Wind generated waves on fabric structures

Chris J K Williams University of Bath

We want to design very light structures with the minimum of prestress.

We will accept ‘large’ displacements under wind provided that the structure is not damaged and is serviceable.

Pontiac Silverdome Effect of wind and snow

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Paper : Williams

Paper

Travelling waves and standing waves on fabric structures C. J. K. Williams, MA School of Architecture & Building Engineering, University of Bath

Synopsis the fluid. At the particular instant of time at which the ‘snapshot’ in Fig The classical theory of travelling and standing waves is applied 1 was taken, the line joining A and B was horizontal and the line joining to the case of wind blowing across a fabric structure. Results A and C was vertical. The anticlockwise angular velocity of the line joining are obtained which are of practical importance for the design A and B is equal to of fabric structures. vertical velocity of B-vertical velocity of A - (v + *v) - V

- Introduction This paper discusses the theory of the stability of fabric structures such as tents, sails and air-supported roofs in a wind. The theory is based on the work of Kelvin’ and Rayleigh’ which was published more than 100 years ago. This classical work is fully discussed in Lamb3, and more recent authors 442 have used essentially the same starting point and ever more complicated analytical and numerical methods.

Despite the fact that these methods are well known in aeronautical circles, it seems that structural engineers responsible for the design of fabric structures are largely unaware of their existence. In addition, the mathematical complexity of much of the work makes it somewhat inaccessible. The aim of this paper, therefore, is to describe the theory in relatively simple terms and to discuss the implications for the practical design of fabric structures.

All the authors cited above discuss the flow of wind past a surface which is basically flat, but which contains small irregularities or disturbances. These disturbances modify the wind flow and hence the air pressure. The change in air pressure will, in turn, cause the surface to deflect if it is flexible. This will cause further changes in air pressure, and so on. Thus we have a problem in aero-elasticity, the interaction of aerodynamic and elastic forces.

As the wind speed increases, the changes in air pressure caused by the irregularities increases and at a critical wind speed the surface may lose static stability, a process known as divergence, or it may begin to oscillate orflutter. Dynamic behaviour may take the form of waves travelling across the surface or there may be stationary or standing waves as on a plucked guitar string.

It may be argued that fabric structures are not flat-indeed, a great deal of design effort is expended to ensure that they are not flat. However, provided that the wavelength of a disturbance is reasonably small compared to the radius of curvature of the surface as a whole, the effect of the overall curvature is relatively unimportant.

Of far greater consequence is the fact that we will make the assumption that the air flow is irrotational, as did all the authors cited above. Flow is irrotational if the vorticity is zero, and the concept of vorticity can be explained using Fig 1 in which U and v are the fluid velocities in the x and y directions. U and v will in general be functions of x and y and also time, if the flow is unsteady. A, B and C are three fluid ‘particles’ moving with

’ t t“ I T I 6 y l t‘

I Fig 1.

distance AB *X . . . . ( l )

- _ - _ - *v - a’ as 6x tends to 0. 6~ ax

Similarly, the clockwise angular velocity of the line joining A and C is -. The vorticity, or mean angular velocity about a line parallel to ay the z axis, is

au

The rate of shear strain is

1 !!+!E. 2 lax aYl

. . . . (3)

(Note that engineering texts often leave out the !h in the definition of the rate of shear strain.) In 3-dimensional flow vorticity is a vector (or more precisely a pseudovector) with components of angular velocity about three axes.

A perfect fluid is a fluid (liquid or gas) with no viscosity, and if the vorticity of the flow of a perfect fluid is initially zero, it will always remain so. This is demonstrated in any reasonably advanced book on fluid mechanics. Thus viscosity is necessary to generate vorticity and since the viscosity of air is very low, it seems reasonable to assume that the flow of air is irrotational (i.e. has zero vorticity). However, the intense rates of shear strain which occur when a fluid flows past a solid surface, means that vorticity is generated in the boundary layer and if the flow separates from the surface, the vorticity is shed into the main flow to form a turbulent wake.

