Abstract:
 

An analysis of wind induced wave setup and circulation patterns is undertaken with numerical modeling provided in the RMA2 package within the Surfacewater Modeling Systems (SMS) software. Graphics are provided depicting changes in water surface characteristics for associated changes in wind direction for a given wind speed. Though this is the focus of the project RMA2 methodology will also be discussed.

Introduction:
 

Circulation patterns and wind induced setup in water bodies have been something of intrigue in the hydrodynamics world for quite some time. Numerical models have recently been implemented to help scientists and engineers better understand these processes. RMA2 is one such model that is a two-dimensional depth averaged finite element hydrodynamic numerical model. It computes water surface elevations and horizontal velocity components for subcritical, free surface flow in two-dimensional flow fields. Utilizing its calculations of water surface elevation, depth, and two-dimensional velocity in the horizontal plane a clearer picture of natural environments can be attained. The practical attributes of these types of analyses stretch from evaluation of existing coastal structures to tracking extremely sensitive transport of water insoluble contaminants. Surfacewater Modeling Systems (SMS) provides pre- and post processing for models like RMA2.

Motivations:
 

Between the two issues of circulation patterns and wind induced setup, most notably storm surge, an extrapolation can be made to most water quality and coastal engineering problems. For example, ascertaining the magnitude of wind induced setup is paramount in designing coastal structures. It is similar to the importance of foundation design for high-rise buildings, with the foundation being the storm surge and the building being the wave. Often times the public perceives that the destructive mechanism in coastal environments is the wave, but in reality, especially in oceanic coastal areas, storm surge is the major culprit. Simply put, assuring a stable coastal area requires understanding wind induced setup.

Some of the major issues requiring understanding circulation patterns include retention time calculations and contaminant transport. Retention time is loosely defined as the time a particle of water spends in an impoundment. However, this calculation is often times extrapolated to the time a contaminant particle spends in a water body. This is typically a conservative estimate so understanding circulation may lead to tracing a contaminant’s retention time with numerical models. This, in turn, may help avoid over design for a given project.

The quantification of contaminant transport is, in many ways, a more complicated problem, but it is an easier problem to conceptualize. Scientists and engineers simply want to find out what areas of a water body are more likely to see higher concentrations of contaminants. These areas of higher concentrations are often times solely due to the physics of the system, and surface water circulation patterns can have a very significant contribution to the systems characteristics.
 

The objective of this project is to observe how changes in wind and basin characteristics affect both wind induced setup and circulation patterns in water bodies.

Methods:

RMA2 computes a finite element solution of the Reynolds form of the Navier-Stokes equations for turbulent flows. Friction is calculated with the Manning’s or Chezy equation, and eddy viscosity coefficients are used to define turbulence characteristics. Both steady and unsteady state problems can be analyzed. To do this, the model solves the depth integrated equations of fluid mass and momentum conservation in two horizontal directions. The conservation of mass is seen as equation 1, and momentum conservation is seen in equations 2 and 3.

    (1) 

     (2) 

      (3) 
 
 
 

                              Table 1:Key to terms in equations 1 through 3

x, y	Cartesian directions
u, v	velocities in the x and y directions, respectively
h	water depth
a	bottom elevation
E	eddy viscosity coefficient
n	Manning's roughness coefficient
Va	Wind Speed
y	wind direction
w	Earth’s rotation
	Wind Shear Coefficient
F	local latitude

In looking at the factors seen in Table 1 it is clear that this model considers setup due to Coriolis Force (w, F) also, but for smaller inland lakes the primary driving force is wind. The finite element method solves the mass and momentum equations using the Galerkin Method of weighted residuals. The solution is fully implicit and the set of equations is solved simultaneously by Newton-Raphson nonlinear iteration. To better illustrate what this method of iteration entails a quadratic function may be seen in figure 1. The idea is to find a solution to the equation as close to the root as possible.

On the x axis, x₁ is the initial guess at a value which will yield the optimum solution. The solution with x₁ is the point marked “initial solution”. A line (tangent line 1) which is tangent to the curve at this point is computed. The place where this line crosses the x axis becomes x₂; the second guess used to solve the problem. A new solution is calculated from x₂, and another tangent line (tangent line 2) is computed. The point where this tangent line crosses the x axis becomes the next guess, x₃. And so on, until the difference in value along the x axis, between two successive solutions, becomes less than the a pre-defined convergence criterion. At this point, the solution has converged.
 

