Hybrid technique for verifying the interactions

Computational

The Fourth International Wind Engineering(CWE2006),

Symposium on Yokohama, 2006

Hybrid technique for verifying the interactions between oscillating square prism and surrounding 2D flow Rei Okada a, Yuka Isono b, Makoto Kanda c,Eizo Maruta d a COEpost doctoral researchfellow, CUEE,TokyoInstituteof Technology,4259 Nagatsuta, Midori-ku,Yokohama, Kanagawa,Japan bGraduateStudent,Departmentof Architectureand ArchitecturalEngineering,Collegeof IndustrialTechnology, NihonUniversity,Chiba,Japan cAssociateprofessor, Departmentof Architectureand ArchitecturalEngineering,CollegeofIndustrial Technology, NihonUniversity,Chiba,Japan dProfessor , Departmentof Architectureand ArchitecturalEngineering,Collegeof Industrial Technology, NihonUniversity,Chiba,Japan KEYWORDS: Hybridtechnique,Interaction,2Doscillatingsquareprism,uniformflow ABSTRACT: In this paper,the experimentalsystemusingthe NHAT(NewHybridAerodynamic vibrationTechnique)is introduced. Thistechniquecanverifythe interactionsbetweenthe oscillating 2D squareprismand the surroundingflow. Moreoverthe accuracyverificationsof dynamic behaviorsfor the mechanismof NHATwithoutthenon-lineareffectdue to the aerodynamicforceis conducted,and then,the aerodynamiccharacteristicsof the stasisand the oscillatingcylinders(i.e. characteristicsof pressuredistributionsand aerodynamicforces,the dampingand mass ratio dependentof the responsedisplacement','the lock-in', and `thehysteresisof the responsedisplacement') are verifiedusingNHAT. The experimentalresultsare comparedwith the resultsof the previouswindtunneltests to arguethe validitiesof the resultsusingNHAT. On NHAT,the dynamiccharacteristics of the squareprismsuchas themass,the damping,and the stiffnesscanbe set easilyand precisely. Especially,in case of the non-linearrestoringforceis set on the equationof motionof the squareprism,It is possibleto simulatethe aerodynamicbehaviorsof the elastoplastic structure,too. The interactionsbetweenthe elastic structureand the surrounding2D flow are simulatedhere. 1.INTRODUCTION Thehybridtechniqueis the combinationof pluraldistinct-simulation techniquessuchas numericalsimulationsand experiments1). In this technique,it makespossibleto get the best of everyapproach. Kandaet al. developedthe hybridtechniqueto simulatethe complexinteractionsbetweenthe structureandthe air flow2). It is called`NewHybridAerodynamic vibration Technique;NHAT'. In this technique,the modelis set in the air flow preparedby the wind tunnel equipment. The externalwind force is measuredby differentialpressureinstruments (64ch),and the valuesof the forceare passedalongthe computerand theresponsedeformation is calculatedby the numericalcalculationon the PC. Andthen,basedon the resultsof the calculations,the modelis movedforcibly. Finally,the interactivephenomenabetweenstructural modeland air flow can be reproducedon our technique. On the otherhand, the dynamicparametersof the structuresuch as mass, damping,and stiffnessare set numericallyon the PC. As theresults,thepressuredistributions and aerodynamicforcescanbe measuredsystematically. This techniqueis able to becomethe effectivetools for this problemin the wind engineering field.

―709―

2. CONCEPT OF NEW HYBRID AERODYNAMIC VIBRATION TECHNIQUE (NHAT) 2.1 Outline of NHAT NHAT consists of two elements. The first one is the experimental equipment including the wind tunnel device. The second one is the PC for calculating the response of the virtual structure. The measured data of the air pressure and the displacement are transmitted to PC via A/D board. Based on the data of the air pressure, the drag and lift force, and the torque acting on the cylinder are derived. These three forces are treated as the external force for the equations of motion. From these equations, the response velocities are derived. These values are sent to the shaking mechanism, and then the structural model which is based on these values in three directions is moved. One cycle of this procedure as shown in Fig. 1 is 4msec.

Fig.

1 Flowchart

2.2

Experimental

The

model

Fig.

the 3a

the

to

the

the

a. Experimental Fig. 2 Experimental

hybrid

air

simulation

(NHAT)

of NHAT prism and ƒÆ

experimental

shows

time

square

direction, up

around

part of

across-flow opened

of real

flow

has

axis; of

model,

configuration

the the of

three torsional wind model

the

2D

degrees

of

direction) tunnel.

freedom as

To

is surrounded

maintain by

(X

shown

the

axis; in

the end

Fig.

flow 2a.

properties plates

as

direction, The of shown

Z

uniform in

Fig.

model

b. End plates and experimental

apparatus part of NHAT

―710―

axis;

apparatus

model

is flow 2b.

b

a. built-in pressure

sensors

and taps

b. Locations

of sensors

Fig.3 Air pressure sensors in experimentalmodel 2.3 Numerical simulation part of NHAT NHAT is the real-time simulation. So there are some requirements on numerical integrate method such as no convergent calculations, explicit time integration method, high level of numerical stability, and high accuracy for high-frequency electronic noise. As a means of qualifying method, a-OS method 3) is employed. 3. VERIFICATIONS FOR DEVELOPED EXPERIMENTAL STSTEM To examine the accuracy of the pressure-measurement mechanism, the results using NHAT system are compared with the results of previous experiments in Figs. 4a. Fig. 4b indicates the aerodynamic coefficients based on the pressure distributions. As shown in these figures, it is good agreement of the results in previous studies 4). Moreover Strouhal number in NHAT is 0.12. As shown in these results, this value coincides with the value of the previous studies.

a. Mean

and fluctuate

components

of pressure

coefficients

. aerodynamic

coefficients

Fig.4 Comparisonwithvalue of previous studies about pressure and aerodynamic force coefficients It is also checked that following capabilities of exciter and the accuracy verifications of dynamic motions for SDOF system in NHAT. Good agreements are confirmed as shown in Fig. 5 Finally, the aerodynamic simulations are shown in Fig. 6. The nonlinearities of aerodynamic phenomenon are reproduced on NHAT properly. In case that the Sc is less than 16.0, once the vortex-induced oscillations occur, the response level of oscillations increases and diverges by the galloping oscillations. In case that the Sc is more than 16.0, the response level increases around the resonant wind velocity (Vr =8.0). As shown in Fig. 6b and c, the hysteresis of response and lock-in phenomena occur in the wind tunnel test using NHAT.

