Domains tab - Embedded 1D Structures

In this subtab, you can provide details about 1D structures embedded into the 2D domain.

Theory

A simple 2D model will utilise topography data represented by a single digital terrain model (DTM or DEM). However, it is likely the modelled area is far more complex than this, with variations in ground and land use types (equated to roughness variations in the model) and additional flow paths, e.g. culverts, narrow pathways, etc, not identified in the terrain data.

Additional flow paths can be incorporated into a model by the introduction of additional topographic features. However, it might be important to understand the specific flows within certain features, e.g. if an overloaded culvert could cause flows to back-up, leading to additional flooding in the upstream area. This information will not be provided if these structures are only considered as topographic features or roughness modifications within the 2D model. A more detailed assessment of flows in such structures is provided by modelling them using bespoke 1D components within 2D model.

Flood Modeller provides two options for linking a 2D model with 1D elements representing structures within the 2D domain:

1. Add embedded structure file(s) to a 2D model – this option can be instead of or in addition to a linked 1D River network. The 1D calculations for this option are incorporated into the 2D solver and are only utilised when water flows through the structure. Hence it is not necessary to maintain a flow through the structure throughout a simulation. This makes the connection easier to define and more stable within simulations.

   Specific tools within the user interface enable embedded structure data to be specified. This is stored in an open text file format with a file extension “.str” (i.e. known as a structure file). In addition, shapefiles are required to define the structure location within the 2D domain. The 2D solver will exchange hydraulic information between the structure and the 2D cells underlying the associated polyline shapefile.  

2. Dynamically link a full 1D River network – this option has been available for a number of years. The linked 1D River model has to be a standard network, i.e. include upstream and downstream boundaries and be capable of running independently (using the 1D River solver). The latter requirement means that the defined 1D elements must always remain wet (1D River solver cannot model a dry channel). Thus, if the modelled structure is located in an initially dry flood plain, it will require an additional “dummy” constant flow to ensure it remains wet throughout a simulation and thus the 1D components remain stable. The dummy flow can hopefully be set to a nominal level so as not to significantly influence results. Alternatively, abstractions can be introduced to remove dummy flows before they spread onto the flood plain.

   Shapefiles are required to define the location within the 2D domain that will link to the 1D River network. The 2D solver will exchange hydraulic information between the 1D River and the 2D cells underlying the associated shapefile (must be a polyline).

   Although this method works and can produce good results, it can prove difficult to configure embedded structures in this way as models tend to be sensitive to stability issues.

Flow modifications for 1D hydraulic structures

A wide range of versions of empirical equations are available to account for the effects of different structure shapes and functionalities (e.g., broad crested vs sharp crested weirs, or orifices vs inverted syphons). For the embedded structure options within the 2D solver, currently only the general cases for weir, orifice and culvert flows have been considered (the range of options will be extended in the future).

Orifice

Four flow modes are considered when modelling flow through an orifice: dry structure, free weir flow, drowned weir flow and orifice flow. The flow mode is identified considering the relations that involve the upstream bed level (z_up), the upstream sill level (z_cup), the invert level of the orifice throat (z_inv), the water levels above the invert level of the orifice aperture upstream (y₁) and downstream of the orifice (y₂) and a modular limit (m). Flow discharge through the structure is obtained as the following Q_dry, Q_free, Q_drowned or Q_ori depending on flow mode:

1. Dry structure (z_up < z_cup):

   Q_dry = 0

2. Free weir flow (y₁ – z_inv ≤ αh, y₂ – z_inv ≤ h and (y₂ – z_cup) / (y₁ - z_cup) < m):

   Q_free = (2/3)^3/2 g^½ C_weir b (y₁ – z_cup)^1.5 (rectangular)

   where g is the gravity acceleration constant, C_weir is the weir discharge coefficient and b is the breadth of the orifice.

   Q_free = C_weir β d^2.5 (circular)

   where C_weir is the discharge coefficient, β is the discharge factor for circular weirs (Bos, 1989) and d is the diameter of the orifice aperture. 

   α = 1.5 (rectangular case) or 1.25 (circular case) and m is the modular limit.

3. Drowned weir flow (y₁ – z_inv ≤ αh, y₂ – z_inv ≤ h and (y₂ – z_cup) / (y₁ – z_cup) ≥ m):

   Q_drowned = F_d Q_free

   where F_d represents the drowning factor obtained as F_d = {[(y₂ – z_cup) / (y₁ – z_cup)] /(1 – m)}^½ or F_d = [1 – (y₂ – z_cup)/(y₁ – z_cup)] / [0.3 (1 – m)] if the first formula gives F_d < 0.3.

4. Orifice flow (y₁ – z_inv > αh, y₂ – z_inv > h):

   Q_ori = C₁ C_full A (2g Δh)^½

   where C₁ = 0.799 (rectangular) or 0.6 (circular), C_full is the orifice discharge coefficient and Δh = min(y₁ – y₂, y₁ – 0.8 h) (rectangular) or min(y₁ – y₂, y₁ – 0.5 d).

