DefenceSeminar

NUMERICAL SIMULATION AND ANALYSIS OF
LOW REYNOLDS NUMBER FLOW PAST
TWO SIDE BY SIDE CYLINDERS
Supradeepan K
under the guidance of
Dr. Arnab Roy
Department of Aerospace Engineering,
Indian Institute of Technology, Kharagpur.
4th March, 2015
Supradeepan K (10AE90R05) Defence Seminar 4th
March, 2015 1 / 72

Overview
Overview
Introduction.
Governing equations.
Objective and scope of present research work.
Initial and boundary conditions.
Results and discussion.
Conclusions.
Publications.
March, 2015 2 / 72

Introduction
Introduction
Flow past 2D and 3D geometies have real time applications
Airfoils, Wings,
Earth fixed structures,
Heat exchangers, Cooling in electronic devices etc.
Analytical solutions of Navier-Stokes equations involve lot of simplifying
assumptions.
Exact solutions are available only for a few selected flow problems.
Numerical solution of these equations have attracted the attention of
researchers.
Large number of numerical solvers have come up for solving a variety of flow
problems using these equations.
CFRUNS is one such solver for incompressible flows.
This presentation is about the developments and applications of CFRUNS.
March, 2015 3 / 72

Governing Equations
Governing Equations
The non-dimensional, conservative form of governing equations in primitive
variables for two dimensional incompressible viscous ﬂow without body forces
are
Continuity Equation:
∂u
∂x + ∂v
∂y = 0
X - Momentum Equation:
∂u
∂t + ∂u2
∂x + ∂uv
∂y = −∂p
∂x + 1
Re
∂2u
∂x2 + ∂2u
∂y2
Y - Momentum Equation:
∂v
∂t + ∂uv
∂x + ∂v2
∂y = −∂p
∂y + 1
Re
∂2v
∂x2 + ∂2v
∂y2
March, 2015 4 / 72

Objective and Scope of the work
Objective and Scope of the work
1 Development and improvements of CFRUNS
2 Application of the solver in 2D ﬂow problems
Flow past two side by side circular cylinders.
Flow past two side by side rotating cylinders.
Flow past two side by side rotationally oscillating cylinders.
Flow past two side by side triangular cylinders.
March, 2015 5 / 72

Brief literature review NS Solver
Review on existing work in CFRUNS
Development of Navier-Stokes Solver
CFR Roy and Bandyopadhyay [2006]
CFRUNS Harichandan and Roy [2010]
Improvements in Derivative calculation
Existing CFR and CFRUNS −→ Taylor Series based
Least Square based gradient reconstruction Mavriplis [June 2003]
Improvements in Temporal Accuracy
Existing CFR and CFRUNS −→ Euler interpolation
Adams-Bashforth second order
Improvements in implementation of boundary condition
Existing CFRUNS −→ Dirichlet and Neumann
Convective outﬂow boundary condition Orlanski [1976]
March, 2015 6 / 72

CFRUNS Numerical Scheme
Numerical Scheme
Consistent Flux Reconstrution for Unstructured Grids (CFRUNS)
Two dimensional
Incompressible solver
Primitive variable formulation
Unstructured collocated mesh
Explicit algorithm
Finite volume discretization
Pressure is calculated from pressure Poisson equation
Improved CFRUNS is a second order spatio-temporal accurate scheme
March, 2015 7 / 72

CFRUNS Initial and B.C’s
Initial and Boundary Conditions
Initial Conditions
u = u∞ = 1.0
v = v∞ = 0.0
p = p∞ = 0.0 Figure 1: Computational
Domain.
Boundary Conditions
Boundary u v p
Inﬂow u = u∞ v = 0.0 ∂p
∂x = 0.0
Outﬂow ∂u
∂t + Uc
∂u
∂x = 0 ∂v
∂t + Uc
∂v
∂x = 0 p = 0.0
Top and Bottom ∂u
∂y = 0 v = 0.0 ∂p
∂y = 0
On the Body u = 0.0 v = 0.0 ∂p
∂n = 0.0
March, 2015 8 / 72

CFRUNS Validation
Validation of CFRUNS
Lid driven cavity.
Unconfined flow past circular cylinder.
Unconfined flow past square cylinder.
March, 2015 9 / 72

CFRUNS Validation
Lid driven Cavity
Figure 2: x-component of
velocity (u) at the mid vertical
plane at steady state
Figure 3: y-component of
velocity (v) at the mid horizontal
plane at steady state
March, 2015 10 / 72

CFRUNS Validation
Flow past circular cylinder at Re = 100
Figure 4: Streamlines and vorticity contours for ﬂow past a circular cylinder.
Figure 5: Time history of force coeﬃcients for a circular cylinder.
March, 2015 11 / 72

