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In this paper, the authors present an analysis of the magnitude of the temporal and spatial acceleration (inertial) terms in the surface-water flow equations and determine the conditions under which these inertial terms have sufficient magnitude to be required in the computations. Data from two South Florida field sites are examined and the relative magnitudes of temporal acceleration, spatial acceleration, and the gravity and friction terms are compared. Parameters are derived by using dimensionless numbers and applied to quantify the significance of the hydrodynamic effects. The time series of the ratio of the inertial and gravity terms from field sites are presented and compared with both a simplified indicator parameter and a more complex parameter called the Hydrodynamic Significance Number (HSN). Two test-case models were developed by using the SWIFT2D hydrodynamic simulator to examine flow behavior with and without the inertial terms and compute the HSN. The first model represented one of the previously-mentioned field sites during gate operations of a structure-managed coastal canal. The second model was a synthetic test case illustrating the drainage of water down a sloped surface from an initial stage while under constant flow. The analyses indicate that the times of substantial hydrodynamic effects are sporadic but significant. The simplified indicator parameter correlates much better with the hydrodynamic effect magnitude for a constant width channel such as Miami Canal than at the non-uniform North River. Higher HSN values indicate flow situations where the inertial terms are large and need to be taken into account.

Surface-water modeling is an important tool to support many types of hydrologic investigations. As is generally the case in modeling, it allows the prediction of future conditions and investigation of causative factors. Surface- water models offer a variety of formulas to represent flow and water level. Formulas that make simplifying as- sumptions by neglecting certain terms provide for solutions requiring fewer calculations and often require less parameterization. Hydrodynamic flow governing equations account for spatial and temporal changes in mo- mentum, whereas simpler formulations retain only the frictional and gravitational forcing terms or combine an empirical flow relationship with mass conservation. Generic criteria are used to determine the formula appropriate for a specific hydrologic situation. For example, a location with dominant tidal fluctuations requires the full hy- drodynamic equations whereas a steadily flowing mountain stream can be represented without the inertial terms. But for more complex and diverse conditions, it is difficult to develop the insight and details required to choose a formula without a thorough examination of the terms in the numerical formulation and how they relate to actual flow conditions.

Comparisons between different surface-water flow formulations have usually relied on simulation results only or comparisons of model results to field water levels or discharge values. Reference [

Field data are almost always used for model comparison and verification, as the purpose of any model is to represent a physical system. Basic parameters of the model output, such as water level, discharge, and constituent concentrations, are compared between the model output and data measured in the field. However, substantially more information can be derived from the field data than a simple comparison of values. Statistics are often de- rived from field data, lending insight into the system’s behavior. Furthermore, sufficient field data exist in many locations to compute the individual mathematical terms in the surface-water flow equations and find their relative magnitude, yielding insight into what formulation is appropriate for a particular situation.

This paper describes an effort to parameterize the flow conditions under which inertial terms are important and are needed in surface-water flow computations. The importance of the inertial terms in surface-water flow com- putations is analyzed through: 1) computation of surface-water flow equation terms from field data; 2) formulation of parameters that incorporate dimensionless numbers to represent the significance of the inertial terms; and 3) comparison of test-case model simulations with and without inertial terms. Two field sites are used for com- parison, a natural river in a tidal environment and a structure-managed coastal canal. The results of the analyses demonstrate the importance of the inertial terms and the conditions in which a full hydrodynamic solution is needed.

The different surface-water flow formulations can be categorized according to complexity, that is, how many of the terms in the conservation equations of mass and momentum are included. Terms that are small relative to the predominant terms can be neglected, allowing for simpler formulations and solution schemes.

The governing hydrodynamic flow equations in one dimension [

(A) (B) (C) (D)

temporal spatial gravity friction

where, Q is discharge, t is time, A is cross-sectional area, x is distance, g is gravitational acceleration, Z is stage, k is the friction coefficient, and R is hydraulic radius (area divided by wetted perimeter). The friction coefficient k is Mannings n squared in SI units or Mannings n squared divided by 2.208 in English units. Equation (1) ex- presses conservation of mass and Equation (2) expresses conservation of momentum.

