In unsteady open channel flow of storm water networks of circular section, besides the hydrographs entering in each link resulting from the continuity equation and excluding weirs or control devices, frequently the boundary conditions considered are the energy equation in the manholes and the critical condition or the uniform flow condition in the case of a free fall downstream or manhole without influence from downstream. A possible approach that is sometimes considered in design situation considers each link separately and the condition of steady uniform flow in the downstream end of each link, for each flow rate reached in the extremity in any instant of time. What is (or may be) the magnitude of the differences reached with this approximation with respect to the real conditions, assuming, in any case, an open channel flow as should be considered in a design situation?
In open channels, the boundary condition is important in calculating the flow profile including the peak flow. For example, you can assume no slip condition (velocity = 0) at the boundary and you can also assume v not equal zero at the boundary and you will get different flow profile for each case. Also, you can solve the problem with and without friction at the boundary of the channel.
Hope this helps answer your question.
Professor Yehia Khalil
Yale University
USA
Simulation of Turbulent Flows
https://web.stanford.edu/class/me469b/handouts/turbulence.pdf
CHAPTER 4
FLOW IN CHANNELS
http://ocw.mit.edu/courses/earth-atmospheric-and-planetary-sciences/12-090-introduction-to-fluid-motions-sediment-transport-and-current-generated-sedimentary-structures-fall-2006/course-textbook/ch4.pdf
Quantifying the relative importance of flow regulation and grain size regulation of suspended sediment transport a and tracking changes in grain size of bed sediment b
https://walrus.wr.usgs.gov/reports/reprints/Rubin_WRR_37.pdf
Limits of the Hydrodynamic No-Slip Boundary Condition
http://groups.mrl.illinois.edu/granick/Publications/PDF%20files/2002/Granick%20group%20-%20139%20-%20Limits%20of%20no-slip.pdf
Thanks for the sympathy of your answer, Dear Yehia that has perhaps the merit of helping me to opening the discussion. I was thinking about the numerical integration of the Saint-Venant equations for one-dimensional gradually varied unsteady open channel flow (for example by finite differences using a 4 points implicit schema) where two boundary conditions (or conditions of frontier) are required in the extremity of each link (sewer between manholes) to complete the set of linear equations required for solving the differential equations numerically in each instant of time based on a solution known on the previous instant of time. I am interested in dendritic networks in open channel flow (directed trees or tree-like systems, not in looped networks and also not pipe flow). SWMM from USEPA, for example, uses also finite differences with an explicit schema and I suppose that it allows both conditions of frontier mentioned in the downstream extremity (uniform flow, mentioned as “normal depth”, and free outfall). The first results that I obtained for a single link (single sewer between manholes) does not seem show substantive differences in the peak flow reached between the uniform flow condition in the extremity, the critical condition in the extremity, or the maximum water flow depth of design at the extremity for a circular section (frequently y=0.82 D, when Manning Equation is considered). It seems that the differences when considering these different conditions in the peak flow reached in any section are merely residual (i.e., small) and only occur practically in a relatively small extension of the link. But I still need to develop or extend fully these initial results, and analysis, particularly for arborescent networks (and not just a link) which I hope to do as soon as possible.
Thanks Dear Jetty for your friendly answer with some good references that may help to make some sort of introduction to some aspects of the extremely wide, wonderful and complex world of Open Channel Hydraulics.
Dear Antonio
it was indeed a good food for thought and Jetty has given a precious tasty answer. A bit of spice I would like to add, on your valuable reactions. Do we attempt finite element configurations while solving governing differential equations of gradually varied flow. May be you get a bit superior end results during calibration/simulation. I would like to add upon my knowledge if you have any reaction on it. Solving equations at unequal grids / spacing looks to be a issue in your case, if I am not much wrong.
Thanks for your good answer, Dear Gaur. I suppose that you are thinking mainly about numerical issues that may be really important in the simulation. When I implemented in computer for the first time the implicit model that I developed in 1995/1996, for example, I defined all real variables as Double Precision (in Fortran Language), in order to allow a very thin grid/mesh of points in the time-space plan.
Yes, IMPLICIT methods of finite differences are recognized as being superior, particularly with respect to stability, and to the steps of time and space required, i.e., the considered mesh in the plan of the time-space. Can eventually the differences numerical introduced by different spacing of the mesh with some numerical models be larger than the differences due to the mentioned boundary conditions?
Currently there are several models available. SWMM from USEPA, for example, is freely available in Internet. See the link bellow and also the 13th edition from 2010 of the User’s guide to SWMM 5, by James, W.; Rossman, L.; James, W.R.C.
https://www.epa.gov/water-research/storm-water-management-model-swmm
Happy to read good points, but the magnitude of the differences reached with this approximation with respect to the real conditions further needs elaboration with examples.
Here is a good resource
https://www.sciencedirect.com/science/article/pii/S0307904X06001181
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