CFRUNS Validation
Flow past circular cylinder
Table 1: Lift, drag coefficient and Strouhal number for flow around a single
circular cylinder at Re = 100 and 200.
Lift coefficient Drag coefficient Strouhal Number
(Cl) (Cd) (St)
Re 100 Re 200 Re 100 Re 200 Re 100 Re 200
Braza et al. [1986] ± 0.25 ± 0.75 1.366 ± 0.015 1.40 ± 0.05 0.160 0.200
Meneghini et al. [2001] - - 1.37 ± 0.010 1.30 ± 0.05 0.165 0.196
Ding et al. [2007] ± 0.287 ± 0.659 1.356 ± 0.010 1.38 ± 0.05 0.166 0.196
Harichandan and Roy [2010] ± 0.278 ± 0.602 1.352 ± 0.010 1.352 ± 0.010 0.161 0.192
Present result ± 0.275 ± 0.652 1.360 ± 0.010 1.42 ± 0.05 0.165 0.198
March, 2015 12 / 72

CFRUNS Validation
Flow past square cylinder at Re = 100
Figure 6: Streamlines and vorticity contours for flow past a square cylinder.
Table 2: Time averaged drag coefficient and Strouhal number for flow around a
single square cylinder.
Drag coefficient (Cd) Strouhal Number (St)
Franke et al. [1990] 1.61 0.154
Davis and Moore [1982] 1.63 0.15
Robichaux et al. [1999] 1.53 0.154
Present result 1.65 0.153
March, 2015 13 / 72

Tools used
Tools used
Vorticity contours.
Force coeﬃcients.
λ2 criterion.
Proper Orthogonal Decomposition (POD).
Instantaneous streamwise normal stress.
March, 2015 14 / 72

Flow past two side by side circular cylinders Literatures on circular cylinders
Review of existing works on circular cylinders
Single Cylinder
Near wake region of the cylinder Bloor [1964]
Experiments to ﬁnd the distribution of velocity and pressure Nishioka
and Sato [1974]
Variety of problems were attempted for Re between 100 and 300
Harlow et al. [1965], Patankar and Spalding [1972], Hirt et al. [1975],
Braza et al. [1986], Breuer [1998]
Two side by side Cylinder
Numerical and experimental investigation has been reported by
Bearman and Wadcock [1973], Zdravkovich [1977], Williamson
[1985], Chang and Song [1990], Kang [2003], Sumner et al. [2005],
Inoue et al. [2006], Ding et al. [2007], Liu et al. [2007], Xu et al.
[2003], Yoon and Yang [2009],
Existing research focuses on force coeﬃcients and various wake patterns.
March, 2015 15 / 72

Flow past two side by side circular cylinders Computational Domain
Computational Domain
Figure 7: Computational Domain.
Reynolds number of the ﬂow (Re = 100).
Centre to centre distance between the cylinders (T) [1.1 D - 8.0 D].
March, 2015 16 / 72

Flow past two side by side circular cylinders Single bluﬀ body periodic regime
Single bluﬀ body periodic regime 1.1 D ≤ T ≤ 1.3 D
Figure 8: Contours of vorticity, λ2 and instantaneous stream wise normal stress.
Figure 9: First and third POD modes.
March, 2015 17 / 72

Flow past two side by side circular cylinders Single bluff body periodic regime
Single bluff body periodic regime 1.1 D ≤ T ≤ 1.3 D
Figure 10: History of force coefficients.
Figure 11: Phase portraits.
March, 2015 18 / 72

Flow past two side by side circular cylinders Aperiodic regime
Aperiodic regime 1.4 D ≤ T ≤ 2.2 D
March, 2015 19 / 72

Flow past two side by side circular cylinders Aperiodic regime
Aperiodic regime 1.4 D ≤ T ≤ 2.2 D
Figure 14: History of lift coeﬃcients.
March, 2015 20 / 72

Flow past two side by side circular cylinders Anti-phase Synchronised Regime
Anti-phase Synchronised Regime 3.2 D ≤ T ≤ 7.9 D
March, 2015 21 / 72

Flow past two side by side circular cylinders Anti-phase Synchronised Regime
Anti-phase Synchronised Regime 3.2 D ≤ T ≤ 7.9 D
March, 2015 22 / 72

Flow past two side by side circular cylinders In-phase Synchronised Regime
In-phase Synchronised Regime T ≥ 8.0 D
March, 2015 23 / 72

Flow past two side by side circular cylinders In-phase Synchronised Regime
In-phase Synchronised Regime T ≥ 8.0 D
March, 2015 24 / 72

Flow past two side by side circular cylinders Transformation Regime
Transformation Regime 2.3 D ≤ T ≤ 3.1 D
T = 3.1 D
Figure 24: Vorticity contours.
Figure 25: Contours of λ2 and instantaneous stream wise normal stress.
March, 2015 25 / 72

