4 Ideal Flow EAS 4101 S11

4 Ideal Flow EAS 4101 S11 - EAS 4101, S11 University of...

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EAS 4101, S11 University of Florida 1/21/11 1/23 Section 4, Ideal Flow 4 Ideal Flow 4.1 I 3 Flow Ideal flow is I3 (Incompressible, Inviscid, Irrotational) Fluid motion in ideal flow: o Incompressibility eliminates dilatation o Inviscid assumption removes the divergence of shear strain o Irrotational assumption removes rotation Incompressible : compressibility effects or forces in the flow are negligible with respect to other effects or forces. o 1 00 D V Dt    o NOTE: a fluid is compressible ( fluid property ), but the role of compressibility forces may not be important to a given flow ( flow property ). For example, even though water (liquids in general) possesses a low compressibility, explosions in water will generate shocks which are compressible flow phenomena.
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EAS 4101, S11 University of Florida 1/21/11 2/23 Section 4, Ideal Flow For example, even though air (gasses in general) is considered to be compressible, low Mach number flows have negligible compressibility forces.
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EAS 4101, S11 University of Florida 1/21/11 3/23 Section 4, Ideal Flow Irrotational: there is negligible rotation of the fluid particles, 0 V   . For incompressible flow, irrotationality usually requires zero shear stress…see HW#4 for exception. Inviscid: viscous forces are negligible compared to other forces. o DV pf Dt     inviscid p f Dt  o This is Euler’s Equation. Typically, this implies that there is zero strain rate: 0 o Note: this in an interesting assumption on many fronts. There must be an imbalance of viscous stresses for viscous forces to be important, 0   . This differs from 0 for a Newtonian fluid by 0  o NOTE: all fluids possess viscosity ( fluid property ), but the role of viscous forces may not be important to a given flow ( flow property ). Illustrative example:
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EAS 4101, S11 University of Florida 1/21/11 4/23 Section 4, Ideal Flow 4.2 Euler’s Equation Streamline coordinates Examine curvature effects Derive Bernoulli’s equation 4.2.1 Euler’s Equation in Streamline Coordinates Consider the coordinate system below.
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EAS 4101, S11 University of Florida 1/21/11 5/23 Section 4, Ideal Flow o s coordinate following the streamline o n coordinate perpendicular to the streamline By definition, 0 Vd s for a streamline. Therefore 0 n V , so (,,) (,) s Vxzt Vsts We can drop the subscript, so (,,) How do we express Euler’s Equations in this coordinate system? Streamline Direction: steady “Euler-s” Equation o 1 p V V ss   o Physical insight regarding pressure/velocity relationship along a streamline… 21 1 p pV V V     or p Vp V   4.2.2 Curvature Effects Normal Direction: steady “Euler-n” Equation o 2 1 p V nR where R is the radius of curvature of streamlines.
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EAS 4101, S11 University of Florida 1/21/11 6/23 Section 4, Ideal Flow 2 n 1 n 21 ~ pp nn o
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This note was uploaded on 09/05/2011 for the course EAS 4101 taught by Professor Sheplak during the Spring '08 term at University of Florida.

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4 Ideal Flow EAS 4101 S11 - EAS 4101, S11 University of...

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