Figure 1: Iteration discussion from RMA2 literature.
 

Surfacewater Modeling Systems (SMS) was chosen to meet the aforementioned objectives. The software provides pre- and post processing for surface water modeling and analysis. It includes two-dimensional finite element, two-dimensional finite difference, three-dimensional finite element and one-dimensional backwater modeling tools. It provides interfaces specifically designed to facilitate the utilization of several numerical models. SMS can develop profiles and cross section plots, two-dimensional vector plots, drogue plots, color shaded contour plots, time variant curve plots, and dynamic animation sequences from solution sets produced by RMA2.

Once these very basic conceptualizations of the focus models have been sewn, a general approach to understanding the manifestations of wind induced setup and circulation patterns must be developed. To that end, a synthetic lake was created. Lake Larry was created as a uniform depth basin with rectangular dimensions. Though easy to envision it may be seen in figure 2.
 
 

Figure 2: Lake Larry dimensions and wind directions used in modeling process. All angles are referenced to the positive x direction.
 

The reader may be questioning how a basin such as this could ever be applied to an actual natural body of water. One of the principle lessons learned during this process was that in trying to utilize a numerical model the user must understand the way the model handles the most basic of problems. So the analysis of Lake Larry represents the first step in fully understanding RMA2.

In addition to the dimensions shown in the figure other key parameters are the Manning’s n value of 0.03 and the eddy viscosity coefficient of 25. Though, clearly, the implementation of an eddy coefficient is not necessary here it was done so as to stay consistent with examples found in the SMS documentation.

In order to proceed boundary conditions had to be established. To maintain a stable and stagnant water surface these were set to 5 cfs of steady flow at the east and west ends of the lake. The north and south boundaries are no flow. Once a still water surface had been established, the wind could be introduced into the system. Five different wind directions were analyzed while keeping the wind speed constant to study the effects of a changing wind direction only. These directions can be seen in figure 2. Setup calculations from RMA2 were then verified with analytical techniques learned in Professor Wu’s Coastal Engineering class. The method of choice was the non-linear storm surge technique. The equation and associated components can be seen in equation 4.

 (4) 

               Where:
 
 

Table 2: Table describing terms in equation 4.

d	water depth
rw	density of water
ts	surface shear stress due to wind
g	gravity
Dx	wet distance over which the wind blows
U10	wind speed
Cd	1.21E-6 if U10 is less than 5.6 m/s E-6 if U10 is greater than 5.6 m/s

The following are plan views of Lake Larry. Each view represents a different wind direction while maintaining a wind velocity of 50 mph. The arrows on each diagram are created by a grid representation of a two-dimensional velocity vector field. The color coded contoured information represents water surface elevation. Please disregard the upstream inflow (40000) and downstream head (20) displayed on the figures. They are meaningless, and are not the true boundary conditions for the actual model simulations.

Table 3: Expected setup given wind direction as calculated with non-linear method

Wind Direction (° cc from positive x-axis)	Expected Setup (ft)
0	0.648
45	0.372
90	0.264
180	0.648
225	0.372

Wind direction equal to 0° from the positive x-axis:

 

Wind direction equal to 45° from the positive x-axis:

Wind direction equal to 90° from the positive x-axis:

Wind direction equal to 180° from the positive x-axis:

 

Wind direction equal to 225° from the positive x-axis:

 
 
 
 
 
 
 

Discussion:
 

After observing the contour mapping of the RMA2 simulations it seems quite reasonable to ascertain that the model can predict wind induced setup. For all of the wind directions the results are quite close to what were calculated with analytical techniques. It seems, though, that the rest of the surface elevation shape is possibly inaccurate. This is especially true when the wind direction is against what the model perceives as the flow direction. For this study the upstream and downstream flows were 5 cfs, an essentially negligible amount of flow. This was thought to have provided a stagnant situation in which wind of the same magnitude from the exact opposite direction would create the same setup. This can be accomplished, but the surface water elevations from the shore of the wind’s origin are obscenely small. As seen in the 180° and 225° wind direction simulations these values are -8.5 and -2, respectively. Discovering the reason for this unequal handling of, essentially, the same wind direction must be understood before an accurate representation of wind induced setup can be surmised.

The true value of the figures displayed in the results section is the setup analysis. Though observing the circulation patterns is a very useful tool, a transient (multiple time steps) solution is more appropriate than the steady state solution shown here. Again, this study represents a first step, and an understanding of the steady state must be acquired before the unsteady scenario is engaged.

