Important Concepts in Fluid Dynamics

Important Concepts in Fluid Dynamics

1. STREAMLINE

   • Is a path traced out by a massless particle as it moves with the flow.
     • Since the streamline is traced out by a moving particle, at every point along the path the velocity is tangent to the path.
     • Since there is no normal component of the velocity along the path, mass cannot cross a streamline
     • We can use Bernoulli's equation to relate the pressure and velocity along the streamline.
     • Since no mass passes through the surface of the airfoil (or cylinder), the surface of the object is a streamline.

2. PATHLINES

   • is the path that is followed by a particle released into the fluid; the actual path traversed by a given fluid particle.
     • For steady flow, streamline and pathline are the same
   • A pathline represents a particle's path
   • For unsteady flow, the pathlines for different fluid elements passing through the same point are not the same

3. ANGULAR VELOCITY

   • By definition, the angular velocity of the fluid element seen in the x y plane is the average of the angular velocities of lines AB and AC
   • The motion of a fluid element along a streamline is a combination of traslation and rotation; in addition, the shape of the element can become distorted
   • The angular velocity of the fluid element in three-dimensional space
     • • This is the equation in terms of derivatives of the velocity field

4. VORTICITY

   • Basically, is twice the angular velocity
   • In a velocity field, the curl of the velocity is equal to the vorticity
   • The rotational flow implies that the fluid elements have a finite angular velocity
   • The irrotational flow implies that the fluid elements have no angular velocity; their motion through space is a pure translation
     • Easier to analyze
     • Called potential flows

5. CIRCULATION

   • Is simply the negative of the line integral of velocity around a closed curve in the flow
   • Is a kinematic property depending only on the velocity field and the choice of the curve
   • Does not necessarily mean that the fluid elements are moving around in circles within this flow field
   • When circulation exists in a flow, it simply means that the line integral is finite
   • The circulation about a curve C is equal to the vorticity integrated over any open surface bounded by C (From Stokes´ theorem)

6. STREAM FUNCTION

   • The equation for a streamline is given by setting the stream function equal to a constant
   • For a steady flow, the mass flow inside a given streamtube is constant along the tube; the mass flow across any cross section of the tube is the same
   • Is equal to the mass flow between the two streamlines
   • Can be used in either rotational or irrotational flow

7. VELOCITY POTENTIAL

   • Is analogous to the stream function in the saense that derivatives of ∅ yield the flow-field velocities
   • ∅ is a scalar function, denote as the velocity potential
   • Is defined for irrotational flow only
   • Apllies to three-dimensional flows, whereas the stream function is defined for two-dimensional flows only

8. LAPLACE´S EQUATION

   • Solutions of Laplace´s equation are called harmonic functions
   • Any irrotational, incompressible flow has a velocity potential and stream function (for two-dimensional flow) that both satisfy Laplace´s equation
     • Any solution of Laplace´s equation represents the velocity potential or stream function (two-dimensional) for an irrotational, incompressible flow
   • Although the governing equation for the different flows is the same, the boundary conditions for the equation must conform to the different geometric shapes, and hence yield different flow-field solutions

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