OCEN 489/689

Special Topics in Mixing and Transport in the Environment(Environmental Fluid Mechanics)

Problem Set #5: Turbulent Diffusion

Date distributed : 03.03.09Date due : 03.13.09 by 12:00 noon.

Return your solution either in class or in my mail box (8th Floor, CE/TTI) by the date shownabove. Please show all your work and follow the rules outlined in the course syllabus.

1 Advection in a stream

To estimate the mixing characteristics of a small stream, an engineer injects 5 g of dye instanta-neously and uniformly over the river cross section (A = 5 m2) at the point x = 0. A measurementstation is located 1 km downstream and records a river flow rate of Q = 0.5 m3/s. In order to designthe experiment, the scientist assumed that the longitudinal mixing coefficient is Dx = 0.1 m2/s.Use this value to answer the following questions.

• The fluorometer used to measure the dye downstream at the measuring station has a detectionlimit of 0.1 µg/l. When does the measuring station first detect the dye cloud (t1)?

• When does the maximum dye concentration pass the measuring station (tmax), and what isthis maximum concentration?

• After the maximum concentration passes the measuring station, the measured concentrationdecreases again. When is the measuring station no longer able to detect the dye (t2)?

• Why is the time span tmax − t1 different from t2 − tmax? Justify your answer in words.

• If the fluorometer has an accuracy of ±0.2 µg/l, did the engineer plan to inject sufficient dyeto get an accurate concentration distribution at the measuring station?

2 Dye study

A small stream has been found to be contaminated with Lindane, a pesticide known to causeconvulsions and liver damage. Groundwater wells in the same region have also been found to containLindane, and so you suspect that the river contamination is due to groundwater inflow. To test

CIVIL College Station, Texas 77843-31363136 TAMU (979) 845-4517 FAX (979) 862-8162

Station 1:

Dye injected @x = 0 mQd = 100 cm3/sCd = 50 mg/l

x = 70 mCd = 10 ug/lCl = 0.5 ug/l

x = 170 mCd = 8 ug/lCl = 0.9 ug/l

Station 2:

Station 3:

River cross-section:W = 1 md = 0.5 m

Figure 1: Dye study to determine the source of Lindane contamination in a small stream.

your theory, you conduct a dye study using a continuous release of dye. Based on the informationgiven in Figure 1, what is the groundwater volume flux and the concentration of Lindane in thegroundwater between Stations 2 and 3? The variables in the figure are Qd the volume flow rateof the dye at the injection, Cd the concentration of the dye at the injection and at downstreamstations, Cl the concentration of Lindane in the river at each station, W the width of the river, andd the depth in the river. (Hint: this is a steady-state problem, so you do not need to use diffusioncoefficients to solve the problem other than to determine whether the dye is well-mixed when itreaches Station 2. Recall that the mass flow rate is

m = CQ (1)

and must remain constant along the river in a steady state problem unless there are inflows oroutflows of mass or volume.)

(G) Due to problems with the pump, the dye flow rate has an error of Qd = 100 ± 5 cm3/s.Assume this is the only error in your measurement and report your measurement uncertainty.

3 Image Sources

The time-scale for a point source injected mid-way between two impermeable boundaries tobecome uniformly mixed across the section is given by the rule-of-thumb

tm =L2

8D(2)

where L is the distance between the two boundaries and D is the diffusion coefficient. Referring tothe sketch below, consider a line of dye released instantaneously along the centerline of a shallowriver so that the problem can be treated as one-dimensional.

• Write a program applying the superposition method to calculate the relative concentration(relative to the centerline concentration of the river C(x = 0, t)) at the river bank x = L/2 as

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a function of time, that is, find the ratio C(L/2, t)/C(0, t). The width of the river is L andthe diffusion coefficient of the dye is D (please email me your final program).

• Plot your result with C(L/2, t)/C(0, t) as the y-axis and t/(L2/D) as the x-axis using thevalues L = 10 m and D = 0.01 m2/s.