•\ 711•\

a. Comparison

between

target

Fig. 5 Following

capability

a. Aerodynamic

response

Fig. 6 Nonlinearity

and measured

of exciter,

value

and accuracy

b. Transfer verifications

b. Two limit cycles

of the aerodynamic

function for SDOF

exist around

dynamic

Vr =8.0

system

c. lock-in

phenomena

4. SURFACE PRESSURE AND AERODYNAMIC FORCE OF OSCILLATING CYLINDER The time histories of displacement, pressure distribution, and aerodynamic force can be measured simultaneously using NHAT. Fig. 7a shows the distributions of stationary and oscillating cylinder. Fig. 7b shows the lift force and its dominant frequency component characteristics. We are going to conduct comprehensive exams on this theme.

a. Mean and fluctuate

cbmponent

of the pressure

b. Dominant-frequencycomponent of liftforce

Fig. 7 Response displacement and lift force VS wind velocity REFERENCES 1) i.e. D. Rittel, A hybrid experimental-numericalinvestigation of dynamic shear fracture, Engineering Fracture Mechanics, 2004 2) M. Kanda, A. Kawaguchi, T. Koizumi, and E. Maruta, A new approach for simulating aerodynamic vibrations of structures in a wind tunnel —developmentof an experimental system by means of hybrid vibration technique, Journal of Wind Engineering and Industrial Aerodynamics,91, pp. 1419-1440,2003 3) Masayoshi Nakashima, Masatoshi Ishida, and Kazuhiro Ando, Integration technique for substructure pseudo dynamic test, Journal of Struct. Engng, 417, pp. 107-116, 1990 4) i.e. Bearman, P. W. and E. D. Obasaju: An experimental study of pressure fluctuations on fixed and oscillating square-sectioncylinders, Journal of Fluid Mechanics, 25, pp. 401-413, 1982

―712―

Computational

Study on Aerodynamic Oscillation Phenomena of High-Rise Tetsuo

Matsuyama

a, Wataru

Nanami

b, Makoto

Including Buildings Kanda



Unstable

C and Eizo Maruta

d

'President , WindStyleCo.,Ltd Niigata, 956-0805, Japan bGraduate Student , Department of Architecture and Architectural Engineering, College of Industrial Technology, Nihon University, Chiba, 275-8575, Japan cAssociate Rofessor, Department of Architecture and Architectural Engineering, College of Industrial Technology, Nihon University, Chiba, 275-8575, Japan dProfessor , Department of Architecture and Architectural Engineering, College of Industrial Technology, Nihon University, Chiba, 275-8575, Japan ABSTRACT: This paper describes the advantage of the New Hybrid Aerodynamic Vibration Technique, which is referred to as the NHAT, and experiments to make use of the advantages in the structural aerodynamic research field. It is proved that some new sights can be found from the result of the experiments, which are to estimate response curves under gradually changing the mass-damping parameters, coefficient of variation of wind force for normalized wind velocity and correlation between air pressure fluctuations at a same section. KEYWORDS: Mass-Damping stable Aerodynamic, Moment tion

Parameter, Coefficient

New Hybrid Aerodynamic Vibration Technique, Unof Variance, External Forces and Responses, Correla-

1 INTRODUCTION About ten years ago, some of the authors proposed the Hybrid Aerodynamic Vibration Technique1)2)3), which is referred to as HAT. Furthermore, the NHAT is being developed. The NHAT is a modified technique of the HAT, and utilizes multi-simultaneous measuring air pressure sensors as the measuring system of wind loading. The NHAT has some advantages which other aerodynamic experimental techniques of structures don't have. One might be able to get some new sights in the aerodynamic vibration research field in order to make use of the advantages of the NHAT. In this paper, the advantages of the NHAT are produced, and the possibility for finding new sights of aerodynamic phenomena is proved. Moreover, experiments to make use of the advantage are performed, and the new sights are discussed.

2 ADVANTAGES OF NHAT NHAT is a technique for simulating aerodynamic phenomena of structures by means of step-bystep integration calculation, to which measured wind external force acting on the model in wind tunnel flow is arranged, in a computer. Analyzing the concept of NHAT, we can find NHAT's advantages which the passed techniques don't have as follows; I. It is possible to estimate not only the fluctuation of each external forces and responses also the correlation of those even under unstable vibration. 2. The advantage in 1 can be applied to also local pressure acting on the model. 3. Experimental efficiency is much higher than the passed techniques and it is possible to simulate the case of low damping or low mass which is impossible to simulate by means of the

—713—

passed technique because vibration parameters can easily and accurately be set up in the computer. 4. Introducing the concept of substructure technique, it is possible to simulate vibration behavior of structures having control devices in a wind tunnel flow. In the above, the advantages of NHAT were produced. In the next chapter, experimental conduct to prove effect of the advantages is expressed.

3

AERODYNAMIC

The

EXPERIMENTS

configuration

m.This

of

model

has

built-in

single-degree-oftunnel

is

is

4

Response

Fig.

1(a)

The

responses

the as

case

EXPERIMENTAL

4.1

ference creasing

at

on

the

greater

is changed 0.3

case

in by can

the

make

in

use

2.0.

which

vibration

performed of

is

mode

the

the

is the

100mmx

wind

advantage

case

100mmx500m

is rocking

across

The mass-damping structural density,

ratio-constant

3,

the

parameter indicates

p

1, and

vibration

direction.

a case

wind

mass-damping6 air

of

with

The

is defined as density and h

damping

parame-

2.

are

means

of

case

in

range the

passed

the and

1 and

Fig.

normalized The

some in

tendency

6=0.5

wide

case

Vr increasing.

response

The

than

Parameters of

around

in the

the

be

mass

curves

there

Vr=8.5•`9.0

of

to

Mass-Damping

with

shown

in

order

increases

in detail,

Vr =6

NHAT

response

but

range

than

for

the

decreases

is

is

prism,

The

RESULTS

suddenly

response a whole,

of

square

2.

Curves

shows

In

a

sensors.

the range from 0.1 to Pm / 3 ,where pm indicates A case

is the

is

pressure simulation

flow. in

damping.

ters-constant

model air

The

smooth

is set up p) •E h ,Pm=

structural

60

freedom.

flow

parameter 8 =(Pm/

experimental

1, but isn't

shown cases.

impossible technique.

lower

the

both

in the

between

range

and

the

curves

to perform The

experiments

hard the

response

Vr=8.

