Full Equations Utilities (FEQUTL) Model for the Approximation of Hydraulic Characteristics of Open Channels and Control Structures During Unsteady Flow

U.S. GEOLOGICAL SURVEY WATER-RESOURCES INVESTIGATIONS REPORT 97-4037

3.2 Channel Expansions and Contractions

The expansions and contractions considered in this section are large enough that critical controls may be present or are located at a junction among branches. The eddy losses at minor variations in channel shape and size can be represented within a branch (see section 5.5.3 in the documentation report for the Full Equations model (Franz and Melching, 1997)). The expansions and contractions considered here may result from both natural and constructed changes of channel cross-sectional shape.

A 2-D table of type 14 is computed in the EXPCON command (section 5.7) in FEQUTL to approximate the flow through a transition in channel cross-sectional shape. The transition is defined by an upstream cross section, a downstream cross section, the respective bottom elevations, the distance between the cross sections, and parameters related to the computation of friction and shock losses. Flow through a channel transition is a complex phenomenon, and evaluation of losses is difficult. The shock losses are approximated in EXPCON by a constant fraction of the difference in the true velocity heads where the constant differs for contracting and expanding flows. This approximation has been frequently used in hydraulic engineering, and most handbooks have recommended ranges for the loss coefficients. The approximation is crude, but no suitable replacement is readily available.

The principal difficulty with flows in transitions is that the velocity distribution changes from section to section in a manner only slightly affected by the channel boundary. This is especially true of expanding flows with possible channel-wall-separation effects and the attendant, inherent instability in the velocity distribution. This unpredictable variation in velocity distribution makes estimation of the losses difficult. Simplifying assumptions are made in the EXPCON command so that flow computation is possible. Some of the major assumptions are given below.

1. The velocity-head (energy-flux) correction factor, , for both the upstream and downstream sections, is a function of depth in the section and is computed in the cross-section table commands in FEQUTL. This means the velocity distribution is determined by the boundary geometry alone. This is only approximately correct and may be substantially incorrect if the transition is an expansion. However, little other choice is available for reasonably determining flow through the transition without making each transition a research project.


2. The pressure distribution at both the upstream and downstream cross sections is hydrostatic. This should be satisfied so long as the sections are away from regions with pronounced vertical or horizontal curvature (or acceleration).


3. The losses resulting from boundary friction can be estimated by a mean friction slope multiplied by the distance between the two sections. Several options are available for computing the mean friction slope. The user also may request that boundary-friction losses be ignored in the computations. Given the uncertainties of the loss estimation, ignoring the boundary-friction losses is often a reasonable option.


4. Control by critical depth will be located at either the upstream section or the downstream section. The possibility that the control is at some intermediate point is ignored. Determination of the location of the actual control point in a transition is difficult, and again the uncertainties inherent in the calculation of flows in transitions do not justify the additional effort and assumptions required to determine the location of a control at an intermediate point. In any case, it is unlikely that the control would remain at an intermediate point over more than a narrow range of flows. Thus, although such a control is possible, it is unlikely to practically affect flows at the level of detail of interest in applications of FEQ.


5. Critical flow is as defined in the cross-section tables computed in FEQUTL. The critical-flow value is affected by the choice of parameters when the cross-section table is computed. Application of the NEWBETA option(sections 5.8 and 5.9) will produce results that differ from those produced without its application. A critical-flow value that reflects the effect of velocity distribution is computed in the NEWBETA option. In all other cases, critical flow is computed as if the velocity distribution were uniform across the cross section and = = 1. Thus, if the channels under consideration are not compact, then NEWBETA should be applied. The critical flow in closed conduits with flow transitions must be limited by application of the QCLIMIT command (section 5.18) before the EXPCON command is invoked.


