Simultaneous Prediction of Solid stress, Heat Transfer and Fluid flow by a Single Algorithm

By Brian Spalding

Invited Lecture presented at 2002 ASME Pressure Vessels and Piping Conference,

August 4-8, 2002, Vancouver, British Columbia, Canada

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Contents

1. The problem

2. A multi-physics example

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3. The mathematics of the method

4. Details of the auxiliary models

5. Discussion of achievements and future prospects

6. References

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1. The problem

(a) Its essential nature

It is frequently required to simulate fluid-flow and heat-transfer processes in and around solids which are, partly as a consequence of the flow, subject to thermal and mechanical stresses.

Often, indeed, it the stresses the major concern, while the fluid and heat flows are of only secondary interest.

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(b) Practical occurrence

Engineering examples of fluid/heat/stress interactions include:

• gas-turbine blades under transient conditions;

• "residual stresses" resulting from casting or welding;

• thermal stresses in nuclear reactors during emergency shut-down;

• manufacture of bricks and ceramics;

• stresses in the cylinder blocks of diesel engines;

• the failure of steel-frame buildings during fires.

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(c) The conventional solution

It has been customary for two computer codes to be used for the solution of such problems, one for the fluid flow and the other for the stresses

Iterative interaction between the two codes is then employed, often with considerable inconvenience.

(d) A better solution

It is however possible for fluid flow, heat flow and solid deformation, and the interactions between them, all to be calculated at the same time.

The method of doing so exploits the similarity between the equations governing velocity (in fluids) and those governing displacement (in solids).

In the present lecture, the results of such a calculation will be shown first.

The explanation of how it was conducted will then follow.

