Introduction

Rigorous analysis of tidal crossings is difficult and is probably unwarranted in most cases (Neill, Guide to Bridge Hydraulics, Second Edition, June 2001). This is due, in part, to the assumptions that must be made in setting up the tidal analysis including:

Nevertheless, the SHA has determined that the TIDEROUT 2 program provides a reasonable Level 2 design approach for estimating storm tide elevations and velocities at bridge sites. Tidal velocities for Maryland estuaries which flow into the Chesapeake Bay are generally low and the TIDEROUT 2 Program typically provides for a conservative (high) estimate of these velocities. For very complex cases, the services of a specialist in tidal hydraulics are obtained.

The TIDEROUT2 Program provides the capability to analyze flow of riverine and tidal hydrographs through culverts and bridges and over roadways. The program output provides a listing of flow depths and velocities versus time. It can be used to evaluate the adequacy of the structure's waterway area, and to provide information necessary to evaluate scour.

Please note that the program uses the term "bridge" to represent culverts, arches or any highway structure over a waterway.

The program provides for great flexibility in use, allowing the user to start and end the analysis at any stage of the tide hydrograph. It accounts for overtopping flow over the roadway/bridge, and provides for various options to account for riverine flow from the upland drainage area. The program can account for reversal of flow during a tide cycle in the event that this condition develops as a result of the various factors affecting flow through the structure. A positive value for flow means the flow is moving downstream out of the tidal basin and a negative value means the flow is moving upstream into the tidal basin.

The typical problem involved in a tidal analysis consists of computing the stream hydrograph flowing into the tidal basin on the upstream side of the bridge, and then routing the combined stream flow/tidal flow through the bridge and over the road into the downstream tidal basin. The user needs to make certain decisions in order to use the program:

The reasoning behind the selection of these factors involves the application of the theory of probability to the two separate, but usually related, hydrologic events. For example, the tidal peak flow and the riverine peak flow for a hurricane may coincide at a bridge over a small tributary near its confluence with the Chesapeake Bay. On the other hand, the riverine peak flow at the Potomac River tidal reach may not occur until several days after the tidal surge has passed.

SHA recommends an empirical "worst case" approach to this problem, at least for the initial computer runs. The time of concentration of the riverine flood peak can be computed with TR-20. The user can then use the TIDEROUT 2 program input values to adjust the relative timing of the riverine peak flow with the tidal peak flows to determine a worst case scenario. Such adjustments are quickly and easily made through a trial and error process.

This tide-routing program is similar to the reservoir routing program in the TR-20 program. The basic flood routing equation, based on the continuity principle, governs the flow through a structure.

QI-dS/dt=QO+QW


Where
  QI = Incoming discharge of the stream
  QO = Discharge through bridge opening
  QW = Discharge overtopping the roadway
  S = Basin storage
  t = time
This equation states that the incoming stream discharge minus the rate of increase in the basin storage (where the rate of change of storage can be either positive or negative) equals the sum of the discharges through the bridge opening and over the roadway.

This equation is modified in the TIDEROUT2 program to account for the effect of tides moving through the structure.

Tidal basin storage rate of change can be written as

dS/dt=As*dHs/dt	
	
Where Hs =Water elevation in the basin (headwater)
      As =basin surface area at Hs
      dHs/dt = rate of change of the basin elevation with time. 
Flow through the bridge opening is modeled as following

When the difference in elevation of the flow from upstream to downstream (hydraulic gradient) of the bridge exceeds one-third of the flow depth upstream of the bridge, critical depth will occur and inlet control will govern. In all other cases, outlet control is assumed to govern.

For the outlet control case, the flow through the bridge opening can be treated as flow through an orifice

QO=Cd*Ac*sqrt(2*g*(Hs-Ht))  when Hs>Ht
QO=-Cd*Ac*sqrt(2*g*(Ht-Hs)) when Hs<Ht

Where  Cd = dischrage coefficient and is assumed to be 0.6 for a typical bridge
       Ac = flow area through bridge opening
       Ht = tidal water elevation downstream of the bridge
       sqrt() stands for square root of the number inside the ()
Note that when Hs is less than Ht the flow reverses, flows into the tidal basin and the discharge becomes negative

The tidal water elevation Ht downstream of the bridge is assumed to fluctuate as a cosine function of time and can be expressed as

Ht=A*cos(2*π*(t-tp)/T)+MEL

Where A =one-half the tidal wave amplitude (half-amplitude)
      tp =the peak time in hour
      T =tidal period normally 24 hour
      MEL =tidal mean water elevation
			       
For the inlet control case, the flow can be calculated at the control section that the flow depth is the critical depth
QO=Cd*Acr*sqrt(g*dcr) =Cd*Acr*sqrt(g*d*2/3)

Where  Acr =critical flow area
       d   =flow depth at upstream section of the bridge or culvert, measured from the invert of the structure.
       dcr =critical depth and is equals to 2/3*d
       g = 32.2 ft/sec2 for English units and 9.81 m/sec2 for metric units
Flow overtopping the roadway can be treated as a broad-crested weir
QW=Cw*L*(Hs-Ht)1.5 when Hs >Ht

QW=-Cw*L*(Ht-Hs)1.5 when Hs <Ht			 	

Where Cw =discharge coefficient and is normally 2.6 for English units and 1.44 for metric units
      L = length of the weir
Note: For the weir equation, Hs or Ht will become the roadway elevation when either one is lower than the roadway. Since the roadway profile may not be level, an integration of the weir flow over the roadway under water is used to obtain the total weir discharge.

To solve the continuity equation mentioned above, an improved Euler's method is used to solve the equation at next time step. In the following equations, the subscripts 1 represents the current known time step, and 2 represents the next time step to be solved. Only positive flow is shown, although the reverse flow case is similar

(QI1+QI2)/2-(As1+As2)/2*(Hs2-Hs1)/dt=QO1-2+QW1-2

for outlet control case 

QO1-2=Cd*(Ac1+Ac2)/2*sqrt(2*g*((Hs1+Hs2)/2-(Ht1+Ht2)/2)

for inlet control case

QO1-2=Cd*(Acr1+Acr2)/2*sqrt(2/3*g*(d1+d2)/2)


for weir flow

QW1-2=Cw*L*((Hs1+Hs2)/2-(Ht1+Ht2)/2))1.5

Note, when Ht<Roadway elevation, the roadway elevation will replace Ht.
 
The only unknwon in this equation is Hs2, since As2 and Acr2 are functions of Hs2. An iteration process is used to solve this non-linear equation.

The accuracy and the stability of the solution of this finite difference equation depend on the time step (dt). The smaller the value of (dt) the more accurate and stable the solution. Normally (dt) should be in the range of 0.1 hour to 0.2 hour. If the output hydrograph shows a serious oscillation, the solution is unstable and the user may need to reduce the time step in order to obtain a stable solution.

The Output Column 7 , Bridge V, lists the velocity of the flow at the control section for the given time step. This is not necessarily the same as the outlet velocity for the structure. A hydraulic analysis will serve to provide a more accurate estimate of the outlet velocity at the structure.

The equations used for in TIDEROUT 2 for critical flow conditions for discharge and velocity are based on the flow depth at the upstream side of the structure, assuming a rectangular channel. For an irregular waterway area under a structure, critical flow conditions of discharge and velocity should technically be based on the hydraulic depth of the flow where hydraulic depth is defined as waterway area/ top width. In most cases, the difference between these two ways of defining the upstream flow depth will be within an acceptable range, consistent with other assumptions used in the tidal routing procedure.