EACWE 5

Florence, Italy19th – 23rd July 2009

Flying Sphere image © Museo Ideale L. Da Vinci

Keywords: POD, CFD, Aerodynamic, Hyperbolic paraboloid

ABSTRACT

Referring to the previous paper Rizzo et al. [1] the aim of this research has been to investigate the aerodynamic behaviour of structures tested in a wind tunnel with Proper Orthogonal Decomposition (POD) technique and with Computational Fluid Dynamic (CFD) analyses. The purpose of CFD analyses is to describe vortex shading and to compare data in order to identify characteristics for each different geometrical shape considered; three dimensional models are used to evaluate the pressure streamlines and vectors. Analysis with POD technique aims instead to describe the action on hyperbolic paraboloid surfaces of randomly fluctuating wind pressure fields and wind tunnel tests data are used to evaluate pressure modes. Moreover POD technique is used to obtain min and max values of pressure coefficients only with few data.

1. PROPER ORTHOGONAL DECOMPOSITION (POD): STATE OF ART

Some of the first researchers who investigated the complex fluctuating pressure pattern on buildings using eigenvector analysis are Armitt during his study on the wind pressure fields on the West Burton cooling tower, Lee for the systematic studies on prismatic bluff bodies, Best and Holmes, Kareem and Cermak, Holmes focalized on applications to isolated low-rise buildings.

Although there are no doubts about the incredible synthesis feature of the POD techniques and the

Contact person: 1st F. Rizzo, Chieti-Pescara University, +39-320-0384186, fabiorizzo79@libero.it

Aerodynamic behaviour of hyperbolic paraboloid shaped roofs: POD and CFD analysis

Fabio Rizzo1, Piero D’Asdia2, Massimiliano Lazzari3, Giuseppe Olivato4 1University “G. D’Annunzio” Chieti-Pescara, viale Pindaro, 42, Pescara, Italy

e-mail: fabiorizzo79@libero.it 2University “G. D’Annunzio” Chieti-Pescara, viale Pindaro, 42, Pescara, Italy

e-mail: pidasdia@tin.it 3University of Padova, Via Marzolo, 9 - 35131 Padova, Italy

e-mail: lazzari@dic.unipd.it 4University IUAV, Dorsoduro 2206 – 30123 Venice, Italy

e-mail: olivato@iuav.it

consequent computational consuming saving, the physical meaning of the eigenvector modes is still a controversial issue. Initially Armitt suggests that there are no reasons to suppose that the spatial variation pressure field due to one cause has to be necessarily orthogonal with respect to that due to other causes; the mathematical constraints of orthogonality are too strong to describe the nature’ behaviour. Another drawback is the relationship between the total number of modes founded in the analysis and the number of pressure cells. It is implied that the number of points or panels considered must be high enough to describe the modal forms in a exhaustive manner, but for the investigation purposes only the lower modes are considered, the higher ones with low energy will just complete the mathematical system of equations.

Nevertheless, Holmes notes that the separated physical causes implies zero correlation between the modes, so the orthogonal techniques are the most indicated to find such uncorrelated forms. For instance, during his study of wind effects on a circular silo, the superposition of the first two proper modes with the mean pressure distribution and its rate of change with angular position are shown. Similar considerations are reported by several authors, Vickery and Gillian et al, who resume all the wind fields by the combination of few modes. Afterwards Holmes partially reviews his position pointing out that any physical interpretation of the proper modes could be misleading or fictitious in many cases.

A very helpful contribute for fixing conflicting opinions is exposed in a very exhausting paper by Tamura et al when they demonstrate that the first proper modes always represent the mean pressure distribution and how this feature creates an heavy constrain for other modes which must be orthogonal to the first one. An authors’ opinion deals to apply the decomposition only on the fluctuating wind pressure nil mean. Using this precaution, the physical interpretation of the POD modes can be discussed but not at all generalized.

Writing about the problem of generalize wind loads, Davenport suggests three space functions which control the magnitude of the responses: the POD, powerful for the synthesis, the influence lines and the natural frequency mode shapes. When the eigenvectors of the covariance matrix (proper modes) are closed to the other representations, they boost some stress resultant (force/moment) diagrams or some instability shape. In those cases the modes can be considered separately (Holmes Tamura et al) otherwise the wind load must be written as the combination of modes.

