MAE 222Mechanics of FluidsPrinceton UniversityAssignment # 10April 22, 1998Due on Wednesday, 2PM, April 29, 1998The second mid-term is scheduled on on Friday (May 1). It willcover everything except Chapter 10 (unless by popular demand!).The Wednesday class will be a review session.Chapter 10. Open Channel Flow. This is also called “free surface flows.”§10.1-10.2, General Characteristics, Surface Waves and Fr. Inthis chapter, attention is mainly confined (except for hydraulicjump) to “shallow water” open channel flows—flows in which thecharacteristic depth of the water is small in comparison to thecharacteristic length in the streamwise direction. For shallow wa-ter open channel flow problems, the “one-dimensional” assump-tion can be justified: the streamwise velocity V is a constant acrossany cross-section of the channel. In other words, V = V (x, t)where x is the streamwise coordinate and t is time.You learn from these sections that the speed of a shallow waterwave is c =√gy where y is the vertical distance from the freesurface to the bottom of the channel. You also learn that Froudenumber Fris defined as the ratio of V to c.§10.3. Energy Considerations. Skip this section.§10.4. Uniform Depth Flow. Skip this section.§10.5. Gradually Varied Flow . Such flows are usually called shal-low water flows. We will see that flows with Fr≤ 1 (subcritical)behaves very differently from supercritical flows (The Froude num-ber is, more or less, the “Mach number” of open channel flows). Iwill go over this in class.1§10.6. Rapidly Varied Flow . The big deal here is the hydraulicjump. Eq.(10.19) is the major result: the ratio of water depthacross a hydraulic jump is a function of the upstream Froude num-ber. The upstream Froude number MUST be larger than unity(supercritical), and the downstream Froude number is always sub-critical. The Bernoulli’s constant changes value across a hydraulicjump; it always decreases!Skip §10.6.2, (Sharp-crested Weirs)§10.6.3, Broad-crested Weirs is interesting stuff! See Smits Notes.Skip §10.6.4, (Underflow gates).Problems at end of Chapter 10, page 461:• Problem 10.2. Totally straightforward.• Problem 10.4. I will discuss this in class. We want to use theformula c =√gy to provide us with a convincing argument thatincoming waves on a sloping beach ought to “break.”• Problem 10.9. The flow is assumed steady and one-dimensional.You have the continuity equation, and you have the Bernoulli’sequation for the free surface streamline. Thus you have two equa-tions for the two unknowns, V2and y2.• Problem 10.14. Watch for this in class. If I forget, remind me.• Problem 10.46. This is a somewhat tricky problem. You needto change your coordinate system so that it is moving with themoving hydraulic jump. In that coordinate system, you can applyeq.(10.19) (using Frin that coordinate system).• Problem 10.51. See Fig. 10.17 on page 457.Use email or the newsgroup to ask