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Simulating Ocean Water

By Jerry Tessendorf

Kwang Hee Ko(modified from the slides by Seo, Myoung Kook)

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Introduce

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Introduce

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Introduction

water

glitter

air

clouds

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Introduce

• Focus on algorithms and practical steps to building height fields for ocean waves.– Gerstner Waves (Gravity Waves): linear waves– FFT based method: statistical approach

Introduction

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Introduction

• Fluid Dynamics Revisited....– To describe the velocity, density, pressure of

fluid…• Lagrangian description• Eulerian description

– In fluid mechanics/dynamics, the Eulerian description method is widely used.

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Mass Conservation

• Mass conservation is applied to an infinitesimal region -> Continuity Equation

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Momentum Conservation

• Momentum Conservation

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Navier-Stokes Fluid Dynamics

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• Force Equation

• Mass Conservation

• Solve for functions of space and time:– 3 velocity components u =(u,v,w)– Pressure(p)– density (ρ) distribution

Equations

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Application to Surface Waves

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Application to Surface Waves

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Application to Surface Waves

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Application to Surface Waves

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• Limitation– single sine wave horizontally and vertically

• a more complex profile by summing a set of sine waves

N

iiiiiii tXKAkKXX

000 )sin()/(

N

iiiii tKKAy

00 )cos(

Application to Surface Waves

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Statistical Wave Models

• To simulate realistic waves in the ocean, we can use statistical models in combination with experimental observations.– The wave height is considered a random variable of

horizontal position and time, h(x,t).– The wave height field is decomposed as a sum of sine

and cosine waves.• The amplitudes of the waves have nice mathematical

and statistical properties, making it simpler to build models.

• Computationally, the decomposition uses FFTs.

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Fourier Analysis

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Statistical Wave Models• The fft-based representation of a wave height field

expresses the wave height h(x,t) at the horizontal position x=(x,z) as the sum of sinusoids with complex, time-dependent amplitudes:

• ĥ is the height amplitude Fourier component, which determines the structure of the surface.

k

xkkx )exp(),(),(~

ithth

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Statistical Wave Models

• In computer graphics, we need to compute the slope vector of the wave height field– to find the surface normal– to find the angle of incidence, etc.

• An exact computation of the slope vector can be obtained by using more ffts:

– Slope computation via the fft in equation 20 is the preferred method.

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Statistical Wave Models

• The fft representation produces waves on a patch with horizontal dimensions Lx X Lz.– Outside the patch the wave surface is perfectly periodic.– The patch can be tiled seamlessly over an area.– The consequence of such a tiled extension is that an

artificial periodicity in the wave field is present.

– As long as the patch size is large compared to the field of view, this periodicity is unnoticeable.

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Statistical Wave Models

• In practice, equation 19 is enough to model the ocean waves.

• The amplitudes ĥ(k,t) are nearly statistically stationary, independent, gaussian fluctuation with a spatial spectrum. Ph(k) = <|ĥ(k,t)|2>, <> denotes the ensemble average.

• We can find an empirical model of Ph(k)

– A useful model for wind-driven waves larger than capillary waves in a fully developed sea: the Phillips spectrum.

- L=V2/g is the largest possible waves from a continuous wind of speed V.

- ω is the direction of the wind.

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Statistical Wave Models• To realize the waves

– We sample Ph(k) in a random manner and compute the ĥ(k,t) using the following equation

– Then use the following to create the Fourier amplitudes of the wave field realization at time t

Propagating waves to the left and to the right

ξ’s are ordinary independent draws from a gaussian random number generator, with mean 0 and standard deviation 1.

Realizations of the wave height field

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Examples

• The reflectivity of the water is a strong function of the slope of the waves, as well as the directions of the light(s) and camera.

• The visible qualities of the surface structure tend to be strongly influenced by the slope of the waves.

A surface wave height realization , displayed in

greyscale.

The x-component of the slope for the wave height realization

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Variation in Wave Height Field

• Increasing directional dependence

Pure Phillips Spectrum

Modified Phillips Spectrum

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Effect of Resolution

• varying the size of the grid numbers M and N• the facet sizes dx and dz proportional to 1/M and 1/N.

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