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Modeling two-dimensional overland flow across complex real-world topography is a challenging problem. Predicting the overland flow variables for various whatif rainfall scenarios can facilitate designing water infrastructure components aimed at preventing inundation and urban flooding. Numerical models that are being used range from those that solve the simplified St. Venant equations to CFD models that solve the complete three dimensional Navier-Stokes equations. In this work, the performance of the USGS Diffusion Hydrodynamic Model (DHM) for a series of overland flow test problems was tested by comparing numerical solutions obtained for an event-driven simulation across various sensitive parameter combinations. The reliability of the model and its ability to incorporate various topographical characteristics in the domain are illustrated.

Advances made in numerical algorithms used to solve the Navier-Stokes equations (which describe the fluid motion) and its simplified variants are pushing the frontiers of knowledge in all engineering disciplines. When these mathematical advances are integrated with the increased computational abilities, researchers can now attempt to solve large scale problems at small spatial grid size that hitherto have been difficult, primarily because of the lack of computational resources by which the modeling tasks can be accomplished in a reasonable amount of CPU time. While the popular algorithms used to solve the flow governing equations are from finite difference, finite element and finite volume methods, over the last few decades new algorithms from the family of spectral methods, Total variation diminishing, Non Oscillatory and Boundary element methods [

The numerical codes that were written in the early years of computers for solving the flow equations were primarily in Fortran language. While some of these codes have been re-written using an object-oriented language, the rest have been integrated with advanced graphical user interface applications to make their execution and output visualization easier. Since hundreds of years of testing have gone into these codes by the audience from around the world, their results tend to act as a benchmark for codes that have evolved over the last two decades. As experimental investigations for complex fluid flow problems is not possible and as theoretical solutions for these problems do not exist, the application of numerical models will continue to become more popular in the coming decades.

In this work, we apply the DHM [

There are many codes that are being used for modeling overland flows. These include FLOW-3D [

physical-based distribution models [

The outline of this paper is as follows. The flow equations that describe the two-dimensional unsteady overland flow solved in DHM are described in the next section. The DHM numerical algorithm characteristics are briefly discussed after which the test problem, the computational grid, boundary flow conditions are detailed. The model results for different grid and roughness conditions are presented, and its performance analyzed.

The two-dimensional flow continuity and momentum equations along the X and Y axis (assuming a constant fluid density without sources or sinks in the flow field and hydrostatic pressure distribution) can be written as [

∂ q x ∂ x + ∂ q y ∂ y + ∂ H ∂ t = 0 (1)

∂ q x ∂ t + ∂ ∂ x [ q x 2 h ] + ∂ ∂ y [ q x q y h ] + g h [ S f x + ∂ H ∂ X ] = 0 (2)

∂ q y ∂ t + ∂ ∂ y [ q y 2 h ] + ∂ ∂ y [ q x q y h ] + g h [ S f y + ∂ H ∂ X ] = 0 (3)

in which q X , q y are flow rates per unit width in the x, y—directions; S f x , S f y represents friction slopes in x, y—directions; H, h, g stand for water surface elevation, flow depth, and gravitational acceleration, respectively; and x, y, t are spatial and temporal coordinates.

The local and convective acceleration terms can be grouped and Equations (2) and (3) are rewritten as

m Z + [ S f z + ∂ H ∂ Z ] = 0 , z = x , y (4)

where m Z represents the sum of the first three terms in Equations (1) and (2) divided by gh. Assuming the friction slope to be approximated by the Manning’s formulae, the flow equation in the U.S. customary units for flow in the x or y directions:

q Z = 1.486 n h 5 / 3 S f z 1 / 2 , z = x , y (5)

Equation (5) can be rewritten in the general case as

q Z = − K Z ∂ H ∂ Z − K Z m Z , z = x , y (6)

where

K Z = 1.486 n h 5 / 3 | ∂ H ∂ S + m S | 1 / 2 , z = x , y (7)

The symbol S in Equation (7) indicates the flow direction which makes an angle of θ = tan − 1 ( q y / q x ) with the positive x-direction. By assuming the value of m to be negligible, the diffusion model can be expressed as:

q Z = − K Z ∂ H ∂ Z , z = x , y (8)

Two-dimensional DHM is formulated by substituting Equation (8) into Equation (1)

∂ ∂ X K x ∂ H ∂ X + ∂ ∂ y K y ∂ H ∂ y = ∂ H ∂ t (9)

If the momentum term groupings were retained, Equation (9) can be written as

∂ ∂ x K x ∂ H ∂ x + ∂ ∂ y K y ∂ H ∂ y + S = ∂ H ∂ t (10)

where

S = ∂ ∂ x ( K x m x ) + ∂ ∂ y ( K x m y )

and K x , K y are also functions of m x , m y respectively.