1 and

response 6

if the

perform

experiment that

most

lower

agree

and

makes

this

use

of

the

case of

difstart

the

infrom

6 are

experiment

2.

peak,

tendency

increasing

values

in

case

the

conspicuous curves

start

the

of

after

2 shows

The

The of

to

curves

Vr=10.0,

cases.

curves

quite

the

case

the

than

response

It is

shows velocity

responses

difference the

1(b) wind

in

the

which that

6

6 is less

advantage

of

performed.

(a)Casel(b)Case2 Figure 1 Aerodynamic

Responses

4.2 Moment Coefficient of Variation The experiment is analyzed to make use of the advantage 1. The moment coefficient of variation is estimated based on the equation as follows: (1)

―714―

Where, M(t) indicates wind force acting on the model, Vr indicates the normalized wind speed, BH indicates reference area and 1 indicates reference length. Fig.2 (a) and (b) show the moment coefficients of variation for the normalized wind velocity. CMrmshas peak around Vr=10, and the tendency doesn't depend on 6. There is difference between the Vr values when the coefficient curves have peaks and the response curves have peaks. If 6 is smaller, the tendency is more significant. In the case 1of mass ratio-constant, CMrmsstarts increasing around Vr=8.5^-9.0 when d is. low. On the other hand, in the case 2 of damping parameter-constant, CMrmsstarts increasing from less than Vr=8.5 if 6 is low. We can find some examples in the passed papers4)5)to estimate the wind force coefficient under unstable vibration, but we can find few example to estimate the wind force coefficient for wide range of 6 under the unstable vibration in the passed papers.

(a)Case l Figure 2 Moment Coefficient

(b)Case2 of Variance

4.3 Structural Response and WindForce under the Unstable Phenomena Fig•3 shows structural response and wind force when d is 0.5. In the figure, the response curve, the moment coefficient of variance, the frequency ratio of vortex generation to natural frequency and the phase between wind force for Vr. The phase is estimated at the frequency which the spectrum of structural response has a peak. In the figure, (a), (b), (c) and (d) indicates Yr=9.2, 10.2 ,11.2 and 14.8, respectively. The values of normalized wind velocity at the peaks of the response .curve and CM,,sdon't agree. The range which the ratio of frequency keeps constant value can be shown, and the rock-in phenomenon of vortex is confirmed. In the range, the normalized wind velocity getting bigger, the phase between the wind force and the structural response gets bigger, and the rock-in phenomenon disappears when the phase angle gets 60°. The range after the disappearance, the phase angle keeps about 60°. The frequency ratio increases with constant proportion out of the range of the rock-in phenomenon, and the proportion agrees with the value which evaluate from the Strouhal Number. The rock-in phenomenon is known as a classical unstable phenomenon, but there are few examples to identify the phenomenon from the measurement of force acting on structures.

Figure 3 Response locity.

Curve, Moment

Coefficient

of Variance, Frequency

•\ 715•\

Ratio and Phase VS. Normalized

Wind Ve-

4.4 Correlation for Local Wind Pressure Fig.4 shows the correlation coefficients of wind pressure at the height of z=25cm. The correlation coefficients of wind pressure are evaluated from the measured wind pressure at the pressure hole in the opposite of center. The correlation coefficients are negative when Vr are 9.2, 10.2 and 11.2, and are positive when Vr is only 14.8. The negative correlation coefficient means that the wind pressure at the opposite position fluctuates in opposite phase, and the wind force acting on the model is strengthened by the pressures in the opposite position. The positive correlation coefficient means that the wind pressure at the opposite position fluctuates in synchronizing, and the wind force acting on the model is weakened by the pressures in the positive position. Paying attention to the absolute value of the correlation coefficients, those wind force coefficients are greater as the order of the absolute value.

Figure 4 Cross Correlation

Coefficient

between Wind Pressures

at Right Opposite Position

5 CONCLUSION The advantages of the NHAT were provided to examine the aerodynamic vibration of structures, and the experiments were performed to make use of the advantages. The wind force and the local pressure acting on the model including the correlation with the structural response could be estimated quantitatively under the unstable vibration of a square prism in three-dimensional flow. Some new sight could be found from the experimental results. We would like to find new sight for the aerodynamic phenomena to examine it more detail.

6

2

3 4 5

REFERENCES 1M.Kanda, E.Maruta,Y. Honma and K.Ueda, Development of Hybrid Experimental System Combined with Random Response Analysis for Unsteady Aerodynamic Vibration of Structure, Proceedings of 9th International Conference on Wind Engineering, 1995, pp.287-298 I.Hirata, T.Matsuyama, M.Kanda and E.Maruta, New Hybrid Vibration Technique for Simulating Aerodynamic Vibration of Structures in a Wind Tunnel, Report of the research institute of industrial technology , Nihon university , 2003 T.Koizumi, T.Yahagi, M.Kanda and E.Maruta, Development of Measuring System of Wind External Force Acting on Base-Isolated High-Rise Buildings and Response Applying New Hybrid Aerodynamic Vibration Technique, AIJ Journal of Technology and Design, No.23,2006,6 Yusuke Maruyama, Yoshihiro Taniike and Hiroaki Nishimura, Unsteady Aerodynamic Pressure on a HighRise Building Oscillating in the Transverse Direction, Journal of Struct. Engng, No.484, 1996, pp. 31-37 Tetsuro Taniguchi and Yoshihito Taniike, Technique for the Estimation of the Unsteady Aerodynamic Force with Wavelet Transform, Proceedings of 15thNational Symposium on Wind Engineering, 1998,pp.239-244

―716―

Computational

Wind

effects

on MDOF

Martinez Pedroa, Rodriguez

structures


using


ANN

Neftalib, Baker Chrisc, Chan Andrewc

a Ph .D. candidate, National Autonomous University of Mexico (UNAM)Instituto de Ingenieria, Avenida Universidad #3000, Coyoacan, Mexico bProfesor Emeritus , Instituto de Ingenieria, UNAM Professor, Birmingham University School of Engineering, Department of Civil Engineering, Edgbaston, Birmingham, UK

ABSTRACT: The use of Artificial Intelligence (Al) in the solution of problems of Civil Engineering has given new perspectives to the subject after being successfully applied in different disciplines. In the present work, wind time series were projected into two-dimensional plots and characterized as short dimension vectors, based on image recognition (IR) techniques, to be used to train a Neural Network (NN) to determine the wind regime in a place characterized by the local average velocity, ground roughness and height over the ground. Complete wind fields were produced with conditional simulation (CS), based on the prediction of the NN, and applied over five different structures to compute their internal forces and displacements to construct and train a neuro-fuzzy system (NFS) to predict the dynamical response of single elements with time series of three minutes of duration, which were finally compared with values using the finite element program LUSAS to asses the suggested algorithm. KEYWORDS:

Wind simulation,

neural

networks,

conditional

simulation.

