Turbulent Flow in Open Channels

Although the turbulent flow in open channels is also modeled using 3-node network elements, the solution procedure is quite different. This is because the flow direction is usually known and hydraulic jumps (Froude number equal to 1; this is the equivalent of sonic conditions in aerodynamic flow) can occur within an element, also when the cross section is constant. This makes the approach of setting up a nonlinear system of equations which is to be solved by repeatedly linearizing the system very difficult. The usual approach to the steady state equations of turbulent flow in open channels is as follows (focussing at first on one channel consisting of one line of elements):

• Determine the upstream element of the channel (usually a sluice gate or weir)
• Assume supercritical flow, i.e. the flow is controled by upstream conditions
• Proceeding from upstream to downstream, calculate the fluid depth.
• If for some reason supercritical flow is not feasible (e.g. the critical depth is reached somewhere along the channel), proceed downstream to the next element and so on until there is again a supercritical solution, e.g. starting at node A. Before going on with this supercritical solution start from node A upstream calculating a subcritical flow until a jump occurs or the upstream element in the channel is reached. Then, continue downstream calculating a supercritical flow starting at node A.
• if the downstream element in the channel is reached (usually a reservoir) check for a fall or jump in the reservoir. Else start a subcritical calculation starting at the entry of the reservoir (raccordation) and proceed upstream until a jump is detected or the upstream element is reached.

Starting this iterative procedure at the upstream element is logical, since it usually allows for the determination of the mass flow (a reservoir doesn't) and if there is some section in which the flow is supercritical it is governed by the upstream conditions. The unknowns are the temperature, the fluid depth (measured in the direction of the gravity vector; the specification of the gravity vector with the *DLOAD card is mandatory for channel calculations) and the mass flow. They correspond with the degrees of freedom shown in Table 24. The critical depth is actually not an independent unknown, it is calculated as soon as the mass flow is known.

Table 24: Degrees of freedom in open channel nodes.

DOF	corner node	midside node
0	temperature	-
1	-	mass flow
2	depth	-
3	critical depth	-

Right now, the fluid element routines are the sluicegate (includes the weir), the straightchannel, the reservoir and the contraction (includes the enlargement and the step) routines. The IO elements (input/output) do not need a routine and are not considered to be “true” elements. Field ineighe(i) gives the number of true elements in end node i. Fields iponoel and inoel allow for the determination of ALL elements (true and IO) belonging to node i. Entry iponoel(i) points to a first element in inoel(1,iponoel(i)), inoel(2,iponoel(i)) points to the next element etc. up to the point where inoel(2,...)=0. The upstream, middle and downstream node of an element nelem are called nup, nmid and ndo, respectively, the upstream and downstream neighbor (assuming there is only one) nelup and neldo. The upstream and downstream depth is hup and hdo.

There are extra routines for the calculation of:

• the critical depth hk (“kritische hoogte” in Dutch) (hcrit.f)
• the normal depth he (“hoogte van eenparige beweging” in Dutch) (hnorm.f)
• the depth h2 after a jump corresponding to the depth h1 before (hns.f; “hoogte na sprong” in Dutch)

The friction can be of type White-Coolebrook (xks $>$ 0) or Manning (Manning coefficient = -xks, xks $<$ 0). For White-Coolebrook the friction coefficient $f$ is calculated from the grain size xks in friction_coefficient.f.

The mode of calculation (from upstream to downstream, i.e. supercritical, or from downstream to upstream, i.e. subcritical) is denoded by the variable mode (character*1) taking 'F' for forward or 'B' for backward, respectively. If the forward mode (frontwater curve) is temporarily interrupted to calculate a backwater curve the switching point (point A above) is stored on a stack: the number of stored switching points is nstack, for switching point i the element upstream of A is stored in istack(1,i), and the node corresponding to A in istack(2,i).

The above iterative procedure leads to the following algorithm:

• Look for a sluice gate or wear element one of the end nodes of which is connected to only one “true” element (i.e. the element itself). This is the starting point of a frontwater curve calculation
• Proceed downstream calculating a supercritical flow. Determine the next downstream element using the actual element and its downstream node (nelup and nup for the downstream element) and fields iponoel and inoel.
• If no supercritical flow is possible, set hup = -1 and move downstream until supercritical flow is feasible again, e.g. at node A. Store the appropriate information of this switching node in fields nstack and istack. Important here is that a negative depth points to the unfeasibility of a frontwater curve starting at that node.
• proceed from node A upstream with subcritical flow. For each element use its number and its upstream node to determine the next upstream element. Continue this until a jump is detected or the originating sluice gate or weir is reached. Then, proceed downstream with the most recently stored information on stack
• when arriving at a reservoir, check for a fall or jump in the reservoir. Else start an upstream calculation as detailed before.

Now, some more detailed information on the true elements is given.