Figure 26: History of drag coeﬃcients for T = 2.7 D
Figure 27: History of drag coeﬃcients for T = 3.1 D
March, 2015 26 / 72

Figure 28: Phase portraits 40<t<200.
Figure 29: Phase portraits 200<t<400.
March, 2015 27 / 72

Flow past two side by side rotating cylinders Review of Literature
Review of existing works on rotating cylinders
Single Cylinder
Lift coeﬃcients increase with increasing rotational velocity Townsend
[1980]
Strouhal number increases with increasing Re for the rotating cylinder
Badr et al. [1989]
Steady solutions at Re = 60 and 100 for rotating cylinders Tang and
Ingham [1991]
Tokumaru and Dimotakis [1993], Chen et al. [1993], Hu et al. [1996], Kang et al.
[1999], Mittal [2001a], Mittal [2001b], Mittal [2003], Padrino and Joseph [2006]
Two side by side Cylinder
Some earlier investigations Ueda et al. [2003], Yoon et al. [2007], Guo et al.
[2009], Yoon et al. [2009], Chan and Jameson [2010], Kumar et al. [2011].
March, 2015 28 / 72

Flow past two side by side rotating cylinders Validation of single rotating cylinder
Validation of single rotating cylinder at Re = 100
Figure 30: Variation of time averaged force coeﬃcients with rotation speed ratio
for single rotating cylinder.
March, 2015 29 / 72

Flow past two side by side rotating cylinders Parameters governing the ﬂow
Parameters governing the ﬂow
Centre to centre distance between the cylinders (T) [1.1 D - 3.5 D].
Rotational speed ratio α [0, 0.5, 1.0, 1.25].
Direction of rotation.
Figure 31: Direction of rotation.
March, 2015 30 / 72

Flow past two side by side rotating cylinders Zones of various regimes
Zones of various regimes
Figure 32: Zones of various
regimes
A −→ Single bluﬀbody
periodic regime
B −→ Aperiodic regime
C −→ Steady state regime
D −→ Periodic oscillation
with unstable wake
E −→ Periodic oscillation
with constant amplitude
F −→ Periodic oscillation
with amplitude modulation
G −→ Anti-phase
synchronised regime
H −→ Transform regime
March, 2015 31 / 72

Flow past two side by side rotating cylinders Single bluﬀbody periodic regime
Regimes discussed in the previous section
Single bluﬀbody periodic Regime [Zone A]
T = 1.1 D to 1.3 D for α = 0.5 and 1.0
Aperiodic Regime [Zone B]
T = 1.5 D to T = 2.4 D for α = 0.5
T = 2.3 D for α = 1.0
Transformation Regime [Zone H]
T = 2.5 D to T = 3.4 D for α = 0.5
Anti-phase Synchronised Regime [Zone G]
T ≥ 3.5 D for α = 0.5 and 1.0
T ≥ 2.6 D for α = 1.25
March, 2015 32 / 72

Flow past two side by side rotating cylinders Steady state regime (Zone C)
Steady state regime (Zone C)
Figure 33: Contours of vorticity and λ2 for T= 1.5 D and α = 1.0.
Figure 34: History of force coeﬃcients for T= 1.5 D and α = 1.0.
March, 2015 33 / 72

Flow past two side by side rotating cylinders Periodic oscillation due to unstable wake (Zone D)
Periodic oscillation due to unstable wake (Zone D)
March, 2015 34 / 72

Flow past two side by side rotating cylinders Periodic oscillation with constant amplitude (Zone E)
Periodic oscillation with constant amplitude (Zone
E)
March, 2015 35 / 72

Flow past two side by side rotating cylinders Periodic oscillation with amplitude modulation (Zone F)
Periodic oscillation with amplitude modulation
(Zone F)
March, 2015 36 / 72

Flow past two side by side rotationally oscillating cylinders Review of Literature
Review of existing works on oscillating cylinder
Drag reduction and synchronisation of wake patterns Chou [1997]
Lock-on regime at Re = 110 Baek and Sung [1998]
Lock-on phenomenon occurs within a band of frequency that
encompasses the natural frequency Mahfouz and Badr [1999]
Identiﬁed four diﬀerent modes Tokumaru and Dimotakis [1993]
Lu and Sato [1996], Mahfouz and Badr [1999], Baek et al. [2001], Cheng et al.
[2001,b], Choi et al. [2002], Lu [2002], Fujisawa et al. [2005], Al-Mdallal and
Kocabiyik [2006], Lee and Lee [2006].
March, 2015 37 / 72

Flow past two side by side rotationally oscillating cylinders Validation
Validation parameters
Modes listed by Choi et al. [2002]
Mode 1: Umax = 2.0, Stf = 0.165.
Mode 2: Umax = 2.0, Stf = 0.4.
Mode 3: Umax = 2.0, Stf = 0.8.
Mode 4: Umax = 0.6, Stf = 0.8.
March, 2015 38 / 72

