advanced computational methods in heat transfer vi, c.a ... · convective and conductive heat...

Computational connective and conductive heat

transfer

B. Sunden

Division of Heat Transfer, Lund Institute of Technology, Lund, Sweden

Abstract

In this paper a brief description of numerical computation methods forconvective and conductive heat transfer problems is presented. In particular thefinite volume method (FVM) is considered. Various discretization schemes,methods for handling the pressure-velocity coupling, turbulence modeling,solution procedure etc. are described. A note on complex geometries is alsoincluded.

1 Introduction

Efficient, accurate and stable numerical methods for solving fluid flow problems,heat and mass transfer processes, chemical reactions, turbulent phenomena are ofgreat importance in many industrial applications. It is nowadays generallyrecognised that computer analysis of complex problems may provide a cost-effective, quick and sufficiently reliable method in many cases. Also sometimesthe computational methods may be an alternative or a complement toexperimental investigations. Despite the tremendous developments andachievements in methodologies, computer capacity and range of applicationsduring the last few decades, still research is needed on many topics, e.g., complexgeometries, turbulence modeling, two-phase flows etc.

Many industries and companies worldwide are using commercially availableso-called CFD-codes (CFD = Computational Fluid Dynamics) for simulation offlow and heat transfer topics in heat exchangers, enhancement of heat transfer,turbulent combustion in gas turbines and combustion engines, electronicscooling, gas turbine blade and combustor wall cooling etc. Among these codesare: FLUENT, CFX, STAR-CD, PHOENICS, CFD2000, FIDAP, ADINA and

Advanced Computational Methods in Heat Transfer VI, C.A. Brebbia & B. Sunden (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-818-X

4 Advanced Computational Methods in Heat Transfer VI

others. However, to successfully apply such codes and interpret the computedresults, it is necessary to understand the fundamental methods of suchcomputations. By computational heat transfer or numerical heat transfer is meantCFD for thermal problems.

This paper gives an introduction and overview of current numerical methodsand then a detailed description of the well-established so-called finite-volumemethod (FVM) follows.

2 Mathematical formulation

All the governing differential equations of mass conservation, momentum, energyand mass fraction of species can be cast into a general partial differentialequation as

dt

where (j) is the general dependent variable (e.g., the velocity components,

temperature etc.). F is the generalized diffusion coefficient, and S is the source

term for 0. The general differential equation consists of four terms. From the

left to the right in equation (1), they are called the unsteady term, the convectionterm, the diffusion term and the source term.

2.1 Turbulence modeling

Many flows of engineering significance are turbulent. The instantaneous massconservation, momentum and energy equations form a closed set of fiveequations with five unknowns u, v, w, p and t. However, the computingrequirements (in terms of resolution in space and time) for the direct solution ofthe time-dependent equations of fully turbulent flows at high Reynolds numbers(DNS calculations) are enormous and major developments in computer hardwareare needed. Meanwhile practising engineers need computational proceduressupplying information about the turbulent processes, but avoiding the need topredict effects of every eddy in the flow. This calls for information about thetime-averaged properties of the flow and temperature fields (e.g., meanvelocities, mean stresses, mean temperature etc.). In performing the time-averaging operation on the governing equations using the so-called Reynoldsdecomposition (every variable is written as the sum of a time-averaged value anda superimposed fluctuating value), one obtains six additional unknowns in themomentum equations and another three unknowns in the temperature fieldequation. The general variable 0 is written as

(2)


Advanced Computational Methods in Heat Transfer VI 5

The unknowns appearing in the differential equations have the form

<K (3)

where the prim means fluctuating quantities and the bar means time-averaged

value. In eq. (3), t' means the fluctuating temperature and 0 ' represents the

velocity components. Thus the complete term in eq. (3) is proportional to theturbulent heat fluxes.

The task of turbulence modeling is to provide procedures to predict theadditional unknowns (turbulent or Reynolds stresses, turbulent heat fluxes etc.)with sufficient accuracy and generality.

2.1.1 Types of modelsThe most common turbulence models may be classified as

• zero-equation models• one-equation models• two-equation models• Reynolds stress models• Algebraic stress models• Large Eddy Simulation (LES)

The five first models form the basis of turbulent calculations in commerciallyavailable CFD-codes. The LES is a model where the time-dependent flowequations are solved for the mean flow and the largest eddies while the effects ofthe smaller eddies are modeled. Recently the LES - model has become availablefor isothermal flow in some commercial codes.