Each term on the left-hand side of Equation (2) expresses a force or a change in momentum. Term A is the change in momentum from temporal acceleration. Term B is the change in momentum from spatial acceleration. Term C is the gravity force and Term D is the force due to friction. The sum of the forces and change in mo- mentum equal zero.

If the temporal acceleration term is removed from Equation (2), it represents flow that is steady in time but varies gradually in space. If both the temporal acceleration and spatial acceleration terms are removed, Equation (2) represents steady, uniform flow and it reduces to a simple, Mannings-based, equation for flow [

Equation (3) can be combined with the continuity equation (Equation (1)) to form the diffusion analogy equa- tion. When rearranged in terms of the dependent variable Q, this equation resembles the classic diffusion equation in mathematics [

The ability to evaluate the effects of the inertial (acceleration) terms without resorting to the complex solution of Equations (1) and (2) is of considerable interest. Reference [

One use of Froude-Number-based parameters is to determine when the full hydrodynamic equation should be used instead of the diffusion analogy in a numerical model. Such a parameter is derived below and comparisons to field data and numerical model results are shown.

Both the full hydrodynamic formulation and the diffusion-analogy formulation are used by many numerical model applications. Commonly, models designed for estuaries and coastal rivers use the full hydrodynamic formulation. This includes one-dimensional models such as BRANCH [

Many of these hydrodynamic and diffusion analogy models are used while coupled to a groundwater model [

TheSWIFT2D model, imbedded in FTLOADDS, was used to compare the hydrodynamic and diffusion analogy methods. Simulations of the same scenario are run with the fully hydrodynamic formula and also with the inertial terms (A and B in Equation (2)) removed. Without the hydrodynamic terms, the solution resembles the SWR1 diffusion analogy method. A two-dimensional test problem was used to simulate a rainfall event dissipating in a shallow-water environment. A range of frictional resistance and mean flow values were used to assess the mag- nitude of the hydrodynamic effects on flow velocity, discharge, and stage. The SWIFT2D model with and without inertial terms was also applied to one of the field site conditions.

In order to examine the magnitude of the component terms in the hydrodynamic flow Equation (2), continuous stage and discharge data were used from two sites in South Florida. Each site had two measurement stations, al- lowing the computation of the water level slope and the spatial changes in discharge. The two stations on the North River, southwest coast of Whitewater Bay, are located about 2550 meters apart and continuously collect stage measurements and acoustically-measured discharge. The Miami Canal site is located upstream of hydraulic con- trol structure S-26 (

The terms in the hydrodynamic flow equation (Equation (2)) can be expressed in finite-difference form to compute their values from the field data. At the North River location, the terms referenced in Equation (2) take the form:

Term A (temporal acceleration) =

Term B (spatial acceleration) =

Term C (gravity term) =

Term D (friction term) =

where Δt is the data-collection time interval (15 minutes), Δx is the spacing between the stations (2550 meters), B is the channel top width (approximating the wetted perimeter), the superscripts indicate the collection time (t is the current time), and the subscripts indicate the station location (upstream or downstream). Note in

The magnitudes of hydrodynamic equation component terms at North River in 2008, shown in

The analysis of the North River data indicates that the inertial terms can be important in their absolute magni- tude and percent of time they dominate. North River is in a coastal environment with a tidal signature and is fed at its upstream end from the Shark Slough (

A somewhat different hydrologic condition exists on the east coast of South Florida where canal conveyance of surface water is highly regulated using control structures. The Miami Canal location (

The magnitude of each term for the 2008 time series at Miami Canal, shown in

Term in Equation 2 | Temporal Solution | Spatial Solution |
---|---|---|

Term A (temporal acceleration) | Forward difference | Computed separately for each location |

Term B (spatial acceleration) | Present | Upstream/Downstream difference |

Term C (gravity term) | Present | Computed separately for each location |

Term D (friction term) | Present | Computed separately for each location |

at North River. The water-surface slope in the Miami Canal tends to be very low, reaching no slope when the gate is closed, making the hydrologic conditions conducive to hydrodynamic dominance.

