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Flow control of bluff bodies using Genetic Algorithms: rotary oscillation
of circular cylinderby
Venkata Kaali Rupesh TelaproluY3101043
Thesis Supervisors:
Prof. Tapan K Sengupta Prof. Kalyanmoy Deb Department of Aerospace Engineering Department of
Mechanical Engineering
Indian Institute of Technology, KanpurIndia
Introduction
Necessity for flow control
Structural vibrations
Acoustic noise or resonance
Increased unsteadiness
Pressure fluctuation
Enhanced heat and mass transfer
Earlier methodologies of flow control
Simple geometric configurations:
Splitter plate
Use of second cylinder
Inhomogeneous inlet flow
Oscillatory inlet flow
Localized surface excitation by suction and
blowing
Vibrating cylinder
Why rotary oscillation ?
Can be employed for bodies with non-circular cross-section.
Promotes drag-crisis at significantly lower Reynolds numbers
as compared to that triggered by surface roughening.
Problem definition• The computational simulations for two-dimensional flow
past a circular cylinder that is executing rotary oscillation for a range of Reynolds numbers, peak rotation rates and frequency of oscillation, are performed and studied.
• Flow control by rotary oscillation for a circular cylinder is governed by three major parameters.
• Reynolds number, • Maximum rotation rate (Ω1) and • Forcing frequency (Sf)
where, is the translational speed of the cylinder d is the diameter of the cylinder ν is the kinematic viscosity Amax is the dimensional physical peak rotation rate f is the dimensional forcing frequency
Problem definition (contd) All equations have been solved in non-dimensional form
with d as the length and as the velocity scales. A time scale is defined from these two and the pressure is non-dimensionalized by .
For the dynamic problem, a novel genetic algorithm based optimization technique has been used, where solutions of Navier-Stokes equations are obtained using small time-horizons at every step of the optimization process, called a GA generation. The objective function is evaluated, followed by GA determined improvement of decision variables.
where, TH is the time-horizon for one GA generation.
Literature survey S. Taneda (1978)
Flow visualization results for 30 ≤ Re ≤ 300 have been reported.
For Re = 40 and 11.5 π < Sf < 27π, vortex shedding was completely eliminated.
A. Okajima et al (1981) Forces acting on a cylinder, in the range 40 ≤ Re ≤ 160 and 3050 ≤ Re ≤ 6100, were measure for 0.2 ≤ Ω1 ≤ 1.0 and
0.025 π ≤ Sf ≤ 0.15 π.
P. T. Tokumaru and P. E. Dimotakis (1991) Carried our experimental studies for Re = 15000,
calculated drag based on wake survey. Reported drag reduction by more than 80% for Re = 15000.
J. R. Filler et al (1991) Reported alteration of primary Karman vortex shedding by
rotational oscillation of cylinder in Reynolds number range of 250 and 1200, peripheral speed due to rotational oscillation was between 0.5 and 3% of free stream speed.
Literature survey (contd) X.-Y. Lu and J. Sato (1996)
Finite difference simulations of Navier-Stokes equations, by a fractional step method for Re = 200, 1000 and 3000, 0.1 ≤ Ω1 ≤ 3.0 and 0.5 π ≤ Sf ≤ 4π.
S. C. R. Dennis et al (2000) Solved 2-D Navier-Stokes equations using stream function-
vorticity formulation for Re = 500 and 1000 by spectral-finite difference method.
Time-varying grid that becomes less fine with growing shear layer in time is used.
Presence of co-rotating vortex pair and a time variation of drag coefficient that switches frequency abruptly at a discrete time for Re = 500, Ω1 = 1 and Sf = π/2, has been reported.
D. Shiels and A. Leonard (2001) 2-D flows for Re = 15000 using high resolution viscous vortex
method have been studied. Multi-pole vorticity structures revealing bursting
phenomenon in boundary layer, causing large drag reduction during particular cases of rotary oscillation have been noted.
Literature survey (contd) J.-W. He et al (2000)
Gradient-based classical optimization for 200 ≤ Re ≤ 1000 was performed.
Finite element discretization was used and cost function gradient was evaluated by adjoint equation approach.
30 to 60% drag reduction reported. B. Protas and A. Styczek (2002)
Rotary control of cylinder wake at Re = 75 and 150 using optimal control approach with adjoint equations over a time interval is reported.
Advantage of velocity-vorticity formulation with usage of more localized and compact vorticity variable was noted.
M. Milano and P. Koumoutsakos (2002) Drag optimization for flow past circular cylinder using two
actuation strategies- belt type and apertures on cylinder, was studied.
