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Jökull - 01.12.1968, Page 26

Jökull - 01.12.1968, Page 26
(3) 3 (• — yc T dR 3t J K and this must equal the heat added through conduction and advection: k div grad T — yc v • grad 1’ — yc div (T'v') 3T = YC - 3t /, k grad T • dA v • dA 3 3t YC T dR The term yc div (T'v') is the turbulent dif- fusion and it must be connected to the time- averages of T and v in order that we can handle it. In technical fluid dynamics it is assumed that The instantaneous values of T and v can be written down as sums of time-averages and fluctuations: T = T + T'; v = v + v'. The time-averages of the fluctuations are equal to zero: TV = -ktx- 3x and corresponding for the other components. It must be emphasized that the diffusion coeffi- cients, Kt, are not constant like the conduc- tivity. With this and writing out in coordinates and rearranging we get: T' =0; v' = 0. Inserting this and taking time-averages we get: f k grad T • dA — f yc T v • dA J A *Z A _ J yc TV' • dA = f yc -J- dR (1) /, 3t The surface integrals can be transformed to volume integrals: | k div grad T dR — f yc div (T v) dR •z r R - f yc div (TÝ) dR = f yc dR. (2) J E J R 3t By making use of the continuity equation for an incompressible fluid, div v = 0, we have div (T v) = T div v + v • grad T = v • grad T; '3T 3T 3T 3T 'iT + Vx "3^ + Vy + Vz 3 / k \ 3T 3x Uc + Tx/ 3x + 3y \ yc k '*-)? 3T 3z 3 / k + , ' ( + KTz dz \ yc (4) Here the bars have been omitted as all values are time-averages. This equation will now be studied closer for the simplest case: A broad rectangular channel where the temperature is uniform transversely across the channel. With x in the flow direct- ion and z as the ordinate (z = 0 at the bottom) we have 3T/3y = 0, vy = vz = 0 and vx = v(z). It is further assumed that the velocity field is independent of the temperature field; this is bclieved to hold in swift rivers. For fully developed turbulence the velocity distribution in a broad channel is given by the well-known equation (Prandtl-Nikuradse (Brett- ing 1960)): and since equation (2) must hold for any volume the integrands must vanish: 360 JÖKULL 18. ÁR

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