genetic algorithm for fluid dynamics

8/10/2019 genetic algorithm for fluid dynamics

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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 DebDepartment of Aerospace Engineering Department of Mechanical

Engineering

Indian Institute of Technology, KanpurIndia


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Introduction


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Necessity for flow control

Structural vibrations

Acoustic noise or resonance

Increased unsteadiness

Pressure fluctuation

Enhanced heat and mass transfer


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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


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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.


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Problem definition

The computational simulations for two-dimensional flow past a circularcylinder 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 threemajor parameters.

Reynolds number,

Maximum rotation rate (1) and

Forcing frequency (Sf)

where, is the translational speed of the cylinderd is the diameter of the cylinder

is the kinematic viscosity

Amax is the dimensional physical peak rotation rate

f is the dimensional forcing frequency


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Problem definition (contd) All equations have been solved in non-dimensional form with das the length

and as the velocity scales. A time scale is defined from these two and thepressure is non-dimensionalized by .

For the dynamic problem, a novel genetic algorithm based optimizationtechnique has been used, where solutions of Navier-Stokes equations areobtained using small time-horizons at every step of the optimization process,

called a GA generation. The objective function is evaluated, followed by GAdetermined improvement of decision variables.

where, TH is the time-horizonfor one GA generation.


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Literature survey S. Taneda (1978)

Flow visualization results for 30 Re 300 have been reported. For Re = 40 and 11.5 < Sf < 27, 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 1 1.0 and0.025 Sf 0.15 .

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 rotationaloscillation of cylinder in Reynolds number range of 250 and 1200, peripheralspeed due to rotational oscillation was between 0.5 and 3% of free streamspeed.


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Literature survey (contd) X.-Y. Lu and J. Sato (1996)

Finite difference simulations of Navier-Stokes equations, by a fractional stepmethod for Re = 200, 1000 and 3000, 0.1 1 3.0 and 0.5 Sf 4.

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 isused.

Presence of co-rotating vortex pair and a time variation of drag coefficientthat switches frequency abruptly at a discrete time for Re = 500, 1= 1 andSf= /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 boundarylayer, causing large drag reduction during particular cases of rotary oscillationhave been noted.


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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 controlapproach with adjoint equations over a time interval is reported.

Advantage of velocity-vorticity formulation with usage of more localized andcompact 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.


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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 benchmarkproblems.

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 beendiscussed.


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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.)


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Working principle of a Genetic Algorithm


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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.


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Governing Equations

&Numerical Method


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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,


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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,


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Boundary and Initial conditions

No-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.


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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 diffusionterm 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,


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Compact schemes

In the present investigation, the OUCS3 scheme is used. In the periodicdirection, 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 forj = 3 to N-2.

For boundary closure, have beenused. To control aliasing and retain numerical stability an explicit fourthorder dissipation term is added at every point with


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Compact schemes compared with CD2 scheme


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The region marked in the (kh-t) plane where the numerical groupvelocity matches physical group velocity in solving linear wave

equation within 5% tolerance


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GA formulationSFE, VTE and PPE along with boundary conditions, define the system tobe 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.


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GA solution procedure

Randomly generate population for thefirst generation in allowed decisionvariable space.

Evaluate the cost function of themembers for a user-defined time-horizon, measured from an initial time.

Apply GA to the initial population forG number of iterations and the bestsolution is recorded.

Using this solution as the initial solution,another GA generation is started to findbest control strategy for the next time-horizon.

This procedure is continued till the bestcontrol strategy of consecutivegenerations are similar to each other or amaximum number of generations isreached.


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Results and Discussions


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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 iscalculated by,

wherep is surface pressure,ixis viscous tensor on surface of cylinder,

ni is unit normal vector in ith direction


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Experimental results of Tokumaru and Dimotakis


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Time variation of CD

and CL

for Re = 15000, Sf

= 0.9

(CD)Avg for 1= 1.5, is 0.7878(CD)Avg for 1= 2.0, is 0.4712

(CL)Avg for 1= 1.5, is 0.4101(CL)Avg for 1= 2.0, is 0.6164


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Time variation of CDand CL for Re = 500, Sf = /2,




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Streamline contours for the initial conditions used by (a)

Dennis et al(2000) and (b) present computation


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Time variation of CD and CLfor Re = 1000, Sf = /2,




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Vorticity contours for Re = 1000, Sf = /2, 1 = 0.5


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Vorticity contours for Re = 1000, Sf = /2, 1 = 1.0


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Streamline contours for Re = 1000, Sf = /2, 1 = 0.5


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Streamline contours for Re = 1000, Sf = /2, 1 = 1.0


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Fourier transform of CDin log-log scale


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Fourier transform of CLin log-log scale


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Streamline contours for Re = 15000, Sf = 0.9, 1 = 1.5


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Vorticity contours for Re = 15000, Sf = 0.9, 1 = 1.5


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Streamline contours for Re = 15000, Sf = 0.9, 1 = 2.0


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Vorticity contours for Re = 15000, Sf = 0.9, 1 = 2.0


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Vorticity contours animated, for Re = 15000, Sf = 0.9, 1 = 2.0


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Disturbance energy plots for Re = 15000, Sf = 0.9, 1 = 2.0, 0 = 0


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Time variation of CD and CLfor Re = 1000, Sf = /2, 1 = 1.0, 0 = 0.5

(CD)Avg = 0.9068 (CL)Avg = 1.336


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Streamline contours for Re = 1000, Sf = /2, 1 = 1.0, 0 = 0.5


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Vorticity contours for Re = 1000, Sf = /2, 1 = 1.0, 0 = 0.5


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Disturbance energy plots for Re = 1000, Sf = /2, 1 = 1.0, 0 = 0.5


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Variation of 1of best member with time


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Variation of Sf of best member with time


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For c = 5; m= 10,


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For c = 5; m= 60,


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For c = 5; m= 100,


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For c = 10; m= 100,


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For c = 2; m= 100,


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Variation of 1andSfof best member

with time, for multiple GA iterations


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For c = 2; m= 50, with multiple GA iterations,


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Conclusions


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Time averaged drag and lift coefficients for all computed cases

Case Re (CD)avg (CL)avg

1 500 0.25 /2 1.3040 0.08341

2 500 0.50 /2 1.2590 0.09089

3 1000 0.50 /2 1.3630 0.07691

4 1000 0.50 1.3360 0.1150

5 1000 1.00 /2 0.8917 0.2219

6 15000 1.50 0.9 0.7878 0.4101

7 15000 2.00 0.9 0.4712 0.6164

Time 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.


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Average drag coefficients for different GA simulations

Case c m (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.


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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.


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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.


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Thank You

genetic algorithm for fluid dynamics

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