I INTRODUCTION This paper presents a methodology for using AI techniques within wind engineering. Wind time series were characterized as short vectors using Image Recognition (IR) techniques, which were in turn used to train a neural network (NN) to generate wind time series at a number of points. The technique of conditional simulation (CS) was then used to predict the wind characteristics at intermediate locations. These algorithms are based on a combination of NN [7], IR [4, 5] and CS [3], following previous applications of NN and IR [I, 2, 6]. These basic concepts are reviewed in sections 2 to 4 and the methodology to simulate turbulent wind fields is presented in section 5. Five plane frames of different height submitted to simulated wind loading were analyzed with the program LUSAS and its results used to train a Neural-Fuzzy System (NFS) to predict the dynamic response of single elements (sections 6 and 7). Finally, in section 8 an example of simulation is compared with data generated by LUSAS and general conclusions are drawn over the work. 2 NEURAL NETWORKS (NN) Neural Networks are a relatively modern tool which aims to simulate the natural process of human learning. Normally NNs are formed by single neurons laid in layers which are interconnected; each connection is characterized by a weight (scalar) which is susceptible to

•\ 7I7•\

change during the learning process [7]. Learning is based on the optimization of weights orientated to eliminate the difference between the output produced by the net and real data; when this difference is minimized, the process is stopped, and the NN is ready to be used. 3 IMAGE RECOGNITION (IR) TECHNIQUE This tool allows the representation of a one-dimensional wind time series as two-dimensional plot called Recurrence Plot (RP) [4], through a procedure that uses the discrete components of time series v(t) to obtain Euclidean distances among them and display the numerical results as image in two dimensions, figure 1. According to [6], the features displayed on the RP, together with the statistical properties of the original series, can be used to characterize the plot in terms of a short dimension vector Q. The numerical version of several RPs can be assembled to form a single space, where RPs share characteristics like the number of points displayed and the distribution of shades. Simulated time series can be reconstructed based on the prediction of the NN for the vector £2, once the IR algorithm is reversible [4], as confirmed in [6].

Figure 1. Wind time series as RP.

4 CONDITIONAL SIMULATION This algorithm allows the inference of power spectrum of time series in the region between two fixed stations were wind regime is known. It is based on the assembly of the covariance matrix of the Fourier coefficients associated with the power spectrum at each location. The definition of the spectrum in terms of Fourier coefficients is given in (1). Conditional simulation of these coefficients in intermediate points is achieved using (2). (1) (2a) (2b) (3)

Equation 2 implies decomposition of the inverse and transpose of the covariance matrix, its Cholesky decomposition, and the use of multivariate linear prediction for each frequency involved [3]. Time series Z(t) in intermediate points, between fixed stations, can be computed using the inverse FFT (3).

•\ 718•\

5

SIMULATION

OF WIND

FIELDS

Using time series estimated by the NN in figure 2, with input: Vm(the mean velocity) at height H, and K, the surface roughness parameter the power spectrum at the top and bottom of each building were computed, and then CS was applied to infer time series in intermediate points. The resulting time series were correlated, once the spatial correlation law was used to assemble the covariance matrix referred in section 4.

* Input

to specify

an element

in a specific

type of building

Figure 2. NN to compute vector fZ (left), and NFS to estimate the response in the structural frames

6

STUDY

OF THE USE OF NFS TO PREDICT

STRUCTURAL

RESPONSE

A step by step analysis was carried out in plane frames of different heights: 25, 30, 50, 60, and 75 m, 6 different levels of local average velocity (Vm):0.5, 10, 20, 30, 40 and 50 m/sec, and 4 different categories for soil roughness, using simulated time series of three minutes of duration acting laterally over the structure. The geometry and material in the frames was that of common structures (regularity). From a LUSAS analysis, moments, shear and axial forces were obtained, as well as joint displacements. These records were processed using IR, as described in section 3, and a NFS was trained to reproduce time series of the dynamic response of beams and columns, using an input vector formed by Vm,K, the type of building, and the level of the element. 7

RESULTS

The dynamic response of beams and columns was computed with the NFS described in 6, and compared with data obtained using LUSAS. Table 1, and Figures 4 show the comparison between simulated and LUSAS data, for a place with Vm=30 m/s, plane soil, in a building 50m height.

•\ 719•\

Table 1. Statistical prouerties

Figure 4. Power spectrum

8

of real and simulated

of simulated

structural

data (bending

moment

response compared

with LUSAS data

CONCLUSION

The combination of Al, IR, and CS showed an alternative way to model wind fields, and reproduce the dynamic response of elements within plane frames. The study made evident compatibility between NN and IR, and the flexibility of Al to incorporate CS. In future work, algorithm outlined here will be extended to study the structure of disturbed wind fields, with incorporation of knowledge came up from other fields, like chaos theory.

9

to the the the

ACKNOWLEDGMENT

To Alban, research

Conacyt,

UNAM,

and

Birmingham

University

for their

support

to develop

this

REFERENCES 1. Romo M. P., Forecasting of shear wave velocities from CPT resistances by means of ANN, Proceedings of the Geotech-Year 2000: 27-30. 2. Shuoxian W., Jiping Z., The application of neural networks to the prediction of traffic noise, International Journal of Acoustic Vibration, 2000; 5 (4):179-82. 3. Vanmarcke E., Heredia-Zavoni E., Fenton G., Conditional simulation of spatially correlated ground motion, Journal of Engineering Mechanics, November 1993; 119 (11), pp. 2333-2352. 4. Eckmann J P, Kamphorst O S, Ruelle D, Recurrence plots of dynamical systems, Europhysics Letters, Nov 1987. 5. Turk M., Pentland A., Eigenfaces for recognition, Journal of Cognitive Neuroscience; 3 (1), 1991, pp. 71-86. 6. Martinez P., Rodriguez N. Wind Field reproduction, using Neural Networks and Conditional Smulation, - Article under Review in Engineering Structures- Elsevier. Date of submission: January 11th2005 7. Muller B, Reinhardt J, Strickland M T. Neural Networks, 55. Springer-Verlag, 1995.

