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