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MAE 5130: VISCOUS FLOWS Lecture 2: Introductory Concepts
August 19, 2010 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk
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IMPORTANT RELATIONSHIPS
If the curl of the velocity field is zero Flow is irrotational Velocity can be written as the gradient of a scalar function, f If the divergence of the velocity field is zero Flow is incompressible If both are true Laplace equation The curl of the gradient of a scalar function is zero The divergence of the curl of a vector is zero
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SUMMARY OF VECTOR INTEGRALS
Gradient Theorem Vector equation involving a scalar function, a Limits of integration such that surface encloses the volume n points normal outward Divergence (Gauss’) Theorem Scalar equation Stokes Theorem Direction of n is given by right hand rule
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CONSEQUENCE: ENGINE INLET VORTEX
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EXAMPLE: VORTICITY AND STOKES’ THEOREM
W a V=Wa Vorticity has to do with the local rate of rotation Consider a plane flow with a small cylinder of fluid rotating with angular velocity, W Apply Stokes’ theorem Magnitude of vorticity is twice the local rate of fluid rotation Physical interpretation: If a small sphere of fluid where to be instantly solidified with no change in angular momentum, local vorticity would be twice the angular velocity of the sphere Purely a kinematic statement Apply divergence theorem Applies for any fluid, compressible or incompressible, viscous or inviscid Implications: Net flux of vorticity over any closed surface = 0 Vortex lines can not end in fluid
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KINEMATIC PROPERTIES: TWO ‘VIEWS’ OF MOTION
Lagrangian Description Follow individual particle trajectories Choice in solid mechanics Control mass analyses Mass, momentum, and energy usually formulated for particles or systems of fixed identity (ex., F=d/dt(mV) is Lagrangian in nature) Eulerian Description Study field as a function of position and time; not follow any specific particle paths Usually choice in fluid mechanics Control volume analyses Eulerian velocity vector field: Knowing scalars u, v, w as f(x,y,z,t) is a solution
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