WATER RESOURCES RESEARCH GRANT PROPOSAL

**Title: **Two-Dimensional Hydrodynamic Model for the
ACT and ACF River Basins

**Focus Categories: **MOD, FL, HYDROL

**Keywords: **Numerical Analysis, Rivers, Fluid Mechanics, Flood Control, Hydraulics.

**Duration: **3/2000 to 2/2001

**Federal funds requested:** $17,996

**Non-Federal matching funds: **$40,278

**Principal Investigator: **

Dr. Fotis Sotiropoulos

School of Civil and Environmental
Engineering

Georgia Institute of Technology, Atlanta, GWRI.

**Congressional District: **5

During the duration of this research, the numerical model will be calibrated for and applied to segments of both the ACT and ACF River Basins. Simulations will be carried out for various inflow hydrographs to understand the transient response of the two basins to discharge variations due to hydropower release events and/or flooding events. The results of the 2D model will also be employed to evaluate and further refine the predictive capabilities of simpler one-dimensional river routing models, which are used in existing operational river management systems. Therefore, the proposed research is a critical prerequisite for developing a reliable and efficient state-of-the-science model for addressing the river management and flood forecasting needs of the State of Georgia. The proposed model will also provide the computational framework for developing a future more general numerical tool capable of tackling sediment transport and water quality issues in the ACT and ACF River Basins. For that reason the model will be formulated so that it can be readily extended in the future to solve the additional transport equations needed to tackle such issues.

**Nature, Scope, and Objectives of the Research**

**Task 1****:** *Development of the Numerical Method*.
A numerical model will be developed for solving the unsteady, depth-averaged,
open-channel flow equations in generalized curvilinear coordinates. The model will be formulated so that it can be readily applicable to
complex, natural river systems. Special
care will be exercised to ensure that the numerical methodology can take full
advantage of massively parallel computational platforms so that it can serve
as a practical engineering tool.

** Task 2**:

** Task 3**:

** Task 4**:

**Methods and Procedures**

In the following sections, we present the depth-averaged open-channel flow equations, discuss the issue of turbulence closure, describe the numerical method we will employ to solve the governing equations, and outline the grid generation procedures we will employ to generate computational meshes for the ACT and ACF River Basins.

*12.1 Governing Equations*

The numerical model will rely on the unsteady, depth-averaged, open-channel flow equations. These equations are obtained by integrating the three-dimensional, incompressible, Reynolds-averaged Navier-Stokes (RANS) equations from the river bed to the water surface under the following assumptions: i) uniform velocity distribution in the depth direction; ii) hydrostatic pressure distribution; iii) small-channel bottom slope; iv) negligible wind shear at the surface; and v) negligible Coriolis acceleration. Although this averaging procedure simplifies the equations considerably, information concerning the vertical variations of the velocity distribution is lost. Nevertheless, the depth-averaged equations have been shown to yield very good predictions for flows in shallow channels and provide a far more practical, from the computational standpoint, engineering alternative to the full, three-dimensional RANS equations. We should note that we propose to depth-average the three-dimensional RANS equations, rather than the instantaneous Navier-Stokes, because we would like the model to be applicable to turbulent open-channel flows. The issue of turbulence closure for these equations is discussed in the subsequent section.

The depth-averaged open-channel flow equations (continuity and momentum) formulated in Cartesian coordinates read as follows (see Kuipers and Vreugdenhil (1973) for the details of the integration):

*Continuity equation*:

(1.a)

x-momentum

(1.b)

*y-momentum*

(1.c)

In the above equations *h* is the water depth, *U*
and *V* are the mean depth-averaged velocity components in the *x*-
and *y*-directions, respectively, *g* is the acceleration of gravity,
*z _{b}* is the channel bottom elevation,

(2.b)

(2.c)

where *u* and *v* are the time-averaged velocity
components, *u’* and *v’* are the fluctuating components, and *m* is the viscosity of the water. In the above integrals, the first terms represent
the viscous stresses, the second terms are the components of the Reynolds-stress
tensor (arising from time-averaging the instantaneous Navier-Stokes equations),
while the last terms are the so-called dispersion stresses (see subsequent
section).