—720--

Computational

Wind load evaluation system for the design of spatial structures Yasushi

Uematsua

and Raku



of cladding

Tsuruishib

aNew Industry Creation Hatchery Center, Tohoku University, Sendai, Japan bDepartment of Architecture and Building Science, Tohoku University, Sendai, Japan ABSTRACT: The paper describes a computer-assisted wind load evaluation system for the design of cladding of spatial structures, such as spherical domes, using artificial neural network (ANN) and time-series simulation technique. The system can be applied to a risk consistent design as well as to a fatigue design of the cladding. KEYWORDS: Spherical dome, Cladding, Wind tunnel experiment, Artificial neural network, Time-history simulation, Fatigue design 1 INTRODUCTION In Japan, roof cladding is usually designed based on the worst peak pressure coefficients irrespective of wind direction. Furthermore, the specifications are usually based on the expected values of the peak pressure coefficients. Neither the probability distribution of the peak pressure coefficients nor the peaks other than the largest one are considered. Therefore, they are not suitable for fatigue and risk-consistent designs. The present study proposes a computer-assisted wind load evaluation system for the design of roof cladding of spherical domes, as schematically illustrated in Figure 1. This system provides wind loads for the design of cladding and its fixings without carrying out any wind tunnel experiment. An aerodynamic database, artificial neural network and time-series simulation technique are employed in the system.

Figure 1. Wind load evaluation

system for the roof cladding of spherical domes

2 AERODYNAMIC DATABASE 2.1 Windtunnel experiment The experiments were carried out in a closed-circuit-type wind tunnel at Kajima Technical Research Institute. Two kinds of turbulent boundary layers were generated, which simulated natural winds over typical open-country and urban terrains (Flows I and II ). The geometry of

•\ 721•\

thewindtunnelmodelis schematicallyillustratedin Figure2(a).Eachmodelis equippedwith 433pressuretapsof 0.5 mmdiameter,as shownin Figure2(b).Thedetailsof the experimental arrangements andproceduresarepresentedin Hongo(1995). 2.2 Database Thedatafromthe simultaneous pressuremeasurements are storedon a computerin the formof pressure coefficient; the pressurecoefficientCpis definedin termsof the velocitypressureqH(= 2p UH, with p and UHbeingthe airdensityandthewindvelocityatthe meanroofheight 1/ H, respectively).Then, the statisticalvalues of pressurecoefficients,i.e. meanC, standard deviationC', maximumand minimumpeaks, Cp.and Cmin, duringa full-scaleperiodof 10 min,skewnessSk,kurtosisKuand powerspectrumSp(f), withf beingthe frequency,are computed. The averaging time for evaluating the peak pressure coefficients is varied from 0.25 to 2 seconds. The distributions of Cp , Cp',Cpmax,Cpmin , Sk and Ku in the circumferential

direction aresmoothed byusinga cubicsplinefunction. Thisprocedure mayeliminate noisy errors included in the experimental data. Sample results onCp are shown in Figure 3. The power spectrum Sp (f) can be stored in the database as it is. However, it requires a huge amount of computer memory. Hence, the power spectrum is approximated by the following equation: (1) where o p is the standard deviation of pressure fluctuation; a1 and a2 are the position constants and c1 and c2 are the shape constants. The values .of the four constants can be determined by the least-squares method. Figure 4 shows sample results on the power spectra. It is seen that the approximation by Eq (1) is relatively good. The values of a1, a2, Cl and c2 at the pressure taps on the dome's centerline are computed for all the cases tested and stored in the database.

(a)Geometry (side view)

(b)Location of pressure taps (top view)

Figure 2. Wind tunnel model and coordinate

Before smoothing

Figure 3. Distribution of Cp

system

Figure 4. Measured and fitted wind pressure spectra (flD = 0.1, h/D = 4/16, Flow I)

―722―

After smoothing

3

ARTIFICIAL

Although

NEURAL

the

spatial

wind

(ANN) used

to

to

the

for

Figure

5

Regarding

the

Approximately

Cp

and

still

of IuH

each

of

of

the

and %

Cp'

four

of

results. This

is because

The

Learning input and

predicted

data

the

skewness

However, the

viewpoint

wind

pressures.

the skewness

of

,

Cp'

, Sk and

experiment pressure is

and

is

Cladding

an

artificial

neural

Fahlman and five parameters, y) of height

is

network

Lebiere that

pressure H. Each

the

1990) is is, two

tap, and network

the is

Ku.

a range

kurtosis

points,

design.

(x,

and

kurtosis,

agreement

hundreds

roof

prediction

coefficients,

within and

several cladding

Hence,

h/D), the coordinates flow at the mean Cp

deviation

at

the

Network (CCLN, vector consists of

between

standard the

fluctuating

parameters,

. Regarding

simultaneously

from

dome (f/D the approach

a comparison

mean

for

the of

measured

limited

variation

shows

experimental Cp'.

were

be

Cascade Correlation the resolution.

95

0.05

the

spatial

parameters intensity

constructed

and •}

may

based on improve

geometric turbulence

of

pressures

resolution

sensitive

NETWORK

of the

somewhat exhibit

by

the the

the

agreement target

ANN poor,

is

value •}

captures

very

ANN

the

0.2 general

compared large

for

fairly

values

with

Cp

.

good. for

C p trend

that

for

in magnitude

inrelativelysmallareas.Furthermore, theirvariationin theseareasis alsoremarkable.