The three first models in the list above account for the turbulent stresses andheat fluxes by introducing an eddy viscosity (turbulent viscosity) and an eddyconductivity (turbulent conductivity). The eddy viscosity is usually obtained fromcertain main parameters of the fluctuating motion itself. In two-equation models,these parameters are determined by solving two additional differential equations.

Here such a model, namely the k - £ model, is examplified. The turbulence is

characterized by the turbulent kinetic energy k of the fluctuating motion and itsdissipation rate £ . These variables are governed by the following differentialequations:

d pk d , d \ V + jilt dk } ^-JL.+ pku = \ * +G- (4)ot dXj * dX: 0,, dx:



d d \ // + /X, d£

In eqs. (4) and (5), /J, is the turbulent viscosity given by

M,=C,p^ (6)

G in eqs. (4) and (5), is the rate of generation and involves the turbulent

stresses and the mean rate of strain tensor. <J^,CTg,Cj, C^,C^ are empirical

constants.By comparing eqs. (4) and (5) with eq. (1) it is obvious that eqs. (4) and (5)

have the common general form. The constants and the exact form of thegeneration term may be found in refs. [1,2]. In these references also more generaland detailed information about various models can be found.

The models using the eddy viscosity and eddy diffusivity approach areisotropic in nature and cannot evaluate non-isotropic effects. Then higher ordermodels, like Reynolds stress models of various kind, are needed. A discussion ofsuch models are beyond the scope of this paper and the reader is referred toAmano [3].

2.1.2 Wall effectsThere are two standard procedures to account for wall effects in numericalcalculations of turbulent flow and heat transfer. One is to employ a low Reynoldsnumber modelling procedure, and the other is to apply the wall function method.In general the wall function method is efficient and requires less CPU time andmemory size but it becomes inaccurate at low Reynolds numbers. Theformulation given in eqs. (4) - (6) is essentially a high Reynolds numberformulation although the molecular viscosity has been added in the diffusiveterms. Thus this formulation has to be combined with the wall function method.Further details can be found in [1-3].

3 Methods for numerical solution of partial differential

equations

Nowadays there are some methods available to numerically solve the governingequations of fluid flow and heat transfer problems. These are: the finitedifference method (FDM), the finite volume method (FVM), the finite elementmethod (FEM), the control volume finite element method (CVFEM) and theboundary element method (BEM). A brief description will be given to all thesemethods and then the finite volume method will be considered more extensively.



3.1 FDM

The finite difference method is the oldest method and is also the easiest methodto apply for problems with simple geometries.

The computational domain is covered by a grid. Taylor series expansion orpolynomial fitting is used to approximate the derivatives of the variables withrespect to coordinates at each grid point. Algebraic equations are achieved ateach grid point. Further information is available in, e.g., [4].

3.2 FVM

In the finite volume method the domain is subdivided into a number of so-calledcontrol volumes. The integral form of the conservation equations are applied toeach control volume. At the center of the control volume a node point is placed.At this node the variables are located. The values of the variables at the faces ofthe control volumes are determined by interpolation. The evaluation of thesurface and volume integrals is carried out by quadrature formulas. Algebraicequations are obtained for each control volume. In these equations values of thevariables for neighbouring control volumes appear.

The FVM is very suitable for complex geometries and the method isconservative as long as surface integrals are the same for control volumes sharingboundary.

The FVM is a popular method particularly for convective flow and heattransfer. It is also applied in several commercial CFD-codes.

Further details can be found in [5,6].

3.3 FEM

The finite element method was when it appeared mainly applied to problems insolid mechanics. However, gradually it has been developed to a general tool forsolving partial differential equations. It has become popular also for fluid flowand heat transfer problems but it is not as popular as the FVM.

There are similarities between the FEM and the FVM. The domain is dividedinto finite elements (volumes). The distinguished feature of the FEM is that theconservation equations are multiplied by a weight function before the integrationover the domain. Within each element the numerical solution is approximated bya shape function (often linear) in such a way that continuity across the boundaryis guaranteed. The weight function is commonly of the same form.

The FEM has a good ability to handle complex geometries.Further details can be found in Fletcher [7], and in Reddy and Gartling [8].

3.4 CVFEM

The CVFEM is a hybrid method between the FEM and FVM methods. In a two-dimensional case the domain is divided into triangular elements. The nodes are



located at the vertices. Any variable is assumed to vary linearly within theelement.

