MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 6: DIMENTIONAL ANALYSIS

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MECH 221 FLUID MECHANICS(Fall 06/07)Chapter 6
DIMENTIONAL ANALYSIS

• Instructor Professor C. T. HSU

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6 Dimensional Analysis

• The objective of dimensional analysis is to
  obtain the key non-dimensional parameters that
  govern the physical phenomena of flows.
• Variables like etc. are combined to
  form the key parameters that have no physical
  units (dimensionless). The non-dimensional
  parameters include both the geometric and dynamic
  parameters.

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6.1 Similarity

• Geometric Similarity
• Dynamic Similarity
• Two flows are said to be similar if they have the
  same geometric and dynamic dimensionless
  parameters.

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6.1.1 Geometrical Similarity

• Two body are geometrically similar, if the
  geometry of one can be obtained from another by
  scaling all dimensions by the same factor.

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6.1.2 Dynamic Similarity

• Flows are said to be dynamically similar if by
  scaling the dependent and independent variables
  they yield the same non-dimensional parameter.
• This is more difficult to achieve than the
  geometric similarity.
• What are these non-dimensional parameters?
• How can they be found?

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6.2 Buckingham Pi Theorem

• Dimensionless product is the product of several
  dimensional quantities that render the product
  dimensionless
• The rank of matrix Nm is the minimum dimensions
  that leads to non-zero determinant, which is also
  the minimum dimensions of the quantities that
  describe the physics.

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6.2 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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6.2.1 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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6.2.1 Example

• Dynamics similarity

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6.2.1 Example

• Both sides must have the same dimensions!

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6.2.1 Example

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

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6.2.1 Example

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

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6.2.1 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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6.2.1 Example

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

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6.3 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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6.3 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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6.3 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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6.3 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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6.3 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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6.3 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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6.3 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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6.3 Normalization Method

• The dynamic of fluid motion then depends solely
  on the magnitudes of these non-dimension
  parameters, i.e., pressure coefficient and
  gravitational body-force coefficient.
• Flows are dynamically similar if they have the
  same Re and Fr

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6.3 Normalization Method

• Remark
• New non-dimensional parameters can also emerge
  from the non-dimensional analysis on the boundary
  conditions which is not deliberated here.
• The dimensional analysis reduces experimentalists
  the need of carrying out measurements for
  different U, D, etc.

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6.4 Characteristics of Non-Dimensional Parameters

• Reynolds Number
• Froude Number

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6.4.1 Reynolds Number

• Lets for simplicity consider the case where the
  gravitational force has no consequence to the
  dynamic of the flow, i.e. the case where
  or the contribution of is only to the static
  pressure. Then, the pressure P represents the
  dynamics pressure.
• The normalized momentum equation becomes

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6.4.1 Reynolds Number

• where is the viscous diffusion length in
  an advection time interval of .
• Here, measures the time required
  for fluid travel a distance L.

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6.4.1 Reynolds Number

• High Reynolds Number Flow, Regtgt1
• Intermediate Reynolds Number Flow, Re1
• Low Reynolds Number Flow, Reltlt1

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6.4.1.1 High Reynolds Number Flow

• When , inertia force is much greater
  than viscous force, i.e., the viscous diffusion
  distance is much less than the length L.
• Viscous force is unimportant in the flow region
  of
• , but can become very important in the
  region of
• near the solid boundary.
• This flow region near the solid boundary is
  called an boundary layer as first illustrated by
  Prandtl.

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6.4.1.1 High Reynolds Number Flow

• Flow in the region outside the boundary layer
  where viscous force is negligible is inviscid.
  The inviscid flow is also called the potential
  flow.

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6.4.1.1 High Reynolds Number Flow

• The normalized dimensionless equation to the
  first order approximation is
• Clearly, the proper pressure scale should be
  chosen such that is of O(1), and for
  simplicity, can be set as
  is the proper pressure scale for high
  Reynolds number flow.
• Flows in the boundary layer are governed by
  boundary layer equations that need to be derived
  separately with different approach

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6.4.1.1 High Reynolds Number Flow

• For inviscid, incompressible, steady flow, the
  governing equations in terms of dimensional
  variable to the first order approximation are
  written as,
• If the gravitational force is retrieved, then we
  have
• where is the steady Euler equations for
  incompressible fluid as given in Chapter 2.

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6.4.1.2 Intermediate Reynolds Number Flow

• When , inertia forces and viscous forces
  are of equal importance. The flow is viscous in a
  region of surrounding the body since
  
• . No approximation can be done.

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6.4.1.2 Intermediate Reynolds Number Flow

• The governing equations remain as
• whose solutions satisfying proper boundary
  conditions can usually only be obtained
  numerically.

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6.4.1.3 Low Reynolds Number Flow

• When , the inertia force is very much
  smaller than the viscous force. The viscous
  diffusion length is much larger than L.
• The flow is viscous for almost the entire region
  except at vary far away from the solid boundary.

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6.4.1.3 Low Reynolds Number Flow
U
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6.4.1.3 Low Reynolds Number Flow

• Since implies , the
  inertia force is negligible and the pressure
  force has to balance the viscous force.
  Therefore, the proper scale for P is such that
  
• The governing equation for low Reynolds number
  flows, without the gravitational force, can be
  approximated to the first order by

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6.4.1.3 Low Reynolds Number Flow

• If the gravitational force is recapped, the
  governing equations to the first order
  approximation becomes
• Flows of low Reynolds number are called the
  Stokes flows, or creeping flows.
• One such example is the settling of small
  particles in water. Lava flows from volcanic
  eruption are also typical low Reynolds number
  flows although the viscosity is non-Newtonian.

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6.4.1.4 Dynamically Similar Reynolds Number

• Flows with the same Reynolds number are
  dynamically similar.
• For example, flight of very small insects in air
  can be studied much more easily on large models
  in very viscous fluid (liquids).
• Similarly, flows for large object such as train,
  airplane, tall buildings, etc., can be studied
  experimentally with small models.
• (Note They should be geometrically similar)

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6.4.2 Froude Number

• The Froude number measures the gravitational
  effect on the flows. It depends on the problem
  encountered.
• For instance in free surface flows, is
  the phase speed c of shallow water gravity waves
  when L is water depth. The Froude number becomes

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6.4.2 Froude Number

• If U represents the speed of a ship moving on a
  flat water surface, the wave patterns generated
  by the ship, called ship wakes, then depend on
  whether the Froude number is
  .

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6.4.2 Froude Number

• Another example is the rising of a hot-air
  balloon, which is characterized by the net
  buoyancy force. The natural convection flows
  generated by a vertical heated surface represents
  another example.

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6.4.2 Froude Number

• The ratio of the net buoyancy force
  (per unit volume) to the inertia force
  (per unit volume) is
• where Ri is the Richardson number. The
  Richardson number is the key parameter in dealing
  with advection of fluid of different density.
• Another example is the propagation internal waves
  in stratified fluids.

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6.5 Map based on Non-Dimensional Parameter Flow
ln Fr
Re1
Fr1
ln Re
Fr0
Re0