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

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38 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation 2.4.2.1 Kinematic Bottom Condition The kinematic bottom condition is derived by balancing mass fluxes on a thin fluid layer immediately above the bottom of the estuary (fig. 2.6). The result is u ∂h ----- v ∂x ∂h + ----- + w = 0 on z = – h( x, y) , (2.59) ∂y which is similar to the result at the surface except that the location of the bottom z = – h (x, y), unlike the free surface, is not a function of time and therefore ∂h/∂t = 0. In this case, h represents a vertical distance measured positive downward from a datum (such as a mean sea level height) above the bottom. Equation 2.59 is sometimes referred to as the tangential flow condition because it requires the flow on the boundary to be tangent to the boundary and not to separate from it. 1 u -- 2 u ∂ – ----- Δ x ∂x 1 Δ z -- 2 h ∂ – ----- Δ x ∂x 1 Δ z -- 2 h ∂ + ----- Δ y ∂y 1 v -- 2 v ∂ + ---- Δ y ∂y y Δ y z Datum h(x,y) w Δ z x 1 -- 2 w ∂ + ------ Δz ∂z Δ x 1 v -- 2 v ∂ – ---- Δ y ∂y Control volume 1 Δ z -- 2 h ∂ + ----- Δ x ∂x 1 u -- 2 u ∂ + ----- Δ x ∂x 1 Δ z -- 2 h ∂ – ----- Δ y ∂y Figure 2.6. Sketch showing velocities for mass flux balance on a bottom control volume in three dimensions. 1 Expressions in the form u -- represent the velocities on each control volume face. Expressions in the form 2 represent the heights of each vertical control volume face. ∂u + -----Δx ∂x 1 Δz -- 2 ∂h + -----Δx ∂x
2.4.2.2 Dynamic Bottom Condition 2. Governing Equations and Boundary Conditions 39 The dynamic bottom condition is derived by applying the equation of motion to the thin fluid layer on the bottom (fig. 2.7). The equation of motion reduces to a balance between the stresses exerted by the boundary on the fluid and the stresses in the fluid exerted on the boundary. The stress vector exerted by the boundary is τ b and has components τ xb and τ yb acting along the boundary. The boundary slope in any direction is assumed sufficiently small so that the cosine of the vertical angle with the horizontal plane is unity. The balance of stresses results in and τ xb τ yb ∂h ∂h = – τ ----- – τ ----- + τ xx ∂x xy ∂y xz (2.60) τ ----- ∂h ∂h = – – τ ----- + τ (2.61) yx ∂x yy ∂y yz for the x- and y-directions, respectively. It is common practice to neglect entirely the slopes of the bottom boundary, ∂h/∂x and ∂h/∂y, so the first two terms on the right sides of the above equations can be eliminated. Then the dynamic equations become τ xb = τ and τ xz yb = τ , (2.62) yz which are the counterparts to equations 2.56 used at the surface. To evaluate equations 2.62, the fluid stresses (τ xz, τ yz) can be defined using equations 2.57 applied at the bottom of the estuary. It remains then to define the boundary stresses (τ xb, τ yb). τ xx τ yy 1 -- 2 τ ∂ xx – -------- Δ x ∂ x τ yx τ xy y 1 -- 2 τ ∂ yy + -------- Δ y ∂ y 1 -- 2 τ ∂ yx – -------- Δ x ∂ x 1 -- 2 τ ∂ xy + -------- Δy ∂ y z Datum h(x,y) τ xb τ yz τ xy x 1 -- 2 τ ∂ yz + -------- Δ z ∂ z τ yb 1 -- 2 τ ∂ xy – -------- Δ y ∂ y τ xz 1 -- 2 τ ∂ xz + -------- Δ z ∂ z τ yx τ yy 1 -- 2 τ ∂ yx + -------- Δ x ∂ x τ xx 1 -- 2 τ ∂ yy – -------- Δ y ∂ y Control volume 1 -- 2 τ ∂ xx + -------- Δ x ∂ x Figure 2.7. Viscous stresses on a bottom layer of a water body with boundary stress components τ xb and τ yb. Expressions in the form 1 τ -- represent the viscous stresses on each control volume face. yz 2 ∂τ yz + ---------- Δz ∂z
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A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

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