The bottom shear stresses can be calculated by using the following formulas:

_{}
, _{ }
(3)

where *C _{f}* is the friction coefficient
that needs to be determined as part of the model calibration (see related
section below).

The above equations are presented, for the sake of simplicity,
in Cartesian coordinates. In the proposed
numerical model, however, these equations will be transformed into generalized
curvilinear coordinates by invoking the partial transformation approach _{}, which maintains
the Cartesian velocity components as the dependent variables. This step is essential in order to ensure that the model can simulate
accurately arbitrarily shaped natural River channels. Due to space limitations the transformed equations are not included
herein (see Molls and Chaudry (1995) for details).

*12. 2 Turbulence Closure*

Closure of the depth-averaged equations requires the determination of the components of the effective stress tensor. Since our emphasis is on high Reynolds-number flows, we will neglect the viscous stresses from eqns. (2). We will also neglect the depressive stresses as these are difficult to quantify and model. It is common practice in the literature to neglect these stresses, a simplification which does appear to have a significant impact on the accuracy of the model. The remaining depth-averaged Reynolds stresses will be modeled using the Boussinesq eddy-viscosity concept:

_{} (4)

where *i,j* = 1,2, *n _{t}*
is the depth-averaged eddy-viscosity,

There are several alternatives to obtain the eddy-viscosity
needed to close the governing equations. The simplest approach is to assume a constant
eddy-viscosity along the entire basin. The value of this constant can be determined by calibrating the
model equations with available data. Although
this treatment is computationally very attractive, due to its simplicity and
computational expedience, it constitutes an oversimplifying treatment of the
very complex turbulent transport processes within natural rivers. A far more rigorous approach, albeit somewhat more expensive computationally,
is to determine the eddy-viscosity distribution by solving two additional
transport equations for the turbulence kinetic energy, *k*, and its rate
of dissipation e, the so-called *k-*e model. These
equations provide turbulence velocity and length scales that can be used to
calculate the eddy-viscosity as follows:

_{}
(5)

where *C** _{m}*
is a model constant (

In the proposed research, we will evaluate both turbulence
closure options. Implementing the
constant eddy-viscosity model is, of course, rather trivial but will, as already
indicated, require considerable calibration with field data. We will also implement the *k-**e* model and compare its performance relatively
to the constant-eddy viscosity treatment.

A very important issue when modeling turbulent flows is the approach adopted to resolve the flow near solid boundaries (the channel banks, in the present case). Given the scales the proposed model is intended for, the only practical approach for modeling the near-wall flow is to employ the so-called wall-functions approach. This approach makes use of the logarithmic law of the wall to bridge the gap between the fully turbulent region and the laminar sublayer and allows the simulation of flows at very high Reynolds numbers with rather coarse computational meshes. An important advantage of the wall functions approach is that it can be readily extended, using appropriately modified expressions for the logarithmic velocity profile, to account for roughness effects, which are very important and need to be accounted for in natural rivers (see Rodi (1980) for details).