There are two attributes suggesting the Miami Canal is a better site to analyze hydrodynamic effects. Upstream and downstream measurement points at Miami Canal are closer than at North River (366 meters versus 2550 meters) so computed spatial-difference terms are more reliable. Also, the manmade cross section is more uniform at Miami Canal, which better satisfies the assumptions of one-dimensional flow. One reasonable indicator of the analysis reliability is to sum the terms in Equation (2), which should equal zero, and average them over the time period. Another is the standard deviation of the sum of the terms, which indicates how much the sum fluctuates from its mean value. At North River, the mean of the sum of the computed terms is −0.623 × 10^{−3} cubic meters per second squared, 45.3 percent of the gravity term’s mean value of −1.376 × 10^{−3} cubic meters per second squared. At Miami Canal, it is assumed that the spatial acceleration term would be negligible enough so that the mean of the sum of the computed terms is representative, yielding a value of 0.195 × 10^{−3} cubic meters per second squared, which is 8.1 percent of the gravity term’s mean value of −2.415 × 10^{−3} cubic meters per second squared. This lower discrepancy at Miami Canal is reasonably attributable to its spatially uniform cross section and closer up- stream and downstream stations. The standard deviation of the sum of computed terms is 1.40 × 10^{−}^{2} cubic meters per second squared for North River and 0.85 × 10^{−}^{2} cubic meters per second squared for Miami Canal, demon- strating a lower variability for Miami Canal also.

In order to quantify the significance of the component inertial terms, a formula was developed in terms of di- mensionless quantities.

The Froude Number is defined as:

where, v is the flow velocity, g is gravitational acceleration, and h is depth of water. Differentiating the Froude Number squared with respect to time:

Expanding and multiplying through by v:

Dividing by

The continuity equation (Equation (1)) can be simplified for a unit width of flow and relatively uniform depth as:

Combining Equations (7) and (8):

Rearranging:

Equation (7) can be rearranged to:

Combining Equations (10) and (11):

where the right-hand side of Equation (12) is the temporal and spatial acceleration (inertial) terms in the mo- mentum equation expressed in terms of velocity v rather than discharge Q as in Equation (2).

A similar approach is taken when differentiating the Froude Number with respect to spatial location x:

Expanding and multiplying through by v:

Dividing by

Equation (8), the continuity equation, can be rearranged:

Combining Equations (15) and (16):

Multiplying through by v and rearranging:

Rearranging:

Modifying the equation so the right-hand side of the equation is in terms of stage, Z, instead of depth, h:

where ε is the bottom bed elevation. A rearrangement yields:

As

The right-hand side of Equation (22) is the gravity term in the momentum equation.

The comparison of Equations (12) and (22) can indicate the properties of the dimensionless numbers when inertial terms dominate:

Rearranging and transforming this into finite-difference form: the conditions when Equation (12) (inertial terms) is substantially larger than Equation (22) (gravity term) can be expressed:

where the absolute value signs indicate that magnitude, not sign, is the important factor. The first term on the left-hand side and the last term on the right-hand side represent the effects of temporal and spatial variations in Froude Number, whereas the last term on the left-hand side and the first term on the right hand side represent the effects of temporal variations in depth and the channel slope Δε/Δx. This can be visualized as two sets of rela- tionships, dependent on each other, one dealing with the temporal and spatial change in the Froude Number squared:

and the other with the relationship of temporal depth changes to the bottom slope:

Equations (25) and (26) cannot be considered independent of each other, but they do give a framework to define important quantities. Equation (25) includes

As the variable ^{−1}. The same relationship is plotted for the Miami Canal data in ^{−1} at a ratio of terms value 0.5. One alternative might be to replace _{r}^{2}, a non-dimensional parameter, but doing this with the North River data produces a poorer relationship to the ratio of inertial terms to gravity terms, as seen in

The results in ^{−1} strongly indicate than inertial terms are less than half of the gravity terms. However, values of ^{−1} seem to occur at both high and low inertial/gravity values. So this parameter can only confirm that inertial forces are not important.

