EGM 6812, F11, University of Florida, M. Sheplak1/74 Section 9, Ideal Flow 9 “Ideal” Flow (I3) Now we will focus on a certain class of flows termed “ideal” because they possess negligible compressibility, rotationality, and viscous effects Incompressible: compressibility effects in the flow are negligible with respect to other effects 100DVDt 5 conditions discussed in class Irrotational: there is negligible rotation of the fluid particles, 0V. For incompressible flow, irrotationality usually requires zero divergence of shear stress (i.e., inviscid flow). We can see that for a Newtonian, incompressible flow, the divergence of shear stress and vorticity are related by 2VV V. Inviscid:viscous forces are negligible compared to other forces bDVpfDt inviscidbDVpfDt which is Euler’s Equation. Often, this implies that there is zero strain rate: 0ij. Strictly speaking, 0ijis NOT required. There must be an imbalance of viscous stresses for viscous forces to be important, 0. For a Newtonian fluid with constant viscosity, this means 0, which is a broader statement than 0ij. Illustrative example:
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EGM 6812, F11, University of Florida, M. Sheplak2/74 Section 9, Ideal Flow 9.1 Helmholtz Decomposition of Vector Field Before studying ideal flows, it is beneficial to review the Helmholtz decomposition. Helmholtz’s theorem states that any vector field F, which is finite, uniform, continuous, and vanishes at infinity, can be uniquely expressed as the sum of the gradient of a scalar potential and the curl a divergence free vector potential A, (Morse and Feschbach, 1953, Karamcheti 1966) ; 0FAA (0.1) Constraining Ato be solenoidal is permissible since it is indeterminate to the extent of the gradient of a scalar of position and time. This is not a unique decomposition. We could still add an arbitrary gradient of a potential and still get the same answer. This will be useful later. In the case of the velocity field vector in fluid mechanics, this decomposition has the following physical meaning, VVV(0.2) where Vis the rotational and Vis the irrotational part. It follows mathematically that VV (0.3) and 0V(0.4) where is the vorticity vector. If Vis constrained to be solenoidal (incompressible), 0V(0.5)