MECH 221 FLUID MECHANICS (Fall 06/07) Tutorial 7

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MECH 221 FLUID MECHANICS(Fall 06/07)Tutorial 7
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Outline

1. Conservation of Momentum(Navier-Stokes Equation)
2. Dimensional Analysis
3. Buckingham Pi Theorem
4. Normalization Method

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1. Conservation of Momentum (Navier-Stokes
Equations)

• In Chapter 3, for inviscid flows, only pressure
  forces act on the control volume V since the
  viscous forces (stress) were neglected and the
  resultant equations are the Eulers equations.
  The equations for conservation of momentum for
  inviscid flows were derived based on Newtons
  second law in the Lagrangian form.

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1. Conservation of Momentum (Navier-Stokes
Equations)

• Eulers equation
• This is the Integral Form of Eulers equation.

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1. Conservation of Momentum (Navier-Stokes
Equations)

• This is the Differential Form of Eulers
  equation.

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1. Conservation of Momentum (Navier-Stokes
Equations)

• Here we should include the viscous stresses to
  derive the momentum conservation equations.
• With the viscous stress, the total stress on the
  fluid is the sum of pressure stress(
  , here the negative sign implies that tension is
  positive) and viscous stress ( ), and is
  described by the stress tensor given by

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1. Conservation of Momentum (Navier-Stokes
Equations)

• Here, we generalize the body force (b) due to all
  types of far field forces. They may include those
  due to gravity , electromagnetic force,
  etc.
• As a result, the total force on the control
  volume in a Lagrangian frame is given by

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1. Conservation of Momentum (Navier-Stokes
Equations)

• The Newtons second law then is stated as
• Hence, we have

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1. Conservation of Momentum (Navier-Stokes
Equations)

• By the substitution of the total stress into the
  above equation, we have
• which is integral form of the momentum
  equation.

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1. Conservation of Momentum (Navier-Stokes
Equations)

• For the differential form, we now apply the
  divergence theorem to the surface integrals to
  reach
• Hence, V?0, the integrands are independent of V.
  Therefore,
• which are the momentum equations in differential
  form for viscous flows. These equations are also
  named as the Navier-Stokes equations.

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1. Conservation of Momentum (Navier-Stokes
Equations)

• For the incompressible fluids where
  constant.
• If the variation in viscosity is negligible
  (Newtonian fluids), the continuity equation
  becomes , then the shear stress tensor
  reduces to .

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1. Conservation of Momentum (Navier-Stokes
Equations)

• The substitution of the viscous stress into the
  momentum equations leads to
• where is the Laplacian operator
  which in a Cartesian coordinate system reads

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2.1 Buckingham Pi Theorem

• Given the quantities that are required to
  describe a physical law, the number of
  dimensionless product (the Pis, Np) that can
  be formed to describe the physics equals the
  number of quantities (Nv) minus the rank of the
  quantities, i.e., NpNv Nm,

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2.1 Example

• Viscous drag on an infinitely long circular
  cylinder in a steady uniform flow at free stream
  of an incompressible fluid.
• Geometrical similarity is automatically satisfied
  since the diameter (R) is the only length scale
  involved.

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2.1 Example

• Dynamics similarity

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2.1 Example

• Both sides must have the same dimensions!

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2.1 Example

• where is the kinematic viscosity.
• The non-dimensional parameters are

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2.1 Example

• Therefore, the functional relationship must be of
  the form
• The number of dimensionless groups is Np2

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2.1 Example

• The matrix of the exponents is
• The rank of the matrix (Nm) is the order of the
  largest non-zero determinant formed from the rows
  and columns of a matrix, i.e. Nm3.

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2.1 Example

• Problems
• No clear physics can be based on to know the
  involved quantities
• Assumption is not easy to
  justified.

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2.2 Normalization Method

• The more physical method for obtaining the
  relevant parameters that govern the problem is to
  perform the non-dimensional normalization on the
  Navier-Stokes equations
• where the body force is taken as that due to
  gravity.

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2.2 Normalization Method

• As a demonstration of the method, we consider the
  simple steady flow of incompressible fluids,
  similar to that shown above for steady flows past
  a long cylinder.
• Then the Navier-Stokes equations reduce to

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2.2 Normalization Method

• If the proper scales of the problem are
• Here the flow domain under consideration is
  assumed such that the scales in x, y and z
  directions are the same

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2.2 Normalization Method

• Using these scales, the variables are normalized
  to obtain the non-dimensional variables as
• Note that the non-dimensional variable with
  are of order one, O(1).
• The velocity scale U and the length scale L are
  well defined, but the scale P remains to be
  determined.

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2.2 Normalization Method

• The Navier-Stokes equations then become
• where is the unit vector in the direction of
  gravity which is dimensionless.

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2.2 Normalization Method

• The coefficient of in the continuity
  equation can be divided to yield
• Dividing the momentum equations by in
  the first term of left hand side gives

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2.2 Normalization Method

• Since the quantities with are of O(1), the
  coefficients appeared in each term on the right
  hand side measure the ratios of each forces to
  the inertia force. i.e.,
• where Re is called as Reynolds number and Fr as
  Froude number.

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2.2 Normalization Method

• Conservation of mass
• Conservation of momentum

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2.2 Example

• Given that
• Velocity U U8cos(?t)
• Pressure P
• Length L

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2.2 Example

• Momentum conservation equation with viscous
  effect

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2.2 Example

• Normalized parameters

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2.2 Example

• Normalization

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2.2 Example

• Normalization

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2.2 Example

• Since

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The End