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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-component 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-analysis 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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1. distortions of a sea-bed structure by ocean waves

,

Final examples

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Final examples

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