Figure 5. Comparison

between experiment

and ANN prediction

for Cp

4 TIME SERIES SIMULATION A simulation technique proposed by Kumar and Stathopoulos (2001) is applied to the present problem. A simple stochastic model with a single parameter b is used for simulating the phase. The computation of b is accomplished by minimizing the sum of the squared errors in skewness and kurtosis. Table 1 shows comparisons between experiment and simulation for the statistical values of pressure fluctuations at two points. A good agreement can be seen in the table. Table 1. Comparison between experiment ona dome with f/D=0.2 and h/D=4/16in

and simulation FlowI

for the statistics of wind pressure coeffciients

―723―

at two points

These results indicate that the Kumar and Stathopoulos's method can be used effectively for simulating the fluctuating wind pressures on spherical domes and thereby for evaluating the design wind loads by combining the database of the statistics of wind pressures. The most troublesome and time-consuming procedure is the determination of the optimum value of b. It is found that the values of Sk and Ku increase monotonically with an increase in b. In practice, the value of b is not very large and the values of Sk and Ku are less sensitive to b. Therefore, the variation of Sk and Ku can be approximated by a simple function of b with a small number of data points. Using such a function, the optimum value of b can be calculated easily. 5 APPLICATION TO A RISK CONSISTENT DESIGN AND A FATIGUE DESIGN The proposed methodology can provide peak pressure coefficients according to a predetermined risk level by combining the extreme value analysis. Furthermore, by introducing a load cycle counting method, such as the rainflow count method, the system can provide the wind load cycles for fatigue design. Figure 6 shows the probability of non-exceedence for Cemin(thick solid line) calculated from a set of 200 extremes that the system predicted. The result is different from that predicted from a set of 6 experimental data by using BLUE. In addition, Figure 7 shows a frequency distribution obtained from the rainflow count method.

Figure 6. Distribution function of Cpmin (f/D=0.2,h/D=4/16, Flow I,point 212) 6 CONCLUDING

Figure 7. Number of loa( (f/D=0.2,h/)=4/16, Flow I,point 001)

REMARKS

The present paper describes the components of the load evaluation system proposed. Although there are some problems to be investigated further, the results of the present paper indicate that the proposed system is promising. Further studies are planned to improve the accuracy of prediction

as well as to simplify

the time series simulation

technique.

REFERENCES 1) T. Hongo, Experimental study of wind forces on spherical roofs, Ph.D. Thesis, Tohoku University,1995 2) Y. Uematsu et al, Wind Load Evaluation System for Cladding of Spherical Domes using Aerodynamic Database, Neural Network and Simulation Proc.6th Asia-Pasific Conf. on Wind Eng., Seoul, 2005 3) Fahlman. S.E. and Lebiere. C, Advances in Neural Information Processing Systems II, 524-532,1990 4) K. S. Kumar and T. Stathopoulos, Generation of local wind pressure coefficients for the design of low building roofs Wind and Structures, Vol.4,No.6, 2001 5) Julius Lieblein, Report NBSIR 74-602, National Bureau of Standards, Washington, 1974

―724―

Computational

The wind

features

of Taipei

101 Financial



Building

L. C. Chena, C. C. Chena, C. Y. Chena, S. W. Yehb, S. K. Zenc and J. H. Choua'd a Department of Engineering Science, National Cheng Kung University Tainan, Taiwan 70101 bArchitectural Building Research Institute , MOI, Taipei, Taiwan Department of Architecture, Cheng Shiu University, Kaushuang, Taiwan d Corresponding Author: Professor Jung -Hua Chou Department of Engineering Science, National Cheng Kung University Tainan, Taiwan 70101 ABSTRACT: Wind features of the 508 m height Taipei 101 Financial Building are explored in an atmospheric turbulent boundary layer wind tunnel using a 1/500 model scale. The inverted-trapezoidal sections of the building tend to localize the wind effects on the building by retarding the pressure increasing rate and cause surface pressures to oscillate and exhibit jump behaviors. The effects of surrounding buildings can be large depending on their relative positions with respect to the 101 Building. KEYWORDS: building aerodynamics, wind tunnel test, surface pressure, unsteady flows 1. INTRODUCTION As one of the highly populated countries in the world, Taiwan has quite a few tall buildings. Among them, the Taipei 101 Financial Building is the most recent and tallest one. With its height of 508 m, it is also currently the highest building in the world. On the other hand, due to the unique geographical location, the buildings and structures of Taiwan need to withstand typhoons and earthquakes. For tall buildings, such as Taipei 101 Financial Building, current building codes require wind tunnel tests in the design stage to ensure building's safety and people's comfort. On November 1, 2005, typhoon Longwang swept through Taiwan and caused considerable damage to the eastern counties of Taiwan. The tuned-mass-damper of the 101 building also swung noticeably. But there was neither damage nor discomfort reported from the building administrator. It is undoubtedly that the building's wind-resist design is sufficient. This kind of tall building wind-resist capability is quite common(' as wind tunnel tests are always required to access the design(2,3). In this study, the wind features of a 1/500 scale model of the Taipei 101 Financial Building are explored by wind tunnel tests. Thus its wind field can be understood and the data can be used for the purpose of CFD validation.

―725―

2. Experimental

approach

Experiments

were

Building

Research

Cheng or

Kung

University.

as

two

test

sections,

test

sectional 30

this a

environment

layer

thickness

Two

model

top

influx

m,

for

air

and

the

Tunnel

located flow For

for

latter

Architectural

Campus,

can

and

tests,

of

Guei-Zen

and

building

second the

Facility

at

type

pollution

whereas

120

be

either

bride

closed-return

tests,

it is open-return.

bridge

6 m •~

tests. 2.6

it

is

The

The

m.

National

tunnel

former

The

always

has

has

a cross

maximum

speed

is

pressures 3200

was

were

the

building

and

The

money,

a city

in

the

first

building with

test

has

the

post

section,

type

with

measured transducers

RTD

examined. with

mean

quantities were

pressure

out

section

eight

maximum is

a

inverted width

about

a power

using

8,

law

of

1/500

scale

trapezoidal 10.5

typical

for

exponent

of

cm.

sections,

The

a tall 0.36

model

aspect

ratio,

building. and

The

a boundary

cm.

The

Turbulent

cm. of

configurations was

carried

telecommunication

respectively.

thermometers

Interior,

objective.

building 2.6

96.7

adopted of

other

probes.

RAD

of

countless the

Surface

test while

for m •~

of

Wind

is a continuous

the

were

height

wind

1(b),

4

experiments

total

excluding

the

on

first

of

Atmospheric

m/s.

symbolizing

and

tunnel

tunnel,

the area

study,

with

The

a close-return

the

Department

depending

running

about

in

Institute,

open-return,

In

conducted

other

One

measured by

pressure

by

plastic

the

surrounding

turbulent were

was

buildings

velocity by

was a constant

tabs tubes.

Taipei

on

the Tunnel

101 as

Financial shown

measured

by

temperature model

surfaces

temperatures

Building in

figures

Pitot-static hot

wire

and were

itself 1(a)

and

pressure anemometry.

connected

to

monitored

the by

sensors.

