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9 Ideal Flows - EGM 6812 F11 University of Florida M Sheplak 9"Ideal Flow(I3 Now we will focus on a certain class of flows termed"ideal because

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EGM 6812, F11, University of Florida, M. Sheplak 1/74 Section 9, Ideal Flow 9 “Ideal” Flow (I 3 ) 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 1 00 D V Dt    5 conditions discussed in class Irrotational : there is negligible rotation of the fluid particles, 0 V  . 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   2 VV       V     . Inviscid: viscous forces are negligible compared to other forces b DV pf Dt      inviscid b Dt     which is Euler’s Equation. Often, this implies that there is zero strain rate: 0 ij . Strictly speaking, 0 ij is 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 0 ij . Illustrative example:
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EGM 6812, F11, University of Florida, M. Sheplak 2/74 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) ; 0 F A A       (0.1) Constraining A to 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,   V V V   (0.2) where V is the rotational and   V is the irrotational part. It follows mathematically that VV    (0.3) and   0 V (0.4) where is the vorticity vector. If V is constrained to be solenoidal (incompressible), 0 V  (0.5)
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EGM 6812, F11, University of Florida, M. Sheplak 3/74 Section 9, Ideal Flow then it may be represented by the curl of a vector potential (you can show that 0 B  ), VB   (0.6) If B is further constrained to be solenoidal 0 B  , then   V B B       2 B  (0.7) or 2 B   . (0.8) Therefore, the vector potential of the velocity field is governed by Poisson’s equation where the source is the vorticity field. Since   V
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This note was uploaded on 01/17/2012 for the course EGM 6812 taught by Professor Renweimei during the Fall '09 term at University of Florida.

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9 Ideal Flows - EGM 6812 F11 University of Florida M Sheplak 9"Ideal Flow(I3 Now we will focus on a certain class of flows termed"ideal because

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