Incompressible NavierStokes equations

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Incompressible Navier-Stokes equations
Conservation of momentum
Conservation of mass (continuity)
Material derivative
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Four means of solution

• Direct numerical solution/simulation (DNS)

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Four means of solution

• Direct numerical solution/simulation (DNS)
• Approximation

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Four means of solution

• Direct numerical solution/simulation (DNS)
• Approximation
• Statistical methods

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Four means of solution

• Direct numerical solution/simulation (DNS)
• Approximation
• Statistical methods
• Dimensional analysis

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Common approximations

• Reynolds averaging
• Boundary layer approximation
• Ideal flow
• Potential flow
• Hydrostatic approximation
• Boussinesq approximation

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Reynolds averaging (RANS)
Assumption
Results in the following rules
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Reynolds-averaged Navier-Stokes equations (RANS)
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Boundary layer approximation
Assumption Lx, Ly gtgt Lz
1-D wall-bounded boundary layers

Additional assumption (transverse velocity)
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Ideal flow
Assumption v (viscosity) 0

• Not particularly helpful

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Potential flow

• Ideal flow assumption
• Kelvins transport theorem states if
  initially, the flow will remain irrotational.
• So if we make the substitutions (for a 2D flow)

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Hydrostatic approximation
Assumption
Which yields
If gravity is only body force
Trivial, but powerful result.
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Boussinesq approximation

• Assumes that density variations only matter in
  the body force term, not the inertial terms.

Substituting the result from the hydrostatic
approximation (i.e., neglecting density
variations in the inertial terms)
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Approximations key points

• Reynolds averaging
• Generates Reynolds stress tensor, which is a
  statistical quantity
• Allows for solution of NS at arbitrary length
  scales
• Boundary layer
• Ubiquitous, eliminates only a few terms, but
  allows conversion to ODEs
• Potential flow
• Limited to viscous flows, but powerful
• Boussinesq
• Reduces nonlinearity, applicable for slowly
  varying flows