R. Mittal and S. Balachander (1995) 2-D flow at Re = 200 simulated using Navier-Stokes solver on
staggered grid, using CD2 method in generalized co-ordinates. 50% drag reduction for low Re, for single parameter combination
case.
Literature survey (contd) R. W. Morrison (2004)
Discussed the capability of evolutionary algorithms (EAs) to find solutions for dynamic models.
Quantification of attributes to improve detection and response.
J. Branke (2001) Surveyed evolutionary approaches available and applied to
various benchmark problems.
R. K. Ursem et al (2002) Practical problem of greenhouse control is tried using
evolutionary algorithms. Role of control-horizons from direct online control point of
view has been discussed.
Genetic Algorithms (GA)s
Essential components of GAs
A genetic representation for potential solutions to the
problem.
A way to create the initial population of potential
solutions.
An objective (evaluation) function that plays the role of
the environment, rating solutions in terms of their fitness.
Genetic operators that alter the composition of children
during reproduction.
Values of various parameters that the genetic algorithm
uses (population size, probabilities of genetic operators
etc.)
Working principle of a Genetic Algorithm
Aims of present investigation
Study the two dimensional simulation of
rotationally oscillating circular cylinder.
Study the disturbance energy creation/exchange
mechanism in an incompressible flow framework.
Study the effects of design parameters on the drag
acting on the body and explore the possibility of
using Genetic Algorithms to implement the
investigated problem physically.
Governing Equations
&Numerical Method
Stream Function-Vorticity Formulation
Navier-Stokes equations, in non-dimensional form are given as,
Flow is computed in the transformed orthogonal grid plane, where
Grid is stretched smoothly in the radial direction by the transformation,
where,
Navier-Stokes equations in transformed plane
Stream function equation (SFE) is given by,
Vorticity transport equation (VTE) is given by,
Pressure-Poisson equation (PPE) is given by,
Boundary and Initial conditionsNo-slip boundary condition on the cylinder wall,
Convective boundary condition on radial velocity at outflow,
The initial condition: impulsive start of cylinder in a fluid at rest.
Solving procedure
Stream function equation (SFE) and PPE are solved using Bi-
CGSTAB variant of conjugate gradient method.
ILUT pre-conditioners used to make Bi-CGSTAB converge fast.
Vorticity transport equation (VTE) is solved by discretizing
diffusion term by second order central difference scheme and
time-derivative by four-stage Runge-Kutta scheme.
Convection terms of VTE are evaluated using compact schemes.
Neumann boundary conditions on the physical surface and in
the far-stream, required to solve PPE, are given by,
Compact schemes In the present investigation, the OUCS3 scheme is used. In
the periodic direction, to evaluate first derivates, following form is used.
In the non-periodic direction, additional boundary closure schemes for j = 1 and j = 2 are used, along with the above equation for j = 3 to N-2.
For boundary closure, have been used. To control aliasing and retain numerical stability an explicit fourth order dissipation term is added at every point with
Compact schemes compared with CD2 scheme
The region marked in the (kh-θΔt) plane where the numerical group velocity matches physical group velocity in solving linear wave equation within 5% tolerance
GA formulation SFE, VTE and PPE along with boundary conditions, define the
system to be controlled with input as and the output is minimized.
Selection operator: Tournament selection with participation size of two.
Crossover operator: Simulated Binary Crossover (SBX)
operator.
Mutation operator: Polynomial mutation operator.
GA solution procedure Randomly generate population
for the first generation in allowed decision variable space.
Evaluate the cost function of the members for a user-defined time-horizon, measured from an initial time.
Apply GA to the initial population for ‘G’ number of iterations and the best solution is recorded.
Using this solution as the initial solution, another GA generation is started to find best control strategy for the next time-horizon.
This procedure is continued till the best control strategy of consecutive generations are similar to each other or a maximum number of generations is reached.