(a) 101 Building Fig. 1

(b) 101 Building + Surrounding Buildings Wind tunnel

model

test configurations

3. Results and discussion Surface pressure distributions are shown in Fig. 2. At 0-degree wind direction, Fig. 2(a), (b), (c) and (d) depict the surface pressures on the windward face (0-degree face), leeward face (180-degree face), 90-degree side face and 270-degree side face, respectively.

In each

figure, two sets of results corresponding to the two model configurations mentioned above are presented; namely, one set for the 101 Building alone, the other set for the 101 Building ―726―

with surrounding buildings. From Fig. 1(a), it can be seen that on the windward face, surface pressures increase with building height as expected. With the surrounding buildings, the surface pressures are larger than those at the corresponding locations without surrounding buildings. Comparing to a typical rectangular surface, the trapezoidal geometry shows two distinct features. First, surface pressures tend to oscillate as the height increases and their rate of increase is hindered by the discontinuous trap. Second, there is a relatively large jump in pressures at the mid-height of the building.

This relatively large jump is likely caused by a

combination of the retarding effect of geometric discontinuity and increasing dynamic pressures as the height increases. These two features are quite different from the fairly smooth increase in surface pressures of a rectangular building.

Fig. 2

(a)

(b)

(c)

(d)

0-degree wind-direction surface pressures coefficient distributions (0, 90, 180 and 270 degrees surfaces)

The pressure oscillations can also be observed at the other three faces shown in Fig . 2(b) through 2(d), indicating a persistent influence of the geometric discontinuity on the flow fields. As for the pressure distributions between the 101 Building with and without surrounding buildings, a large difference exists at the leeward face (Fig. 2(b)). Thus, the wind load in the along-wind direction is larger for the configuration of the 101 Building with ―727―

surrounding buildings. As a further comparison, the fluctuation pressure coefficients for the above four faces are shown in Fig. 3 as a function of the normalized building height. It can be seen that on the 0-degree surface, pressure fluctuation levels are higher for the configuration with surrounding buildings, primary due to some low-rise buildings upstream of the 101 Building. In contrast, the difference between the model test configurations is smaller on the 90- and 180-degree faces. A large difference also occurs for the 270-degree face, especially in the lower portion of the 101 Building, due to groups of surrounding buildings on this side of the 101 Building. Furthermore, all of the faces, except the 180-degree face, exhibit a relatively large jump in fluctuation surface pressure coefficients near the mid-height of the 101 Building, a trend consistent with that shown in Fig. 2. In other words, the geometrical discontinuity of the trapezoidal shape has special effect not observed in other types of buildings.

Fig

3.

Fluctuation

surface

Re=8.83••c,

0-,

pressure 90-,

180-,

coefficients 270-degree

(0-degree surfaces,

wind-direction, H=building

height)

4. Conclusions Experiments were conducted for the Taipei 101 Financial Building by wind tunnel model tests. Key results show that the discontinuities of the trapezoidal sections of the 101 Building cause oscillation and jump in surface pressures. The effects of surrounding buildings depend strongly on their relative positions to the 101 Building. References 1. Q. S. Li, Y. Q. Xiao, C. K. Wong, Full-scale monitoring of typhoon effects on super tall building, J. Fluids and Structures, 20(2005):697-717. 2. H. Liu, Wind Engineering, Prentice-Hall, New Jersey, 1991, pp.154-15 8 3. Q. S. Li, J. Q. Fang, A. P. Jeary, C. K. Wong and D. K. Liu, Evaluation of wind effects on a supertall building based on full-scale measurements, Earthquake Eng. Struct. Dyn., 29(2000):1845-1862

―728―

Computational

Reducing

flow induced C. C. Chen,

building

C. C. Hsu**,

Department of Engineering **Department

vibration L. C. Chen



by streamlining

and J. H. Chou*

Science, National Cheng Kung University

Tainan, Taiwan 70101 of Architecture , National Cheng Kung University

*Corresponding

Tainan, Taiwan 70101 author: Email: iungchou@mail

.ncku.edu.tw

Abstract: In this study, experiments are conducted to explore the possibility of flow streamlining to reduce flow induced vibrations of building and structures by placing a smaller circular cylinder downstream of a larger one. A water table flow visualization facility is used in this study. The diameter ratio varies from 1.25 to 6.25 and the Reynolds number based on the diameter of the larger cylinder is either 356 or 890. The results show that flow streamlining of the larger cylinder can be achieved and vortex shedding from the upstream cylinder can be shifted to the downstream cylinder. Furthermore, the Strouhal number of the upstream cylinder can be reduced. The extent of shedding frequency reduction depends on the diameter ratio. In addition, the asymmetric vortex shedding pattern of an isolated cylinder can be altered to a symmetric shedding pattern. be further reduced.

Thus, flow-induced vibration can

Key words: streamlining circular cylinders, reducing vortex shedding, flow visualization 1. INTRODUCTION Flow-induced building vibration is of great concern to building comfort and safety in windy conditions. The collapse of Tacoma Narrows Bridge (Washington, USA) in 1940 and Cooling Towers (Ferrybridge, UK) in 1965 are two lessons learned from wind-induced damages. As buildings are bluff bodies Bergeaerodynamically, shedding vortices are a natural consequence [1, 2]. Bluff bodies of simple geometries, such as circular cylinders, rectangular cylinders and tapered cylinders, have been studied for their vortex shedding behaviors. The parameter commonly adopted for describing the vortex shedding behavior is the Strouhal number, e.g. [3]. Because of the potential damages for buildings and structures associated with shedding vortices, methodologies have been developed to reduce the wind-induced vibrations. For example, in practical applications, spiral stakes have been used to reduce flow-induced vibrations for off-shore structures [4, 5] and chimneys. For circular cylinders, Zdravkovich[6] categorizedthree types of suppressionmethods,includingsurfaceprotrusionsto change separationpoints and shear layers, surroundinggrids, and splitterplates or guide vanes. By the same token, Wong and Kokkalis17]furthersuggestedstrakesas vortexsuppressors;same as Chou,et al [8]. For flows withlow ―729―

Reynolds numbers, Strykowski and Sreenivasan [9] employed a nearby daughter cylinder to suppress vortex shedding from the mother cylinder. In this study, a streamlining

mechanism

is used to reduce flow-induced

vibrations

of building

and structures.