2. A multi-physics example

(a) Description: The task is to calculate the stresses in radiation-heated solids cooled by air.
```
20 deg C| air
|             80 deg C
| V |/////// hot radiating wall ///////////|
|   ----------------------------------------
|                       duct              ----->  exit
|-------------                 -------------
|// steel ///|     cavity      |/// steel /|
|------------------------------------------- ? temperature ?
|////////////// aluminium /////////////////|
|-------------------------------------------```
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Details of the calculation are:

1. The Reynolds number (based on the inflow velocity and horizontal duct width) is 1000.
Therefore the LVEL model is used for simulation of the turbulence.

2. The radiative heat transfer is represented by the conduction-type IMMERSOL model, which is:
1. economical and
2. fairly accurate
for such situations.

The absorptivity of the air is taken as 0.01 per meter;
the scattering coefficient as 0.0;
and the solid surface emissivity as 0.9 .

3. Both LVEL and IMMERSOL make use of the distributions of:
1. distance from the wall (WDIS) and
2. distance between walls (WGAP), both of which are calculated by solving a scalar equation for the
3. LTLS variable.

4. The stresses within the metals result primarily from the differences in their thermal-expansion coefficients. namely:
• 2.35 e-5 for aluminium, and
• 0.37 e-5 for steel.

(b) Vector and contour plots

• Vectors

The vectors displayed in Fig.1 show simultaneously:

1. the nature of the air-flow pattern (upper part of the domain), and
2. the displacements of the solid material (lower part of the domain).

Both are calculated at the same time. Of course, the two sets of vectors have different scales, and indeed dimensions. namely m/s for velocity and m for displacements.

The solids are supposed to be confined by a stiff-walled box, but are allowed to slide relative to its walls. This is why the displacement vectors are vertical near the confining-box walls.

They are however not allowed to slide relative to each other; this is what causes the concentrations of stress at their contact surfaces.

• Stresses and strains

The stresses in the x- (horizontal) and y- (vertical) directions are displayed in Fig.2 and Fig.3 respectively.

They have been deduced from the strains shown in

Fig.4 for the x-direction , and in

Fig.5 for the y-direction,

The strains have been deduced from the displacements by differentiation. The displacements are however not here displayed as contour plots because these do not easily display small variations.

That is the end of the stress-strain results.

There will now be shown some of the other variables which had to be computed.

• Temperature fields

Fig. 6 displays contours of temperature in the air and the solid, and reveals that:

• the air is heated by contact with:
1. the 80-degree-Celsius top wall, and
2. the metal blocks, which have been receiving heat by radiation from the top wall;

• temperature differences within the high-conductivity solids are too small to be discerned visually.

It is interesting to compare Fig. 6 with Fig. 7.

This displays the distribution within the air space of:

• the "radiation temperature", T3, which:

• is computed by IMMERSOL, and

• is defined as the temperature which would be taken up by a probe which was affected only by radiation.

Obviously, and understandably, T3 and TEM1 have very different values, unless the absorptivity is very great (as in solids).

The solid temperature influences the stresses and strains, of course, primarily through the agency of the temperature-dependent thermal-expansion distribution.

However, its variations with position, within a single material, are too slight to be revealed by a contour diagram, as inspection of Fig. 8 will reveal.

The IMMERSOL model, of which the solution of the T3 equation is the major feature, enables the radiant heat fluxes in the coordinate directions to be established by post-processing.

The results are displayed in
Fig. 9 for the x-direction. and by
Fig. 10 for the y-direction.

The values and patterns displayed, if studied and interpreted in physical terms, will be found to be plausible.

Where calculation by hand is easy, namely for the parallel surfaces, they will be found to be correct.

• Contours of auxiliary quantities used by IMMERSOL

A crucial feature of the IMMERSOL model is its use of the distribution of the "distance between the walls", WGAP.

This quantity, which has an easily-understood meaning when the walls are near, and nearly parallel, is computed from the solution of the "LTLS" equation;
this will be explained later in the lecture.

The distributions of these two quantities are shown by

Fig. 11 for the former, and by

Fig. 14 for the latter.

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It will be seen that WGAP has a uniform value in the region of between the top of the duct and the tops of the upper metal slabs, between which the actual distance is 0.008 meters.

Further, it has approximately twice this value near the convex corners; and it becomes zero in the concave corners.

• Contours of auxiliary quantities used in the fluid-flow calculation

The flow field was calculated by means of the LVEL turbulence model, which makes use of the wall-distance (WDIS) field.

This, like WGAP, is also derived from the LTLS distribution.

The contours of WDIS are displayed in Fig. 13.
which exhibits:

• the expected maximum of 0.004 between the parallel horizontal walls, and
• a somewhat greater value near the cavity, where the true distance from the wall depends on the direction in which it is measured.

LVEL, like IMMERSOL, is a "heuristic" model, by which is meant that it is incapable of rigorous justification, but is nonetheless useful.

WDIS is calculated once for all, at the start of the computation. From it, and from the developing velocity distribution, the evolving distribution of ENUT, the effective (turbulent) viscosity is derived.

The resulting contours of ENUT are shown in Fig. 14.

Since the laminar viscosity is of the order of 1.e-5 m**2/s, it is evident that turbulence raises the effective value, far from the walls, by an order of magnitude.

(c) How the stresses were calculated

• As will be shown below, the equations governing the displacements are very similar to those governing the velocities.

• The CFD code PHOENICS, like many others, can calculate velocities in fluids; but this ability is not needed in the solid region; so such codes are usually idle there.

• However, PHOENICS can be "tricked" into calculating what it "thinks" are velocities everywhere; whereas what it actually calculates in the solid regions are displacements.

• The details of the "trickery" now follow.

(a) Similarities between the equations for displacement and velocity

The similarities already referred to are here described for only one cartesian direction, x; but they prevail for all three directions.

1. The x-direction displacement, U, obeys the equation:

[del**2]* U + [d/dx]* [ D*C1 - Te*C3 ] + Fx*C2 = 0

where:
• Te = local temperature measured above that of the un-stressed solid in the zero-displacement condition, multiplied by the thermal-expansion coefficient;
• D = [d/dx]* U + [d/dy]* V + (d/dz]* W
which is called the "dilatation";
• Fx = external force per unit volume in x-direction;
• V and W = displacements in y and z directions;
• C1, C2 and C3 are functions of Young's modulus and Poisson's ratio.
• When the viscosity is uniform and the Reynolds number is low, so that convection effects are negligible, the x-direction velocity, u, obeys the equation:

[del**2]* u - [d/dx]* [ p*c1 ] + fx*c2 = 0 ,

where

• p = pressure,
• fx = external force per unit volume in x-direction,
• c1 = c2 = the reciprocal of the viscosity.

Notes:

1. The two equations are now set one below the other, so that they can be easily compared:

• [del**2]* U + [d/dx]* [ D*C1 - Te*C3 ] + Fx*C2 = 0

• [del**2]* u - [d/dx]* [ p*c1 ] + fx*c2 = 0 ,

2. The equations can thus be seen to become identical if:

• p*c1 = D*C1 - Te*C3
which implies:
D = [p*c1 + Te*C3 ] / C1

• and
fx * c2 = Fx * C2

3. The expressions for C1, C2 and C3 are:

• C1 = 1/(1 - 2*PR)
• C2 = 2*(1 + PR) / YM
where
• PR = Poisson's Ratio. and
• YM = Young's Modulus
and
• C3 = 2 *(1 + PR)/(1 - 2*PR)

4. A solution procedure designed for computing velocities will therefore in fact compute the displacements if:

1. the convection terms are set to zero within the solid region: and

2. the linear relation between:
• D ( ie [d/dx]* U + ...) and
• p
is introduced by inclusion of a pressure- and temperature-dependent "mass-source" term.

(b) Deduction of the associated stresses and strains

The strains (ie extensions ex, ey and ez) are obtained from differentiation of the computed displacements.

Thus:

ex = [d/dx]* U

ey = [d/dx]* V

ez = [d/dx]* W

Then the corresponding:

• normal stresses, sx, sy, sz, and
• shear stresses tauxy, tauyz, tauzx,
are obtained from the strains by way of equations such as:

sx = {YM / (1 - PR**2)} * {ex + PR*ey} and

tauxy = {YM / (1 - PR**2)} * {0.5 * (1 - PR)*gamxy}

where:

• gamxy = [d/dy]*U - [d/dx]*V

(c) The "SIMPLE" algorithm for the computation of displacements

PHOENICS employs (a variant of) the "SIMPLE" algorithm of Patankar & Spalding (1972) for computing velocities from pressures, under a mass-conservation constraint.

Its essential features are:

• All the velocity equations are solved first, with the current values of p.
• The consequent errors in the mass-balance equations are computed.
• These errors are used as sources in equations for corrections to p.
• The corresponding corrections are applied, and the process is repeated.

All that it is necessary to do, in order to solve for displacements simultaneously, is, in solid regions, to treat the dilatation B as the mass-source error and to ensure that p obeys the above linear relation to it.

Therefore a CFD code based on SIMPLE can be made to solve the displacement equations by:

1. eliminating the convection terms (ie setting Re low); and
2. making D linearly dependent on p and temperatureT.

The "staggered grid" used as the default in PHOENICS proves to be extremely convenient for solid-displacement analysis; for the velocities and displacements are stored at exactly the right places in relation to p.

(a) IMMERSOL: summary

• The solved differential equation is:
div( effective_conductivity * T3 ) + source = 0

• effective conductivity =
0.75 * sigma * T3**3 / (abso + scat + 1/WGAP)

• source = abso * sigma * ( T1**4 - T3**4 )

• in solids, abso = large, so T3 --> T1

• surface resistances account for non-unity emissivities

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Notes:

• The main novelty is the inclusion of WGAP, ie the distance between the walls, in the formulae.

• This enables a conduction-type model to be used even with non-participating media.

• Of course, an economical means of calculating WGAP is needed.

• This is provided by the LTLS equation (see below).

• IMMERSOL gives quantitatively correct predictions in geometrically simple circumstances and plausible ones in complex ones.

• It is economical enough to be generalised for wavelength-dependent radiation.

(b) WGAP, WDIS and LTLS

• The solved differential equation is:

The boundary conditions are:
LTLS = 0 , in all solids.

• The solution for plane channel flow is:
LTLS = WDIS * ( WGAP - WDIS ) / 2 , and
grad LTLS = WGAP / 2 - WDIS

• These relations are supposed to prevail also in two- and three-dimensional circumstances.

• That is all!

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Notes:

• The LTLS equation is very simple, and therefore easy to solve.

• Its solution yields values of LTLS and grad LTLS at all points in the field.

• WDIS and WGAP are then deduced from them.

• Their values are quantitatively correct predictions in geometrically simple circumstances and plausible in complex ones.

• The method is especially useful, indeed the only practicable one, when the space in question contains many solids of arbitrary shapes.

(c) LVEL: summary

• LVEL generalises a continuous formula for the "law of the wall", namely:
```
y+ = u+ + (1/E) * [ exp(K*u+) - 1 - K*u+ -
