|
|
| Line 1: |
Line 1: |
| A '''flownet''' is a graphical representation of two-[[dimensional]] steady-state [[groundwater]] flow through [[aquifer]]s.
| | I woke up the other day and [http://Www.Adobe.com/cfusion/search/index.cfm?term=&realised+-&loc=en_us&siteSection=home realised -] I've been single for some time today and after much bullying from buddies I today locate myself registered for on line dating. They [http://Www.Reddit.com/r/howto/search?q=assured assured] me that there are lots of sweet, normal and [http://lukebryantickets.neodga.com luke brian concerts] entertaining folks to meet up, therefore [http://www.museodecarruajes.org luke brian tickets] the pitch is gone by here!<br>My pals and family are magnificent and hanging out with them at bar gigabytes or dishes is definitely a necessity. As I see that one can not own a good conversation against the sound I have never been into nightclubs. Additionally, I have 2 quite cunning and unquestionably cheeky puppies who are almost always ready to meet up new individuals.<br>I endeavor to maintain as physically fit as possible staying at luke bryan tickets ([http://lukebryantickets.pyhgy.com try this]) the fitness center many times per week. I enjoy my sports and attempt to perform or view as numerous a possible. Being wintertime I am going to frequently at Hawthorn fits. Notice: Supposing that you will considered buying an athletics I really do not brain, I've seen the carnage of wrestling fits at stocktake revenue.<br><br>Visit my weblog :: [http://www.hotelsedinburgh.org luke bryan where is he from] |
| | |
| Construction of a flownet is often used for solving groundwater flow problems where the geometry makes analytical solutions impractical. The method is often used in [[civil engineering]], [[hydrogeology]] or [[soil mechanics]] as a first check for problems of flow under hydraulic structures like [[dam]]s or sheet [[Deep foundation|pile]] walls. As such, a grid obtained by drawing a series of equipotential lines is called a flownet. The flownet is an important tool in analysing two-dimensional irrotational flow problems.
| |
| | |
| ==Basic method== | |
| The method consists of filling the flow area with stream and equipotential lines, which are everywhere [[perpendicular]] to each other, making a [[curvilinear grid]]. Typically there are two surfaces (boundaries) which are at constant values of potential or hydraulic head (upstream and downstream ends), and the other surfaces are no-flow boundaries (i.e., impermeable; for example the bottom of the dam and the top of an impermeable bedrock layer), which define the sides of the outermost streamtubes (see figure 1 for a stereotypical flownet example).
| |
| | |
| Mathematically, the process of constructing a flownet consists of [[contour line|contour]]ing the two harmonic or [[analytic function]]s of potential and [[stream function]]. These functions both satisfy the [[Laplace equation]] and the contour lines represent lines of constant head (equipotentials) and lines tangent to flowpaths (streamlines). Together, the potential function and the stream function form the [[complex potential]], where the potential is the real part, and the stream function is the imaginary part.
| |
| | |
| The construction of a flownet provides an approximate solution to the flow problem, but it can be quite good even for problems with complex geometries by following a few simple rules (initially developed by [[Philipp Forchheimer]] around 1900, and later formalized by [[Arthur Casagrande]] in 1937) and a little practice:
| |
| * streamlines and equipotentials meet at [[right angle]]s (including the boundaries),
| |
| * diagonals drawn between the cornerpoints of a flownet will meet each other at [[right angle]]s (useful when near singularities),
| |
| * streamtubes and drops in equipotential can be halved and should still make squares (useful when squares get very large at the ends),
| |
| * flownets often have areas which consist of nearly [[parallel (geometry)|parallel]] lines, which produce true squares; start in these areas — working towards areas with complex geometry,
| |
| * many problems have some [[symmetry]] (e.g., [[wikt:radial|radial]] flow to a [[water well|well]]); only a section of the flownet needs to be constructed,
| |
| * the sizes of the squares should change gradually; transitions are smooth and the curved paths should be roughly [[ellipse|elliptical]] or [[Parabola|parabolic]] in shape.
| |
| | |
| ==Example flownets==
| |
| The first flownet pictured here (modified from Craig, 1997) illustrates and quantifies the flow which occurs under the [[dam]] (flow is assumed to be invariant along the axis of the dam — valid near the middle of the dam); from the pool behind the dam (on the right) to the [[tailwater]] downstream from the dam (on the left).
| |
| | |
| There are 16 green equipotential lines (15 equal drops in hydraulic head) between the 5 m upstream head to the 1m downstream head (4 m / 15 head drops = 0.267 m head drop between each green line). The blue streamlines (equal changes in the streamfunction between the two no-flow boundaries) show the flowpath taken by water as it moves through the system; the streamlines are everywhere tangent to the flow velocity.
| |
| | |
| [[Image:flownet pumping well.png|thumb|right|200px|Example flownet 2, '''click to view full-size'''.]]
| |
| The second flownet pictured here (modified from Ferris, et al., 1962) shows a flownet being used to analyze map-view flow (invariant in the vertical direction), rather than a cross-section. Note that this problem has symmetry, and only the left or right portions of it needed to have been done. To create a flownet to a point sink (a singularity), there must be a recharge boundary nearby to provide water and allow a steady-state flowfield to develop.
