dynamicwave

dynamicwave

Dynamic Wave equation

ResultQ = dynamicwaveq(Ldd, Qin, Hin,
                       ChannelBottomLevel, ChannelRoughness,
                       ChannelLength, ChannelBottomWidth,
                       ChannelDepth, ChannelForm,
                       FloodplainWidth,
                       TimeStepInSeconds, NrOfTimeSlices,
                       Structures, StructureA, StructureB, StructureCrestLevel)

ResultH = dynamicwaveh(Ldd, Qin, Hin,
                       ChannelBottomLevel, ChannelRoughness,
                       ChannelLength, ChannelBottomWidth,
                       ChannelDepth, ChannelForm,
                       FloodplainWidth,
                       TimeStepInSeconds, NrOfTimeSlices,
                       Structures, StructureA, StructureB, StructureCrestLevel)

ResultH,ResultQ = dynamicwaveq, dynamicwaveh(Ldd, Qin, Hin,
                       ChannelBottomLevel, ChannelRoughness,
                       ChannelLength, ChannelBottomWidth,
                       ChannelDepth, ChannelForm,
                       FloodplainWidth,
                       TimeStepInSeconds, NrOfTimeSlices,
                       Structures, StructureA, StructureB, StructureCrestLevel)

Argument	Type	[Units] Remarks
Ldd	ldd, spatial	Channel network
Qin	scalar, spatial	[m3/timestep] lateral inflow into the channel
Hin	scalar, spatial	[m] >= 0, water surface level at begin of timestep
ChannelBottomLevel	scalar, spatial	[m]
ChannelRoughness	scalar, spatial	> 0, if chezy: [m1/2sec-1] if manning: [m1/3sec-1]
ChannelLength	scalar, spatial	[m] > 0
ChannelBottomWidth	scalar, spatial	[m] >= 0
ChannelDepth	scalar, spatial	[m] > 0
ChannelForm	scalar, spatial	[-] >= 0
FloodPlainWidth	scalar, spatial	[m] >= ChannelBottomWidth + ChannelDepth*ChannelForm
TimeStepInSeconds	scalar, non-spatial	[sec] > 0
NrOfTimeSlices	scalar, non-spatial	[-] > 0
Structures	boolean, spatial	[-] False/True values expected on defined Ldd cells
StructureA	scalar, spatial	[-] values expected where Structures is True
StructureB	scalar, spatial	[-] values expected where Structures is True
StructureCrestLevel	scalar, spatial	[-] values expected where Structures is True
ResultQ	scalar, spatial	[m3/timestep] Dynamic wave flow along the defined LDD
ResultH	scalar, spatial	[m] New water level Dynamic wave flow along the defined LDD. The next timestep, this resultH can be used as Hin.

Definition of the channel arguments

Global options:

The algortithm is influenced by the options --chezy (default) or --manning, to select the dynamic flow equation (see below for these equations).

Operation

For an extended discussion of the Dynamic Wave equations is referred to textbooks on hydraulic modelling and hydrology (Chow and Chanson). The most important equation used is to calculate for each iteration step the dynamic flow as a function of water level and slope in water surface (Chezy equation):

\[ResultQ = ChezyCoeff * Aact * \sqrt{Sf} * \sqrt{R}\]

or, in case of the Manning equation:

\[ResultQ = 1.49/ManningCoeff * Aact * \sqrt{Sf} * R^{1.5}\]

In which Aact is the wetted surface, calculated from the ChannelBottomWidth, ChannelForm and Hin (water level), R is the wet perimeter, calculated from ChannelBottomWidth, ChannelForm and Hin, and Sf is the slope in water surface, so the waterlevel upstream minus waterlevel downstream cell.

The volume of water in each cell is updated for the outflow of that cell and the inflow from above cells. Based on the new volume, a new water level H is calculated.

This set of equations is evaluated through the entire channel network, for iteration steps which are equal to TimeStepsInSeconds/NrOfTimeSlices.

The approach taken for structures is a generic approach, which defines outflow from or overflow over the structure using the formula

\[\begin{split}Qstructure = \begin{cases} StructureA*((Hin-StructureCrestLevel)**StructureB) \text{ if $Hin > $StructureCrestLevel},\\ 0 \text{ otherwise}. \end{cases}\end{split}\]

Boundary conditions

Boundary conditions for the dynamic wave should be modelled as fixed water levels. In the situation of a fixed water level, within the dynamic section the points with fixed water levels should overwrite the results of the water level before using them in the next timestep.

Example:

initial
  FixedWaterLevel = Level.map;
  FixedPoints = Points.map;
  Hin = InitialWaterLevel;
dynamic
  …
  DynWaveQ, DynWaveH = dynamicwaveq,dynamicwaveh(LDD,Qin, Hin, ………);
  Hin = if(FixedPoints then FixedWaterLevel else DynWaveH);
  …