Flow past two side by side rotationally oscillating cylinders Validation
Validation
Figure 41: Vorticity contours for
Mode1
Mode2
Mode3
Mode4
March, 2015 39 / 72

Flow past two side by side rotationally oscillating cylinders Parameters governing the flow
Parameters governing the flow
Centre to centre distance between the cylinders (T)[1.2 D and 1.5 D].
Maximum rotational velocity Umax .
Frequency of oscillation (Stf ).
Phase difference between oscillation (ϕ).
March, 2015 40 / 72

Flow past two side by side rotationally oscillating cylinders Proximity eﬀect on two Mode1 cylinders
Proximity eﬀect on two Mode1 cylinders ϕ = 0
Figure 45: Vorticity contours for T = 1.2 and 1.5 D
Figure 46: Cl for History for T = 1.2 D and 1.5 D
March, 2015 41 / 72

Proximity eﬀect on two Mode1 cylinders ϕ = π
Figure 48: Cl for history for T = 1.2 D and 1.5 D
March, 2015 42 / 72

Flow past two side by side rotationally oscillating cylinders Proximity eﬀect on two Mode2 and Mode3 cylinders
Proximity eﬀect on two mode2 and mode3 cylinders,
T = 1.2 D
Figure 49: Vorticity contours for Mode 2 and Mode 3 cylinders
Figure 50: Cl for history for Mode 2 and Mode 3 cylinders
March, 2015 43 / 72

Flow past two side by side rotationally oscillating cylinders Proximity eﬀect on two Mode2 and Mode3 cylinders
Proximity eﬀect on two mode2 and mode3 cylinders,
T = 1.5 D ϕ = 0
Figure 51: Vorticity contours for Mode 2 and Mode 3 cylinders
Figure 52: Cl for history for Mode 2 cylinders and ϕ = π
March, 2015 44 / 72

Proximity eﬀect on two Mode4 cylinders, ϕ = 0
Figure 54: Cl for history for T = 1.2 D and 1.5 D
March, 2015 45 / 72

Flow past two side by side triangular cylinders Review of Literature
Review of existing works on onset of vortex shedding
Onset of vortex shedding by FEM Jackson [1987]
contribution of pressure and viscous forces on the drag coeﬃcient
near the onset of vortex shedding Henderson [1995]
Onset of vortex shedding of a square cylinder by linear stability
analysis Kelkar and Patankar [1992]
Onset of vortex shedding of a triangular cylinder by global mode
analysis Zielinska and Wesfreid [1995] and De and Dalal [2006a]
March, 2015 46 / 72

Flow past two side by side triangular cylinders Validation
Validation
Table 3: Drag coefficient and Strouhal number for flow past a triangular cylinder
at Re = 50, 100 and 150.
Drag coefficient Strouhal Number
(Cd) (St)
Re 50 Re 100 Re 150 Re 50 Re 100 Re 150
De and Dalal [2006b] 1.5420 1.7607 1.8750 0.1505 0.1982 0.2015
Dhiman and Shyam [2011] 1.5257 1.7316 1.8937 0.1455 0.1916 0.2041
Chatterjee and Mondal [2012] 1.5334 1.7546 1.9037 0.1515 0.1968 0.2029
Present result 1.5395 1.7489 1.8901 0.1532 0.1979 0.2030
March, 2015 47 / 72

Global mode Analysis Validation
Figure 55: Amplitude of oscillations for
u and v components of velocity at
y = 0.
Figure 56: Normalised global modes for
y = 0.
March, 2015 48 / 72

ReCr for Triangular cylinder
Figure 57: Re vs Amax for u and A2
max
for v.
Table 4: ReCr for ﬂow past a triangular
cylinder.
ReCr
De and Dalal [2006a] 39.9
Duˇsek et al. [1994] 39.6
Zielinska and Wesfreid [1995] 38.3
Present result 39.71
March, 2015 49 / 72

ReCr for circular cylinder
Figure 58: Amplitude of oscillations for
y = 0.
Figure 59: Re vs Amax for u and A2
max
for v.
Present result 47.72
March, 2015 50 / 72

Flow past two side by side triangular cylinders Two side by side triangular cylinders
Two side by side triangular cylinders
Figure 60: C1.
Figure 61: C2.
Figure 62: C3.
March, 2015 51 / 72

Re = 100, G = 0.2 D
Figure 63: Vorticity and λ2 contours for conﬁgurations C1 and C2
March, 2015 52 / 72

Re = 100, G = 0.2 D
Figure 64: First POD mode for conﬁgurations C1 and C2
March, 2015 53 / 72

Re = 100
Figure 65: Vorticity and λ2 contours for conﬁgurations C1 and C2, G = 0.4 D
Figure 66: Vorticity and λ2 contours for conﬁguration C3, G = 0.8 D
March, 2015 54 / 72