The control volumes are formed around the nodes by joining the centroids ofthe elements and mid points on element edges. The conservation equations inintegral form are applied to the control volumes as outlined for the FVM.

Further details are given by, e.g., Masson et al. [9].

3.5 BEM

The boundary integral method basically transforms the governing equations toboundary integrals which are to be solved numerically. For heat conductionproblems it is well suited but becomes more complicated for convective flow andheat transfer.

More information is available in, e.g., Power and Wrobel [10].

4 The finite volume method for a two-dimensional case

Consider the general equation (1) in two dimensions. With a Cartesian coordinatesystem a rectangular grid looks like as in Fig. 1.

N

w -w

n

£_

Figure 1: Grid and a typical control volume.

The grid points are denoted by upper case letters while the control volumefaces appear with lower case letters.

Equation (1) is now integrated over the control volume. By also applyingthe mass conservation equation (0 = 1, F = 0, S = 0), the discretized form of

eq. (1) becomes:

+ b (7)

The coefficients ap, ... , a% depend on the chosen difference scheme tohandle the convection - diffusion terms. In general one may write:


max -


(8)

) (9)

(10)

(11)

and

a - (12)

(13)

(14)

The D:s, F:s and P:s in eqs. (8) - (11) are:

F, = (pw), Ay , F^ = (pw )w A} , Fn = (pv )n A^c, F, = ( pv )g A%

The function A(|P|) has different forms depending on the selected differencescheme. Table 1 presents some examples.

Table 1. The function A(|P|).

A

1 -

Max (0,

Max (0, (

(PI)

0.5|P|11 -0.5|P|)

1-0.1|P|5)

Difference scheme

CDS - central difference schemeUDS - upstream scheme

Hybrid schemePower law scheme

If higher - order accurate schemes like QUICK, van Leer etc. are used morecomplicated formulas then eqs. (8) -(14) appear, i.e., additional grid points showup and the expressions for the coefficients change accordingly.



4.1 Source term

The source term S may depend on the variable 0 . In the discretized equation it is

desirable to account for such a dependence. Commonly the source term is

expressed as a linear function of 0

At the grid point P, S is then written as

To not cause divergence it is generally requested that Sp is negative.The linearisation procedure above was used as the coefficients in eq. (7) were

described.

4.2 Solution of the discretized equations

The discretized equations have the form of eq. (7) with the 0 - values at the grid

points as unknowns. For boundaries not having fixed (f) - values, the boundary

values can be eliminated by using given or fixed conditions of the fluxes at suchboundaries. Gauss elimination is a so-called direct method to solve algebraicequations. For one-dimensional cases the coefficients form a tridiagonal matrixand an efficient algorithm called the Thomas algorithm or the tri-diagonal matrixalgorithm (TDMA) is achieved. For two-dimensional and three-dimensionalcases, direct methods require large computer memory and computer time.Iterative methods are therefore used to solve the algebraic equations. A popularmethod is a line-by-line technique combined with a block correction procedure.The equations along the chosen line are solved by the TDMA. Iterative methodsare also needed because the equations are non-linear and sometimes interlinked.

In many situations, e.g., turbulent forced convection, the change in the valueof 0 from one iteration to another is so high that convergence in the iterative

process is not achieved. To circumvent this and to reduce the magnitude of thechanges, underrelaxation factors (between 0 and 1) are introduced.

4.3 The pressure in the momentum equations

In the momentum equations, a pressure gradient term appears in each coordinatedirection (i.e., a source term S). If these gradients were known, the discretizedequations for the flow velocities would follow the same procedure as for anyscalar. However, in general the pressure gradients are not known but have to befound as part of the solution. Thus the pressure and velocity fields are coupledand the continuity equation (mass conservation equation) has to be used todevelop a strategy.

There are also other related difficulties in solving the momentum andcontinuity equations. It has been shown that if the velocity components and the



pressure are calculated at the same grid points in a straightforward way, somephysically unrealistic fields, like checker-board solutions, may arise in thenumerical solution. A remedy to this problem is to use staggered grids. Thevelocity components are then given staggered or displaced locations. Theselocations are such that they lie on the control volume faces that are perpendicularto them. All other variables are calculated at the ordinary grid points. Anotherremedy is to use a non-staggered or collocated grid where all variables are storedat the ordinary grid points. A special interpolation scheme is then applied tocalculate the velocities at the control volume faces. Most commonly the so-calledRhie-Chow interpolation methods is applied, see [11].

