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

More documents

Recommendations

Info

22 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation In some cases, the temperature variations in an estuary may be small, in which case the effect of temperature variations on density will be minimal. Under these conditions it is acceptable to specify either a constant or slowly varying temperature distribution rather than compute the temperature distribution using the model. In the model described herein, the temperature distribution is assumed to be constant for the entire water body. Temperature predictions can easily be added to the model by including an equation for the conservation of thermal energy similar in form to equation 2.10, but with Θ ˜ substituted for s˜ and appropriate terms for heat sources and sinks. 2.3 Mean-Flow Equations Although equations 2.4, 2.9, and 2.10 describe the fluid motion and salt transport for both laminar and turbulent flows, it is not practical to use them in numerical calculations of turbulent flow. Turbulence in estuaries occurs over a wide range of eddy sizes; the largest scales are comparable to the basin dimensions and the smallest scales are determined by the fluid viscosity. The length, time, and velocity scales of the smallest eddies are referred to as the Kolmogorov microscales; for the open ocean, Fischer and others (1979, p. 58) estimate these scales to be about 0.1 cm, 1 s, and 0.1 cm/s, respectively. In an estuarine flow, these scales should be close to the same values. For numerical predictions, it is not practical on even the fastest computers to use such small spatial and temporal increments to determine the motions of the smallest three-dimensional eddies. Fortunately, the details of the fluctuating turbulent motion are rarely of interest in estuarine modeling studies. It is therefore common practice to average the instantaneous-flow and -transport quantities to separate the essentially stochastic (random) motions of the turbulence from the deterministic motions of the mean flow. One rigorous approach to the averaging of instantaneous quantities is to resort to the statistical method of ensemble averaging as described by Pritchard (1971a) for the 3-D estuarine model equations and as used in formal textbooks on the theory of turbulence (Monin and Yaglom, 1971, 1975). Ensemble averages have the advantage of encompassing all time scales of the stochastic motion, but they have the disadvantage that they cannot be measured in the field or laboratory. Bedford and others (1987) have introduced generalized filtering methods that incorporate high-order averages of the instantaneous quantities over both the space and time dimensions. These generalized filtering methods are theoretically appealing, but they introduce many additional terms into the governing equations that are not conveniently included in a model formulation; further research demonstrating the improvements to real predictions by using generalized filtering methods is needed before their use can be justified. The averaging that is used here is the conventional time averaging suggested in the late nineteenth century by Osborne Rey- nolds (Reynolds, 1894). It represents a field situation in which measurements are made at a fixed point in a water body and time averaged over some period (usually minutes) to determine a mean quantity. Reynolds suggested decomposing the instantaneous quantities into mean and fluctuating parts. For the instantaneous velocity, pressure, density, and salinity, the decomposition is ũ i = u i + u′ i , p˜ = p + p′ , ρ˜ = ρ + ρ′ , s˜ = s + s′ , (2.12)
where the mean quantities are defined by these time averages: 2. Governing Equations and Boundary Conditions 23 t+ T t + T t+ T t+ T u i = ∫ -- 1 ũ T i t dt ∫ 1 , p = -- p˜ dt T t ∫ 1 , ρ = -- ρ˜ dt T t ∫ 1 , s = -- s˜ dt T t . (2.13) The size of the averaging interval T is difficult to prescribe exactly, but in general it must be long compared to the time scale of the individual fluctuations but small compared to the time scale at which the unsteady mean flow is varying. In estuaries where the astronomical tides are a primary source of unsteadiness, the mean flow can vary significantly in just 15 minutes. The minimum required value for T will vary between flows and can truly be determined only by experiment. Indeed, a criticism of the time average is that T cannot always be chosen to include all of the time scales of the stochastic (turbulent) motion. An argument also can be made that an average should be taken in space as well as time to average the instantaneous quantities over the spatial turbulent eddy structure. The Reynolds time average, however, still is used extensively in practice and has not yet been shown to be inadequate in a significant way. Substituting the relations 2.12 into equations 2.4, 2.9, and 2.10 and time averaging each term in the way indicated by 2.13 yields the mean-flow and transport equations (For details, see most texts on fluid mechanics or turbulence such as Sabersky, Acosta, and Hauptmann, 1971; Hinze, 1975; and Tennekes and Lumley, 1972): Continuity equation, Momentum equation, Salt transport equation, Equation of state, ∂u i ------ = 0 ; (2.14) ∂x i ∂u i ∂u i ρ ------ 0 ρ ∂t 0 uj ------ ∂p ∂ ⎛ ∂u i ⎞ + = – ------ – 2ε Ω ∂xj ∂x ijk jρ 0 uk + ρg i + ρ 0 ⎜ν------ – u ∂x i j ∂x i ′ uj ′ ⎟ ; (2.15) ⎝ j ⎠ ∂s ∂u i s ∂ ∂s ---- + --------- = ⎛λ------ – u ∂t ∂x ∂x i i ∂x i ′ s′ ⎞ ; (2.16) ⎝ ⎠ i ρ = f( s, Θ) . (2.17) These equations are identical in form to the instantaneous equations except for the extra terms u i ′ u j ′ and u i ′ s′ that have been introduced by the Reynolds averaging process (the overbar represents the Reynolds time average). These terms originate with the nonlinear advection terms and are non-zero because correlations exist between the fluctuating velocities and between the velocities and salt fluctuations. In a physical sense, these terms, when multiplied by density, represent fluxes of momentum and salt associated with the turbulent fluctuations. The momentum flux acts as an effective stress on the fluid that is analogous to the viscous stress in gases caused by momentum flux associated with molecular fluctuations. The presence of the heretofore undetermined turbulent stress and salt transport terms in the mean flow equations make the number of unknowns greater than the number of equations. Resolving this imbalance between equations and unknowns is the problem of turbulence closure.
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 24 and 25: 10 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 86 and 87:
72 A Semi-Implicit, Three-Dimension
Page 88 and 89:
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
100 A Semi-Implicit, Three-Dimensio
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?