Glenn E. Moglen Department of Civil & Environmental Engineering Virginia Tech Course Introduction & Begin Specific Energy CEE 4324/5984 –Open Channel Flow – Lecture 1

Course Prerequisites: “CEE 3304: Fluid Mechanics, CEE 3314: Water Resources Eng.” Call Roll Discuss syllabus Wikipedia Assignment (CEE 5984 only) Moglen’s Windmill: Significant Figures Specific Energy Definition Example 1: One energy, two depths Example 2: Minimum energy (critical depth) Today’s Agenda

Discuss Syllabus Main Points: Homework due on date indicated Class Attendance Inform Dr. Moglen BEFORE planned absences Contacting Dr. Moglen Academic Honesty

Introductory Information Class web sites: Scholar (as usual, submit work at dropbox) Open to public: Book: Please order a copy of the book: Chaudhry: Open Channel Flow (2 nd Edition) ISBN:

Wikipedia Assignment Applies only to students in CEE 5984 Explanatory document at web site There are existing pages you may examine: First deadline is Sept. 1 (group formation)

QUESTIONS?!

Rule 1: Exact numbers have infinite significant digits. Ex: A dozen eggs is exactly 12 eggs. Rule 2: Zeros used just to position the decimal point are not significant figures. Trailing zeros are not significant unless a decimal point is explicitly indicated. Ex: has 3 significant figures Ex: 20 has 1 significant figure Ex: 20. has 2 significant figures Ex has 3 significant figures For Starters: Significant Figures

Rule 3: In multiplication and division, an answer contains no more significant figures than the least number of significant figures used in the operation. Ex: For Starters: Significant Figures (cont.)

Rule 4: In addition and subtraction, the last digit retained in the sum or difference is determined by the first doubtful digit. Ex: For Starters: Significant Figures (cont.)

Fact 1: Car generally gets between 40 and 50 mpg depending type of driving and season Fact 2: Fuel efficiency is reported on dashboard to the tenth’s place (e.g mpg) Observation: Although efficiency is reported to the tenths place, certain efficiency values are never seen. For instance, the progression of reported values between 44 and 45 mpg is: 44.2, 44.4, 44.6, 44.9 (you’ll never see 44.1, 44.3, 44.5, 44.7, 44.8) Explain? Significant Digits: Real World Example: Dr. Moglen’s Honda Civic Hybrid

Dr. Moglen’s theory: Fuel efficiency is actually calculated internally in units of km/liter and truncated to 1 decimal place (e.g 18.6 km/liter). The conversion factor from km/liter to mpg is 2.352…. = 2.4 (to two significant digits) Thus: Significant Digits: Real World Example: Dr. Moglen’s Honda Civic Hybrid (cont’d.) Question: if using 2.4 conversion factor, how many sig. figs. should be used to report fuel efficiency?

Significant Figures – WARNING! Significant figures are important. Hydrology is a science with considerable uncertainty (small precision). I reserve the right to subtract percent from any answer with an inappropriate number of significant figures. Use significant figures you report to convey your certainty about a value.

Types of OCF Problems that can be solved Let: y : depth of flow t : time x : distance measured along direction of flow In general: Steady-flow (depth not a function of time) Uniform flow (depth not a function of distance)

Types of OCF Problems that we’ll consider Under almost all circumstances in this class, we’ll be focusing on steady flow Mostly we’ll focus on uniform flow, except near special in-stream features (e.g. sluice gate, constriction, step, hydraulic jump) Later in the semester we’ll take on gradually varied flow (non-uniform flow)

Remember fluid mechanics? Name two important equations you learned… Continuity: Bernoulli:

How is that we can measure energy in units of feet ([=] length)? In physics I learned E [=] ML 2 T -2 Divide by mg Call E’  E and h  y

The basic energy equation for open channel flow or, actually we usually equate energy between an upstream location (1) and downstream location (2): where h L is headloss between 1 and 2

What’s happened to the pressure term from Bernoulli’s equation? Pressure isn’t zero. It’s simply that we are studying open channels and therefore the pressure at location 1 and 2 is the same – atmospheric pressure.

Analogy to Pipe Flow – the Dual Role of Flow Depth >

“Specific” or “Unit” Discharge Discharge is Q [=] L 3 /T Consider a rectangular channel of width, w. Specific or unit discharge ( q ) is the discharge per unit width of channel: Specific or unit discharge is q [=] L 2 /T q is ONLY defined in the context of a rectangular channel

Alternative Expression for Specific Energy By continuity: Therefore: and:

Alternative Expression for Specific Energy Substituting in specific energy equation: This second form of the equation is: ONLY defined for rectangular section Useful because it expresses E as a function of y only (all other terms are constants)

Example 1: Let Q = 10 ft 3 /s, w = 1 foot So q = 10/1 = 10 ft 2 /s We could let y vary over a range of values (say 0.4 feet to 10 feet). The resulting relationship between E and y looks like:

Example 1(cont.): This is the E-y or specific energy diagram

Example 1(cont.): Consider E =5 feet

Example 1(cont.): How can there be two depths that possess the same energy? E is sum of kinetic and energy sources. One depth has high kinetic, low potential energy One depth has high potential, low kinetic energy Which is which? How do we find these two depths?

Example 1 (cont): Take-away facts There are 2 depths (called “alternate depths”) which have the same energy for a given discharge. Depths can be calculated by trial and error For a RECTANGULAR channel if we know one depth, y 1, we can calculate the other, y 2 :

Example 2: What is the smallest energy that can be associated with q =10 ft 2 /s?

Example 2 (cont): What is the smallest energy that can be associated with q = 10 ft 2 /s? >

Example 2 (cont): Take-away facts Critical depth in a rectangular channel: Critical energy in a rectangular channel: Definition of Froude number in a rectangular channel:

Example 2 (cont): Take-away facts Froude number at critical depth Regions of super-critical or sub-critical flow. Sub-critical Region Super-critical Region

Example 3: Sluice Gate Settings: q = 10.0 ft 2 /s Incoming flow depth is y 1 =4.94 feet Gate opening, y g is set to y g =1.30 feet Fact: Energy is conserved at a sluice gate. What is depth downstream of gate? >

Example 3 (cont): Take-away facts Flow accessibility: Super-critical flow downstream ONLY if gate opening is less than or equal to critical depth. Depth downstream is alternate to depth upstream (generally less than gate opening). What if y g > y c ? What if y g < y 2 ? (Example 4)