CE 576 Environmental Flows Spring 2012 Homework 7 *33. A river has a width of 30 m, a slope of 2.5 × 10−4 , a Manning coeﬃcient of 0.036, and a discharge of 3.6 m3 /s. The geometry is shown below. a. Compute the depth in steady uniform ﬂow with Manning’s equation. (Because we have not discussed this calculation in class, let me know if you need guidance.) b. Will sediment with a diameter of 0.1 mm move in this ﬂow? c. What is the largest sediment that will move in this ﬂow? 1m 20 m 10 m *34. Problem 33 dealt with the initiation of motion of sediment. Now you will compute a proﬁle of suspended sediment concentration in a channel ﬂow. a. In equilibrium, sinking of sediment particles (with velocity ws ) is balanced by turbulent mixing with an eddy diﬀusivity Ks . Conservation of sediment mass yields d dC dC = −ws Ks , dz dz dz where z is the coordinate normal to the channel bottom. Argue that this equation is correct for this situation. b. A simple approach to estimating the eddy diﬀusivity is to assume that it is proportional to the eddy viscosity: Ks = ανt . In class, we took νt = κu∗ z. Here take z νt = κu∗ z 1 − , h where h is the water depth. Explain why this formulation might be more realistic than the one we used in class. c. Take C(za ) = Ca and show that C = Ca z (h − za ) za (h − z) where P = ws /κu∗ is the Rouse number. − Pα , d. Compute the Rouse number for the sediment in problem 33b. e. Plot C/Ca vs. z/h for P = 0.1, 0.5, and 1; α = 1; and za /h = 0.01. Provide a physical explanation for the eﬀect of the Rouse number P . 35. Yet another way to estimate the shear velocity in channel ﬂow is to use the turbulent kinetic energy. Find the paper by Kim, Friedrichs, Maa, and Wright from the Journal of Hydraulic Engineering in 2000. a. Explain the TKE method in general as discussed on p. 400. b. Explain why Kim et al. (2000) use the vertical velocity variance (i.e., w 2 ). c. Brieﬂy summarize the authors’ assessment of the TKE method. d. Apply the TKE method to the data from the lab exercise (i.e., problem 32). Does the vertical velocity variance work better than the full TKE? 36. Derive the equation for the turbulent kinetic energy: ∂k ∂u ∂u 1 g ∂ ∂k i = −ui uj − ρ u3 − p uj + 12 ui 2 uj − νui i − , + uj ∂t ∂xj ∂xj ρ0 ∂xj ρ0 ∂xj where ν = 2 ∂uj ∂ui + ∂xj ∂xi ∂uj ∂ui + . ∂xj ∂xi *37. Munk (1996, Deep-Sea Res.) estimated that explaining the historic temperature proﬁles in the deep ocean would require vertical mixing with an eddy diﬀusivity on the order of 10−4 m2 /s. Much of the last 45 years of physical oceanography has been devoted to ﬁnding mixing strong enough to support Munk’s estimate. For example, oceanographers infer the vertical eddy diﬀusivity K from measurements of the dissipation and a simpliﬁed version of the equation for turbulent kinetic energy. a. Osborn (1980, J. Phys. Oceanogr.) devised a way to measure the buoyancy ﬂux b, a measure of vertical mixing. He assumed that the turbulence is stationary and homogeneous (i.e., that the statistics vary neither in space nor time), deﬁned the ﬂux Richardson number as Rf = b/P (where P is production), and used the TKE equation to ﬁnd b = Γ, where Γ = Rf /(1 − Rf ). Derive that result. b. Show that a gradient transport assumption yields K = Γ 2. N c. Assume that Γ takes its maximum estimated value of 0.2 and use Fig. 5 of Moum (1997), which is on the website, to determine the types of ﬂows for which K > 10−4 m2 /s. d. Discuss whether the assumptions of stationarity and homogeneity are likely to be valid for turbulence in the ocean. 38. The k- model uses the following model for the Reynolds stress: −ui uj = νt ∂uj ∂ui + ∂xj ∂xi 2 − kδij 3 a. Write down an expression for −u1 u3 b. Show that the model reproduces the correct turbulent kinetic energy. c. The Wikipedia page for “Turbulence kinetic energy” states, “k- models assume isotropy of turbulence whereby the normal stresses are equal: u 2 = v 2 = w 2 . Is this statement correct?