*12.3 The Numerical Method*

The depth-averaged continuity, momentum, and turbulence closure equations will be formulated in strong-conservation form and discretized using the finite-volume approach. Second-order accurate central-differencing will be employed to discretize all spatial derivatives. Due to the purely dispersive errors introduced by the central-differencing of the convective terms, artificial dissipation terms need to be explicitly introduced for stability. We will employ third-order, fourth-difference artificial dissipation terms scaled appropriately with the eigenvalues of the Jacobian matrices of the convective flux vectors (see Lin and Sotiropoulos 1997, for a three-dimensional implementation of various artificial dissipation schemes for incompressible flow equations). To facilitate the resolution of regions of steep gradients of the water depth, as well as to ensure that the proposed model can also be applied to simulate traveling hydraulic jumps, we will implement a non-linear artificial dissipation model by extending to depth-averaged flows techniques developed for capturing shock waves in compressible flows (see Jameson et al. 1981). More specifically, the artificial dissipation terms will consist of both first and third-order terms, each multiplied by appropriate functions designed to switch-on or off certain terms based on the local intensity of the spatial gradients of the water elevation. Thus, in regions of steep water-surface gradients the first-order terms are automatically turned on by the model to preserve the robustness and monotonicity of the numerical model. In smooth regions of the flow, however, only the third-order terms are activated, thus, preserving the second-order accuracy of the numerical method.

The discrete governing equations will be integrated in time using an explicit, multi-stage, Runge-Kutta algorithm designed to ensure second-order temporal accuracy. In spite of the fact that explicit integration algorithms typically impose more severe time-step limitations, as compared to fully implicit procedures, they are preferable because they can take full advantage of parallel computer architectures. Such algorithms do not involve matrix inversion operations and can, thus, be parallelized very effectively and in a rather straightforward manner. The resulting computer code should be able to run very efficiently on multi-processor computer architectures and, thus, the additional cost due to the increased number of time steps required to simulate a given time interval (because of the stability restriction of the explicit algorithm) can be greatly offset by the drastic reduction of CPU time per time step.

*12.4 Grid Generation Procedure *

The ACT and ACF River Basins are both very complex systems, each consisting of three major rivers and several tributaries. The use of curvilinear grids is, therefore, essential in order to ensure that the various topographical features of these systems are adequately resolved. In the proposed research, we will model directly only the three major rivers of each system. The effects of the tributaries on the flow will be accounted for by specifying inflow or outflow-type boundary conditions at the locations where the various tributaries intersect the main river.

Using available topography data (from the Army Corp of
Engineers), we will re-construct the geometry of the riverbanks and the variation
of the bed elevation, *z _{b}=z_{b}(x,y)*, that appears
in the depth-averaged equations. Curvilinear
computational grids will be constructed using the GRIDGEN commercial grid
generation software, available at the Computational Fluid Dynamics laboratory
of CEE at Georgia Tech. Spline interpolation
will be used to obtain the values of

*12.5 Model Calibration*

In general, there are two model constants that need to
be calibrated with input from field observations: the eddy viscosity coefficient
*n _{t}*, and the bed friction coefficient

The bed friction coefficient results from the depth-averaging
of the governing equations and can only be determined by calibrating the numerical
model. Following Wenka et al. (1991),
*C _{f}* is related to the friction coefficient

**Related Research**

Depth-averaged models have been employed in the past to simulate flows in man-made open channels as well as in natural environments, such as river systems and estuaries. Since the emphasis of this proposal is on natural river basins, we present here a brief review of those previous studies that focused on shallow flows in natural environments. It should be pointed out that what follows is not intended to be an extensive review of related literature. We rather focus on few studies that underscore the progress made in modeling open channel flows with 2-D models during the past decade.

Among the most notable early studies is the work of Tingsanchali
and Rodi (1986) who developed a depth-averaged *k-**e* model for simulating suspended sediment
transport in the Neckar River in Germany. Wenka et al. (1991)
employed a similar *k-**e*
model to simulate flooding over a stretch of the river Rine in Germany. They reported good agreement between the predicted
flow patterns and those observed in aerial photographs of the river during
the flooding event. Muin and Spaulding
(1996) developed a two-dimensionsal model for simulating tidal circulation
patterns in the Providence River. Their
model neglects all viscous and turbulent stresses, except, of course, the
stresses due to bed friction. Hu and
Kot (1997) developed a depth-averaged model for studying tides in the Pearl
River estuary in China. They employed
the constant eddy-viscosity assumption for turbulence closure. Shankar and Cheong (1997) employed various
formulations of a 2-D numerical model to study tidal motions in Singapore
coastal waters. They also adopted
the constant eddy-viscosity assumption.

