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 48, 2002, Vancouver, British Columbia, Canada
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Abstract
 It is often believed that FLUIDFLOW and SOLIDSTRESS problems MUST be
solved by DIFFERENT methods and DIFFERENT computer programs.
 This is NOT TRUE, if the solidstress problems are formulated in terms of
DISPLACEMENTS.
 The lecture exemplifies and explains how both DISPLACEMENTS and VELOCITIES
can be calculated AT THE SAME TIME.
 ALSO described, incidentally, are economical methods of simulating:
 thermal RADIATION between solids immersed in fluids; and
 TURBULENT CONVECTION at low Reynolds numbers in the same situation.
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Contents
 The
problem
 Its
essential nature
 Practical
occurrence
 The
conventional solution
 A
better solution
 A
multiphysics example
 Stresses
resulting from radiation, conduction and convection
 Vector
and contour plots
 How
the stress calculations were performed
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 The mathematics of the method
 Similarities
between the equations for displacement and velocity
 Deduction
of the associated stresses and strains
 The
"SIMPLE" algorithm for the computation of displacements
 More
details of the equations
 Details of the auxiliary models
 IMMERSOL,
for radiation
 WGAP,
WDIS and LTLS, for radiation and turbulence
 LVEL, for
turbulence
 Discussion
of achievements and future prospects
 References
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1. The problem
(a) Its essential nature
It is frequently required to simulate fluidflow and heattransfer
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:
 gasturbine blades under transient conditions;
 "residual stresses" resulting from casting or welding;
 thermal stresses in nuclear reactors during emergency shutdown;
 manufacture of bricks and ceramics;
 stresses in the cylinder blocks of diesel engines;
 the failure of steelframe 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.
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(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.
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2. A multiphysics example
(a)
Description: The task is to calculate the stresses in radiationheated
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:
 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.
 The radiative heat transfer is represented by the conductiontype IMMERSOL model,
which is:
 economical and
 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 .
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 Both LVEL and IMMERSOL make use of the distributions of:
 distance from the wall (WDIS) and
 distance between walls (WGAP), both of which are calculated by solving a
scalar equation for the
 LTLS variable.
 The stresses within the metals result primarily from the differences in
their thermalexpansion coefficients. namely:
 2.35 e5 for aluminium, and
 0.37 e5 for steel.
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(b) Vector and contour plots

Vectors
The vectors displayed in Fig.1 show
simultaneously:
 the nature of the airflow pattern (upper part of the domain), and
 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.
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The solids are supposed to be confined by a stiffwalled box, but are
allowed to slide relative to its walls. This is why the displacement vectors
are vertical near the confiningbox walls.
They are however not allowed to slide relative to each other; this is what
causes the concentrations of stress at their contact surfaces.
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 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 xdirection , and in
Fig.5 for the ydirection,
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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 stressstrain results.
There will now be shown some of the other variables which had to be
computed.
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 Temperature fields
Fig. 6
displays contours of temperature in the air and the solid, and reveals that:
 the air is heated by contact with:
 the 80degreeCelsius top wall, and
 the metal blocks, which have been receiving heat by radiation from the
top wall;
 temperature differences within the highconductivity solids are too
small to be discerned visually.
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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).
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The solid temperature influences the stresses and strains, of course,
primarily through the agency of the temperaturedependent thermalexpansion
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.
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 Radiationflux contours
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 postprocessing.
The results are displayed in
Fig. 9 for the
xdirection. and by
Fig. 10 for the
ydirection.
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.
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 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 easilyunderstood 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.
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 Contours of auxiliary quantities used in the fluidflow
calculation
The flow field was calculated by means of the LVEL turbulence model,
which makes use of the walldistance (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.
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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.e5 m**2/s, it is evident
that turbulence raises the effective value, far from the walls, by an order of
magnitude.
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(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.
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3. The mathematics of the method
(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.
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 The xdirection displacement, U, obeys the equation:
where:
 Te = local temperature measured above that of the unstressed
solid in the zerodisplacement condition, multiplied by the
thermalexpansion coefficient;
 D = [d/dx]* U + [d/dy]* V + (d/dz]* W
which is called the
"dilatation";
 Fx = external force per unit volume in xdirection;
 V and W = displacements in y and z directions;
 C1, C2 and C3 are functions of Young's modulus and
Poisson's ratio.
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 When the viscosity is uniform and the Reynolds number is low, so that
convection effects are negligible, the xdirection 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 xdirection,
 c1 = c2 = the reciprocal of the viscosity.
