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By Dimitris Drikakis, William Rider

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Additional info for High-Resolution Methods for Incompressible and Low-Speed Flows (Computational Fluid and Solid Mechanics)

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8) depend on nine joint invariants of A1 and A2 , where trA1 = 0, and are usually determined experimentally. • Models of the rate type: These models consider that the stress tensor T(t) and the tensor of the history of the deformation gradient F(t) satisfy the differential equation ˙ . . , T(n−1) ; F, F, ˙ . . 60) T(n) = S(T, T, where S is a tensor function. The above equation defines media of the rate type of order n, the latter being determined by the order of the derivative of T. The Oldroyd constitutive equation [401, 402] for a viscoelastic fluid falls in this category.

92) is more appropriate for low Reynolds numbers (Re → 0). For Re = 0 we obtain the Stokes flow which is dominated solely by viscous effects. 93) where u and l are the characteristic velocity and length scales of the largest eddies in the turbulent flow; Rel is the corresponding Reynolds number. In engineering and geophysical applications, Rel is usually of the order of O(106 ) − O(108 ) and thus the number of grid points needed to perform simulations would be of the order of O(1013 ) − O(1018 ). Therefore, direct numerical simulation (DNS) of flows at such high Reynolds numbers is beyond the projecting future capacity of parallel computers, unless there is a major breakthrough in computers technology.

Examples of flows in which the Froude number can be important include the waves generated by a ship, open channel flows, jet flows. 5 Nondimensionalization of the Governing Equations 41 where σ is the coefficient of surface tension (measured in SI units as Newtons per meter). The Weber number is the ratio of inertia forces to surface tension and is used when surface tension is important. Applications in which the Weber number may be important include droplets, capillary and free surface flows. 86) Eu = ρU 2 where ∆p is a reference pressure difference.

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