2. EXPERIMENTALLY The streamlining

APPROACH

effect is explored

experimentally

diameter ratios ranging from 1.25 to 6.25. visualization

The water table is a continuous

30 cm wide and 16 cm high of water. techniques

are used to observe

with

The cylinders are installed in a water table flow

facility in an in-line configuration

smaller one.

using a pair of circular cylinders with the larger cylinder

upstream

of the

flow type with a dimension of 140 cm long x

Dye flow visualization

the flow flows.

Flow structures

and limiting

streamlines

are recorded

by a digital

video camera for further analysis. The upstream

larger cylinder has a diameter

Reynolds number, while the downstream mm, respectively.

Thus, the Reynolds

of either 10 cm or 25 cm, depending

on the

cylinder has diameters of 2 mm, 4 mm, 6 mm and 8 number based on the diameter of the larger cylinder

is either 356 or 890. The gap between the two cylinders, denoted by explored.

The diameters

is varied so that wake interactions

of the larger and smaller cylinders

are represented

can be

by D and d,

respectively.

3. RESULTS Figures

AND DISCUSSION

1 and 2 depict the distributions

2.5, respectively.

of Strouhal number versus G/d for D/d of 1.25 and

The Reynolds numbers for both figures are 356.

Several features can be

deduced from the results shown in these two figures.

First, when the gap is very small, the

larger cylinder becomes a shelter to the downstream

cylinder

only for the upstream cylinder.

and vortex shedding

occurs

Then, as the gap increases beyond the sheltering range, the

two cylinders act as a single cylinder and vortex shedding occurs only from the downstream cylinder. increases increase.

In this range, the Strouhal number decreases. as D/d increases.

In addition,

Finally, as G/d increases

the extent

The reduction of Strouhal number of this range also increases

farther, both cylinders

shed vortices

as D/d

and the Strouhal

number of the upstream cylinder returns to the value of an isolated cylinder. In addition to the reduction phenomena a general trend of elongation

described above, the vortex formation length exhibits

by the presence

of a downstream

true when vortex shedding occurs from only the downstream

―730―

cylinder.

cylinder.

This is especially

Fig. 1

Another which

Strouhal number distribution (D/d=1.25)

interesting

phenomenon

the Reynolds

from

a circular

presence

cylinder

in Fig.

shown

time occupied

symmetric

the ratio oscillates presence

frequency

from

of

pattern

can avoid

interest

to practical

shedding

a downstream

cylinder

vibration

shedding

G/d is larger only

can

than about

reduce

can be altered.

along

can be modified.

5.7.

upstream

direction,

The

to switch

the ratio of

It can be seen that

As symmetric

the cross-wind

tends

Fig. 4 shows time.

In the

symmetric.

The flow structure

to the total observation

not

pattern

3 for

shedding

occurs.

is fairly

patterns.

in Fig.

that vortex

vibration

cylinder

flow pattern.

but also the pattern

flow-induce

vortex

the upstream

20% and 50% when

is shown

It is well-known

and asymmetric

pattern

pattern

flow-induced

the asymmetric

shedding

the symmetric

between

can be reduced

Fig. 3

cylinder,

Thus,

in Fig. 3 is an instantaneous between

shedding

is 890 and D/d is 6.25.

3, vortex

back-and -forth

the

the vortex

is asymmetric.

of a downstream

As shown picture

number

about

Fig. 2 Strouhal number distribution (D/d=2.5)

In other words, vortex vortex

shedding shedding

this observation

is of

applications.

Symmetric

shedding

pattern

Fig. 4

Time span ratio of symmetric

vortex shedding

4. CONCLUSION In this study, streamlining

flow visualization to reduce

experiments

flow induced

are conducted

to explore

of building

and structures

vibrations

―731―

the possibility by placing

of flow a smaller

circular

cylinder

flow upstream

cylinder

them.

diameter

shedding

cylinder

can

be

shifted

to

can

be

E.

and

Strouhal

will can

cylinder

vibration

one.

larger

the

reduction

a larger

the

ratio

isolated

of

of

Furthermore,

larger

an

downstream

streamlining

not be

further

the number

only

be

give

of

a larger In

altered

results

be

to

summarized and

cylinder the

reduction

symmetric

adjusting

also

the can

a larger

asymmetric

First, from

gap

be

range

vortex

shedding

follows:

shedding

cylinder

but

the

as

vortex

by

upstream

addition,

a

are

achieved

downstream

achieved

can

Key can

between

reduced. of

Thus,

A

gap

shedding

pattern.

the

where

pattern

of

flow-induced

reduced.

References 1. Berger, pp.314

Willie,

R,

Periodic

flow

phenomena,

Annual

Review

of

Fluid

Mechanics,

Vol.4,

(1972)

2. Zdravkovich, 3. Roshko,

M. A.,

On

M., the

Flow drag

around and

circular

shedding

cylinders,

frequency

Oxford

University

Press,

of two-dimensional

bluff

(1997).

bodies,

NACA

TN

3169

(1954). 4. Panton,

R. L.,

5. Dyrbye, 6.

7.

C. and

M.M., means

Aerodynamics.,

pp.145-189

Wong,

Chou,

H.Y.

2nd ed.,

Wind

loads

"Review for

and vortex

p. 382, on

Wiley

structures,

and

suppressing

John

p.

& Sons,

137,

John

classification vortex

of

shedding.",

Inc.,

1996.

Wiley

& Sons,

various J.

Inc.

(1997)

aerodynamic

Wind

Eng.

and

and Industrial

(1981)

Kokkalis, induced

A.,

"A

comparative

oscillation.",

Journal

of

study

of

Wind

Eng.

three

aerodynamic

And

devices

Industrial

for

Aero.,

Vol.10,

flow

induced

(1982) J.H.,

vibration pp.113-118, Strykowski, `shedding'

flow, S. 0.,

hydrodynamic

pp21-29

9.

Hansen,

Zdravkovich,

suppressing

8.

Incompressible

for

Chao, a

S.Y.,

circular

ASME P.T. at low

Chiu,

S.S

cylinder

by

and

Miau,

strakes",

J.J,

"The

PVP-Vol.206,

mechanism Flow

of

reducing

Induced

Vibration

and

Wear,

(1991) and Reynolds

Sreenivasan, numbers

K.R., .",

"On

J. Fluid

―732―

the Mech.

formation Vo1.208,

and

suppression

pp.71-107

(1990)

of

vortex •e