6. Critical flow is meaningful in a flow transition, and the water-surface curvature introduced may invalidate the assumption of hydrostatic pressure distribution. However, the transitions often are short and the often-substantial water-surface curvature resulting from critical flow in the transition may not affect the computations in FEQUTL. A further problem is that critical flow may result at the downstream section even though the flow is expanding in a situation where the flow area at the critical section is larger than the flow area upstream. If the cross-section shapes were the same, then this situation would not be possible. However, changes in cross-section shape also may be simulated in EXPCON. When the cross-section shape changes in the transition, critical flow can result at the section with the larger area in FEQUTL computations.


7. Computation of the losses in the transition in channel cross-sectional shape as a fraction of the difference in velocity heads between upstream and downstream sections applies in the limit as flow becomes critical. The formula of head loss as a function of velocity-head difference was derived for subcritical flows not close to critical because most designs of transitions in cross-sectional shape avoid flows near critical.


8. The loss formula can be smoothed to represent losses when the flow areas are nearly the same but the cross-section shapes are not. The uncorrected formula results in a zero loss at that point. Some losses must result because the change in shape can be large even though the flow areas are the same. This illustrates a deficiency of applying any formula that depends on velocity or velocity head differences alone. Again, little alternative is available to this approach, and smoothing over this region as outlined in the following section results in losses that are consistent with the simple loss formula.

3.2.1 Governing Equations for Expansions and Contractions

The principal equation used to estimate the flows in a transition is

(72)

,

where z_m is the elevation of the minimum point, is the mean conveyance, x is the location of the section along the channel, and k_ec is the loss factor depending on the sense of the transition. The sense of the transition denotes whether the flow is expanding or contracting. Another convenient and perhaps preferable description is to define the flow in the transition as either accelerating or decelerating. The flow is accelerating in a contraction and decelerating otherwise. The subscripts denote the section with section 1 the upstream section and section 2 the downstream section of the transition.

Equation 72 is applied to critical flow despite the potential problems outlined previously. The need to compute critical flow results primarily from the requirement for a consistent flow in FEQ simulations. Therefore, the flow relations must include the possibility of critical control even though the user believes such flow will not result. The purpose of modeling is to understand the behavior of a proposed or real stream system and to make reasoned estimates for conditions for which no data are available (for example, design conditions and extreme conditions for planning scenarios). Simulation of these conditions may unintentionally impose flows and depths that result in critical control at one or more locations where such a control is possible. If FEQ does not include that possibility, the computations either will fail for some unknown reason or will yield an unreasonable result because the control present in the structure is not simulated. Furthermore, if such a control appears unavoidable, some change in the physical structure should be made so that critical flow can be estimated more accurately. If this is not possible, a physical model must be constructed to define the flow and its characteristics at that location in the stream.

Consistent and reasonable estimates of critical flow, within the bounds of 1-D analysis, are required. For those cases in which critical flow is difficult to define, assumptions must be made as needed to produce consistent relations for the flow through the transition. The assumptions applied in FEQUTL computations of flow in transitions have been described in this section.

3.2.2 Expansion-Contraction Losses

The computation of expansion-contraction losses (eddy losses or shock losses) is approximate and uncertain. In general, the loss fraction for expanding flows commonly is considerably larger than for contracting flows. No problems result if the sense of the transition never changes. However, when the sense of the transition changes, patterns of flow may result such that the velocity head change is small, but the flow undergoes a major change in section shape. A computational point may result in the flow simulation where the velocity heads on each side of the transition are the same and no energy loss will be estimated with equation 72. The user has the option of requesting smoothing of the loss formula near this point to provide some loss at this point. The loss term can be exactly computed as

(73)

,

where Vis the average velocity in the cross section (=Q/A). The function, f_ec(X), is defined as Xk_A when X 0 and as -Xk_D when X< 0, where k_A is the loss coefficient for accelerating (contracting) flows, k_D is the loss coefficient for decelerating (expanding) flows, and X is the difference in the product of the velocity and the energy-flux correction coefficient from the downstream section to the upstream section (difference in the rescaled velocities). For equation 73 to be valid for both transition senses, f_ec (X) must be greater than or equal to zero. No information is available on the loss coefficients when the kinetic-energy-flux correction factor is large. In equation 73, it is assumed that the loss should be proportional to the difference in the true velocity head and not in the nominal velocity head. The sense of the loss is assumed to change at the zero difference point between the rescaled velocities, the argument to the function f_ec. Smoothing this function when the argument is near zero will result in a loss when the difference in velocity heads is zero.

