Wake of oblate spheroids and flat cylinders

M. Chrust, G. Bouchet and J. Dusek

Institut de Mecanique des Fluides et des Solides, Universite de Strasbourg/CNRS,France

Parametric study of the transition scenario in the wake of oblate spheroidsand flat cylinders

Introduction Numerical Method Results Conclusions

Summary

1 IntroductionGoal statementTransition scenario of a sphere and a flat disk

2 Numerical MethodMathematical FormulationNumerical MethodComputational domainBoundary conditions

3 ResultsOblate spheroidsFlat cylinders

4 Conclusions

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Summary

1 IntroductionGoal statementTransition scenario of a sphere and a flat disk

2 Numerical MethodMathematical FormulationNumerical MethodComputational domainBoundary conditions

3 ResultsOblate spheroidsFlat cylinders

4 Conclusions

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Summary

1 IntroductionGoal statementTransition scenario of a sphere and a flat disk

2 Numerical MethodMathematical FormulationNumerical MethodComputational domainBoundary conditions

3 ResultsOblate spheroidsFlat cylinders

4 Conclusions

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Summary

1 IntroductionGoal statementTransition scenario of a sphere and a flat disk

2 Numerical MethodMathematical FormulationNumerical MethodComputational domainBoundary conditions

3 ResultsOblate spheroidsFlat cylinders

4 Conclusions

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Summary

1 IntroductionGoal statementTransition scenario of a sphere and a flat disk

2 Numerical Method

3 Results

4 Conclusions

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Goal statement

Introduction

Study :

flow around oblate spheroids and flat cylinders.Goal :

establishment of a link between transition scenarios of thatof a sphere and a flat diskAplications :

tool allowing to access the expected asymptotic state forany numerical or experimental configuration involvingoblate spheroids or flat disks

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Transition scenario of a sphere and a flat disk

Transition scenario of a sphere and an infinitel flat diskSphere

Flat disk

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Transition scenario of a sphere and a flat disk

Sphere

Steady axisymmetric (A)

Characteristics

steadyaxisymmetricthe only non-zero mode is m=0lift is equal to zero

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Transition scenario of a sphere and a flat disk

SphereSteady non-axisymmteric (a)

Visualisation

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Transition scenario of a sphere and a flat disk

SpherePeriodic state with planar symmetry (c)

Visualisation

(c), χ = 1.25, Re = 268

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Transition scenario of a sphere and a flat disk

SphereQuasi-periodic state (f)

Visualisation

χ = 1.25, Re = 268

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Transition scenario of a sphere and a flat disk

SphereChaotic states (g)

Visualisation

(g), χ = 1.5, Re = 310

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Transition scenario of a sphere and a flat disk

Flat diskPeriodic state without planar symmetry (b)

Visualisation

χ = 6, Re = 145

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Transition scenario of a sphere and a flat disk

Flat diskPeriodic state with a zero mean lift (d)

Visualisation

χ = 6, Re = 183

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Summary

1 Introduction

2 Numerical MethodMathematical FormulationNumerical MethodComputational domainBoundary conditions

3 Results

4 Conclusions

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Mathematical Formulation

3-d Navier Stokes equations are non-dimensionalized withrespect to the inflow velocity U and the diameter of thetransverse cross section dsolved in a cylindrical coordinate system (z, r , θ)Re = Ud/ν ; χ = d/a where d is the transverse diameterand a the length of the streamwise axis of the spheroid

∇ · v = 0 (1)

∂v∂t

+ (v · ∇) · v = −∇p +1

Re∇2v (2)

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Numerical Method

Discretization

the equations are discretized using thespectral-spectral-element discretizationFourier expansion in the azimuthal directionspectral-element discretization in the (z, r)-plane by Kstandard high-order Lagrangian finite elements with NGauss-Lobato-Legendre collocation points in eachdirection within elementtime integration by semi-implicit method with a fully explicitthird order accurate treatment of the non-linear couplingterms and with a full inversion of the Stokes problem in theindividual azimuthal eigen-subspaces.

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Computational domain

Discretization

FIGURE: Spectral element discretization of the computational domainof an ellipsoid of χ=2.

