. 25
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