(K*u+)**2/2  - (K*u+)**3/6 - (K*u+)**4/24 ]```
which implies another for dimensionless effective viscosity, namely:
```
v+ = 1 + (K/E) * [ exp(K*u+) - 1 -
K*u+ - (K*u+)**2/2 - (K*u+)**3/6 ]```
• These relations enable the effective viscosity to be calculated at every point for which are known:
• the distance from the wall, y
• the fluid velocity, u, and
• the kinematic viscosity

• They are known at all points; so the model is complete.

• The LVEL model presumes that the same relation holds, regardless of the geometry of the solid surface.

• That is all!

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Notes:

• The LVEL model is very simple, and therefore easy to implement.

• The predicted effective viscosities are quantitatively correct in geometrically simple circumstances and plausible in complex ones.

• The method is especially useful, indeed often the only practicable one, when the space in question contains many solids of arbitrary shapes.

• LVEL handles the complete Reynolds-number range: laminar, transitional and fully turbulent.

• LVEL can be easily extended so as to improve its accuracy in locations far from walls.

(a) Preliminary conclusions

The following conclusions appear to be justified:

1. Simultaneous simulation of solid-stress, heat transfer and fluid flow is indeed practicable and economical.

2. As compared with the alternative, namely the use of distinct methods for each phenomenon with iterative interaction between them, the simultaneous-solution method is very attractive.

3. It therefore seems possible that, when its attractiveness is fully recognised, SFT (i.e. Solid-Fluid-Thermal) analysis may become as popular as CFD.

The author now proposes several answers to this question, as follows:
1. First, it requires long-held convictions to be questioned and then abandoned. Many people find this uncomfortable.

2. Secondly, those who are in principle willing to try new methods prefer to do so only after very many others have led the way.

"Never do anything first" commends itself to those who have observed how seldom pioneers actually prosper.

3. Thirdly, in the present case, The pioneers ( the author and his colleagues) have been reprehensibly backward in documenting, illustrating and generally promoting the SFT method.

True: it has been a part of the PHOENICS package for severalf years, but with too many limitations to be easily usable.

4. Lastly, the method described above has proved to have two deficiencies, namely:
1. the SIMPLE algorithm converges very slowly when both the flow and solid-stress situations are complex and intertwined; and
2. additional 'source terms' must be provided in order that 'bending' phenomena can be properly represented.

(c) The future of SFT

CHAM's current work on SFT is therefore focussed on the following:
1. attaching the appropriate 'convergence accelerator' ( already accomplished);

2. including the 'bending' feature ( in progress);

3. generalising the coding so that arbitrary numbers of materials, each with different temperature-dependent properties, can be handled ( in progress);

4. providing copious practically-interesting demonstrations ( as rapidly as possible).

(d) Research opportunities

Single-algorithm SFT presents numerous opportunities, both to numerical analysts and to practically-minded researchers, for example, to:
1. extend the algorithm to curvilinear grids and to cartesian grids cut by curved solid-fluid interfaces;
2. extend to transient phenomena;
3. allow deformations which are large enough to affect the flow;
4. permit plastic as well as elastic deformation of the solid material; and
5. allow the fluid to be multiphase.

An offer

CHAM will be happy to offer free-of-charge PHOENICS licences to academic researchers willing to collaborate in this important field.

END of LECTURE

6. References

1. The differential equations governing displacements, stresses and strains in elastic solids of non-uniform temperature can be found in numerous textbooks, for example:

• CT Yang
Applied Elasticity
McGrawHill, 1953

• BA Boley and JH Weiner
Theory of Thermal Stresses
John Wiley, 1960

• PP Benham, RJ Crawford and CG Armstrong:
Mechanics of Engineering Materials
Longmans, 2nd edition, 1996

It has not been common to choose the displacements as the dependent variables in numerical-solution procedures. However, this has been done by:

• JH Hattel and PN Hansen
A Control-Volume-based Finite-Difference Procedure for solving the Equilibrium Equations in terms of Displacements
Applied Mathematical Modelling, 1990

Their numerical procedure differ from that used here, which was that of

• SV Patankar and DB Spalding
"A Calculation Procedure for Heat, Mass and Momentum Transfer in Three-Dimensional, Parabolic Flows"
Int J Heat Mass Transfer, vol 15, p 1787, 1972

2. The first use of the present method for solving the solid-displacements and fluid-velocity equations simultaneously appears to have been made by CHAM, late in 1990.

Reports describing the early work include:

• KM Bukhari, HQ Qin and DB Spalding
Progress Report (to Rolls-Royce Ltd) on the Calculation of Thermal Stresses in Bodies of Evolution
CHAM Ltd, November, 1990

• KM Bukhari, IS Hamill,HQ Qin and DB Spalding
Stress-Analysis Simulations in PHOENICS. CHAM Ltd, May, 1991

From that time onwards, the solid-stress option was made available as a (little-advertised) option in successive issues of PHOENICS,

3. Open-literature and conference publications have been few, but include:

• DB Spalding
Simulation of Fluid Flow, Heat Transfer and Solid Deformation Simultaneously
NAFEMS 4, Brighton 1993
• D Aganofer, Liao Gan-Li and DB Spalding
The LVEL Turbulence Model for Conjugate Heat Transfer at Low Reynolds Numbers
EEP6, ASME International Mechanical Engineering Congress and Exposition, Atlanta, 1996
• DB Spalding
Simultaneous Fluid-flow, Heat-transfer and Solid-stress Computation in a Single Computer Code
Helsinki University 4th International Colloquium on Process Simulation, Espoo, 1997
• DB Spalding
Fluid-Structure Interaction in the presence of Heat Transfer and Chemical Reaction
ASME/JSME Joint Pressure Vessels and Piping Conference, San Diego, 1998