The idea of comprehending the basics also fostered the design of Lake Larry, but a more appropriate idea would have been to construct a channel rather than a basin.  This would have allowed for a more educational circulation output. With the current basin, it is almost impossible to get a clear picture of how the model analyzes circulation because of the reflective characteristics of the basin itself.

In addition to these issues, because RMA2 is only a two-dimensional model, it doesn't have the ability to model the affects of stratification in a water body. Temperature stratification in lakes causes a density differential within the body itself. These differentials cause vertical circulation patterns, and these vertical patterns induce a subsequent circulation in the horizontal plane. The third dimension that RMA2 doesn't consider is the vertical one.

Lastly, I don’t feel that SMS provides a straightforward interface in which RMA2 can be used. Many of the useful tools of RMA2 aren’t readily available in the SMS interface and require manual input using text file representations of programming cards. Though the pre- and post processing tools are very nice in SMS, if the user doesn’t have easy access to the more useful features of the numerical models it houses these processing techniques are useless. I also feel that the literature provided by SMS is horribly insufficient. If one plans to use the software I would highly recommend taking a short course or getting involved with an individual with prior SMS modeling experience.
 

As was noted in the previous section, this study was a first step. Interestingly, the next several steps will be nearly as basic as the current one. The next analysis will be in evaluating setup and circulation with changing wind speed for the wind directions used for this study. When the nuances of these scenarios are understood a variety of changes in basin roughness will be applied. Finally, I will be able to study a basin with a complicated bottom topography.

The steps taken in increasing complexity are in an effort to eventually develop a reliable and sound model of Lake Kegonsa, WI. This lake is the southernmost lake of the Madison chain, which is fed by the Yahara River. This basin is of interest to me because it has a very significant nutrient contaminant problem, and I grew up recreating on it. Both an aerial photo and a bathymetric map can be seen in figures ???? and ????.  In this case, I would be much more interested in the circulation patterns in an attempt to better quantify retention time for such things as insoluble phosphate groups. Typically, the major culprits in agricultural lakes like this are soluble contaminants, but understanding circulation patterns in natural systems will prove to be very useful in my career. I look forward to expanding on my knowledge of numerical models, wind induced setup, and circulation patterns.
 

Figure 3: Aerial photograph of Lake Kegonsa, WI

Figure 4: Hydrographic map of Lake Kegonsa, WI with depth in meters
 

Frankly, I haven’t enjoyed the majority of the hours I have spent working on this project. I let my own grandiose view of engineering design put me in a very difficult position. The major factor in my discomfort, however, was underestimating the incredible amount of hours required to understand a new model. This effect is accentuated when the user doesn’t have a lot of modeling experience. Though the extra time spent with storm surge and shallow water numerical method theories helped fill in some of the spaces in what I learned in Professor Wu’s class, the greatest lessons that I will take away from this project are of the more general variety. I learned that an idea of the big picture should never be sacrificed and that once the engineer loses this perspective he finds himself tongue-tied at the simplest of inquiries. More importantly than the embarrassment is that if you don’t constantly remind yourself of the grander issues and applications you can’t learn. I also learned the importance of understanding the simple before engaging the complicated. I suppose I knew this before Professor Wu repeated to me 4 to 5 times in succession, but it is an easy principle to look past. It is the idea that spawned innovation, and neglecting it will lead to certain failure. Thanks Chin for being persistent and making time for your students.

References:
 

Boss International. Surfacewater Modeling System (SMS) Overview Guide. www.bossintl.com.

Boss International. Surfacewater Modeling System (SMS) Tutorial. www.bossintl.com.

Chow, Ven Te. Open Channel Hydraulics. 1959. McGraw-Hill. Boston.

Hahn, C.T., Barfield, B.J., Hayes, J.C. Design Hydrology and Sedimentology for Small Catchments. 1994. Academic Press. San Diego.

Kamphuis, William J., Introduction to Coastal Engineering and Management, 2000, World Scientific, London.

RMA and WES, User Guide to RMA2 Version 4.3

Hoopes, Dr. John. University of Wisconsin-Madison Department of Civil and Environmental Engineering.

Wu, Dr. H. Chin. University of Wisconsin-Madison Department of Civil and Environmental Engineering.

Yang, Jian. University of Wisconsin-Madison.

http://limnology.wisc.edu/