Extract from lecture at ICHTC, Sydney, 2006

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Extract from lecture at ICHTC, Sydney, 2006

• The numerical methods used for heat-conduction
  problems can also be extended to the calculation
  of stresses and strains in solids.
• There are many ways of doing so but probably the
  simplest is to solve the equations for the
  displacement components.
• The Figure and Equation shown below are a little
  more complex than those for temperature but not
  much.

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Control-volume for vertical displacement v
First the Figure
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Extending Numerical Heat Transfer

• then the equation
• The slight complication of the displacement-compon
  ent problem is that there are three sets of
  equations ( for U, V and W) and they are linked
  together in special (but easily-formulated) ways.

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Solving the equations

• I now show some results of solving the equations
  by the same successive-substitution method as is
  used for heat conduction.
• It is applied to the case of a square-sectioned
  beam having a square hole, filled with fluid,
  along its axis.
• Contours and vectors of displacement are shown.

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1/4 of square beam with fluid in square hole

• When the outer-wall temperature is raised

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1/4 of square beam with fluid in square hole

• When the inner-duct pressure is raised

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1/4 of square beam with fluid in square hole

• When both changes are made simultaneously.

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Consequential stresses

• From the displacement fields may be deduced the
  distributions of the direct stresses in the
  horizontal direction...

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Consequential stresses

• and in the vertical direction.

Comparison with solutions made by the
finite-element code Elcut showed close
agreement, of course for the finite-volume and
finite-element methods solve the same
differential equations.
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The research opportunities

• The computer time needed for solving the 3
  displacement equations is more than 3 times that
  needed for the temperature equation. The reason
  is that the equations for the 3 displacement
  components are inter-linked.
• Naive sequential solution procedures may
  (depending on geometry) converge rather
  slowly.More refined procedures are needed, and
  are being developed but there is still much to
  do.
• Researchers seeking little-exploited territories
  may therefore find them here and the world still
  awaits compilation and publication of the
  definitive textbook.Why? The numerical-stress-ana
  lysis field was devastated in the 1960's by the
  finite-element tsunami. Recovery takes time.

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Summary PUM_2006

• Earlier versions of solid-stress features had
  some limitations.
• These have now been removed.
• Anyone wishing to solve SFT (i.e.
  solid-fluid-thermal) problems with PHOENICS can
  now do so.
• However, consultancy help from CHAM is advisable
  at first.

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SFT

• The end

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3. Extending Computational Fluid Dynamics to SFT

• 3.1 Essential Ideas
• When Numerical Heat Transfer concerns itself with
  convection as well as conduction, it becomes a
  part of CFD..
• This also came into existence in the late 1960s.
• It uses equations similar to those governing heat
  conduction, shown above, with additional
  features, namely

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The additional features of the CFD equations

• the dependent variables include the components of
  velocity
• the coefficients (aN, aS, etc). account for
  convective as well as diffusive interactions
  between adjacent control volumes
• the sources include pressure gradients, gravity,
  centrifugal and Coriolis forces and
• the effective transport properties vary with
  position over many orders of magnitude.
• The CFD equations is thus more complex than the
  thermal-stress problem yet satisfactory
  iterative solution procedures have been in
  widespread use since the early 1970s.

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Use of CFD procedures for solid-stress problems

• CFD solution procedures have been successfully
  applied to solid-stress problems. Both Steven
  Beale and I independently showed this in 1990, as
  did Demirdzic and Mustaferija soon after.
• Mark Cross's group at Greenwich University has
  also made significant use of such methods for
  fluid-solid-interaction problems.
• Since the fluids and the solids occupy
  geometrically separate volumes, a single computer
  program can predict the behaviour of both solids
  and fluids simultaneously.
• This possibility has not been widely exploited
  because of the popular misconception that
  solid-stress problems must be solved by
  finite-element methods.
• It is therefore high time that CFD should enlarge
  to become SFT, i.e. Solid-Fluid-Thermal.

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3.2 A simple example

• Let us consider a primitive counterflow heat
  exchanger, consisting of two concentric tubes.
• Let us also suppose that because of
• natural convection in the cross-stream plane, or
• non-uniformity of external surface temperature,
  or
• turbulence-promoting baffles within one or both
  of the tubes ,
• the distributions of temperature and pressure,
  and therefore also of stress and strain in the
  tubes, are not axisymmetrical.

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The concentric-tube heat exchanger

• How are the stresses and strains to be computed?
• Numerically, of course and, if (misguided !)
  common practice is followed, one computer code
  will be used for the fluids and another for the
  solids.
• Then means must be devised for transferring
  information between them.
• How much more convenient it will be to use one
  computer code for the whole job!

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Extending CFD to SFT

• A true SFT code can do just that by
• solving for velocities and pressure in the space
  occupied by fluid
• solving for displacements and strains in that
  occupied by solid
• solving simultaneously for temperature in both
  spaces.
• The following images relate to the heat exchanger
  in question, with the radial dimension magnified
  four-fold.

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Concentric tube heat exchanger

• 1. Pressures in the two fluids causing mechanical
  stresses

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Concentric tube heat exchanger

• 2. The temperature distribution, causing thermal
  stresses.

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Concentric tube heat exchanger

• The circumferential variation of temperature
  imposed on the outer surface has produced 3D
  variations of temperature, stress and strain, as
  follows

3. radial-direction strains (positive being
extensions, negative compressions)
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Concentric tube heat exchanger

• 4. circumferential-direction strains

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Concentric tube heat exchanger

• 5. radial-direction stresses (positive being
  tensile, and negative compressive)

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Concentric tube heat exchanger

• 6. circumferential-direction stresses

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Concentric tube heat exchanger

• 7. axial-direction stresses.

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Extending CFD to SFT

• Three questions
• 1. Are the predictions correct?
• Probably, because
• the code produces the analytically-derived exact
  solutions for all cases in which these exist
• the displacement equations, are, after all, very
  simple.
• 2. Did solving for stress and strain increase the
  computer time?
• Not noticeably. Calculating finite values of
  displacement is not much more expensive then
  setting velocities to zero and convergence of
  the velocity and pressure fields dictated how
  many iterations were needed.
• 3. Could the same result have been achieved by
  coupling a finite-volume and a finite-element
  code?
• Certainly, but with much greater difficulty so
  why bother?

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3.3 A choice to be made

• Which forms the better method for SFT?
  Finite-volume or finite-element?
• The printed version of the lecture discusses the
  question at length. Here I summarise thus
• The general-purpose SFT codes needed by
  heat-transfer engineers could be based on
  finite-element methods). But..
• The highly-demanding F part of SFT, is handled so
  much better by finite-volume methods than
  finite-element ones
• Why else did Ansys buy Fluent and CFX?,
• that the best SFT codes are likely to be
  FV-based.
• Early arguments that FE methods are better for
  awkward geometries lost their force more than
  twenty years ago.
• It is only mental and commercial inertia that
  keeps the finite-element juggernaut in motion.

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Final examples

• 1. distortions of a sea-bed structure by ocean
  waves,

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Final examples

• 2. flapping of a wing, courtesy of K Pericleous