Page 2 of 3. Problem 1.

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1 Page of 3 Problem. Derive formulas for the a nb coefficients in the Second-Order Upstream scheme (SOU) for the convection-diffusion equation for the concentration of diatom type algae with fall velocity w. Assume a D width-averaged orthogonal grid with cell sizes dx and dy, and a uniform water velocity U. Assume a constant eddy-viscosity Γ. Problem. Draw a structured non-orthogonal D grid with between 00 and 50 quadrilateral cells for the channel given in the figure below. The dotted lines indicate inflow/outflow, and the solid lines are side walls. Do not use outblocking. You may draw on the figure on the exam paper, and hand this in. Problem 3. In 989 I visited a professor in California. We discussed CFD and he said he had some stability problems with his finite element computations. I said I planned to use a finite volume method. He then said I would have less instability problems because of false diffusion in the finite volume method. Discuss his last statement. Problem 4. Explain the following words: - eutrophic - aphotic - oligotrophic - littoral - mesotropic - dimictic Problem 5. Derive the equation for the velocity of the D kinematic wave.

2 Page 3 of 3 Problem 6. A 00 meter deep lake in Norway has a water inflow of 00 m 3 /s in the summer. A map of the lake is given below. The temperature of the inflowing water is 0 0 C. There is no wind above the lake. For the thermocline to be completely horizontal, how far below the water surface must it be? In r=000 m Width = 00 m Formulas and tables: Out U 3 A = r n h I r h = M = P n M = 6 - y 3 Ey q + - = 0 g 6 k s a = fu = 4π - sinφ, f=.6*0-4 (Norway) T U a w c w + a e c e + a n c n + a s c s c p = a p gi ( 0 I f ) g dy - U du du = + dx dx dt ( U c) h + gh ( U c) = h + gh Table of water density as a function of temperature: Temp ( 0 C) Density (kg/m 3 )

3 Oppgave. Utled formler for a nb koeffisientene for. ordens oppstrøms skjemaet (SOU) for konveksjonsdiffusjons ligningen for konsentrasjon av diatom algaer med fallhastighet w. Anta et D grid, i vertikal retning og strømretningen. Anta at gridet er ortogonalt med cellestørrelse dx og dy, og en uniform vannhastighet U. Anta en konstant turbulent viskositet Γ. Oppgave. Tegn et strukturert ikke-ortogonalt D grid med mellom 00 og 50 firkantede celler for kanalen gitt i figuren under. De stiplede linjene indikerer innstrømning og utstrømning, og de heltrukne linjene er vegger. Ikke bruk utblokking. Du kan tegne på figuren i oppgavearket, og levere dette inn som besvarelse. Oppgave 3. I 989 besøkte jeg en professor i California. Vi diskuterte CFD og han sa at han hadde noen stabilitetsproblemer med sine elementmetodeberegninger. Jeg sa jeg planla å bruke en endelig volummetode. Han sa da at jeg ville få mindre stabilitesproblemer på grunn av falsk diffusjon i den endelige volummetoden. Diskuter hans siste utsagn. Oppgave 4. Forklar de følgende engelske faguttrykkene: - eutrophic - aphotic - oligotrophic - littoral - mesotropic - dimictic Oppgave 5. Utled ligningen for hastigheten av en D kinematisk bølge.

4 Oppgave 6. En 00 meter dyp innsjø i Norge har en vanninnstrømning på 00 m 3 /s om sommeren. Et kart over innsjøen er gitt under. Temperaturen i det innstrømmende vannet er 0 0 C. Det er ingen vind over innsjøen. For at termoklinen skal være fullstendig horisontal, hvor langt under vannoverflaten må den være? Inn r=000 m Bredde = 00 m Formler og tabeller: Ut U 3 A = r n h I r h = M = P n M = 6 - y 3 Ey q + - = 0 g 6 k s a = fu = 4π - sinφ, f=.6*0-4 (Norge) T U a w c w + a e c e + a n c n + a s c s c p = a p gi ( 0 I f ) g dy - U du du = + dx dx dt ( U c) h + gh ( U c) = h + gh Tabell med tetthet av vann som funksjon av temperatur: Temp ( 0 C) Tetthet (kg/m 3 )

5 Solution: Problem. The derivation is given in Chapter 5.6 of the class notes. The answer is: 3 A a w U A w Γ w w = + + U w dx e A e a ww = U A w w A e a e = Γ e - dx a ee = 0 3 A a n U A n Γ n - n = + + U A n dy s s a nn = U A n n A s a s = Γ s - dy a ss = 0 Problem. (5.6.7) (5.6.8) (5.6.9) (5.6.0) (5.6.) (5.6.) (5.6.3) (5.6.4) A sketch of the grid is given below. Note that the grid should be as non-orthogonal as possible, and that no outblocking should be used. Also, all cells have to have four sides. Problem 3. Two topics are discussed: False diffusion and stability. False diffusion is a numerical effect, causing inaccurate results. Stability is often a problem in CFD computations. For some schemes, for example the central scheme, one has to have a certain amount of diffusion to get convergence. One can then add artificial diffusion to get convergence. In principle the professor is mixing the two topics, and this is the correct answer to the question. It is possible that false diffusion also will improve stability, but this is not stated in the class notes or the lectures.

6 Problem 4. eutrophic: lake with high inflow of nutrients, and high amount of orgamic growth aphotic: zone of deep water with too little sunlight to get organic growth oligotrophic: lake with little inflow of nutiiensts, and therefore also little organic growth littoral: shallow parts of a lake mesotropic: state between eutrophic and oligotropic dimictic: lake with two overturns pr. year Problem 5. The derivation of the equation for the kinematic wave is given in the class notes, in Chapter.5. The Manning s formula is derivated with respect to time. The wave speed is 5/3 times the water velocity. Note that this must not be confused with the dynamic wave. Problem 6. The Coriolis accelleration, a = fu, is set equal to the centrifugal accelleration: a =. The r values of f and r are given. The velociity then becomes equal to rf or 0.5 m/s. The continuity equation then gives the thickness of the layer above the thermocline as: Q 00 h = - = = 4m UB 0 5x00 U