Equation (12) can be divided by Equation (22) in finite-difference form to get the ratio of the inertial to gravity terms:

Multiplying numerator and denominator by Δt and rearranging:

Defining

where HSN is the Hydrodynamic Significance Number, which can be defined with six dimensionless numbers:

C_{r} is the advective Courant Number

Equation (29) can be recalculated with the numerator only representing the temporal acceleration Term A in Equation (2) to yield a modified Hydrodynamic Significance Number:

Note the sign change in the numerator between Equations (29) and (30). This corresponds to the formulation in [

Miami Canal has good hydrologic conditions for evaluating Equation (29), but lacks downstream discharge meas- urements that are needed to compute the spatial acceleration term. North River has both upstream and downstream measurements of discharge and depth, but large variations in channel width, 13 meters at the up-stream site and 80 meters at the downstream site. The unit width form of the continuity equation (Equation (8)) was used for deriving the Hydrodynamic Significance Number, which assumes a constant width over the distance Δx. This assumption applies to a uniform numerical grid, as in a numerical model, but is obviously contradicted in the case of North River field data.

Given these limitations in both North River and Miami Canal field data, the spatial acceleration terms are neglected for comparisons to these data. The modified HSN_{t} in Equation (30) is compared to the temporal ace- leration (A in Equation (2)) divided by the gravity term (C in Equation (2)) for upstream North River and upstream Miami Canal. Six-hour average values are plotted in ^{2} value of 0.003, but a good relationship is shown in ^{2} value of 0.847. The most obvious explanation for this big difference between the two locations is that North River has quite a variable width whereas Miami Canal has a relatively constant width. This appears to be a factor even when not considering the spatial acceleration term. The other observation from _{t} values are consistently smaller than the ratio of terms for both sites. Neglecting the spatial acceleration terms may be a factor in this, but a more likely cause is the large Δx distances between upstream and downstream stations, which violates the small increment assumption for Δx, creating a larger value of C_{r} and a smaller HSN_{t}. Despite these limitations,

Applying the Hydrodynamic Significance Number to numerical model simulations avoids the bias due to variable channel width, as the cell to cell widths are the same. This provides an indicator of when and where the flow conditions warrant a full hydrodynamic solution. In order to examine the importance of the inertial terms, several simulations were performed with the SWIFT2D hydrodynamic code and the SWR diffusion-analogy code.

A test problem was constructed using the SWIFT2D simulator and represented a 3.05 kilometer section of the Miami Canal upstream of hydraulic gate/structure S-26 (

The model is composed of 128 columns and 7 rows of 25-meter-square grid cells. The upstream boundary is a time-varying stage boundary and a flow boundary, based on the measurement station, is established downstream

to simulate gate operations. The active model can be divided into two sections. A 122 column long by 1 row wide segment is located through the center of the model with the last cell containing the downstream flow boundary. In addition, a 4 column by 5 row pool is added to the upstream section in order to maintain more stable specified stage levels and provide adequate boundary inflows (

Bottom elevations were fixed at −5.00 meters for the widened pool section and were set to −2.64 meters for the simulated canal section. The vertical datum used was the North American Vertical Datum of 1988 (NAVD88). For stability, a constant time step of 18 seconds was required. In order to isolate the hydrodynamic effects, precipita- tion, evapotranspiration and wind were not included in any of the simulations. All cells were given an equal Manning’s n value of 0.02, which is within the common range of 0.02 - 0.03 for straight, uniform dredged/ex- cavated channels [

The stage boundary input data were linearly extrapolated to the boundary 3 kilometers upstream of the structure using stage data measured at S-26 and at the field station 366 meters upstream of S-26 (

The Hydrodynamic Significance Number (HSN) was computed using the time series output from the Miami Canal numerical model and Equation (29). A 300-minute period of time was chosen which covers the gate opera- tions creating the highest peak in the friction term seen on June 20, 2008 (

Note that the HSN values in

A purely synthetic test problem was also created using the SWIFT2D simulator to examine the importance of inertial terms and the use of the Hydrodynamic Significance Number. Ten scenarios were simulated with different flow rates and frictional resistances. Each test case was initiated with a common initial stage and designed to drain and reach a static state of 1-meter-deep flow to the east down a sloped surface. Each simulation lasted 7 days and the transient period from each case was plotted and compared.

