Lecture 3

Flow PropertiesTools to describe flows

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Velocity and velocity profile

• The velocity perpendicular to a surface is the flow per unit of area

• In a generic reference, velocity is a vector with 3 components.

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Spatial variability: 1D, 2D, 3D flow

• 3D flow: velocity varies in all the directions of the space.• 2D flow: velocity varies in two space directions.• 1D flow: velocity varies in one direction of the space.• Uniform flow: the velocity is uniform, independent of

directions.• Steady & Uniform (escoamento tampão): uniform and

steady.

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Velocity Profile

Steady/permanent (estacionário/permanente) Flow: the velocity does not change in time.

Uniform

Parabolic

Boundary layer type (tipo camada limite)

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Steady flow, imcompressible flow and inviscid flow

• Steady flow (escoamento estacionário): flow properties do not change over time (the partial derivative of any property in order to time is null) at any point in the system.

• Incompressible flow (incompressivel): density of a fluid element does not change during its motion (liquids and gas if the gas velocity is less than 30% of the speed of sound of the gas).

• Inviscid flow (de fluido ideal): is the flow of an ideal fluid that is assumed to have no viscosity.

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Streamlines

Streamlines are lines tangent to the velocity vector.

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Pathline and streakline

• Pathline (trajetória) is the actual pathway of a fluid particle.

• Streakline (linha de emissão) is the locus of particles which have earlier passed through a prescribed point (e.g. smoke from a chimney seen from far away).

• Streamlines, pathlines and streaklines are identical in steady flow

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Streamline, pathline and streakline(Linha de corrente, trajectória e linha de emissão)

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• http://www.youtube.com/watch?v=rDhSdtMjSpA&feature=relmfu

• http://www.youtube.com/watch?v=BsGbOgVFoS0

• http://www.youtube.com/watch?v=nuQyKGuXJOs&feature=related

• http://www.youtube.com/watch?v=DOUfyDHxkYQ&feature=related

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Information given by the streamlines

• How much is the flow passing across a streamline? – Zero, because the velocity is parallel to streamlines!

• How does the flow rate vary between two streamlines from one cross section to another?

– Remains constant. • How does the velocity vary between two non-parallel streamlines?

– Increase when they approach and decrease when they move away (if ρ constant).

• How the resultant of the forces varies between two streamlines?– It is against the flow when streamlines move away and is in favor when they

approach. According to Newton's Law the resultant force is equal to mass times acceleration. If the pressure is the most important force, we conclude that it decrease when velocity increase.

• How the pressure varies from one side to the other of a streamline with curvature?

– Increases to the outside. When we have curvature we gain velocity (and momentum) perpendicular to the streamline, to the inside of the curve and so we have to have an applied force.

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Basic principles of Fluid mechanics

• Conservation of mass

• =>

• Newton law:

• =>

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Eulerian and lagrangian description

E se o escoamento fosse não estacionário?

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Flow analysis techniques