Re = 100
Figure 67: First POD mode for configurations C1 and C2, G = 0.4 D
Figure 68: Energy content of POD modes for periodic and aperiodic flow of
streamwise velocity field.
March, 2015 55 / 72

Flow past two side by side triangular cylinders Onset of Vortex Shedding
Conﬁguration C1
Figure 69: Amplitude of oscillations for v component of velocity at y = 0, G =
0.2 D and 0.4 D
Figure 70: Re vs A2
max for v for G = 0.2 D and 0.4 D
March, 2015 56 / 72

Conﬁguration C2
Figure 71: Amplitude of oscillations for v component of velocity at y = 0, G =
0.2 D and 0.4 D
Figure 72: Re vs A2
max for v for G = 0.2 D and 0.4 D
March, 2015 57 / 72

ReCr
Table 5: Critical Reynolds (Recr ) number for different configurations
Configuration C1 Configuration C2
G = 0.2 D 14.53 18.35
G = 0.4 D 22.80 31.86
March, 2015 58 / 72

Flow past two side by side triangular cylinders Steady Flow
Re = 13
Figure 73: Streamlines for configuration C1,G = 0.2 D and 0.4 D.
Figure 74: Streamlines for configuration C2, G = 0.2 D and 0.4 D.
Figure 75: Streamlines for configuration C3 and G 0.8 D for Re = 13.
March, 2015 59 / 72

Conclusions Conclusions
Conclusions
1 A second order spatio-temporal accurate Navier-stokes solver is
developed based on CFRUNS with impovements.
2 The above solver is used to
Characterise the flow past two side by side circular cylinders
Characterise the flow past two side by side rotating circular cylinders
Study the lock on characteristics of the flow past two side by side
rotationally oscillating cylinders
Determine the onset of vortex shedding for flow past two side by side
triangular cylinders of various configurations.
March, 2015 60 / 72

Publications Publications
Publications
1 Supradeepan, K., and Roy, A., 2014. Characterisation and analysis of flow
over two side by side cylinders for different gaps at low Reynolds number: A
numerical approach. Physics of Fluids (1994-present) 26(6).
2 Supradeepan, K., and Roy, A. Low Reynolds number flow characteristics for
two side by side rotating cylinders. ASME Journal of Fluids Engineering
(Accepted).
3 Supradeepan, K., and Roy, A., 2014. Numerical study on low Reynolds
number flow past two side by side triangular cylinders. Proceedings of 44th
AIAA Fluid Dynamics Conference, Atlanta.
4 Supradeepan, K., and Roy, A., 2013. Numerical investigation of convective
heat transfer from a hot wall assisted by a vortex generator geometry.
Proceedings of 22nd
National and 11th ISHMT-ASME Heat and Mass
Transfer Conference, IIT Kharagpur.
5 Supradeepan, K., and Roy, A., 2013. Flow past two side by side rotationally
oscillating cylinders at low Reynolds number. ICTACEM 2014, IIT
Kharagpur.
March, 2015 61 / 72

Future scope of research Future scope of research
Future scope of research
1 Three dimensional version of the present solver may be developed and applied
to study flow past various geometries like three dimensional cylinders, spheres
etc.
2 In order to enhance the scope of the solver and study real life problems, it
needs to be equipped to solve high Reynolds number turbulent flows. For
this, a suitable turbulence model as well as higher order upwind scheme for
convective terms would have to be incorporated.
3 Implement more features in the solver to enhance its versatility, for example,
by introducing suitable source terms to solve buoyancy driven flows, flows of
electrically and magnetically conducting fluid under an applied field, etc.
March, 2015 62 / 72

Thank You
March, 2015 63 / 72

References
Al-Mdallal, Q. M., Kocabiyik, S., 2006. Rotational oscillations of a
cylinder in cross-flow. International Journal of Computational Fluid
Dynamics 20 (5), 293–299.
Badr, H., Dennis, S., Young, P., 1989. Steady and unsteady flow past a
rotating circular cylinder at low Reynolds numbers. Computers & Fluids
17 (4), 579 – 609.
Baek, S.-J., Lee, S. B., Sung, H. J., 9 2001. Response of a circular
cylinder wake to superharmonic excitation. Journal of Fluid Mechanics
442, 67–88.
Baek, S.-J., Sung, H. J., 1998. Numerical simulation of the flow behind a
rotary oscillating circular cylinder. Physics of Fluids (1994-present)
10 (4), 869–876.
Bearman, P. W., Wadcock, A. J., 10 1973. The interaction between a pair
of circular cylinders normal to a stream. Journal of Fluid Mechanics 61,
499–511.
Bloor, M. S., 6 1964. The transition to turbulence in the wake of a circular
cylinder. Journal of Fluid Mechanics 19, 290–304.
March, 2015 63 / 72

