Session K.pdf - Clarkson University

17th International Symposium on IceSaint Petersburg, Russia, 21-25 June 2004International Association of Hydraulic Engineering and ResearchTHE NUMERICAL MODELING OF HYDRO-ICE-THERMALPROCESSES IN RESERVOIRSV.I. Klimovich 1 , V.A. Prokofiev 1 , I.N. Shatalina 1 , G.A. Tregub 1ABSTRACTMathematical modeling of hydro-ice-thermal processes has been considered in a numberof papers (e.g. Wake and Rumer, 1979; Alexandrov et al, 1992). In this paper atime-dependent 2D (plan) model of ice-thermal processes is viewed, in which for theprocess of ice cover formation the first approximation of uneven temperature profile (indepth) is taken into account. The algorithm of numerical solution of hydro-ice-thermalproblem was worked out and its verification was carried out. The numerical results ofthe hydro-ice-thermal regime and their comparison with the in-situ data of the Volga-Don NPP reservoir-cooler are given.MATHEMATICAL MODELThe conservative form of the shallow water equations with due regard for the friction onthe bottom, the wind influence and Coriolis’ forces looks like∂ h∂ t+∂ Q∂ x+∂ Qx y=∂ yq; (1)⎛2∂Q ⎞x ∂⎛ ⎞⎜Q ∂ QxQxy⎟∂Hτ∂h+ ⎜ ⎟bx τ wx 1 τ xx 1+= -gh + Ω + + +∂ ∂cQy-t x ⎝ h ⎠ ∂y⎝ h ⎠ ∂xρwρwρw∂xρw⎛2∂Q ⎞y ∂ ⎛ QxQy⎞ ∂ ⎜Qy⎟ ∂Hτ by τ wy 1 ∂hτyx 1+ ⎜ ⎟+ = -gh − ΩcQx-+ + +∂t∂x⎝ h ⎠ ∂y⎜ h ⎟ ∂yρ ρ ρ ∂xρ⎝ ⎠w w wwQx+ Qy1 1Qx = hux, Qy= hu y , τ bx = ρwC Qx, τ by = ρwC Qy, Ccur= g, C = h6,2 2C h n2 −2⎛z ⎞ 2τ wτ wx = τ w cos θw,τ wy = τ w sin θw,τ w = ρaκ ln ⎜ ⎟, z0= 0, 0144zW⎝ 0 ⎠ρagThe averaged in depth shear stresses τ xx , τ xy , τ yx , τ yy can be represented in the form (theindexes i, j correspond here to x, y)22∂hτ∂y∂hτxy∂yyy(2)(3)1 All-Russian Research Institute of Hydraulic Engineering (VNIIG), Gzhatskaya str., 21, St. Petersburg,195220, Russia344