Zhi Long Liu, Daniel Shen. Instructor: Alex Bayen

Transcription

1 Student: Zhi Long Liu, Daniel Shen Instructor: Alex Bayen Advisor: Raja Sengupta

2 Background Problem Statement Simulations Error Analysis Analytical Model Conclusion Future Works

3 Interaction with ocean water & wind (velocity fields) shoreline (boundary conditions) Processes Involved: advection horizontal diffusion evaporation ocean current field random motion (discussed later) dissolution emulsification change in density & kinematic viscosity etc.

4 SPILL LOCATION: (south side of Portugal) ⁰ N 8.50 ⁰ W DIFFUSION D = 10 m 2 /sec ADVECTION Current Velocity Field data from NOAA (National Oceanic & Atmospheric Administration) duration: 4 days space accuracy: 1/8 ⁰ time accuracy: 6 hours

5 GNOME (General NOAA Operational Modeling Environment) MATLAB Analytical Solution of DE w/ the Mapping Toolbox Level Set Toolbox by Ian Mitchell Computer Science, UBC The Analytical Level GNOME Set Model

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9 Diffusion matches well Diffusion Advection behave differently Advection + Diffusion 1. different interpolation technique of current field 2. small deviation in map file (boundary conditions)

10 Diffusion => Const Normal Expansion Advection => Convection

11 Advection + Diffusion

12 Diffusion Diffusion Const Expansion 1⁰ Longitude 1⁰ Latitude distorted Diffusion Advection no direct comparison should be fine Advection + Diffusion Point Source Instability not a point

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14 The ADE: advection component diffusion + v C = D 2 C component W/O the advection part v = 0 the Diffusion Equation (DE) is v C = 0 = D 2 C

15 Problem Statement: no no boundary oil is a t = 0 mass is constant over all volume = D 2 C x 2 C ±, t = 0 C x, 0 = M/A δ x V C x, t dd = M (PDE) (BC) (IC) (mass cons.)

16 STEP 1: Vashy-Buckingham (π) Theorem. STEP 2: Use the Chain Rule & Plug into the DE PDE. STEP 3: Boundary Condition STEP 4: Initial Condition STEP 5: Mass Conservation

17 STEP 1: Vashy-Buckingham (π) Theorem. A = M L T C M/A D x t N rrrr = 2 π 1 = π 2 = C M/ x A DD DD Set similarity variable η = π 2 = x DD C = M A DD f x DD then C = M A DD f η

18 STEP 2: Use the Chain Rule & Plug into the DE PDE. C = M f η A DD C = M A DD = D 2 C x 2 chain rule f η = = M 2AA DD f η + η f η 2 C x 2 = chain rule M A DD f η = = M AAA 2 f DD η 2 d 2 f dη f η η dd dd = 0 ODE

19 STEP 3 & 4: Boundary Conditions & Initial Conditions Boundary Conditions C = M A DD f x DD C ±, t = 0 Initial Conditions C = M A DD f x DD C x, 0 = M/A δ x M A DD f x DD f ± = 0 x=± = 0 M A DD f x DD f x DD f ± = 0 BC s & IC s are the same. t=0 t=0 = δ x DD = M/A δ x

20 STEP 5: Mass Conservation V C x, t dd = M η = x DD x = η DD dd = DD dd V C x, t dd = = = M f η dd + M f η AAA A DD M DD f η DDdd f η dd = 1 = M

21 The ODE Problem d 2 f η dη f η f ± = 0, η dd η dd = 0, + f η dd = 1 Solution to the ODE C x, t = M/A 4πππ exp x2 4DD

22 Solution to the 1-D Diffusion Equation C x, t = M/A 4πππ exp x2 4DD Let σ 2 = 2DD, then the 1-D Diffusion Equation becomes C σ, x M/A = 1 2πσ exp x2 2σ 2 Gaussian distribution with variance σ 2. MATLAB command normrnd( μ, σ ) is used to model diffusion μ = 0 mean σ = 2DDD standard deviation Diffusion => Normal distribution

23 Solution to the 1-D Diffusion Equation C x, t = M/A 4πππ exp x 2 d 4DD For the 1-D Advection-Diffusion Equation: shifted by a velocity term C x, t = M/A 4πππ exp x a&d v x t 2 4DD x d = x a&d v x t x a&d = x d + v x t Advection => position increments

24 Particle Movement previous position current position diffusion component advection component convert factor from meter to degree

25 Boundary Condition at coastline Δd = vδt Δd 0 means Δt 0 (Small time increments )

26 Surface Movement

27 Boundary Condition at coastline High resolution binary mask Δx 0 & Δy 0 (small space increments)

28 Analytical Oil Spill Lagrangian Particles Diffusion Random Motion (Gaussian Distribution) Advection Velocity Field B.C. High resolution in time (small Δt) Level Set Oil Spill Stretchable Surface Diffusion Const Normal Expansion Advection Velocity Field B.C. High resolution in space (small Δx & Δy)

29 Level Set Diffusion Coefficient Const Diffusion Expansion Rate Diffusion Scaling in Vertical Dimension Possibly Moving Source?

30 [1]. Socolofsky, S., Jirka, G., Advective Diffusion Equation. Special Topics in Mixing and Transport Processes in the Environment 2 : [2] General NOAA Operational Modeling Environment (GNOME) Technical Documentation. Office of Response and Restoration. Emergency Response Division. [3]. Performing a Numerical Simulation of an Oil Spill. Mathworks. [4]. A Toolbox of Level Set Methods. Ian M. Mitchell. UBC Department of Computer Science Technical Report TR (June 2007).

31 The Advection-Diffusion Equation (ADE) : + v C = advection component diffusion D 2 C component where: 0 (incompressible for oil) v C + v C = concentration/density of fluid (oil) v = velocity of current field D = diffusion coefficient For incompressible fluid, v = 0, by the chain rule + v C = D 2 C