References
Braza, M., Chassaing, P., Minh, H. H., 3 1986. Numerical study and
physical analysis of the pressure and velocity fields in the near wake of a
circular cylinder. Journal of Fluid Mechanics 165, 79–130.
Breuer, M., 1998. Large eddy simulation of the subcritical flow past a
circular cylinder: numerical and modeling aspects. International Journal
for Numerical Methods in Fluids 28 (9), 1281–1302.
Chan, A. S., Jameson, A., 2010. Suppression of the unsteady vortex wakes
of a circular cylinder pair by a doublet-like counter-rotation.
International Journal for Numerical Methods in Fluids 63 (1), 22–39.
Chang, K.-S., Song, C.-J., 1990. Interactive vortex shedding from a pair of
circular cylinders in a transverse arrangement. International Journal for
Numerical Methods in Fluids 11 (3), 317–329.
Chatterjee, D., Mondal, B., 2012. Forced convection heat transfer from an
equilateral triangular cylinder at low reynolds numbers. Heat and Mass
Transfer/Waerme- und Stoffuebertragung 48 (9), 1575–1587.
Chen, Y. M., Ou, Y. R., Pearlstein, A. J., 8 1993. Development of the
wake behind a circular cylinder impulsively started into rotatory and
rectilinear motion. Journal of Fluid Mechanics 253, 449–484.
March, 2015 63 / 72

References
Cheng, M., Chew, Y., Luo, S., 2001b. Numerical investigation of a
rotationally oscillating cylinder in mean flow. Journal of Fluids and
Structures 15 (7), 981 – 1007.
Cheng, M., Liu, G., Lam, K., 2001. Numerical simulation of flow past a
rotationally oscillating cylinder. Computers & Fluids 30 (3), 365 – 392.
Choi, S., Choi, H., Kang, S., 2002. Characteristics of flow over a
rotationally oscillating cylinder at low reynolds number. Physics of
Fluids (1994-present) 14 (8), 2767–2777.
Chou, M.-H., 1997. Synchronization of vortex shedding from a cylinder
under rotary oscillation. Computers & Fluids 26 (8), 755 – 774.
Davis, R. W., Moore, E. F., 3 1982. A numerical study of vortex shedding
from rectangles. Journal of Fluid Mechanics 116, 475–506.
De, A. K., Dalal, A., 2006a. Numerical simulation of unconfined flow past
a triangular cylinder. International Journal for Numerical Methods in
Fluids 52 (7), 801–821.
De, A. K., Dalal, A., Jun. 2006b. Numerical Study of Laminar Forced
Convection Fluid Flow and Heat Transfer From a Triangular Cylinder
Placed in a Channel. J. Heat Transfer 129 (5), 646–656.
March, 2015 63 / 72

References
Dhiman, A., Shyam, R., 2011. Unsteady heat transfer from an equilateral
triangular cylinder in the unconfined flow regime. ISRN Mechanical
Engineering.
Ding, H., Shu, C., Yeo, K. S., Xu, D., Jan. 2007. Numerical simulation of
flows around two circular cylinders by mesh-free least square-based finite
difference methods. International Journal for Numerical Methods in
Fluids 53, 305–332.
Duˇsek, J., Gal, P. L., Fraunié, P., 4 1994. A numerical and theoretical
study of the first hopf bifurcation in a cylinder wake. Journal of Fluid
Mechanics 264, 59–80.
Franke, R., Rodi, W., Schnung, B., 1990. Numerical calculation of laminar
vortex-shedding flow past cylinders. Journal of Wind Engineering and
Industrial Aerodynamics 35, 237 – 257.
Fujisawa, N., Asano, Y., Arakawa, C., Hashimoto, T., 2005.
Computational and experimental study on flow around a rotationally
oscillating circular cylinder in a uniform flow. Journal of Wind
Engineering and Industrial Aerodynamics 93 (2), 137 – 153.
March, 2015 63 / 72

References
Guo, X.-H., Lin, J.-Z., Tu, C.-X., Wang, H.-L., 2009. Flow past two
rotating circular cylinders in a side-by-side arrangement. Journal of
Hydrodynamics, Ser. B 21 (2), 143 – 151.
Harichandan, A. B., Roy, A., 2010. Numerical investigation of low
Reynolds number flow past two and three circular cylinders using
unstructured grid CFR scheme. International Journal of Heat and Fluid
Flow 31 (2), 154 – 171.
Harlow, F. H., Welch, J. E., et al., 1965. Numerical calculation of
time-dependent viscous incompressible flow of fluid with free surface.
Physics of Fluids 8 (12), 2182.
Henderson, R. D., 1995. Details of the drag curve near the onset of vortex
shedding. Physics of Fluids (1994-present) 7 (9), 2102–2104.
Hirt, C., Nichols, B., Romero, N., Jan 1975. SOLA - a numerical solution
algorithm for transient fluid flows. Los Alamos Scientific Laboratory,
Rep. LA-5852.
Hu, G., Sun, D., Yin, X., Tong, B., 1996. Hopf bifurcation in wakes behind
a rotating and translating circular cylinder. Physics of Fluids
(1994-present) 8 (7), 1972–1974.
March, 2015 63 / 72

