L06-flow_nets

# L06-flow_nets - 1 6 FLOW NETS 6.1 Introduction Let us...

This preview shows pages 1–5. Sign up to view the full content.

6. FLOW NETS 6.1 Introduction Let us consider a state of plane seepage as for example in the dam shown in Figure 1. Drainage blanket Phreatic line Unsaturated Soil Flow of water z x Fig. 1 Flow through a dam For an isotropic material the head satisfies Laplace's equations, thus analysis involves the solution of: 0 2 2 2 2 = + z h x h subject to certain boundary conditions. 6.2 Representation of Solution At every, point (x,z) where there is flow there will be a value of head h(x,z). In order to represent these values we draw contours of equal head as shown on Figure 2. Flow line (FL) Equipotential (EP) Fig.2 Flow lines and equipotentials 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
These lines are called equipotentials. On an equipotential (EP). by definition: constant ) , ( = z x h (1a) it thus follows h x dx h z dz + = 0 (1b) and hence the slope of an equipotential is given by dz dx h x h z EP = - / / (1c) It is also useful in visualising the flow in a soil to plot the flow lines (FL), these are lines that are tangential to the flow at a given point and are illustrated in Figure 2. It can be seen from Fig. (2) that the flow lines and equipotentials are orthogonal. To show this notice that on a flow line the tangent at any point is parallel to the flow at that point so that: [ ] [ ] dx dz v v x z : : (2a) it follows immediately that: dx dz v v now from Darcy s law v k h x v k h z thus dx dz h x h z FL x z x z FL = = - = - = ' / / (2b) and so dx dz dx dz FL EP = - 1 (3) and thus the flow lines and equipotentials are orthogonal in an isotropic material. 2
6.3 Some Geometric Properties of Flow Nets Consider a pair of flow lines, clearly the flow through this flow tube must be constant and so as the tube narrows the velocity must increase. Suppose now we have a pair of flow lines as shown in Figure 3. Q X y z t T Y Z X FL FL Q h h+ h h+2 h EP Fig. 3 Equipotentials intersecting a pair of Flow Lines Suppose that the flow per unit width (in the y direction) is, Q, then the velocity v in the tube is given by v Q yx = (4a) Also let us assume that the potential drop between any adjacent pair of equipotentials is h then it follows from Darcy’s law: v k h zt = (4b) It thus follows that: Q k h yx zt = (4c) using an identical argument to that used in developing equation(4c) it can be shown that: Q k h YX ZT = (4d) and hence that: 3 x

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
yx zt YX ZT = (5) Thus each of the elemental rectangles bounded by the given pair of flow lines and a pair of equipotentials (having an equal head drop) have the same length to breadth ratio.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern