THE NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF HYDRAULIC AND ENVIRONMENTAL ENGINEERING

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1 Page of 4 THE NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF HYDRAULIC AND ENVIRONMENTAL ENGINEERING Scientific contact during the exam: Department of Hydraulic and Environmental Engineering Professor Nils Reidar B. Olsen, Tel EXAM FOR TVM 455 NUMERICAL MODELLING AND HYDRAULICS Tuesday nd of December 008 Time: 4 hours Aid code: D: Specific hand-held calculator with empty memory allowed, according to list from NTNU (HP0S). No printed or hand-written material allowed, except a dictionary checked by Studieavdelingen. There are five problems. Each problem will count 0 % towards the final grade. The grades will be determined before Christmas.

2 Page of 4 Problem. A dam break causes a flood wave to propagate down a valley. The valley has a longitudinal slope of :00. Assume a valley width of 00 meters and a rectanglar cross-setion. Also, assume a Manning-Stricklers M value of 5 and a discharge of m /s. A village is located 0 km downstream a of the dam. Give an estimate on how long time it will take for the flood wave to reach the village. Problem. a) Describe the grid in the figure to the right. b) What is the main problem with this grid? c) Suggest an alternative grid with a better solution of the problem in question b). Problem. Estimate the bed form height in a channel with depth. meters, longitudinal slope :00 and a discharge. m /s, which is covered with a thick layer of uniform sediments that has an average diameter cm. Problem 4. A water intake is located 0 meters below the water surface in a 0 m deep lake. The water density is varying linearly from the water surface to the deepest part of the lake. The water surface holds 0 degrees and the temperature is 0 degrees at the deepest part of the lake. The intake gate measures x meters in horizontal and vertical direction. What is the maximum disharge we can draw through the intake before we get water from the surface? Problem 5. a) Why would algae of type diatoms want to be located close to the water surface? b) Describe the process causing this type of algae to be located close to the water surface. c) How can this process be modelled numerically?

3 Page of 4 Formulas and tables: U r n h I r A h M P n M 6 6 d 90 a fu 4π - sinφ, f.6*0-4 (Norway) T U Table of water density as a function of temperature: a c w c w + a e c e + a n c n + a s c s p a p gi ( 0 I f ) g dy - U du + du dx dx dt τ c 0 ρ a U a cxt (, ) c 0 L e πγt ( x Ut) - 4Γt Temp ( 0 C) Density (kg/m ) c U i x i Γ c x i x i kc Γ Q IB - U i U i + U t j x j - ( Pδ ρ ij ρu i u j ) ρu i u j ρν U i U T + j ρkδ j x j x i ij K sinφsinα + tanθ sinφsinα tanθ cos φ tanφ + tanθ τ τ c 0. D τ c e τ τ c 5 k h h τ c s D e 5 λ c 0.*0 ρ a. kg/m cy - ( ) c a h y a - z y h a z - w κu* τ ρgyi c U± gy

4 Page 4 of τ* R* Shields diagram. Vertical axis: τ* τ c ρ w g( ρ s )d R* u * d ν d τ ρ ν q s 0.05ρ s U d - 50 g ρ s - ρ w - τ g( ρ s ρ w )d 50 q s ρ w gri 0.047g ( ρ s ρ w )d 50 g ρ s ρ w 0.5ρ w ρ s k. d k q - N g N ρ dρ - Q d k dz - B hn

5 Page 5 of 4 Solution: Problem. First, compute the uniform velocity from Manning s formua: U My I Continuity gives the depth: y Q - BU The two formulas give: U Q 5 MI B m s - The speed of a kinematic wave is: c 5 5 U x6.7. m - s The time T for the wave to cover the distance of 0 km is: S 0000 T - 89 U. The time is 89 seconds or around 5 minutes. Checking the water depth: y Q/BU 7.5 m. The widt/depth ratio is :7, meaning it is ok to use the water depth instead of the hydraulic radius in Mannings formula. Problem a) The grid is: Structured, non-orthogonal, has quadrilateral cells. b) Problem: Corner angles close to 80 degrees. Should be close to 90 degrees. c) Thanks to student no for providing the grid in the figure on the right. This grid still has quadrilaeral cells. It is also possible to make a grid with triangular cells or outblocking, which will give triangular cells. However, the triangular cells give poorer accuracy and convergence properties, so the best grid is the one to the right.

6 Page 6 of 4 Problem The shear stress is: τ ρghi 000x9.8x.x Pa The particle Reynolds number is: Re d τ ρ ν 0 6 The Shields coefficient is given from Shields diagram as: 0.06 The critical shear stress is then: τ τ*g(ρ s ρ w )d 0.06x9.8x( )x Pa The dune height is given from the following formula: 0. D e h h τ τ c τ c 5 τ τ c τ c x( 0.) x9.7 e The dune height is 60 cm. Problem 4 Density gradient is (998. kg/m kg/m) / 0 m kg/m N g ρ dρ x ( 0.05) dz 000 Q d k - Q B hn 0m. xx Q.0 m/s. Problem 5 a) The algae needs sunlight for photosyntesis. Since the water is not completely clear and dampens the sunligt, the algae needs to be close to the surface. b) The algae needs eigher areas of upwelling, where there is positive vertical water velocities, or turbulence.

7 Page 7 of 4 c) Upwelling can be modellled by solving the Navier Stokes equations for the lake. The solution of the Navier-Stokes equation can be described by solvding the different terms: transient term, convective term, diffusive term and the pressure (SIMPLE method). Turbulence is modelled by a turbulence model. Some more details should be given: classifications in zeo, and equation models, Reynolds stress models and large eddy simulation.

8 Page 8 of 4 Problem. Et dambrudd gjør at en flombølge beveger seg nedover en dal. Dalen har en helning i lengderetningen på :00. Anta en bredde på dalen på 00 meter og et rektangulært tverrprofil. Anta også en Manning-Stricklers M faktor på 5 og en vannføring på m /s. En landsby befinner seg 0 km nedstrøms dammen. Gi et estimat for hvor lang tid det tar før flombølgen når landsbyen. Problem. a) Beskriv gridet på figuren til høyre. b) Hva er hovedproblemet med dette gridet? c) Foreslå et alternativt grid med en bedre løsning på problemet fra oppgave b). Problem. Beregn høyden på bunnformene i en kanal med dybe. meter, helning :00 i lengderegningen og en vannføring på. m /s, som er dekket av et tykt lage med sedimenter av lik størrelse, med gjennomstnitts diameter cm. Problem 4. Et vanninntak befinner seg 0 meter under vannoverflaten i en 0 m dyp innsjø. Vannets tetthet varierer linært fra overflaten til bunnen. Vanntemperaturen ved overflaten er 0 degrees og på det dypeste nivået er temperaturen 0 grader. Inntaksluken måler x meter i horisontal og vertikal retning, henholdsvis. Hva er maksimum vannføring vi kan ta ut før vi får inn vann fra overflaten? Problem 5. a) Hvorfor vil alger av familien diatomer ønske å befinne seg nær vannoverflaten? b) Beskriv prosessen som får disse algene til å befinne seg nær vannoverflaten. c) Hvordan kan denne prosessen modelleres numerisk?