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Notes:
 The two equations are now set one below the other, so that they can be
easily compared:
 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
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 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)
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 A solution procedure designed for computing velocities will therefore in
fact compute the displacements if:
 the convection terms are set to zero within the solid region: and
 the linear relation between:
 D ( ie [d/dx]* U + ...) and
 p
is introduced by inclusion of a pressure and
temperaturedependent "masssource" term.
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(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
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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
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(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 massconservation constraint.
Its essential features are:
 All the velocity equations are solved first, with the current values of
p.
 The consequent errors in the massbalance 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.
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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 masssource 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:
 eliminating the convection terms (ie setting Re low); and
 making D linearly dependent on p and
temperatureT.
The "staggered grid" used as the default in PHOENICS proves to be extremely
convenient for soliddisplacement analysis; for the velocities and displacements
are stored at exactly the right places in relation to p.
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4. Details of the auxiliary models
(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 nonunity 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 conductiontype model to be used even with
nonparticipating 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 wavelengthdependent
radiation.
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(b) WGAP, WDIS and LTLS
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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.
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(c) LVEL: summary
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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 Reynoldsnumber range: laminar, transitional and
fully turbulent.
 LVEL can be easily extended so as to improve its accuracy in locations far
from walls.
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5. Discussion of current achievements and future prospects
(a) Preliminary conclusions
The following conclusions appear to be justified:
 Simultaneous simulation of solidstress, heat transfer and fluid flow is
indeed practicable and economical.
 As compared with the alternative, namely the use of distinct methods for
each phenomenon with iterative interaction between them, the
simultaneoussolution method is very attractive.
 It therefore seems possible that, when its attractiveness is fully
recognised, SFT (i.e. SolidFluidThermal) analysis may become as popular as
CFD.
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(b) Why has this recognition not already become widespread ?
The author
now proposes several answers to this question, as follows:
 First, it requires longheld convictions to be questioned and then
abandoned. Many people find this uncomfortable.
 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.
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 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.
 Lastly, the method described above has proved to have two deficiencies,
namely:
 the SIMPLE algorithm converges very slowly when both the flow and
solidstress situations are complex and intertwined; and
 additional 'source terms' must be provided in order that 'bending'
phenomena can be properly represented.
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(c) The future of SFT
CHAM's current work on SFT is therefore focussed
on the following:
 attaching the appropriate 'convergence accelerator' ( already
accomplished);
 including the 'bending' feature ( in progress);
 generalising the coding so that arbitrary numbers of materials, each with
different temperaturedependent properties, can be handled ( in
progress);
 providing copious practicallyinteresting demonstrations ( as
rapidly as possible).
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(d) Research opportunities
Singlealgorithm SFT presents numerous
opportunities, both to numerical analysts and to practicallyminded researchers,
for example, to:
 extend the algorithm to curvilinear grids and to cartesian grids cut by
curved solidfluid interfaces;
 extend to transient phenomena;
 allow deformations which are large enough to affect the flow;
 permit plastic as well as elastic deformation of the solid material; and
 allow the fluid to be multiphase.
An offer
CHAM will be happy to offer freeofcharge
PHOENICS licences to academic researchers willing to collaborate in this
important field.
END of LECTURE
References,
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6. References
 The differential equations governing displacements, stresses and strains
in elastic solids of nonuniform 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 numericalsolution procedures. However, this has been done by:
 JH Hattel and PN Hansen
A ControlVolumebased FiniteDifference
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 ThreeDimensional, Parabolic
Flows"
Int J Heat Mass Transfer, vol 15, p 1787, 1972
 The first use of the present method for solving the soliddisplacements
and fluidvelocity 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
RollsRoyce Ltd) on the Calculation of Thermal Stresses in Bodies of
Evolution
CHAM Ltd, November, 1990
 KM Bukhari, IS Hamill,HQ Qin and DB Spalding
StressAnalysis
Simulations in PHOENICS. CHAM Ltd, May, 1991
From that time onwards, the solidstress option was made
available as a (littleadvertised) option in successive issues of PHOENICS,
 Openliterature 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 GanLi 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 Fluidflow, Heattransfer and
Solidstress Computation in a Single Computer Code
Helsinki
University 4th International Colloquium on Process Simulation, Espoo,
1997
 DB Spalding
FluidStructure Interaction in the presence of Heat
Transfer and Chemical Reaction
ASME/JSME Joint Pressure Vessels and
Piping Conference, San Diego, 1998