Results and Discussions
Details of present study
Reynolds numbers range - 500 to 15000. Orthogonal grid of size 150 X 450 is used. Outer boundary located at 40 diameter from centre of
cylinder. Surface pressure is obtained from total pressure and drag
at any instant is calculated by,
where p is surface pressure, τix is viscous tensor on surface of cylinder,
ni is unit normal vector in ith direction
Experimental results of Tokumaru and Dimotakis
Time variation of CD and CL for Re = 15000, Sf = 0.9
(CD)Avg for Ω1 = 1.5, is 0.7878(CD)Avg for Ω1 = 2.0, is 0.4712
(CL)Avg for Ω1 = 1.5, is 0.4101(CL)Avg for Ω1 = 2.0, is 0.6164
Time variation of CD and CL for Re = 500, Sf = π/2,
(CD)Avg for Ω1 = 0.25, is 1.3040(CD)Avg for Ω1 = 0.50, is 1.2590
(CL)Avg for Ω1 = 0.25, is 0.08341(CL)Avg for Ω1 = 0.50, is 0.09089
Streamline contours for the initial conditions used by (a) Dennis et al (2000) and (b) present computation
Time variation of CD and CL for Re = 1000, Sf = π/2,
(CD)Avg for Ω1 = 0.5, is 1.3630(CD)Avg for Ω1 = 1.0, is 0.8917
(CL)Avg for Ω1 = 0.5, is 0.07691(CL)Avg for Ω1 = 1.0, is 0.2219
Vorticity contours for Re = 1000, Sf = π/2, Ω1 = 0.5
Vorticity contours for Re = 1000, Sf = π/2, Ω1 = 1.0
Streamline contours for Re = 1000, Sf = π/2, Ω1 = 0.5
Streamline contours for Re = 1000, Sf = π/2, Ω1 = 1.0
Fourier transform of CD in log-log scale
Fourier transform of CL in log-log scale
Streamline contours for Re = 15000, Sf = 0.9, Ω1 = 1.5
Vorticity contours for Re = 15000, Sf = 0.9, Ω1 = 1.5
Streamline contours for Re = 15000, Sf = 0.9, Ω1 = 2.0
Vorticity contours for Re = 15000, Sf = 0.9, Ω1 = 2.0
Vorticity contours animated, for Re = 15000, Sf = 0.9, Ω1 = 2.0
Energy creation mechanism Navier-stokes equation in rotational form for
incompressible flows is given by,
The quantity , has been identified as mechanical energy (E) of the flow and its instantaneous distribution can be described by,
Splitting physical quantities into primary and disturbance components by identifying them with subscripts m and d respectively, the distribution of disturbance energy component of mechanical energy is given in its linearized form by,
Disturbance energy plots for Re = 15000, Sf = 0.9, Ω1 = 2.0, Ω0 = 0
Time variation of CD and CL for Re = 1000, Sf = π/2, Ω1
= 1.0, Ω0 = 0.5
(CD)Avg = 0.9068 (CL)Avg = 1.336
Streamline contours for Re = 1000, Sf = π/2, Ω1 = 1.0, Ω0 = 0.5
Vorticity contours for Re = 1000, Sf = π/2, Ω1 = 1.0, Ω0 = 0.5
Disturbance energy plots for Re = 1000, Sf = π/2, Ω1
= 1.0, Ω0 = 0.5
Variation of Ω1 of best member with time
Variation of Sf of best member with time
For ηc = 5; ηm = 10,
For ηc = 5; ηm = 60,
For ηc = 5; ηm = 100,
For ηc = 10; ηm = 100,
For ηc = 2; ηm = 100,
Variation of Ω1 and Sf of best member with time, for multiple GA iterations
For ηc = 2; ηm = 50, with multiple GA iterations,
Conclusions
Time averaged drag and lift coefficients for all computed cases
Case Re (CD)avg (CL)avg
1 500 0.25 π/2 1.3040 0.08341
2 500 0.50 π/2 1.2590 0.09089
3 1000 0.50 π/2 1.3630 0.07691
4 1000 0.50 π 1.3360 0.1150
5 1000 1.00 π/2 0.8917 0.2219
6 15000 1.50 0.9 0.7878 0.4101
7 15000 2.00 0.9 0.4712 0.6164Time averaged drag coefficient for uncontrolled case for Re = 15000 is 1.3546.For steady rotation coupled with rotary oscillation case, (CD)avg = 0.9068 and (CL)avg = 1.336.
Average drag coefficients for different GA simulations
Case ηc ηm (CD)avg
1 2 100 0.3827
2 5 10 0.4092
3 5 60 0.4108
4 5 100 0.4007
5 10 100 0.4735
For the case with multiple GA iterations, (CD)Avg
= 0.3543.
Summary of results Computational procedure is calibrated by comparing the results with
experimental results of Tokumaru and Dimotakis, for Re = 15000.
Ability of the numerical method for DNS of bluff body flows by two-
dimensional flow models has been testified.
Rotary oscillation is shown to be equivalent to tripping the wall
boundary layer aerodynamically
A large drag reduction has been achieved, by shear release
mechanism on one side of the cylinder, at comparatively low
Reynolds number (Re = 1000).
Efficacy of GA-based optimization strategy, capable of arriving at
near-optimal solutions for a dynamic problem, has been emphasized
in the present work.
Scope for future work
Investigate whether rotary oscillation brings a phase shift on
resultant force experienced by the cylinder.
Control strategy of steady rotation coupled with rotary
oscillation.
Studying the multi-objective framework of the current problem,
using reduction of flow unsteadiness as a second objective.
Thank You