The flow modes as output to the Embedded Structure summary time series file are as follows:

Weirs

Three flow modes are considered when modelling flow through a weir: dry structure, free flow and drowned flow. The flow mode is identified considering relations that involve the water levels at upstream (y₁) and downstream of the weir crest (y₂), the weir crest level (z_c), the water level differences h₁ = y₁ – z_c, h₂ = y₂ – z_c, and a modular limit (m). Flow discharge over a weir can expressed as the following Q_dry, Q_free or Q_drowned depending on the flow mode (for the general type of weir):

1. Dry structure (y₁ < z_c and y₂ < z_c):

   Q_dry = 0

2. Free flow (y₁ > z_c, h₁ > h₂ and h₂/h₁ ≤ m):

   Q_free = C_d C_v (2/3)^3/2 g^1/2 b h₁^e

   where C_d and C_v are the coefficients of discharge and approach velocity respectively, g is the gravity acceleration constant, b is the breadth of the weir crest and e is an exponent that depends on the shape of the control section.

3. Drowned flow (y₁ > z_c, h₁ ≥ h₂ and h₂/h₁ > m):

   Q_drowned = C_d C_v (2/3)^3/2 g^1/2 b h₁ [(h₁ – h₂) / (1 – m)]^1/2

The flow modes as output to the Embedded Structure summary time series file are as follows:

Culverts

Flow through culverts is subjected to head losses due to flow expansion/contraction effects at the inlet/outlet ends, change of flow direction, barrel slope and friction. Culverts act as waterways transferring 2D flow from one end to the other.

Flow through the culverts is predominantly determined by the solution to the Saint Venant equations; in case one end is not filled with water, this transitional state is simulated by solving the Saint Venant equations discretized with the shock-capturing, monotonic upstream centred scheme (MUSCL; Toro, 2001).

Head losses may also be modelled at their respective locations along the culvert due to inlet, outlet and bend losses.

When both ends are filled with water, three flow modes are considered: free culvert flow, drowning culvert flow. surcharged flow. Two types of flow control are then identified: inlet and outlet control according to the location of the hydraulic control section. This leads to the expansion of this classification considering inlet and outlet sub-modes for each of these three modes. A detailed description of each flow mode is given in the Culvert Design Guide of CIRIA (Construction Industry Research and Information Association,1997, 2010).

The flow modes as output to the Embedded Structure summary time series file are as follows:

Links between 2D flow regime and 1D elements: Embedded structures approach

This approach introduces an alternative to defining “complete” 1D River networks to link with a 2D model. Instead it allows just the key 1D elements to be defined in isolation and embedded within the 2D domain. The approach is limited to the three structure types described above. It is based on the following assumptions:

• Polyline shapefiles are used to (a) identify cells of the 2D domain that are linked to 1D flow models that are represented by the embedded structures and (b) identify the flow direction through the structure with respect to the coordinate system of the associated 2D domain:

  • Polylines that represent culvert and orifice links describe the direction of the 1D flow (flow through the structure) - hence, the first and last polyline points represent the inlet and outlet sections of the structure, respectively.

  • In the case of embedded weirs, the polyline identifies the location of the overflow section (or notch section), 1D flow is perpendicular to the direction of the polyline. For embedded weirs, the order of the polyline points is important for defining the direction of the 1D flow (flow through the structure): the first point of the polyline represents the reference point of a left-handed coordinate system where x-axis (first finger) represents the direction of the 1D flow and y-axis (index finger) represents the overflow section; the last point of the polyline indicates where the positive direction of the y-axes lies.

• Fluid motion through structures such as conduits, weirs and orifices is modelled using empirical equations for discharge as for example given in equations above.

• Given that empirical models, such those given above, apply for 1D flow conditions, the size of the link areas should be, relatively, smaller than the size of the 2D domain, i.e., only a few cells of the 2D grid.

The (1D) empirical models are used to determine any changes to the 2D fluid mass due to the flow through the structure at a pseudo time step between two consecutive time steps of the 2D solver. The values of water elevation at a time instant t_n of the 2D flow solution are used to quantify discharge through the structure between time instants t_n and t_n + Δt. This discharge is then translated to a rate of change of the water depth and applied to the 2D flow at the time instant t_n + Δt by enforcing mass sources and sinks on the 2D shallow water equations at the 2D cells that are assigned to the flow sections of the embedded structures (i.e., inlet/outlet sections and weir overflow section). This discharge translation to mass source/sink terms is applied considering the difference in size between the width of a structure section and the length of the line formed by the 2D cell(s) assigned to that section when transferring information between the 1D and 2D flow models. The application of a mass sink takes account of the availability of water at the upstream 2D cells (to avoid errors in the mass balance of the 2D flow). For embedded weirs the application of a mass source takes account of bed elevation at the downstream 2D cells (to avoid flow being transferred against gravity to downstream cells where bed is higher than the weir crest level).