Target values
1.21 Kinematic Routing Method
General
Two different routing methods have been incorporated into MIKE 11 HD. At a given time level, the routing methods are used to determine the spatial varia-tion of the discharge at Kinematic Routing branches. The implemented meth-ods are:
Muskingum method
Muskingum-Cunge method
Similarly, two methods have been incorporated into the model, which are used to compute the water level at Kinematic Routing branches:
User-defined discharge-elevation method
Resistance method
Definition of a Kinematic Routing Element
Figure 1.50 shows an example of a Kinematic Routing branch, at which two Kinematic Routing points are defined. Each point is located at a discharge point and the temporal transformation of the discharge takes place at this grid point. The quantification of the discharge at a Kinematic Routing point is obtained by considering the inflows to the Kinematic Routing element in question, cf. Figure 1.50. The inflows constitute the discharge at the
upstream elevation point (including lateral inflow at that point) and the sum of the lateral inflows at all interior elevation points of the considered Kinematic Routing element.
The extent of each Kinematic Routing element is governed by the number of Kinematic Routing points specified at each Kinematic Routing branch. The upstream boundary of an element is determined as the larger of the following:
Chainage of upstream node (elevation point)
Chainage of elevation point located just downstream of closest upstream Kinematic Routing point (elevation point)
The definition of a Kinematic Routing element ensures that the entire branch is described correctly regardless of the number of routing elements specified by the user. Note that a Kinematic Routing element is located from one eleva-tion point to the other, while the Kinematic Routing point is located at a dis-charge point.
Figure 1.50 Definition sketch of MIKE 11 Routing branch, Kinematic Routing ele-ments and transformation points.
Figure 1.51 Successive computation of discharge at Kinematic Routing branches.
The numbers denote the branch calculation order for a Kinematic Rout-ing branch network.
Discharge Computation
At Kinematic Routing branches, the discharge is computed in the direction of positive flow. In order to ensure that water is discharged correctly from fur-thest upstream in the river network the discharge computation starts at the upstream end of those Kinematic Routing branches located furthest upstream in the river network. At a given discharge point this concept ensures that the
discharge values used as input to the Kinematic Routing method at that point have been updated beforehand. In effect, this means that the Kinematic Routing branches are updated in accordance with Figure 1.51.
The Muskingum method is a hydrologic routing method used to handle a var-iable discharge-storage relationship. The method models the storage volume of flooding in a river by combination of wedge and prism storages. During the advance of a flood wave, inflow exceeds outflow, producing a wedge of stor-age. During the recession, outflow exceeds inflow, resulting in a negative wedge. The Muskingum method is written:
(1.154)
where indices i and j, respectively, refer to the considered grid point and time level. The variables, C1 - C4, are given by
(1.155)
(1.156)
(1.157)
(1.158)
In the Muskingum method the computational input parameters, K and x, are fixed in time and space. Since in MIKE 11 HD the time step is fixed, too, Eqs.
(1.155) - (1.158) need to be evaluated only once.
At a Kinematic Routing points the discharge variable is updated from one time level to the next on the basis of the previous and new discharges at the location of the upstream boundary of the considered Kinematic Routing ele-ment as well as the previous discharge at the Kinematic Routing point in question, cf. Figure 1.50. Moreover, the total lateral inflow to the Kinematic Routing element (denoted Qlat) is evaluated as the mean of the previous and new total lateral inflows for the considered Kinematic Routing element.
In contrast to the Muskingum method, the Muskingum-Cunge method does not take any input parameters, since these are computed at each Kinematic Routing point and at each time level.
The Muskingum-Cunge method is based on the diffusion wave model, which neglects local and convective acceleration terms but includes the pressure Qij++11 = C1Qij+1+C2Qij+C3Qij+1+C4
term. Thus, the Muskingum-Cunge model approximates the diffusion of a nat-ural flood wave.
The Muskingum-Cunge computation uses the same equations as the Muskingum method, but the parameters K and x are functions of time and space. They are given by
(1.159)
(1.160)
(1.161)
In Eqs. (1.159) - (1.161), x is the length of the considered Kinematic Routing element and S0 is the bed slope. The variables B, A and Q, respectively, are the width, the cross sectional area and the discharge, all of which represent the Kinematic Routing element in question. The variables x and t are space and time variables, respectively.
The variables, B, A and Q are evaluated at the elevation point just upstream of the considered Kinematic Routing point. Since at this water level point the computation of the water level depends on the new discharge computed at the Kinematic Routing point, the discharge at each Kinematic Routing point is obtained by iteration. Typically, about 10 iterations are required to obtain an accuracy better than 10-4.
Computation of Water Level
Having computed the discharge at the Kinematic Routing branch under con-sideration, all the interior elevation points of that branch are computed. Since Kinematic Routing points are defined as Q-points, computation of the water level may require interpolation between upstream and downstream informa-tion. Consequently, both an upstream search and a downstream search for a Kinematic Routing point is done. In the case of both an upstream and a downstream Kinematic Routing point, the water level is computed at both Kinematic Routing points and a linear interpolation is made to obtain the water level at the water level point in question.
Should the search efforts fail to find an upstream and a downstream Kine-matic Routing point, the Manning equation is used to compute the water level.
Regardless of the method used to compute the water level, a check is made to ensure that the computed water level is not located below the top of the slot.
At nodes of Kinematic Routing branches, the water level is computed using the same procedures as currently used in MIKE 11 HD. This concept ensures that the water level is continuous from one branch to the other.
If, at a Kinematic Routing point, the user has chosen a QH-relation the dis-charge computed by use of a Kinematic Routing method is used to interpo-late in the QH-table at the elevation point in focus.
If, at a Kinematic Routing point, the user has chosen the Manning resistance method the discharge is computed by employment of the Manning equation.
It is written
(1.162)
in which M is the Manning resistance number, R is the hydraulic radius, S0 is the bed slope, and A is the cross sectional, wetted area. Since M, R, and A are functions of the water level, the cross sectional area is found by iteration.
Once the area has converged to a satisfactory accuracy (10-3), the water level is looked up in the cross section table of MIKE 11 HD.