Submerged-Flow Bridge Scour under Maximum Clear- Water Conditions (II): Theory

A theoretical model for the maximum scour depth under submerged bridge flow was proposed based on the mass and energy conservation laws as well as the recent flume experimental data. It is shown that the ma-ximum scour depth can be described by a scour number and an inundation index. In general, for submerged flow the scour number increases with the inundation index, which is equivalent to the maximum scour depth increases with deck inundation level, decreases with increasing sediment size, and is independent of bridge girders. The proposed method is expected to be applicable to prototype flows without scaling effects since it was derived from the conservation laws. An application procedure was also suggested for bridge foundation design or field scour evaluation. INTRODUCTION In Guo et al. (2010) , a series of flume data have been collected at the FHWA J. Sterling Jones Hydraulics Laboratory. The experimental conditions and the results are summarized in Tables 1 and 2, respectively, where Vue = approach velocity at upstream of bridge, Q = experimental discharge, Rh = hydraulic radius, Re = Reynolds number, Fr = Froude number, dso = median sediment size, Cg = coefficient of gradation, Cu = coefficient of uniformity, hb = bridge opening height based on the original bed, Ys = maximum scour depth, a = block depth of bridge deck, I = inundation index, and (hb + Ys) / (hb + a) = scour number. The experimental study showed that the horizontal scour range of a submerged flow depends on the width of bridge deck, and the design of a scour profile needs the maximum scour depth y" which cannot be reasonably estimated by the existing methods since the Arneson and Abt (1998) method predicted an adverse tendency with the test data, the Umbrell et al. (1998) method in general overestimated the present data, and the Lyn (2008) method underestimates most of the present data. The purpose of this paper is to propose a theoretically based method for predicting the maximum scour depth under bridge-submerged flows. Specifically, a hypothesis based on the mass and energy conservation laws is first formulated, which is then tested with the collected data in Table 2; if the hypothesis is confirmed by the collected data, an application procedure is then presented for practical design.


INTRODUCTION
In Guo et al. (2010) , a series of flume data have been collected at the FHWA J. Sterling Jones Hydraulics Laboratory. The experimental conditions and the results are summarized in Tables 1 and 2, respectively, where Vue = approach velocity at upstream of bridge, Q = experimental discharge, Rh = hydraulic radius, Re = Reynolds number, Fr = Froude number, d so = median sediment size, C g = coefficient of gradation, C u = coefficient of uniformity, hb = bridge opening height based on the original bed, Ys = maximum scour depth, a = block depth of bridge deck, I = inundation index, and (hb + Ys) / (hb + a) = scour number. The experimental study showed that the horizontal scour range of a submerged flow depends on the width of bridge deck, and the design of a scour profile needs the maximum scour depth y" which cannot be reasonably estimated by the existing methods since the Arneson and Abt (1998) method predicted an adverse tendency with the test data, the Umbrell et al. (1998) method in general overestimated the present data, and the Lyn (2008) method underestimates most of the present data.

HYPOTHESIS ON MAXIMUM SCOUR DEPTH
Bridge flows are divided into three cases (Picek et al. 2007): (1) the upstream low chord of a bridge is partially submerged while the downstream low chord is unsubmerged, which is not discussed here; (2) both upstream and downstream low chords are partially submerged; and (3) a bridge is totally submerged. The last two cases are analyzed in this paper.
The problem is stated with Figure 1 where a bridge is over a steady river flow with clear water, the bridge deck is modeled with a rectangular box, Vue = critical velocity of approach flow at upstream of the bridge, Vue = effective velocity corresponding to the flow through the bridge, hu = depth of the flow at the upstream of the bridge before scour, hb = bridge opening height before scour, a = effective thickness of deck blockage where the corresponding stagnation streamline divides the flow into two parts, b = physical thickness of the deck blockage, Vbs =  average velocity at equilibrium maximum scour cross-section that is close to the outlet of the bridge flow according to the present experiments, and hd = depth of the tailwater before scour. Find the equilibrium submerged-flow scour depth, Ys> by considering a unit river flow.
To solve the problem, one can select a control volume consisting of crosssections 1-1 and 2-2 , the upstream stagnation streamline, the left and low faces of the bridge deck, and the scoured river bed. Applying the energy conservation law to the control volume and neglecting the friction loss (due to the short distance) , one can write (1) where hu = hydraulic head at cross-section 1-1 based on the original bed, hd = hydraulic head at cross-section 2-2 , (}j and (}2 are the energy correction factors, 9 = gravitational acceleration, and Kb = the bridge energy loss coefficient that is related to the difference of the upstream and downstream flow depths, huh d , the effective velo city, Vue: and the gravitational acceleration, g. By dimensional analysis , one can assume (2) where Ao and m are two empirical parameters. Eq. (2) makes sure Kb = 0 when hu = h d · On the other hand, applying the mass conservation law to t he control volume gives (hb + a) Vue = (hb + Ys) Vb., Solving for hb + Ys from Eqs. (1)- (3) gives (3)   (4) where the left-hand side is called a scour number. Unfortunately, the downstream flow depth hd is usually unknown. For an approximation, it is hypothesized where (3 is a fitting paramet er. The factor of (huh b ) in Eq. (5) expresses an inundation level that, take a partially submerged flow for example, increases huhd (since the energy loss increases with inundation) and makes huhd = 0 at hu = h b . The fraction (hb + a)jhu is introduced because huhd should be independent of the upstream flow depth hu for significant inundation while it should be 1 for partially submerged flows.
Referring to F igure 1 where the horizontal dashed line divides the approach flow into an overflow above the bridge and a pressure flow under the bridge, the unit flowrate, q] , under the bridge in both cases can be estimated by (6) where for approximately uniform flows of the present experiments n = 0 and for fully developed turbulent flows n = 1/7, and a = effective thickness of deck blockage. Theoretically, the value of a can be determined by a Bernoulli's equation in the dividing streamline if the fluid is ideal, where b = physical thickness of deck blockage, and Va = approach velocity in the dividing streamline. Practically, Eq. (7) gives a small value of a because of the neglect of fluid viscosity. Nevertheless, for a thin deck with overhang shown in Figure 5 of Guo et al. (2010), the overhang forces the dividing streamline approximately at the elevation of the deck surface. Therefore, this study assumes a=b (8) for totally submerged flow. Obviously, for partially submerged flow one has (9) For convenience, Eqs. (8) and (9) can be combined by Considering Eq. (6)

