A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

More documents

Recommendations

Info

64 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation For a bottom layer, the expression using 3.34 can be shown to be 1 ----ρ0 hkmp ∂ km -------------------- + ∂x pkm + 1⁄ 2 ----------------- zkm 1 ∂ + ⁄ 2 ------------------- ρ 0 ∂x z km – 1⁄ 2 ∫ z km – 1⁄ 2 ∫ hkm -------ρ0 p ∂ km hkm ----------- -------- ∂x ρ0 gρkm ------------ 2 h ∂ km g – ----------- -------------- ∂x ρ0hkm h ∂ km = + ----------- ( ρ – ρ ∂x km)dz′dz . (3.37) The two double integral terms in equations 3.36 and 3.37, containing the three-dimensional density ρ, are zero if the density is assumed to be uniform vertically across the surface and bottom layers. An approximation is introduced if these terms are neglected when the vertical density varies continuously within the surface and bottom layers. The error from the approximation is small if the surface and bottom layer heights are small enough to avoid a large change in density across the layers. If the near-surface and bottom waters of an estuary are vertically mixed by surface wind and bottom friction, respectively, the error of the approximation is eliminated. These terms are neglected here, as is customary. In summary, the x-momentum pressure gradient term for the surface, middle, and bottom layers is approximated by the following expressions: Surface layer, Middle layers, Bottom layer, 1 ---ρ0 1 ---ρ0 1 ---ρ0 ζ ∫ z3 ⁄ 2 ∂p ∂x z k – 1⁄ 2 ∫ dz z km + 1⁄ 2 h1 ∂p gρ ----- 1 1 -------ρ0 ∂x 2 ζ ∂ = ⎛ + ----- ⎞ ; ⎝ ∂x⎠ (3.38) ∂p hk ----- dz ----- ∂x ρ0 p ∂ k = ------- k = 2, 3, …, km – 1 ; (3.39) ∂x z k + 1⁄ 2 z km – 1⁄ 2 ∫ ∂p ∂x z km + 1⁄ 2 For the special case of a single layer representing the entire depth of flow, the result is 1 ---ρ0 ζ ∫ z3⁄ 2 dz ∂p ∂x dz hkm ∂p gρ-------- km km ---------------------ρ0 ∂x 2 h ∂ km = ⎛ – ----------- ⎞ . (3.40) ⎝ ∂x ⎠ z h1 ∂p gρ ----- 1 1 -------ρ0 ∂x 2 ζ ∂ gρ1 ----- -------- ∂x 2 z3 2 ⁄ ∂ = ⎛ + + ------- ⎞ , (3.41) ⎝ ∂x ⎠ where z 3 2 ⁄ represents the location of the bottom. Similar expressions are available for the pressure gradient term in the ymomentum equation.
3. Layer Averaging the Governing Equations 65 The average pressure in any layer can be related to the average pressure in the layer above through the hydrostatic pressure equation, which is approximated by pk = pk 1 + ⁄ – ⁄ . – gρk – 1 h 2 k 1 2 Here hk – 1⁄ is the average of the heights for layers k - 1 and k and represents the vertical distance between the centers of the two 2 layers. By differentiating this equation with respect to x and y, the pressure gradients for one layer can be determined from the layer above. For the x-direction pressure gradient, the result is ∂pk ∂pk – 1 = + ∂x ∂x ∂ -------------- g hk – 1⁄ ρ 2 k – 1⁄ 2 ------------------------------ ∂x ∂p ∂ρ k – 1 k – 1⁄ 2 = -------------- + gh ∂x k – 1⁄ + 2 ∂x ---------------- gρk – 1⁄ 2 ∂hk – 1⁄ 2 ---------------- ∂x ∂p gh k – 1 k – 1 -------------- --------------- ∂x 2 ρ ∂ k – 1 ghk --------------- ------- ∂x 2 ρ ∂ k gρk – 1 ------- --------------- ∂x 2 h ∂ k – 1 gρk -------------- -------- ∂x 2 h ∂ k = + + + + ------- ∂x . (3.42) In the surface layer, the x pressure gradient is approximated by ∂p1 ∂x gρ1 -------- 2 ζ ∂ gh1 ----- -------- ∂x 2 ρ ∂ 1 = + -------- . (3.43) ∂x Successively substituting equations 3.43 and 3.42 into equations 3.38 to 3.40, a general formula for the x pressure gradient term can be seen by induction to be 1 ---ρ0 z k – 1⁄ 2 ∫ ∂p hk ∂ζ gh1 ----- dz ----- gρ ----- -------- ∂x ρ 1 0 ∂x 2 ρ ∂ 1 ghm – 1 -------- ---------------- ∂x 2 ρ ∂ m – 1 ghm ---------------- --------- ∂x 2 ρ k ∂ m = + + ⎛ + --------- ⎞ (3.44) ∑ ⎝ ∂x ⎠ m = 2 z k + 1⁄ 2 where the summation is omitted for k = 1. The corresponding formula for the y pressure gradient term is 1 ---ρ0 z k – 1⁄ 2 ∫ ∂p ----- dz ∂y hk ∂ζ gh1 ----- gρ -----------ρ 1 0 ∂y 2 ρ ∂ 1 ghm – 1 -------- ---------------- ∂y 2 ρ ∂ m – 1 ghm ---------------- --------- ∂y 2 ρ = k ∂ m + + ⎛ + --------- ⎞ ∑ ⎝ ∂y ⎠ m = 2 . (3.45) z k + 1⁄ 2 Now consider the integration of the stress terms in equations 3.2 and 3.3 over a layer. After integration and application of equations 3.9 and 3.11 to the x-component terms, the result is 1 ---ρ0 z k – 1⁄ 2 ∫ ∂τxx 1 ∂τxy 1 ∂τxz --------- dz + ----- --------- dz + ----- --------- dz = ∂x ρ0 ∂x ρ0 ∂x z k + 1⁄ 2 z k – 1⁄ 2 ∫z k + 1⁄ 2 z k – 1⁄ 2 ∫z k + 1⁄ 2 1 ∂( hτxx) ∂( hτ----- k xy) ∂z 1 ⎛ k k – ⁄ 2 -------------------- + -------------------- + – ( τ ρ0 ∂x ∂y xx) --------------- 1 – τ ⎝ k – ⁄ 2 ∂x xy ( ) 1 k – ⁄ 2 ∂zk + 1⁄ 2 – – ( τxx) --------------- – τ k + 1⁄ 2 ∂x xy ( ) k + 1⁄ 2 ∂z 1 k – ⁄ 2 --------------- + τ ∂y xz ( ) 1 k – ⁄ 2 ∂zk + 1⁄ 2 --------------- + τ ∂y xz ( ) k + 1⁄ 2 ⎞ ⎠ . (3.46)
Page 1:
In cooperation with the Interagency
Page 4 and 5:
U.S. Department of the Interior Dir
Page 6 and 7:
This page intentionally left blank.
Page 8 and 9:
vi 4.2 Semi-Implicit One-Dimensiona
Page 10 and 11:
viii Figure 5.6. Graph showing solu
Page 12 and 13:
x Tables Table 2.1. Dimensionless n
Page 14 and 15:
This page intentionally left blank.
Page 16 and 17:
2 A Semi-Implicit, Three-Dimensiona
Page 18 and 19:
Page 20 and 21:
Page 22 and 23:
Page 24 and 25:
10 A Semi-Implicit, Three-Dimension
Page 26 and 27:
12 A Semi-Implicit, Three-Dimension
Page 28 and 29: 14 A Semi-Implicit, Three-Dimension
Page 48 and 49: 2.4.1.2 Dynamic Surface Condition 2
Page 68 and 69: Water surface Water surface Level p
Page 70 and 71: h 1 h 2 h 3 h 4 h 5 h 6 z 1 ⁄ 2 z
Page 114 and 115: 100 A Semi-Implicit, Three-Dimensio
Page 128 and 129:
114 A Semi-Implicit, Three-Dimensio
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Page 190:
show all

A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

Create successful ePaper yourself

Delete template?

Save as template?