MIKE 3
MIKE 3 is a computer program that simulates flows, cohesive sediments, water quality,and ecology in rivers, lakes, estuaries, bays, coastal areas and seas in three dimensions. MIKE 3 was developed by DHI. MIKE 3 provides the simulation tools if you need to model 3D free surface flows and associated sediment or water quality processes. MIKE 3 is widely recognized as the gold standard for environmental and ecological studies. MIKE 3 can be used for assessment of hydrographic conditions for design, construction and operation of structures and plants in stratified waters, environmental impact assessment studies, coastal and oceanographic circulation studies, including fine sediment dynamics, sea ice simulations, lak hydrodynamics, water pollution studies and restoration projects, analysis of cooling water recirculation and desalination, water quality and ecological forecasting, phenomena whenever 3D flow structures is important.
Introduction
MIKE 3 is a generalised mathematical modelling system designed for a wide range of applications in areas such as oceanography, coastal regions and estuaries and lakes. The system is fully three-dimensional solving the momentum equation and continuity equationsin the three Cartesian directions.MIKE 3 simulates unsteady flow taking into account density variations, bathymetry and external forcing such as meteorology, tidal elevations, currents and other hydrographic conditions. MIKE 3 can be applied to oceanographic studies, coastal circulation studies, water pollution studies, environmental impact assessment studies, heat and salt recirculation studies and sedimentation studies. MIKE 3 is composed of three fundamental modules: The hydrodynamic (HD) module, the turbulence module and the advection-dispersion (AD) module. Various features such as free surface description, laminar flow description and density variations are optionally invoked within the three fundamental modules. A number of application modules have been implemented and can be invoked optionally. These are advection-dispersion of conservative or linearly decaying substances, a water quality (WQ)module describing BOD-DO relations, nutrients and hygienic problems, a eutrophication (EU) module simulating algae growth and primary production, and a mud transport (MT) module simulating transport along with erosion and deposition of cohesive material. A Lagrangian-based particle (PA) module can also be invoked for simulating e.g. tracers, sediment transport or the spreading and decay of E-Coli bacteria. The modelling system is based on the conservation of mass and momentum in three dimensions of a Newtonian fluid. The flow is decomposed into mean quantities and turbulent fluctuations. The closure problem is solved through the Boussinesq eddy viscosity concept relating the Reynold stresses to the mean velocity field. To handle density variations, the equations for conservation of salinity and temperature are included. An equation of state constitutes the relation between the density and the variations in salinity and temperature and ñ if the MT calculations are invoked ñ mud concentration. In the hydrodynamic module, the prognostic variables are the velocity components in the three directions and the fluid pressure. The model equations are discretised in an implicit, finite difference scheme on a staggered grid and solved non-iteratively by use of the alternating directions' implicit technique. A phase and amplification analysis neglecting effects of viscosity, convective terms, rotation, density variations, etc. has been performed. Under these circumstances, the finite difference scheme is unconditionally stable.
The transport of scalar quantities, such as salinity and temperature, is solved in the advectiondispersion module using an explicit, finite difference technique based on quadratic upstream interpolation in three dimensions. The finite difference scheme, which is accurate to fourth order, has attractive properties concerning numerical dispersion, stability and mass conservation. The decomposition of the prognostic variables into a mean quantity and a turbulent fluctuation leads to additional stress terms in the governing equations to account for the non-resolved processes both in time and space. By the adoption of the eddy viscosity concept these effects are expressed through the eddy viscosity, which is optionally determined by one of the following five closure models:a constant eddy viscosity; the Smagorinsky sub-grid (zero-equation) model; the k- (one-equation) model; the standard k-ε (two-equation) model; and a combination of the Smagorinsky model for the horizontal direction and a k-ε model for the vertical direction.
Governing Equations
In a three-dimensional hydrodynamic model for flow of Newtonian fluids, the following elements are required:namely mass conservation; momentum conservation; conservation of salinity and temperature; equation of state relating local density to salinity, temperature and pressure as well as to possible mud concentration. Thus, the governing equations consist of seven (possibly eight) equations with seven (eight) unknowns.
Advection-dispersion
MIKE 3 is applicable to flow problems in which density variations and turbulence are important features. The mathematical modelling of such flows requires the solution of partial differential equations of the advective-diffusive type. The flow modelling will require the solution of the transport equation for salinity, temperature (heat),turbulent kinetic energy (k-equation),dissipation of turbulent kinetic energy (ε-equation). The latter two equations form the well-known k and k-ε turbulence models. For the k-model and the standard k-ε model, the non-linear transport equations are solved by explicit UPWIND scheme. The one-dimensional (vertical) k-ε model essentially forms two one-dimensional diffusion equations, which are efficiently solved by an implicit scheme. The partial differential equations describing transport of salinity and temperature as well as transport of concentrations of substances, water quality and eutrophication components and mud concentration are all linear advective-diffusive type equations, and accordingly the same solution scheme is applied to all these components. A large number of methodologies for solving the advection-diffusion problem are reported in the literature. However, in order to be consistent with the HD module, a finite difference approach was chosen. The QUICKEST (Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms, Leonard (1979)) is applied. The method is based on a conservative control volume formulation. Upstream interpolation is used to determine higher order derivatives. This procedure avoids the stability problems of central differencing while remaining free of the inaccuracies of numerical diffusion associated with the usual upstream differencing. The extension of this scheme to two and three dimensions is given in Justesen et al. (1989), EkebjÊrg and Justesen (1991) and Vested et al. (1992). For use in situations where resolution of steep fronts are important, the scheme has been further improved by implementation of an exponential interpolation at steep fronts, the so-called QUICKEST-SHARP scheme, see also Leonard (1988). Alternatively, the so-called QUICKEST-ULTIMATE scheme, using operator splitting, may optionally be invoked, see e.g. Leonard (1991). This scheme is advantageous in cases with more than one advection-diffusion component, since in MIKE 3 it has been implemented such that the CPU time consumption is practically independent of the number of components.
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