The model utilized 500-meter-square grid spacing and consisted of 10 active rows and columns resulting in 100 active cells (

A variety of input flow rates and Manning’s n values were chosen to produce a steady-state flow condition with 1-meter-deep flow for all 10 cases. The Manning equation was used to specify the western boundary flow rates for each Manning’s value chosen and is defined as:

where V is the velocity, n is the Manning’s value, R is the hydraulic radius defined as the cross-sectional area of flow divided by the wetted perimeter, and S is the slope of the water surface. Equation (31) shows that velocity and Manning’s n are inversely proportional. With a 1 meter flow depth, a 0.0002 meter/meter slope, and a hydraulic radius of 1 meter, several test problems were constructed with various Manning’s n values and corresponding flow rates (

The Hydrodynamic Significance Number (HSN) was computed from the synthetic test problem output for differing values of Manning’s n. The values at selected columns (identified in

The behavior of the system can be seen from the stage time series as shown in

The flow velocities at the center row for different columns are shown in

Manning’s n | Velocity (m/s) | Total Flow (m^{3}/s) |
---|---|---|

0.03 | 0.471 | 2357.0 |

0.04 | 0.354 | 1767.8 |

0.05 | 0.283 | 1414.2 |

0.06 | 0.236 | 1178.5 |

0.07 | 0.202 | 1010.2 |

0.08 | 0.177 | 883.9 |

0.09 | 0.157 | 785.7 |

0.1 | 0.141 | 707.1 |

0.2 | 0.071 | 353.6 |

0.3 | 0.047 | 235.7 |

A formulation was derived in terms of several dimensionless numbers that represents the ratio of the inertial terms to the gravity term in the conservation of momentum equation. Important parameters were identified by com-

parisons with field data collected at two waterways in South Florida. The formulation was used to develop a Hydrodynamic Significance Number (HSN) that is combined with simulations of field site and synthetic scenarios to indicate flow situations where the inertial terms are important.

The parameter ^{−1}, inertial terms are consistently less than half of the gravity terms. Lower values of

The two field sites represent substantially different conditions, with the Miami Canal site more controlled than North River. The peak hydrodynamic effects are commonly associated with boundary transients such as when the hydraulic control structure on Miami Canal is opened or closed. North River has a large variation in cross section between the upstream and downstream stations, whereas Miami Canal is quite uniform. This difference is the most likely factor causing HSN to show good correlation to the ratio of the inertial and gravity terms at Miami Canal but a bad correlation at North River. A numerical model solution with constant-width grid cells would not have this problem.

The effect of the hydraulic control on the Miami Canal site, relative to North River, can be seen in the behavior of the inertial and other terms in the momentum equation. Whereas the North River data produce more gradually varying terms, the Miami Canal data yield sudden peaks in the inertial and friction terms. However, at more nat-

ural sites like North River the inertial terms can pre-dominate at times when the gravity term is low (near hori- zontal slope).

The synthetic test problem indicated that the inertial terms are more important at lower frictional resistances for similarly configured systems. The difference in results from a hydrodynamic solution and a diffusion-analogy solution depends on the magnitude of the HSN.

The formulations developed can be applied to both field measurements and numerical models to help evaluate when inertial terms are important. Both

A cross-sectional area;

C_{r} advective Courant Number,

F_{r} Froude Number,

g gravitational acceleration;

h depth of water

k Manningsn squared in SI units or Manningsn squared divided by 2.208 in English units;

Q discharge;

R hydraulic radius (area divided by wetted perimeter);

t time;

v flow velocity;

x distance;

Z stage;

Ε bottom bed elevation.