| |
| | |
| ==Flownet results==
| |
| [[Darcy's law]] describes the flow of water through the flownet. Since the head drops are uniform by construction, the gradient is inversely proportional to the size of the blocks. Big blocks mean there is a low gradient, and therefore low discharge (hydraulic conductivity is assumed constant here).
| |
| | |
| An equivalent amount of flow is passing through each streamtube (defined by two adjacent blue lines in diagram), therefore narrow streamtubes are located where there is more flow. The smallest squares in a flownet are located at points where the flow is concentrated (in this diagram they are near the tip of the cutoff wall, used to reduce dam underflow), and high flow at the land surface is often what the civil engineer is trying to avoid, being concerned about [[Soil mechanics#Seepage: steady state flow of water|piping]] or [[Dam#Examples of dams|dam failure]].
| |
| | |
| ==Singularities==
| |
| Irregular points (also called [[mathematical singularity|singularities]]) in the flow field occur when streamlines have kinks in them (the [[derivative]] doesn't exist at a point). This can happen where the bend is outward (e.g., the bottom of the cutoff wall in the figure above), and there is infinite flux at a point, or where the bend is inward (e.g., the corner just above and to the left of the cutoff wall in the figure above) where the flux is zero.
| |
| | |
| The second flownet illustrates a [[water well|well]], which is typically represented mathematically as a point source (the well shrinks to zero radius); this is a singularity because the flow is converging to a point, at that point the [[Laplace equation]] is not satisfied.
| |
| | |
| These points are mathematical artifacts of the equation used to solve the real-world problem, and do not actually mean that there is infinite or no flux at points in the subsurface. These types of points often do make other types of solutions (especially numeric) to these problems difficult, while the simple graphical technique handles them nicely.
| |
| | |
| ==Extensions to standard flownets==
| |
| Typically flownets are constructed for [[wiktionary:Homogeneity|homogeneous]], [[isotropic]] porous media experiencing saturated flow to known boundaries. There are extensions to the basic method to allow some of these other cases to be solved:
| |
| * inhomogeneous aquifer: matching conditions at boundaries between properties
| |
| * [[anisotropic]] aquifer: drawing the flownet in a transformed domain, then scaling the results differently in the principle hydraulic conductivity directions, to return the solution
| |
| * one boundary is a seepage face: iteratively solving for both the boundary condition and the solution throughout the domain
| |
| | |
| Although the method is commonly used for these types of groundwater flow problems, it can be used for any problem which is described by the [[Laplace equation]] (<math>\nabla^2 \phi = 0</math>), for example [[electrical current]] flow through the earth.
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| * Casagrande, A., 1937. Seepage through dams, ''Journal of New England Water Works'', 51, 295-336 (also listed as: Harvard Graduate School Eng. Pub. 209)
| |
| * Cedergren, Harry R. (1977), ''Seepage, Drainage, and Flow Nets'', Wiley. ISBN 0-471-14179-8
| |
| * {{cite book|last=Chanson|first=H.|authorlink=Hubert Chanson|title=Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows|url=http://espace.library.uq.edu.au/view/UQ:191112|publisher=CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages|year=2009|isbn=978-0-415-49271-3}}
| |
| * Knappett, Jonathan and R.F. Craig, 2012. ''Craig's Soil Mechanics'' 8th edition, Spon Press. ISBN 978-0-415-56126-6
| |
| * Ferris, J.G., D.B. Knowles, R.H. Brown & R.W. Stallman, 1962. ''Theory of Aquifer Tests''. US Geological Survey Water-Supply Paper 1536-E. (available from the [http://water.usgs.gov/pubs/wsp/wsp1536-E/html/pdf.html USGS website as a pdf])
| |
| * Harr, M.E., 1962. ''Groundwater and Seepage'', Dover. ISBN 0-486-66881-9 — mathematical treatment of 2D groundwater flow, classic work on flownets.
| |
| | |
| ==See also==
| |
| * [[Potential flow]] (the flownet is a method for solving potential flow problems)
| |
| * [[Analytic function]] (the potential and streamfunction plotted in flownets are examples of analytic functions )
| |
| | |
| [[Category:Hydrology]]
| |
| [[Category:Soil mechanics]]
| |
| [[Category:Water]]
| |
| [[Category:Fluid mechanics]]..
| |
I woke up the other day and realised - I've been single for some time today and after much bullying from buddies I today locate myself registered for on line dating. They assured me that there are lots of sweet, normal and luke brian concerts entertaining folks to meet up, therefore luke brian tickets the pitch is gone by here!
My pals and family are magnificent and hanging out with them at bar gigabytes or dishes is definitely a necessity. As I see that one can not own a good conversation against the sound I have never been into nightclubs. Additionally, I have 2 quite cunning and unquestionably cheeky puppies who are almost always ready to meet up new individuals.
I endeavor to maintain as physically fit as possible staying at luke bryan tickets (try this) the fitness center many times per week. I enjoy my sports and attempt to perform or view as numerous a possible. Being wintertime I am going to frequently at Hawthorn fits. Notice: Supposing that you will considered buying an athletics I really do not brain, I've seen the carnage of wrestling fits at stocktake revenue.
Visit my weblog :: luke bryan where is he from