Open Boundary Conditions

An open boundary can be identified by a volume free node with only one connecting reach.

Such a node must have a boundary conditions set before the flow equations can be solved.

The following sections describe each of the possible open boundary conditions that are supported.

Closed boundary

A closed boundary specifies that no water flows in or out of the model through this boundary, and corresponds to setting a wall at the end of the reach, hence it is also called a wall boundary.

Inflow boundary and lateral inflow

An inflow boundary specifies an amount of water that flows into or out of the model. The amount of water is specified as a discharge. The sign of the boundary matches the discharge sign in the reach, i.e. inflow must be specified positive in the upstream end and negative in the downstream end, and opposite for outflow.

The inflow boundary sets up the equations as if there was another reach connected at the boundary that had a flow through the reach matching the specified discharge.

As an alternative, you could specify a closed boundary condition and a lateral inflow at the boundary. That would discharge the same amount of water at the same place into the model. The difference between the two is that for an inflow boundary the water enters the model with momentum, while for the lateral inflow the water enters the model without momentum. When the water does not have momentum, it needs to be accelerated before it moves, giving higher water levels around the closed boundary with lateral inflow than the inflow boundary condition would give.

An inflow boundary can be specified with or without AD components. If specified without, a zero value is assumed for all AD components.

An inflow boundary with constant zero inflow corresponds to a closed boundary, see Section 5.1.1.

Water level boundary - the outlet

A water level boundary specifies a water level at the boundary.

Water that runs in/out of the model does so with momentum, as is the case with the inflow boundary.

If water flows out of the model, the model applies a free outflow condition, if the free outflow water level is higher than the water level from the boundary. The condition looks like:

\[ H = \text{max}\left(H_{out}, H_{b} + \text{min}\left(y_{n}, y_{c}\right)\right) \]

Equation: Free outflow boundary condition preserving critical depth or bottom elevation.

where

H_out         : water level from boundary condition y_c         : critical depth for the current flow velocity y_n         : natural depth for the current flow velocity

Otherwise, the model applies the specified water level at the boundary, with the corresponding backwater effect and a possibility for inflow.

The free outflow behaviour can be disabled in the HD parameters

A water level boundary can be specified with or without AD components. If specified without, a zero value is assumed for all AD components.

River (MIKE 11 / MIKE HYDRO)

In MIKE 11, currently the use of the natural/critical free outflow condition is disabled.

QH boundary

A QH boundary is a type of inflow boundary, where the inflow Q is found as a function of the current water level H.

The relation between Q and H is specified in a QH table. For every water level H, the corresponding Q value is linearly interpolated in the QH table and used as the inflow boundary value. A QH boundary can specify inflow as well as outflow for different water levels.

When looking up the value of Q in the QH table, the value of H is only known at the previous time step. To obtain a more accurate value of the discharge, a Taylor expansion is used:

\[ q^{n+1} = Q(H^n) + \left.\frac{\partial Q}{\partial H}\right|^{n} \cdot \frac{H^{n+1} - H^n}{\Delta t} \]

Equation: QH boundary condition using Taylor expansion to improve accuracy with temporal derivative.

where \(Q\left(H^{n}\right)\) and \(\partial Q/\partial H\) are calculated from the QH table for the previous water level \(H^n\) and used when solving the implicit equation system.

A QH boundary can be specified with or without AD components. If specified without, a zero value is assumed for all AD components.

Which boundary condition?

The choice of the boundary condition depends on the physical situation being simulated and the available data.

Typical upstream boundaries could be:

• Discharge or water level of a reservoir.
• A discharge hydrograph of a specific event.

Typical downstream boundaries could be:

• Water level, e.g. in a large receiving water body.
• Tidal cycle water level.
• A QH rating curve, e.g. from a gauging station.