A small, positive value of velocity difference, denoted by V, is specified by the user to define an interval, [- V, V]. Over this interval, a cubic polynomial is fitted such that the value of the function and its derivative is matched by the polynomial and its derivative at the ends of the interval. Under these conditions, the coefficient on the cubic term in the polynomial vanishes and a parabolic transition over the interval is obtained, yielding

(74)

.

This smoothing procedure yields a function with a continuous first derivative over the entire range. Without smoothing, the derivative of f_ec is discontinuous at the origin. At the origin, the point of zero loss without smoothing, the smoothed function yields

.

This means that, when the velocity difference is zero, the loss is estimated to be the same as the loss that would result from a velocity difference of V and a loss coefficient that is one-half the average of the two loss coefficients.

3.2.3 Mean Conveyance for the Expansion or Contraction

The friction losses defined in equation 72 depend on the method for computing a mean conveyance. The mean conveyance is defined as a function of an averaging parameter, a, as

(75)

.

The generalized mean for two values is given by equation 75. The mean values obtained as a varies over its range are

(76)

(77)

(78)

(79)

(80)

.

Equation 77 gives the arithmetic mean, equation 78 gives the geometric mean, and equation 79 gives the harmonic mean of the two end-point conveyances. No established rules are available regarding which mean value to apply. The geometric mean is preferred in many cases and may be a reasonable first approximation.

3.2.4 Outline of Solution Process for Expansions and Contractions

The user specifies a series of downstream piezometric heads to define the 2-D table of type 14. The head is measured from the maximum of the two bottom elevations for the cross sections at each end of the transition. For each downstream head, the upstream piezometric head resulting in critical flow at either cross section must be computed with the EXPCON command (section 5.7). This defines the smallest critical flow that can result for the given downstream head. The upstream piezometric head is computed as a function of downstream head and partial free flow for a series of partial free flows (critical flow is the free flow) until the partial free flow is zero. When the partial free flow is zero the two heads are equal. Each stage of the computations involves an iterative solution.

A control may result at either cross section and the control may shift as the flow levels change. An extensive search is made in EXPCON computations for a control at each of the cross sections, if necessary. The validity of this control is checked in EXPCON computations. For a control to be valid, the flow must be critical at one section and subcritical at the other section. It may be that no control can be found. Normally, this problem can be solved by adjustment of the friction losses. In some cases, the addition of a small friction loss allows a control to be found in the computations. In other cases, the reduction or elimination of friction loss allows a control to be found in the computations.

Special care must be applied if a closed conduit is present in the flow transition; for example, a stream directed underground through a long, closed conduit of substantially different cross-sectional area. Critical flow is undefined when a closed conduit is flowing full. The introduction of a hypothetical slot in the top of the conduit allows a hypothetical, equivalent free surface to be simulated to account for pressurized flow, but leads to a critical flow, which commonly is many times larger than any flow that may result in the conduit. This results in unrealistic upstream heads in the computations. These large heads cause only a small part of the table to be utilized in FEQ simulations. To avoid this unrealistic outcome, the QCLIMIT command (section 5.18) should be applied to the closed-conduit cross section or cross sections before EXPCON is invoked for those sections. The critical flows tabulated in the cross-section table are modified in QCLIMIT so that the maximum value is more realistic, although arbitrary. For accurate and reliable simulation, the maximum flow assigned to the critical flow when the closed conduit is full must be somewhat larger than the maximum flow likely to result in the conduit.

3.2.1 Governing Equations for Expansions and Contractions

3.2.2 Expansion-Contraction Losses

3.2.3 Mean Conveyance for the Expansion or Contraction

3.2.4 Outline of Solution Process for Expansions and Contractions