215-241 spectral elementsLin = 12d ,Lout = 25d ,R = 8d

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Computational domain

Mesh optimzation

Re N=6 N=8Re1 117.17 116.92Re2 125.18 (4) 125.15 (6) 125.12 (4) 125.11 (6)

TABLE: Mesh test results for a flat disk. The values in brackets indicate thenumber of azimuthal modes of a Fourier expansion.

influence of the number of azimuthal modes m on thesecond instability treshold, m=4number of collocation points, N=6

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Boundary conditions

Boundary conditions

inflow on the upstream of the cylindrical domain ; uniformvelocity profile u=(U∞, v = 0,w = 0)

outflow on the lateral and downstream of the domain ;pressure and stresses set to zerono-slip conditions on oblate spheroids/diskssymmetry axis boundary condition using the complexvelocity components of Orszag and Patera

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Summary

1 Introduction

2 Numerical Method

3 ResultsOblate spheroidsFlat cylinders

4 Conclusions

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Oblate spheroids

State diagram - Oblate spherodis

FIGURE: State diagram for oblate spheroids. A stands for axisymmetric state. Thefilled triangles (c), and diamonds (e) represent pre-chaotic states with a subharmonicmodulation. The narrow filled band represents the domain of bi-stability at thesubcritical bifurcation.

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Oblate spheroids

Exemplary calculataion

FIGURE: Projection of the lift coeffitient onto the perpendicular direction. χ =∞,Re = 126.

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Oblate spheroids

State with a non-zero helicity (e)

Characteristicsunsteadyunequal amplitudes ofspiral modes(appearing in theweakly non-linearanalysishelicity yields an ellipticpath of the liftslowlyoscillating/rotatingellipse

Visualisation

χ = 1.25, Re = 283

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Oblate spheroids

Bistability

FIGURE: Oscillation amplitude ∆CL of the lift coeficient as a function of theReynolds number for oblate spheroids of χ =∞, 6, 3 and a flat cylinder of χ = 6 (seethe legend). The stability interval of the steady non axisymmetric state is representedby dashed lines along the horizontal axis. Their linear instability thresholds are plottedas empty circles.

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Oblate spheroids

Subcritical bifurcation

FIGURE: Logarithmic plot of the amplitude of the oscillation of the lift coefficient of aflat disk (χ =∞) at Re = 126 (full line) compared to the purely exponential growth(dashed line). Note the super-exponential growth in the time interval t ∈ [1950, 2350]time units.

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Oblate spheroids

Bistability tresholds

spheroids cylindersχ Re′2 Re2 χ Re′2 Re2∞ [124] 125.2 ∞ [124] 125.26 [136] 137.7 6 [148] 150.13 [154] 155.7 4 [164] 165.6

TABLE: Linear stability thresholds Re2 and lower bounds of bi-stability Re′2(Re′2 < Re2) of the subcritical Hopf bifurcation for oblate spheroids and flat cylinders.(The values in brackets are closest integer upper bounds.)

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Flat cylinders

Comparison of the results with literature

χ Re1 Re2 Re3 Re4 Re5 Re6∞ (1) ≈115 ≈121 ≈140∞ (2) 116.9 125.3 143.7∗∞ (3) 116.92 ([124],125.2) (142,143) (165,170)10 (4) 135 155 172 28010 (3) 129.6 ([136],138.7) 154.4 188.83 (5) ≈159.4 ≈179.8 (184,185) (190,191) ≈215 ≈2403 (3) 159.65 (181,182) (185,190) (195,198) (220,230) (235,240)

TABLE: Bifurcation thresholds. Numbers in brackets indicate authors : (1) Fabre et al.,(2) Meliga et al., (3) present study, (4) Shenoy and Kleinstreuer, (5) Auguste et al. ∗result obtained using asymptotic expansion. At χ = 10, the values of the present studyare obtained by interpolation between χ =∞ and χ = 6.

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Flat cylinders

Flat cylinders - bibliographic data

FIGURE: State diagram for cylinders. Bibliographic data.

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Flat cylinders

Flat cylinders

FIGURE: . State diagram for cylinders. The band of bi-stability at the subcriticalbifurcation is still present for 1/χ ≤ 0.25.

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Summary

1 Introduction

2 Numerical Method

3 Results

4 Conclusions

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Recapitulation

1 The parameter space (χ,Re) has been explored for oblatespheroids and flat disks

2 The secondary bifurcation is subcritical for 1/χ ≤ 0.45 and1/χ ≤ 0.25 for oblate spheroids and flat disks respectively

3 The scenario of a flat disk and sphere are separated by theexistence of the state with non-zero helicity

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Future work

Investigation of the transition scenario of freely falling disksand spheroids

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek

Introduction Numerical Method Results Conclusions

Merci

Wake of oblate spheroids and flat cylinders M. Chrust, G. Bouchet and J. Dusek