Watercourses and ridges on topographic maps: Why the V’s?

*If you’re not familiar with topographic maps, I recommend going over this National Resources Conservation Service (NRCS) article on reading topo maps first.

Why does a “V” point uphill for a watercourse, and downhill for a ridge?
When viewing topographic maps, you’ll notice that valleys or watercourses are always shown in areas where a V’s pointy end is oriented uphill. In contrast, mountain ridges are in areas where the “V” points downhill. “Just look for the V’s” is common advice when looking for these features.

Note: This topographic map is from the USGS’s official website, on their GIS map viewer.

Why’s it got to be a “V”?
To me, this advice is not intuitive at first because valleys and ridges do not always look like a “V”. Sometimes they can look like really flat, subtle “U”s. It’s also easy to mistake one feature for the other if you just look for the “V.” What’s so special about the Vs?

The 90° rule
A better way to interpret topographic/contour (line of constant elevation) maps, and to understand why “V”s are indicative of valleys and ridges, is the following rule: “Water always flows downhill, perpendicular (at a 90° angle) to contour lines.” Seriously, if you can remember this, you can understand how water will flow in any area, with or without any obvious “V.”

Take a look at this contour map below. The left side shows a valley (watercourse) and the right side shows a ridge, or watershed boundary. Now, imagine it’s raining uniformly over this entire area. The blue lines show where raindrops will flow to once they hit the ground and gravity takes over. These blue lines follow the 90° rule:

If you can follow this example, you can figure out the drainage patterns of any topographic map you view, especially since most maps have contours going all over the place. To be fair, even for the less-obvious ridges or valleys you’ll be able to find a “V.” However the “V” may be very wide (flatter), or may have a lot of curves resembling a sine wave.

Here’s a USGS topographic map of an area with some obvious watercourses, and a well-defined ridge (in red) at the bottom. This area slopes downward to the west. Points A and B are for reference. Notice how there is vegetation from east to west. Vegetation often coincides with a well-defined watercourse.

Note: These modified topographic maps and imagery are from the USGS’s official website, on their GIS map viewer.

And for a better view, here’s a Google Earth version (looking easterly):

Map Data: Google,  INEGI

I hope this helps! Remember, the 90° rule can hep you in any rainy-day situation…Bah dum tss..

Applying Manning’s Equation to Pipes

Does Manning’s equation work for pipes, not just “open” channels?
Yes! As I’ve previously discussed, Manning’s equation can also be used for pipes, as long as there is a free, exposed water surface. The area and wetted perimeter are hard to calculate, but doable if you utilize the graphs that relate circular pipe ratios (D/Dfull, Q/Qfull, etc.). Here’s Manning’s equation below:

(English Units)

Note: If the pipe is pressurized, then Manning’s equation should not be used, but there’s one exception. You can actually use Manning’s equation for a pipe flowing just full, but not technically pressurized. The assumption is that the pipe has just barely become full, and that any additional infinitesimal flow would make the pipe pressurized. The discharge of the pipe in this condition is usually called Qfull or Qfull capacity. Qfull is actually extremely important, because a pipe flowing under the Qfull condition has less discharge than a pipe flowing just below the full depth (a circular pipe conveys the most flow at about 94% of its full depth). The reason why Qfull is less than Q94% depth is because even though there is more flow area in the full-condition, there is even more friction (wetted perimeter) gained as a result of the pipe closing in on itself. This additional friction cancels out the additional flow area and slows down the water.

Take a look at this example to see how this concept applies to a nine foot pipe:

If you still can’t believe this is true (because I definitely didn’t at first!), check out this graph and look for where Q/Qfull is maximum:

Note: This graph assumes “n” does not change with depth.

Qfull is actually pretty usefull (pun intended). At work, I use a popular program called Flowmaster to calculate Qfull. During the beginning, planning stages of sizing a pipe, I’ll use Qfull to get an idea of my pipe’s maximum capacity rather than Q94% depth. Qfull is a safer number to use since there’s always a chance the pipe will seal up with water, especially if there’s a clog in the system or backwater effects.

But remember, gravity-drained systems, such as storm drains, should not be designed solely on the basis of Qfull. A more detailed hydraulic analysis, utilizing the energy equation and a whole lot of iterative calculations (standard-step method) is usually needed, especially if there are any transitions to different-sized pipes, tight curves, abrupt changes in the slope, and/or the pipe becomes pressurized.

As a shortcut for the PE exam, here’s the formula for calculating Qfull in a circular stormwater or sewage pipe. If the pipe is not full, use the circular pipe ratio graphs to calculate A and R for use in Manning’s equation:

(Circular pipe; English Units)

Open Channel Flow – Manning’s Equation

(English Units)

Manning’s equation is perhaps the most popular formula for open channel flow. You can calculate the flow and velocity (Q/A) of a channel or non-pressurized conduit, such as a circular pipe, using this equation.

This formula can also be rearranged to solve for the normal depth (yn) of an open-channel, such as a rectangular channel:

(rectangular channel; solve by trial & error)

 Here’s a summary of each term below:

Q: Flow, a.k.a. discharge (cfs)

n: The Manning’s “roughness” coefficient of the channel. This value shows how much resistance is acted upon the water by the channel. A lower n-value means less roughness, and usually implies a higher velocity and smaller depth in the channel (and vice-versa), with all else being equal. Concrete, which is valued for its hydraulic “smoothness”, has an n-value between 0.013-0.015. For comparison, a natural stream with little to heavy vegetation can have an n-value ranging anywhere between 0.025 to 0.150. Now just imagine riding your road bike on concrete vs. a grassy field, and which surface is much easier to ride on. That’s how the water feels.

A: Flow area (ft2) (Note: not necessarily the full area of your channel section!). For example, if a rectangular channel is flowing half-full, the area would be the base x ½ height, not base x height.

w: width (base) of a rectangular channel (ft).

R: Hydraulic radius (ft), or R = A / P. P is the wetted perimeter, or the length of water that is in contact with the physical channel (i.e. receiving friction). For example, in a rectangular channel flowing half-full, wetted perimeter is the base plus the length of both vertical sides touching the water (see equation above).

S: Technically it’s the friction slope (Sf), but for most applications (and on the P.E. exam) it is the channel’s slope, in decimal form (e.g 0.003 or 0.2). Channel slope is assumed because the prime assumption of Manning’s equation is that the channel is flowing under uniform flow. In uniform flow, the gravitational forces (i.e the weight of the water) cancel out the frictional (resisting) forces, which causes the friction slope to equal the channel slope (Sf = Schannel). Do an energy balance calculation between two points on a uniform-flow channel and prove it to yourself (I will cover this in a more nerdy, in-depth discussion of uniform flow in the near future)

yn: Normal depth (ft), or the depth of flow the water would normally take in the channel assuming there are no changes in the channel’s shape, friction, or backwater effects in either the upstream or downstream direction from the channel for a good distance. In other words, this is uniform flow, as will be discussed in a future post.

Manning’s equation can also be used for non-pressurized pipes, or those flowing with an exposed water surface, as I’ve discussed in a separate post.

P.S. A shoutout to Conrad at ReviewCivilPE.com. I don’t think he updates his site anymore, but his article on Manning’s equation inspired mine!