1.1 Orthogonal Decomposition Pressure fluctuations on buildings in natural boundary layer flows produced by storms, have a

complex temporal and spatial structure and its study in a complete way keeping under control the whole physical phenomena is quite difficult. One help in tackling this issue is provided by the Proper Orthogonal Decomposition (POD) technique that offers the double benefit to decompose the flow field in uncorrelated proper modes, that have to be simply summed each other, moreover it extracts the space dependant information, stored in load surfaces, from the time histories, which includes the wind action dynamics.

p x, t p x p x, t (1)

p x, t a t   xM

(2)

The first feature permits to sort all the modes in energetically criteria, the lower modes include most of the signal’s energy and they must be taken into account, while the other can be neglected. This consideration is a very important property because it permits to characterize the phenomena with a physical quantity (total amount of energy) and not by means of modelling discretization (total number of pressure cells).

The second feature allows to focus directly to the pressure modes x which are superimposed

as a load on the structure. Every mode has an independent dynamic represented by the a t vectors. The key aspect of this expression consists into computing the energy stored inside the dynamic E and including it inside the proper mode. In this way the eigenvector is turned in a static pressure surface.

p x E x (3)

E1T a t  dt (4)

where 1 is the signal’s length, is the number of trials with time step . In the discrete calculus the integral is changed by the sum symbol. The coordinate vector is rewritten as

, where are the eigenvalues of the covariance matrix and are the non-dimensional coordinate. With those remarks the (Eq. 3) can be written as:

p x λdtT x

S√N 1

x (5)

In order to visualize this expression an ideal situation is presented. If a proper mode is stressed by a simple harmonic wave  cos  , energy of the signal is half of the square of the wave amplitude /2 , thus the equivalent static pressure (Eq. 5) is the eigenvector multiplied by the 70% of the amplitude: /√2  .

With the latter dimensional expression of the proper modes some comments on the use of the POD can be done. First of all the proper modes are computed as the eigenvectors of the covariance matrix, so they are defined apart from a constant factor. This constant makes the modes be unitary respect their energies but the signs remain undefined. Two different modal compositions are possible in order to find the equivalent static pressure field: algebraic and geometric sum.

In the algebraic composition every pressure modes (Eq. 5) is simply added to the mean pressure distribution. In this case the sign indetermination force the user to consider many combination to rich the conservative solution.

p x p x p xM

(6)

This form can become easily expensive cause there are 2 possible combination, but it is the most safe.

On the other side, the geometric sum implements the square root of the sum of squares (SRSS). In this case the sign does not affect the result. The final combinations are just plus or minus the calculated equivalent pressure distribution:

p x p x p x (7)

The second equation is more handy and direct. It follows the same synthesis concept used to remove the time dependence of the field on the modes merging. The drawbacks are the need of safety coefficient to reach a suitable security level and the sign levelling.

The geometric combination (Eq. 7) is normally suggested on standards and codes with the gust peak factor technique, where the equivalent static wind pressure field is obtained by the mean values

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(a)  (b) 

Figure 3: Proper modes pressure coefficients distributions: modes 1, 2, 3. Wind directions 0° (a), 90° (b).

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Figure 3 shows the proper modes separated. They represent special snapshot of pressure waves in characteristic time. In particular the second modes in both directions show an asymmetric load condition in respect to the wind flow, while the third modes highlight some punctual load and some singularity. These configurations are not represented in the global values distributions (Figure 2) cause the fluctuations are nil mean during all the event. It may be noted that there are several instants when the pressure configurations showed by the modes occur, so the structure had to have the resistance against these loads.

Wind direction 0° Wind direction 90°

(a) (b) 

Figure 4: Proper modes energy. Wind directions 0° (a), 90° (b).

The biggest difference between the two wind direction simulations is the sharp-corner in the

windward surface. The shape increases the fluctuations so the energy of the firsts proper modes. Figure 4 shows the modal and cumulative energy pointing out which modes have to be taken into account for the analysis: in 90 degree wind direction there is almost no fluctuation and all the modes as less than 10% of the global kinetic energy.

2. COMPUTATIONAL FLUID DYNAMIC (CFD)

The purpose of CFD analysis is to describe vortex shading and to compare data in order to identify characteristics for each different geometrical shape considered. CFD analyses were evaluated to simulate a life size configuration. Therefore a logarithmic speed profile was chosen, derived from the profile of the wind tunnel speed but also in line with Italian building regulations (z0, reference height is equal to 0.05 m and wind speed value of 27 m/s for height equal to 10 m).

Model sizes are in meters and have a span of 80 m (structures with square plan), or 80 m and 40 m (structures with rectangular plain). The diameter of structures with a circular plain is equal to 80 m. The fluid-space size around the structures is square and equal to a span of 400 m and a height of 80 m. For each geometry about four million cell models are evaluated.

The method chosen to discretize the spatial domain is the Finite Volume Method (FVM). So, continuous fluid has been discretized into variable size cells. The mesh is regular and size cells change from a spacing value of 0.8 (near to model) to one of 8 in the extreme zones of the model. Different dimensions of FVM is used to split up fluid volume, in order to have a precise real condition simulation but also in order to skip out an oversize model. Analyses are evaluated with a workstation (quad-core processor) and the analysis duration was variable between 3 and 15 days (with the Reynolds Stress model of turbulence).

Turbulent flow produces fluid interaction at a large range of length scales. This problem means that a Navier-Stokes equations numerical calculation is necessary for turbulent flow regime

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By comparison between PIV tests (referring to Figure 1.c) and pressure field vectors shown in Fig. 5.a and b, or Fig. 8.a, it’s clear that the global vortex shedding is perfectly simulated. Moreover, In Figure 8, a qualitative comparison between wind tunnel test and CFD analyses is shown, in order to confirm it. In Figure 8.b wind tunnel pressure coefficient variations are plotted, instead in Fig. 8.c, CFD pressure coefficient variations are plotted. Pressure coefficients qualitative variations shown are very similar.

REFERENCES

Rizzo F. et al. (2009) ‘Aerodynamic behaviour hyperbolic paraboloid shaped roofs: wind tunnel tests’. Arimitt J. (1968), ‘Eigenvector analysis of pressure fluctuation on the West Burton instrumented cooling tower’, Central

Electricity Research Laboratories (UK), Internal Report. Lee B.E. (1975), ‘The effects of turbulence on the surface pressure field of a square prism’, Journal of Fluid Mechanics,

Vol. 69, pp. 17-41. Best R.J and Holmes J.D. (1983), ‘Use of eigenvalues in the covariance integration method for determination of wind load

effects’ Journal of Wind Engineering and Industrial Aerodynamics, Vol. 13, pp. 359-370. Kareem A. and Cermak J.E. (1984), ‘Pressure fluctuation on a square building model in boundary-layer flows’ Journal of

Wind Engineering and Industrial Aerodynamics, Vol. 16, pp. 17-41. Holmes J.D. (1990), ‘Analysis and synthesis of pressure fluctuation on bluff bodies using eigenvectors’ Journal of Wind

Engineering and Industrial Aerodynamics, Vol. 33, pp. 219-230. Vickery B.J. (1993), ‘Wind loads on the Olympic Stadium: Orthogonal Decomposition and Dynamic (Resonant) Effects’

Report BLWT-SS28A. Gilliam X. et al. (2004), ‘Using projection pursuit and proper orthogonal decomposition to identify independent flow

mechanisms’ Journal of Wind Engineering and Industrial Aerodynamics, Vol. 92, pp. 53-69. Holmes J.D. et al. (1997), ‘Eigenvectors modes of fluctuating pressure on low-rise building models’ Journal of Wind

Engineering and Industrial Aerodynamics, Vol. 69-71, pp. 697-707. Tamura Y. Suganuma S. Kikuchi H. Hibi K. (1999), ‘Proper Orthogonal Decomposition of Random Wind Pressure

Field’, Journal of Fluids and Structures, n.13, pp. 1069-1095. Davenport A.G. (1995), ‘How can we simplify and generalize wind loads?’ Journal of Wind Engineering and Industrial

Aerodynamics, Vol. 54/55, pp. 657-669. Loève M. (1955), ‘Probability Theory: Foundations. Random Sequences’, Pub. D. Van Nostrand. Katsumura A. Tamura Y. Nakamura O. (2007), ‘Universal wind load distribution simultaneously reproducing largest load

effects in all subject members on large-span cantilevered roof’, Journal of Wind Engineering and Industrial Aerodynamics, n.95, pp. 1145-1165.

Kho S., Baker C., Hoxey R. (2002), ‘POD/ARMA reconstruction of the surface pressure field around a low rise structure’, Journal of Wind Engineering and Industrial Aerodynamics, n.90, pp. 1831-1842.

Ruan D. He H. Castanon D.A. Mehta K.C. (2006), ‘Normalized proper orthogonal decomposition (NPOD) for building pressure data compression’, Journal of Wind Engineering and Industrial Aerodynamics, n.94, pp. 447-461.

Sengupta T. K., Dey S. (2004), ‘Proper Orthogonal Decomposition of Direct Numerical Simulation Data of By-Pass Transition’, Computer and Structures, n. 82, pp. 2693-2703.

Bruno L. (2000), ‘Aerodynamic behavior of large span bridge’, Dottorato di Ricerca in Ingegneria delle Strutture e in Meccanica dei fluidi, Politecnico di Torino.