An important conclusion that follows from the above brief
review is that the enormous complexities of natural flows necessitate the
development of numerical models that are specifically tailored to the idiosyncrasies
of a specific site. To the best of
our knowledge a two-dimensional numerical model for the ACT and ACF River
Basins, such as the one proposed here, has neither been developed nor is currently
under development elsewhere. Moreover,
and with only exception the works of Tingsanchali and Rodi (1986) and Wenka
et al. (1991), we are not aware of any other depth-averaged model that has
employed a rigorous turbulence closure model, such as the *k-**e* model proposed herein, to simulate flows
in natural rivers (recall that most previous studies have assumed constant
eddy viscosity).

**References**

Hu, S., and Kot, S. C. (1997), “Numerical Model of Tides in pearl River Estuary with Moving Boundary,” ASCE J. Hydr. Eng. 123(1), pp. 21-29.

Jameson, A., Schmidt, W., and Turkel, E. (1981), “Numerical
Simulation of the Euler Equations by Finite Volume Methods using Runge-Kutta
Time Stepping Schemes,” *AIAA* paper 81-1259.

Kuipers, J., and Vreugdenhil, C. B. (1973), “Calculation of Two-Dimensional Horizontal Flow,” Report S163, Part I, Delft Hydraulic Laboratory, Delft, The Netherlands.

Lin, F., and Sotiropoulos, (1997) F., "Assessment
of Artificial Dissipation Models for 3-D Incompressible Flow Solutions,"
*ASME J. Fluids Eng.* 119, 331-340.

McGuirc, J. J., and Rodi, W. (1978), “A Depth-Averaged
Mathematical Model for the Near Field of Side Discharges into Open-Channel
Flow,” *J. Fluid Mech*. 86(4), pp. 761-781.

Molls, T., and Chaudhry, M. H. (1995), “Depth-Averaged
Open-Channel Flow Model,” *ASCE J. Hydr. Eng*. 121(6), pp. 453-465.

Muslim, M., and Spaulding, M. (1996), “Two-Dimensional
Boundary-Fitted Circulation Model in Spherical Coordinates,” *ASCE J. Hydr.
Eng*. 122(9), pp. 512-521.

Rastogi, A. K., and Rodi, W. (1978), “Predictions of Heat
and Mass Transfer in Open Channels,” *ASCE J. Hydr. Div*. 104(3), pp.
397-419.

Rodi, W. (1980), “Turbulence Models and Their Application
in Hydraulics,” *IAHR monograph*, Delft, The Netherlands.

Shankar, J., and Cheong, H.-F. (1996), “Boundary Fitted
Grid Models for Tidal Motions in Singapore Coastal Waters,” *IAHR J. Hydr.
Res*. 35(1), pp. 3-19.

Tingsanchali, T., and Rodi, W. (1986), “Depth-Averaged
Calculation of Suspended Sediment Transport in Rivers,” 3^{rd} Int.
Symposium on River Sendimentation, The Univ. of Mississipi, March 31-April
4.

Wenka, T., Valenta, P., and Rodi, W. (1991), “Depth-Averaged
Calculation of Flood Flow in a River with Irregular Geometry,” proc. of *XXIV
IAHR Congress*, Madrid, Spain, Sept. 9-13, pp. A-225 to A-232.

**Deliverables and** **Information Transfer Plan**

The deliverables of this research will be: i) a computer code for simulating unsteady flows in the ACT and ACF River Basins using depth-averaged equations; ii) a user’s manual describing the numerical method and the computer code; iii) a paper submitted to a refereed journal; and iv) a proposal submitted to an appropriate state agency requesting funds for further development of the numerical model. The deliverables will be made available to the Georgia State Department of Natural Resources.

URL: http://water.usgs.gov/wrri/00grants/GAact.html

Last Updated: Wednesday October 26, 2005 1:09 PM

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