• What is the relative concentration when t = tm?

x = -L/2 x = L/2x = 0

4 (U) Image sources in a pipe

A point source is released in the center of an infinitely long, round pipe (diameter 10 cm, injectedmass 1 g, pipe flow rate 1 l/s). The flow is turbulent, and the lateral and longitudinal coefficientsof turbulent diffusion are 0.00005 m2/s and 0.00009 m2/s, respectively. Do the following:

• Derive the image source needed to account for the pipe walls and write down the image sourcesolution in general form.

• Plot the lateral concentration distribution in the pipe at the downstream locations L = 1, 5,15, 25 and 50 m.

• At what non-dimensional distance downstream does the lateral concentration distributionappear well-mixed (use the length scale σ =

√2Dt to non-dimensionalize distance)?

• When should you start applying the longitudinal dispersion coefficient 0.005m2/s?

5 (G) Turbulent diffusion in the ocean

Ten surface drogues are released into a coastal region at local coordinates (x, y) = (0, 0). Thedrogues move passively with the surface currents and are tracked using radio signals. Their locationsat the end of t1 = 1 and t2 = 20 days are given in the following table.

1. Estimate the advection velocity and the lateral coefficients of diffusion (Dx and Dy) for thiscoastal region.

2. Using the radio links, the positions of all ten drogues can be collected within ten minutes. Sup-pose the radio link were to break down and the positions were instead determined throughvisual observation. Even using a helicopter, it requires nearly four hours to locate all tendrogues. How does this change the accuracy of your data? Can you still consider the mea-surements to be synoptic?

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Table 1: Drogue position data.

Drogue x(t1) y(t1) x(t2) y(t2)

Number [km] [km] [km] [km]

1 2.5 0.2 5.3 8.1

2 4.6 1.4 2.3 1.0

3 2.3 -1.2 6.6 3.9

4 3.1 -0.4 6.7 -2.8

5 1.5 0.8 0.5 4.2

6 1.4 2.1 10.1 3.6

7 4.7 2.1 6.6 -1.4

8 2.7 0.2 6.0 -2.9

9 1.5 2.6 3.2 2.1

10 4.9 2.3 -4.0 1.7

3. Later, a freight ship is caught in a winter storm off the coast where this drogue study wasconducted. High winds and rough seas cause several shipping containers to be washed over-board. One of the containers breaks open, releasing its contents: 29,000 children’s bathtubtoys. Estimate how long it will take for the toys to begin to wash up on shore assumingthe same transport characteristics as during the drogue study and that the spill occurs 1 kmoff the coast. (This really happened in the Pacific Ocean, and the trajectory of the bathtubtoys, plastic turtles and ducks, were subsequently used to gain information about the currentsystem.)

4. Writing assignment (pay attention to grammer, spelling, and style): Read the article fromEOS on Tub Toys in the Pacific Subarctic Gyre referenced in class. Write a one paragraphsummary (typed) of the article and explain how it relates to this work-out problem and tothe determination of mixing and transport rates in the ocean. Provide proper referencing anddo not plagiarize any of the article.

6 (G) Mixing of joining rivers

Referring to the picture below, one river (left) with a high concentration of sediment joinsanother river (right) with a negligible sediment concentration. The width of the low concentrationriver near their union is 40 m while the high concentration river is 80 m wide. Assume the riverwidth and depth do not change much after the union, and both rivers are shallow and have thesame velocity. At one particular day the mean velocity downstream of the union is 1 m/s and thelateral diffusion coefficient is 0.1 m2/s.

• Estimate the time required tmix and the distance downstream xmix until the low sediment con-

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centration river is considered to be well-mixed with the sediment from the high concentrationriver. Use a relative concentration of 95% as the criteria for the well-mixed condition.

• If there is a water intake located on the low-sediment side of the river at 3 km downstreamfrom the river union, do you expect the water taken from the intake to contain a significantamount of sediment? Justify your answer.

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