References
Inoue, O., Iwakami, W., Hatakeyama, N., 2006. Aeolian tones radiated
from flow past two square cylinders in a side-by-side arrangement.
Physics of Fluids 18 (4), 046104.
Jackson, C. P., 9 1987. A finite-element study of the onset of vortex
shedding in flow past variously shaped bodies. Journal of Fluid
Jeong, J., Hussain, F., 2 1995. On the identification of a vortex. Journal of
Fluid Mechanics 285, 69–94.
Kang, S., 2003. Characteristics of flow over two circular cylinders in a
side-by-side arrangement at low reynolds numbers. Physics of Fluids
15 (9), 2486–2498.
Kang, S., Choi, H., Lee, S., 1999. Laminar flow past a rotating circular
cylinder. Physics of Fluids (1994-present) 11 (11), 3312–3321.
Kelkar, K. M., Patankar, S. V., 1992. Numerical prediction of vortex
shedding behind a square cylinder. International Journal for Numerical
Methods in Fluids 14 (3), 327–341.
Kumar, S., Gonzalez, B., Probst, O., 2011. Flow past two rotating
cylinders. Physics of Fluids (1994-present) 23 (1), 014102.
March, 2015 63 / 72

References
Lee, S.-J., Lee, J.-Y., 2006. Flow structure of wake behind a rotationally
oscillating circular cylinder. Journal of Fluids and Structures 22 (8),
1097 – 1112.
Liu, K., jun MA, D., jun SUN, D., yuan YIN, X., 2007. Wake patterns of
flow past a pair of circular cylinders in side-by-side arrangements at low
Reynolds numbers. Journal of Hydrodynamics, Ser. B 19 (6), 690 – 697.
Lu, X.-Y., 2002. Numerical study of the flow behind a rotary oscillating
circular cylinder. International Journal of Computational Fluid Dynamics
16 (1), 65–82.
Lu, X.-Y., Sato, J., 1996. A numerical study of flow past a rotationally
oscillating circular cylinder. Journal of Fluids and Structures 10 (8), 829
– 849.
Mahfouz, F. M., Badr, H. M., Nov. 1999. Flow Structure in the Wake of a
Rotationally Oscillating Cylinder. ASME Journal of Fluids Engineering
122 (2), 290–301.
Mavriplis, D. J., June 2003. Revisiting the least-squares procedure for
gradient reconstruction on unstructured meshes. AIAA Paper 2003-3986.
March, 2015 63 / 72

References
Meneghini, J., Saltara, F., Siqueira, C., J.A.FerrariJr, 2001. Numerical
simulation of flow interference between two circular cylinders in tandem
and side-by-side arrangements. Journal of Fluids and Structures 15 (2),
327 – 350.
Mittal, S., 2001a. Control of flow past bluff bodies using rotating control
cylinders. Journal of Fluids and Structures 15 (2), 291–326.
Mittal, S., 2001b. Flow past rotating cylinders: Effect of eccentricity.
ASME Journal of Applied Mechanics 68 (4), 543–552.
Mittal, S., 2003. Flow control using rotating cylinders: effect of gap.
ASME Journal of Applied Mechanics 70 (5), 762–770.
Nishioka, M., Sato, H., 8 1974. Measurements of velocity distributions in
the wake of a circular cylinder at low reynolds numbers. Journal of Fluid
Orlanski, I., 1976. A simple boundary condition for unbounded hyperbolic
flows. Journal of Computational Physics 21 (3), 251 – 269.
Padrino, J. C., Joseph, D. D., 6 2006. Numerical study of the steady-state
uniform flow past a rotating cylinder. Journal of Fluid Mechanics 557,
191–223.
March, 2015 63 / 72

References
Patankar, S., Spalding, D., 1972. A calculation procedure for heat, mass
and momentum transfer in three-dimensional parabolic flows.
International Journal of Heat and Mass Transfer 15 (10), 1787 – 1806.
Robichaux, J., Balachandar, S., Vanka, S. P., 1999. Three-dimensional
floquet instability of the wake of square cylinder. Physics of Fluids
(1994-present) 11 (3), 560–578.
Roy, A., Bandyopadhyay, G., 2006. A finite volume method for viscous
incompressible flows using a Consistent Flux Reconstruction scheme.
International Journal for Numerical Methods in Fluids 52 (3), 297–319.
Sumner, D., Richards, M., Akosile, O., 2005. Two staggered circular
cylinders of equal diameter in cross-flow. Journal of Fluids and
Structures 20 (2), 255 – 276.
Tang, T., Ingham, D., 1991. On steady flow past a rotating circular
cylinder at Reynolds numbers 60 and 100. Computers & Fluids 19 (2),
217–230.
Tokumaru, P. T., Dimotakis, P. E., 1993. The lift of a cylinder executing
rotary motions in a uniform flow. Journal of Fluid Mechanics 255, 1–10.
March, 2015 63 / 72