Flow will separate from a surface at a sharp corner, but it can also separate from gently curved objects if it does not have sufficient momentum to ‘keep it going’. If the flow separates from the top of an aircraft wing, the lift is greatly reduced, and it is said to have stalled. Certainly, one can expect flow to separate from most fabric structures except, perhaps, hang glider wings and sails when sailing close hauled, i.e. almost directly into the wind.

Why then do we make the assumption of irrotational flow? The answer is that, if we do not make this assumption, it is virtually impossible to make any headway with the theory (even if we were to use the most powerful computers). It therefore means that we can expect the theory to make predictions of only the sort of behaviour that we might expect to occur, and aeroelastic wind tunnel tests must be used for accurate quantitative predictions (this point is made in william^'^).

Aeroelastic wind tunnel tests are tests in which the model is designed to scale the stiffness, mass and damping of the real structure, and deflections, strains, etc., are measured directly. Rigid model wind tunnel tests are a complete waste of time if one is concerned with aeroelastic phenomena such as divergence and flutter.

The theory will also assume that the air is incompressible. This is reasonable since pressures associated with the wind are of the order of magnitude of 1 kN/m2, whereas atmospheric pressure is approximately 100kN/m2 so that volumetric strains will be of the order of 1 To (pressure

432 The Structural Engineer/Volume 68/No.21/6 November 1990

Standing waves or ‘oscillation’ of water. Do fabric structures ‘oscillate’ in a wind or do waves travel across them?

Solid mechanics, including fabric - Easy Fluid mechanics - Difficult

Equations of motion, fabric and fluid: ρ DvDt

= ∇⋅σ

DεDt

= 12

∇v + ∇v( )T( )

⎧

⎨⎪⎪

⎩⎪⎪

Barotropic viscous fluid: σ = − pI+ 2µ Dε

Dt+ µbulk −

23µ⎛

⎝⎜⎞⎠⎟ tr

DεDt

⎛⎝⎜

⎞⎠⎟ I

p = c02

γργ

ρ0γ −1

⎧

⎨⎪⎪

⎩⎪⎪

Linear elastic solid: σ = Ε ⋅⋅ε

ε = 12g −G( )

⎧⎨⎪

⎩⎪

Horace Lamb 1849 – 1934

‘I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic.’

SPH - Smoothed Particle Hydrodynamics

Verlet integration

Horace Lamb,1895 Hydrodynamics

Article 246, including surface tension

Flow over stationary sinusoidal disturbance Velocity potential

z = x + iy

η =ϕ + iψ =V z − Aei2πzλ⎛

⎝⎜⎞⎠⎟

vx − ivy =dηdz

Steady Bernoulli equation p + 12ρv2 = 0

Negative stiffness equal to − 2πρV2A

λ

Oscillating sinusoidal disturbance, no wind Horace Lamb, Hydrodynamics, Cambridge University Press 1895, Article 217

Added mass (both sides) is equal to ρλπ

-2

-1.5 -1

-0.5 0

0.5 1

1.5 2

2.5 3

3.5 4

-1

-0.5

0

0.5

1

U = Wave speed2πTρλ

V = Wind speed2πTρλ

V −U( )2 +U 2 = 2πTρλ

Sinusoidal wave wind on one side of membrane

Downwind wave speed same as wind speed

Upwind wave stationary

No solution!

2πTρλ

= 2π ×10 ×103

1.225 × 50= 32 m/s = 115 km/h

T = 10 kN/mρ = 1.225 kg/m3

λ = 50 m

Millennium Dome (The O2 Arena) Richard Rogers, Buro Happold 2000

Mδii

+Dδi

+Kδ = p

Automatically stable - tension increases with wind speed

United States Pavilion at Expo '70 in Osaka, Japan,Geiger Berger Associates

Change in internal pressure under wind load is as important as change in external pressure