TEST OF HYPOTHESIS
Columns 5 and 6 in Table 2  Vue - The latter condition is similar to that of submerged culvert flow where hu/ hb ::::: 1.2 (Gupta 2008, p779). The dashed line in Figme 2 is just an extension of Eq. (14), which does not have any practical meaning since it corresponds to an orifice flow where the downstream edge of a bridge is unsubmerged. Nevertheless, it helps examine the functional structm e of Eq. (14). Mathematically, when hu approaches to hb ' or the inundation index I approaches to zero, the scom number has an asymptote (hb + Ys) / (hb + a) -> 1, which is true since a = huhb -> 0 and Ys = O. This asymptote shows that the structme of Eq. (14) is physically reasonable. In brief, for submerged flow the scom number increases with the inundation index. The effect of sediment size is through the upstream critical velocity Vue. When sediment size increases, the upstream critical velocity Vue increases, which decreases the inundation index I and then reduces the scom number (hb+Ys)/(h b + a). In other words, scour depth decreases with increasing sediment size.
The present analysis has approximated the approach velocity distribution to be uniform, which implies the effects of the bottom and side wall boundary layers can be neglected when estimating the floWl'ate through a bridge. Furthermore, the deck elevation must be much higher than the bottom boundary layer. Otherwise, the estimated flowrate through a bridge is much larger than the real flowrate, which results in a large effective velocity Vu e in Eq. (4) or a small inundation index in Eq. (13) and, as a result, a very small scour number or depth. This explains why the last two tests in Table 2 are excluded in analyses and why the predictions of the proposed method are much smaller than the measurements where the inundation ratio hu/hb = 3.57 and 10, respectively. In practice there are seldom submerged flows with an inundation ratio hu/hb > 2. For prototype flows with fully developed turbulent boundary layers, it is suggested to use n = 1/7 in Eq. (6) for estimating the flowrate through a bridge and Eq. (13) for inundation index.
In briefly, the proposed theoretical model, Eq. (14), has been confirmed with the present experimental data under threshold conditions, and the model parameters are defined by Eq. (15).

APPLICATION PROCEDURE
Unlike a pure empirical method that is restricted to the range of calibrated data, the proposed method is expected to be applicable to similar prototype bridge flow without scaling effects because it is mainly derived from the mass and energy conservation laws. These laws are true within considered framework whatever are inputs, outputs or system modifications. To apply the proposed method, the following procedures are suggested.
Step 2: Check if the scour is clear water scour. If the upstream velocity, V u , is less than or equal to the critical velocity, Vue, the proposed method is used.
Step 4: Calculate the inundation index, I , hom Eq. (13) where n = 1/7 is used for fully developed turbulent boundary layers.
Step 5: Check if the flow is submerged flow according to Eq. (17).
Column 7 in Table 2 was obtained using the above procedures, and column 8 shows t he absolute errors of predictions. It is seen that except for t he last two tests, all errors are less than or equal to 6.3 mm that is usually wit hin the uncertainties of flume measurements. The large errors of the last two tests originate from the amplified upstream effective velocities in the calculations, by neglecting the effect of boundary layers. The real effective velocities were significantly smaller since the deck elevations were close to the bottom boundary layers. Without the last two tests, the correlation coefficient betwee n the predictions and the measurements is R2 = 0.929.

CONCLUSIONS
The theoretical model, Eq. (14), for the ma;ximum scour depth has been confirmed by t he experimental data in Guo et al. (2010 ), which shows that the maximum scour depth can be described by the scour number and t he inundation index. In general, for submerged flow the scour number increases with inundation index, which is equivalent to the ma;ximum scour depth increases with deck inundation level, decreases with increasing sediment size, and is independent of bridge girders. The proposed method is expected to be applicable to prototype flows without scaling effects, and an application procedure has been suggested for bridge foundation design or field scour evaluation.

ACKNOWLEDGMENTS
This study was fin ancially supported by the FHWA Hydraulics R&D Program with Contract No. DTFH61-04-C-00037. The writers would like to thank Mr. Oscar Berrios for running the tests and preparing some of the figures . The writers are also thankful to Mr. Bart Bergendahl at FHWA for his constructive comments and suggestions.