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Water Waves Modules

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CALCULATOR MODULE : Dimensionless Number   ±

Calculate dimensionless numbers for fluid flow and other physical systems.

Dimensionless numbers are calculated from groups of variables so that the result is dimensionless. Dimensionless numbers can be calculated from any consistent set of units, and will have the same value. Dimensionless numbers can be a very powerful tool for analysing physical systems.

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CALCULATOR MODULE : Dimensionless Keulegan Carpenter Number   ±

Calculate the dimensionless Keulegen Carpenter number or period number.

The Keulegen Carpenter number approximates the ratio of drag forces to inertia forces acting on a structure in oscillating flow (typically wave flow).

`Kc = V T / (OOD) = V^2 / (A* OOD) `
`A* = V / T `

where :

Kc = Keulegan Carpenter number
V = velocity amplitude
T = oscillation period
OOD = structure outer diameter or characteristic length
A* = approximate acceleration amplitude

For small Keulegen Carpenter numbers inertia forces dominate. At large Keulegen Carpenter numbers drag forces dominate. The Keulegen Carpenter number can also be applied to structures oscillating in a stationary fluid.

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CALCULATOR MODULE : Dimensionless Wave Number   ±

Calculate common dimensionless and dimensional ocean wave numbers.

Ocean wave numbers include :

`kw = (2 pi) / L = 2 pi(fw) / c `
`fw = 1 / T `
`Ur = h l^2 / d^3 = (h/d)^3 / (l/d)^2 `
`H* = H / (g t^2) `
`d* = d / (g t^2) `

where :

kw = wave number (dimesion 1/length)
fw = wave frequency (dimension 1/time)
Ur = dimensionless Ursell number
H* = dimensionless wave height
d* = dimensionless water depth
L = wave length
f = wave frequency
c = wave celerity or propagation speed

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CALCULATOR MODULE : Dimensionless Ursell Number   ±

Calculate the dimensionless Ursell number.

The Ursell number is a measure of the non linearity of ocean waves.

`Ur = h L^2 / d^3 = (h/d)^3 / (L/d)^2 `

where :

Ur = Ursell number
h = wave height
L = wave length
d = water depth

The Airy wave is suitable for Ur < 1. Stokes wave should be used for Ur < 40. Cnoidal wave should be used for Ur > 40.

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CALCULATOR MODULE : Water Hammer Transient Pressure   ±

Calculate water hammer transient pressure and pressure wave velocity.

Water hammer is caused by a sudden reduction of flow rate in liquid pipelines. Water hammer commonly occurs in water pipes, but it can occur in any liquid piping system. The transient pressure is reduced if gas is present in the liquid, or if the effective shut off time is greater than the maximum shut off time. The maximum shut off time is the time taken for the pressure transient to travel to the pipe inlet, and back again.

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CALCULATOR MODULE : Transient Pressure Wave Velocity   ±

Calculate water hammer transient pressure wave velocity.

A sudden reduction of velocity in a liquid pipeline initiates a pressure wave which travels to the pipe inlet, and then back. The wave velocity increases with pipe stiffness. Any gas present in the liquid reduces the pressure wave velocity. The maximum shut off time is the time taken for the pressure transient to travel to the pipe inlet, and back again.

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CALCULATOR MODULE : Airy Linear Gravity Wave   ±

Calculate Airy wave velocity, acceleration and surface profile. The Airy linear gravity wave theory is a first order model of freshwater and seawater gravity waves. The Airy wave is assumed to have a simple sinusoidal (first order harmonic) profile which is a reasonable approximation for small amplitude deep water waves. As the wave amplitude increases and or the water depth decreases the waves tend to become more peaky and are no longer a simple sinusoidal shape. The Airy wave model is then less accurate for analysing water particle motions. For large amplitude waves, or shallow water waves other wave models such as Stokes wave or Cnoidal wave should be used. The recommended wave type is displayed below the calc bar.

Check that the convergence is close to or equal to one. The wave period should be measured at zero current velocity to avoid Doppler effects.

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    CALCULATOR MODULE : Stokes Fifth Order Wave   ±

    Calculate Stokes wave velocity, acceleration and surface profile using Skjelbria and Hendrickson's fifth order wave method.

    Stokes wave model is suitable for waves with short wavelength or small amplitude. The calculators include the correction to the sign of the c 8 term in the C2 coefficient (changed from + to -2592 c 8 ). Check that the convergence is close to or equal to one. The wave period should be measured at zero current velocity to avoid Doppler effects.

    Note : The Stokes wave theory uses a truncated infinite series. The truncated series is only valid for certain conditions. For shallow water waves the cnoidal wave is recommended. The recommended wave type is displayed below the calc bar.

    Reference : Lars Skjelbria and James Hendrickson, Fifth Order Gravity Wave Theory

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      CALCULATOR MODULE : Cnoidal Fifth Order Wave   ±

      Calculate Cnoidal wave velocity, acceleration and surface profile using Fentons 1999 fifth order wave method.

      The Cnoidal wave is defined by the elliptic modulus m, the wave trough depth w, and the wave alpha parameter α. The Cnoidal wave model is a truncated series and is only valid within certain ranges. The Cnoidal wave theory is not recommended where the wavelength over water depth ratio (Lod) is less than 8. The recommended wave type is displayed below the calc bar.

      Note : The cnoidal wave theory uses a truncated infinite series. The truncated series is only valid for conditions where the series converges (m > 0.8). For deep water waves with small m, the series does not converge (use the Stokes wave instead).

      Check that the convergence is close to or equal to one. The wave period should be measured at zero current velocity to avoid Doppler effects.

      Reference : J D Fenton, The Cnoidal Theory Of Water Waves, Developments in Offshore Engineering, Gulf, Houston, chapter 2, 1999

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        CALCULATOR MODULE : Ocean Wave Shoaling And Wave Height   ±

        Calculate ocean wave shoaling wave height from water depth.

        Shoaling occurs as the water depth decreases or becomes more shallow. the wave length and celerity decrease (the wave becomes slower), and the wave height increases. The wave energy flux is assumed to be constant. For Airy waves the wave energy flux is proportional to c H^2 (the wave celerity times the wave height squared). The same relationship is assumed to also apply to Stokes and cnoidal waves. Use the Result Plot option to compare the initial wave and shoaling wave profiles, or the wave height versus water depth for Airy, Stokes and cnoidal waves. The recommended wave type is displayed below the calc bar.

        Note : The Stokes wave is the most suitable for a transtion from deep water to shallow water waves. The cnoidal wave is not suitable for deep water waves. The Airy wave is not suitable for shallow water waves.

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