Fluid Dynamics and Circulation Theorem Quiz

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Questions and Answers

What is the gravitational attraction exerted by Earth on a unit mass with position vector r called?

  • g_e
  • g* (correct)
  • g_r
  • g

The work done by a conservative force field around a closed path is zero.

True (A)

In a barotropic atmosphere, the density ρ is a function of which variable?

pressure

For a frictionless barotropic flow, the expression rac{d aCa}{dt} = 0 is a direct corollary to _____ theorem.

<p>Kelvin's</p> Signup and view all the answers

Match the following expressions with their significance.

<p>g* = Gravitational attraction by Earth ∇ρ = Density gradient ∫g.dl = Work done by gravity around closed path ∇ × B = Curl of a vector field</p> Signup and view all the answers

What does the circulation theorem state regarding the time rate of change of circulation?

<p>It is equal to the circulation of the time rate of change of the vector field. (A)</p> Signup and view all the answers

Kelvin's Circulation Theorem can only be applied to the relative velocity vector.

<p>False (B)</p> Signup and view all the answers

What is the acceleration of circulation known as in the context of the circulation theorem?

<p>absolute or relative acceleration of circulation</p> Signup and view all the answers

According to the circulation theorem, the acceleration of circulation is equal to the circulation of __________.

<p>acceleration</p> Signup and view all the answers

Match the following concepts with their descriptions:

<p>Circulation theorem = Describes the relationship between circulation and time rate of change Kelvin's Circulation theorem = Circulation theorem applied to absolute velocity Acceleration of circulation = The rate at which circulation changes with time Absolute velocity = Velocity relative to a fixed point or observer</p> Signup and view all the answers

Flashcards

Conservative Force Field

A force field where the work done by the force field around any closed path is zero.

Stoke's Law

Relates the circulation of a vector field around a closed curve to the curl of the vector field integrated over the enclosed surface.

Barotropic Atmosphere

An atmosphere where the density is a function of pressure only.

Kelvin's Circulation Theorem

For a frictionless flow, the change in absolute circulation is solely due to the baroclinicity of the atmosphere.

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Solenoidal Vector

A vector field where the divergence is zero everywhere.

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Jacobian

A mathematical object that maps a change of coordinates from one system to another.

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∇ρ × ∇p = 0

In a barotropic atmosphere, the curl of the density gradient crossed with the pressure gradient is zero.

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∫∫ ∇ × B.nˆ ds

The circulation of a vector field, B, around a closed path (integrated over a surface enclosed by that path).

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Kelvin's Circulation Theorem

The theorem states that the rate of change of circulation of a fluid's absolute velocity is equal to the circulation of the acceleration of the absolute velocity of the fluid

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Circulation Theorem

The rate of change of a vector field's circulation is equal to the circulation of the vector field's time rate of change.

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Absolute Velocity (Va)

The velocity of a fluid particle relative to a fixed frame of reference.

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Relative Velocity (Vr)

The velocity of a fluid particle relative to a moving frame of reference.

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Circulation (Ca)

The line integral of a velocity field around a closed path.

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Acceleration of Circulation

The rate of change of circulation.

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Equation of Absolute Motion

An equation that describes how the absolute velocity of a fluid particle changes over time.

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∇p

Pressure Gradient

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g

Acceleration due to gravity

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ρ

Fluid density

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F

External forces on the fluid

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Study Notes

Circulation

  • Circulation: A macro-scale measure of fluid rotation.
  • Mathematical definition: Line integral of the velocity vector around a closed path.
  • Scalar quantity
  • Sign convention: Positive for anticlockwise rotation in the Northern Hemisphere, negative for clockwise; opposite in Southern Hemisphere.
  • Absolute Circulation (C): Defined for absolute velocities.
  • Relative Circulation (C): Defined for relative velocities.

Vorticity

  • Vorticity: A micro-scale measure of rotation.
  • Vector quantity
  • Direction: Determined by the fluid's rotation direction relative to the plane of rotation.
  • Mathematical expression : ∇ × V = î ζ + ĵ η + k ζ
  • In meteorology: Vorticity is usually only the k component.

Circulation Theorems

  • Circulation Theorems: Explain changes in circulation and causes.
  • Kelvin's Circulation Theorem: Applied to the absolute velocity vector (V) of fluid motion.
  • d/dt(Circulation) = Circulation of rate of change of the vector field (B).
  • Stocked Theorem: Line integral of any vector field B around a closed path is equal to the surface integral.

Barotropic and Baroclinic Atmosphere

  • Barotropic atmosphere: Horizontal temperature gradient is zero; isobars and isotherms are parallel.
  • Baroclinic atmosphere: Horizontal temperature gradient exists; isobars and isotherms are not parallel.

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