Capitulo3.Descripción modeloSWAH

8/18/2019 Capitulo3.Descripción modeloSWAH

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∂η

∂t +

∂ (h + η)u

∂x = 0

∂u

∂t + u

∂u

∂x + g

∂η

∂x +

1

h + η

ηˆ

−h

∂q

∂xdz + cf

u |u|

h + η =

1

h + η

∂ (h + η)τ xx

∂x

t

x

z

u

x

η h

q g

cf τ xx

η

ˆ −h

∂q

∂x

dz = 1

2

(h + η)∂q b

∂x

+ 1

2

q b∂ (η − h)

∂x

q b q b

q |z=η − q |z=−hh + η

= − q b

h + η =

1

2

∂q

∂z

z=η

+ 1

2

∂q

∂z

z=−h

ωs ωb z

∂ωs

∂t =

2q bh −

∂ωb

∂t

ωb = −u∂h

∂x


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∂u

∂x +

ωs − ωbh + η

= 0

ν t(x,y,t)

τ xx = 2ν t∂u

∂x

ν t

∂η

∂t +

∂ (h + η)u

∂x = 0

∂u

∂t + u

∂u

∂x + g

∂η

∂x +

1

2

∂q b

∂x +

1

2

q b

h

∂ (η − h)

∂x + cf

u |u|

h + η =

1

h + η

∂

∂x

(h + η)ν t

∂u

∂x2

ωb = −u∂h

∂x

∂ωs

∂t =

2q bh −

∂ωb

∂t

∂u∂x

+ ωs − ωbh + η

= 0


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|u|t

x ≤ 1.

xi+1/2

xi+1/2 = ix, i = 0,...I

x xi = (xi−1/2 + xi+1/2)/2 u i + 1/2,

η, h, q b, ν t, wb

ws

(n +1/2)t t n tn = nt,

un+1/2i+1/2 − u

n−1/2i+1/2

t + u

n−1/2

i+1/2 (Lxu)

n−1/2

i+1/2 + g

ηni+1 − ηni

x +

q n+1b,i+1 − q n+1b,i

2x +

q Xb,i+1/2

n+1hX

i+1/2

nηni+1 − η

ni − hi+1 + hi

2x + cf

un+1/2i+1/2

un−1/2i+1/2hX

i+1/2

n =

ν t,i+1hni+1

un−1/2i+3/2 − u

n−1/2i+1/2

− ν t,ih

ni

un−1/2i+1/2 − u

n−1/2i−1/2

hX

i+1/2

nx2


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ηn+1i − ηni

t +

ĥni+1/2un+1/2i+1/2 − ĥ

ni−1/2u

n+1/2i−1/2

x = 0

Lx

a

ĥi+1/2

ĥi+1/2 =

ηi + min(di, di+1), if ui+1/2 > 0ηi+1 + min(di, di+1), if ui+1/2 < 0

max(η, ηi+1) + min(di, di+1), if ui+1/2 = 0

t ≤

x2

2ν t

ν t = min

ν t,x2

2t

.

ub =

g

h + η (2ηb − η)

ub ηb

∂u

∂t +

gh∂u

∂x = 0


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∂η

∂t > α

gh

α

α

n

cf = n2g

h1/3

cf =

κ

ln (h/z 0)− 1

2

κ = 0.4 z 0 = ks/30


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kh ≤ 3

kh =


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Capitulo3.Descripción modeloSWAH

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