To maintain continuity in the discussion, while salient aspects of the DHM numerical algorithm are presented here and readers are referred to [

DHM facilitates transient simulation. Predicting transient flow velocities and Froude numbers are important for any engineering designs. The model requires a boundary variable to be specified at the upstream and downstream end. In this application, at the upstream, the inflow hydrograph was specified (discussed in the next section). An unsteady flow model, like DHM, will enable to follow the peak hydrograph flow as it propagates into the flow domain.

Some of the features that were first presented in the DHM model but later carried over to current popular CFD models include: One of the long-recognized challenges in using explicit formulations is in using a constant time step as the solution progresses towards a steady-state solution or to the desired transient period. A constant time step can be computationally intensive and require a significant amount of CPU time. The self-adjustable or self-adaptive variable time step algorithms are powerful tools that are inbuilt into the current codes, which help in choosing an acceptable time step to ensure that the Courant-Friedrich-Lewy (CFL) stability criteria is met. Devoid of such a tool, the user has to enter a best guess, time step (dt) and manually iterate its value until the solution converges. The presence of source terms, which are an important component in the equations to address the physics of flow, coupled with varying bottom roughness coefficient in the domain, will prevent the user from theoretically calculating the optimal time step. Although most of the available commercial software [

The above described DHM model was applied to the flow domain shown in

Boundary conditions are a required component in all numerical models, and the boundary condition used should not only be mathematically consistent, but it should also represent the physics of flow accurately. In this application, for the domain shown in

(

The DHM was applied to predict the maximum flow depth at the north end of the roadway. Models like DHM that can perform transient simulations give more flexibility to researchers for testing various flow combinations.

each computational cell. A manning’s roughness value of 0.03 and 0.015 were used for earth and road. The output was recorded at 0.1 hour period. Overland flow models require smaller cell sizes to reliably predict the preferential flow paths (in the presence of any obstacles) and within the natural watercourse path. In this investigation, we chose a cell size of 30 ft, as the width of the natural watercourse pathway at the upstream is in the order of 30 ft. The optimal cell size is typically influenced by the detailed level that is being predicted and the associated CPU time. An ideal grid size should be small enough to capture the physics of flow and not large so that no additional errors are introduced. Although the currently available computational resources (both memory and processor speed) are enabling researchers to use grids in the order of millions even across a square feet area of the computational domain, the demands of researchers have

also steadily risen. It is not uncommon for large scale application to use days of CPU time across multi-process parallel computers. DHM requires that the grids be uniform size square and it does not have tools like grid stretching or compression.

The results of numerical models tend to be sensitive across different parameters, including grid size, roughness coefficient, and time step. Several parameter sensitivity tests were carried out across the time step, grid spacing and roughness values (for earth and road).

A key component of CFD modeling algorithms is the way they incorporate turbulence terms. These terms are primarily, the fluctuations in pressure and velocity. Both Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) algorithms can well capture these as long as the mesh size is very fine, which will require more computational resources. To address this, Reynolds Averaged Navier-Stokes (RANS) approach, which is based on averaging the flow equations, is being widely used for engineering applications. However, flow turbulence effects can be reasonably modeled by calibration of the friction factor. Additionally, other flow effects can be similarly accommodated, such as flow debris among other factors. In the absence of experimental data for similar applications, the DHM results were compared across various time step, roughness and spatial steps. Results comparing the DHM output with analytical and experimental data for other publications can be found in literature [

In this paper, the performance of the USGS Diffusion Hydrodynamic Model for overland flow application was analyzed. The model has been applied for overland flow over a sloping domain and the focus is on predicting the transient and peak flow variables at the end of the road. The results validate the theoretical basis of the model and the confidence in the values of the predicted variables. Although DHM solves reduced flow equations, it can be reliably used for flood inundation studies. The dominant transient flow processes that govern floodplain inundation are well captured by DHM.

The authors declare no conflicts of interest regarding the publication of this paper.

Hromadka II, T.V. and Rao, P. (2019) Application of Diffusion Hydrodynamic Model for Overland Flows. Open Journal of Fluid Dynamics, 9, 334-345. https://doi.org/10.4236/ojfd.2019.94022