References
Townsend, P., 1980. A numerical simulation of newtonian and visco-elastic
flow past stationary and rotating cylinders. Journal of Non-Newtonian
Fluid Mechanics 6 (34), 219 – 243.
Ueda, Y., Sellier, A., Kida, T., Nakanishi, M., 2003. On the
low-reynolds-number flow about two rotating circular cylinders. Journal
of Fluid Mechanics 495, 255–281.
Williamson, C. H. K., Sept. 1985. Evolution of a single wake behind a pair
of bluff bodies. Journal of Fluid Mechanics 159, 1–18.
Xu, S. J., Zhou, Y., So, R. M. C., 2003. Reynolds number effects on the
flow structure behind two side-by-side cylinders. Physics of Fluids
15 (5), 1214–1219.
Yoon, D.-H., Yang, K.-S., 2009. Characterization of flow pattern past two
spheres in proximity. Physics of Fluids 21 (7), 073603.
Yoon, H. S., Chun, H. H., Kim, J. H., Park, I. R., 2009. Flow
characteristics of two rotating side-by-side circular cylinder. Computers
& Fluids 38 (2), 466 – 474.
March, 2015 63 / 72

Appendix A λ2
Yoon, H. S., Kim, J. H., Chun, H. H., Choi, H. J., 2007. Laminar flow past
two rotating circular cylinders in a side-by-side arrangement. Physics of
Fluids (1994-present) 19 (12), 128103.
Zdravkovich, M. M., Dec. 1977. Review - review of flow interference
between two circular cylinders in various arrangements. ASME Journal
of Fluids Engineering 99 (4), 618–633.
Zielinska, B. J. A., Wesfreid, J. E., 1995. On the spatial structure of global
modes in wake flow. Physics of Fluids 7 (6), 1418–1424.
March, 2015 64 / 72

Appendix A λ2
λ2 criterion
Jeong and Hussain [1995] proposed
λ2 = ∂u
∂x + ∂v
∂y
2
− 4 ∂u
∂x
∂v
∂y − ∂u
∂y
∂v
∂x
λ2 +ve in Shearing region and -ve in swirling region
Figure 76: Contours of λ2 for ﬂow past a circular cylinder at Re = 100.
Figure 77: Vorticity contours for ﬂow past a single circular cylinder at Re = 100.
March, 2015 64 / 72

Appendix B Stress in flow field
Stress in flow field
Stress arrise due to periodic variations in the flow
˜u ˜u = (u − u)2
Figure 78: Contours of instantaneous streamwise normal stress for flow past a
circular cylinder at Re = 100. (˜u ˜umin, ˜u ˜umax , ˜u ˜u) ≡(0.01, 0.2, 0.01).
March, 2015 65 / 72

Appendix C Proper Orthogonal Decomposition(POD)
Proper Orthogonal Decomposition(POD)
Low-dimensional approximate descriptions of a high-dimensional
process.
Basis for the modal decomposition of an ensemble of functions.
The basis functions it yields are commonly called empirical eigen
functions, empirical basis functions, empirical orthogonal functions,
proper orthogonal modes, or basis vectors.
The most striking feature of the POD is its optimality.
The most eﬃcient way of capturing the dominant components of an
inﬁnite-dimensional process.
March, 2015 66 / 72

POD
Method of snapshots has been used in the present study.
Method of residual is followed.
Flow past a square cylinder
Reynolds number of the ﬂow considered 100
March, 2015 67 / 72

POD Convergence
η (N) = Ω ϕN+1
1 (x) − ϕN
1 (x) dx.
Figure 79: Convergence of the ﬁrst eigenmode of streamwise velocity conponent
(u) for single circular cylinder.
March, 2015 68 / 72

POD on square cylinder
Figure 80: First Mode
Figure 81: Second Mode
Figure 82: Third Mode
March, 2015 69 / 72

POD on square cylinder
Figure 83: Fourth Mode
Figure 84: Fifth Mode
Figure 85: Sixth Mode
March, 2015 70 / 72

POD on circular cylinder
Figure 86: First, Third, Fifth and Seventh modes
Figure 87: Energy content of POD modes
March, 2015 71 / 72

POD Validation
Figure 88: Actual instantaneous data.
Figure 89: Reconstructed instantaneous data.
March, 2015 72 / 72

DefenceSeminar

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to DefenceSeminar

Similar to DefenceSeminar (20)

DefenceSeminar