4.3.1 Solution procedure of the momentum equationsAs was mentioned in the preceding section, the velocity and pressure fields arecoupled. Thus a strategy has to be developed in the solution procedure of themomentum equations. The oldest algorithm is the SIMPLE (semi-implicit-method-pressure-linked-equations) algorithm, see [12]. A pressure field p* isguessed and then the momentum equations are solved for this pressure field

resulting in a velocity field u- . Then a pressure correction p' and velocity

corrections u. are introduced. From the continuity equation an algebraic

equation for p' can be obtained. The velocity corrections are related to thepressure corrections by

u|. =d|Ap \ (15)

As the solution proceeds the velocities and pressure are corrected accordingto

p = p* + p' (16)

Ui = u* +u- (17)

Then the momentum equations are solved again but now with the correctedpressure (16) as the guessed pressure. New velocities are obtained and newpressure and velocity corrections are calculated. The whole process is repeateduntil convergence is obtained.

There are other similar algorithms available today. SIMPLEC (SIMPLE-consistent) and SIMPLEX (SIMPLE-extended) are common. They differ fromSIMPLE mainly in the expression for dj in eq. (15), see Anderson et al. [13].

Another algorithm named PISO (pressure implicit splitting operators), seeIssa [14], has become popular recently. Originally it is a pressure-velocitycoupling strategy for unsteady compressible flow. Compared to SIMPLE itinvolves one predictor step and two corrector steps.



Still another algorithm is SIMPLER (SIMPLE - revised). Here the continuityequation is used to derive a discretized equation for the pressure. The pressurecorrection is then only used to update the velocities through the velocitycorrections.

4.4 Convergence

The solution procedure is in general iterative and then some criterion must beused to decide when a converged solution has been reached. One method is tocalculate residuals R as

(18)NB

for all variables. NB means neighboring grid points, e.g., E, W, N, S. If thesolution is converged, R = 0 everywhere. Practically, it is often stated that thelargest value of the residuals [R] should be less than a certain number. If this isachieved the solution is said to be converged.

4.5 Number of grid points and control volumes

The widths of the control volumes do not need to be constant nor do thesuccessive grid points have to be equally spaced. Often it is desirable to have auniform grid spacing. Also it is required that a fine grid is employed where steepgradients appear while a coarser grid spacing may suffice where slow variationsoccur. The various turbulence models require certain conditions on the gridstructure close to solid walls. The so-called high and low Reynolds numberversions of these models demand different conditions.

In general it is recommended that the solution procedure is carried out onseveral grids with different fineness and varying degrees of non-uniformity. Thenit might be possible to estimate the accuracy of the numerical solution procedure.

5 Complex geometries

CFD - methods based on Cartesian or cylindrical coordinate systems havelimitations in complex or irregular geometries. Using Cartesian and/or cylindricalcoordinates means that the boundary surfaces are treated in a stepwise manner.To overcome this problem methods based on body-fitted or non-orthogonal gridsystems are needed. Such grid systems may be unstructured, structured or block-structured. Since the grid lines follow the boundaries, boundary conditions canmore easily be implemented.

There are also some disadvantages with non-orthogonal grids. Thetransformed equations contain more terms and the grid non-orthogonality maycause unphysical solutions. The arrangement of the variables on the grid affectsthe efficiency and accuracy of the solution algorithm.



Grid generation is an important issue and today most commercial CFD-packages have their own grid generators but also several grid generationpackages, compatible with some CFD - codes, are available.

The interaction with various CAD (computer-aided-design) - packages is alsoan important issue today.

Further information on treating complex geometries can be found in refs. [15-16].

6 Results

In this section some examples of recent results of computational heat transfer arepresented.

The first case, see Fig. 2, shows local heat transfer coefficients on a ribbedwall and the corresponding flow field for a rib-roughened rectangular duct with a

large aspect ratio.The flow is turbulent but periodic in the main flow direction. A low Reynolds

number turbulence model was used. The local variation in the heat transfercoefficient is directly related to the flow field. It is obvious that the flow patternis very important for the heat transfer. The recirculation zones downstream andupstream of a rib are clearly visible. The reattachment region between twoadjacent ribs is also evident. Knowledge of the local heat transfer coefficient isessential as the probability of occurrence of so-called hot spots on the wall isinvestigated. Some further details of this investigation are provided in [17, 18]. Ithas been found that for a small ratio between the rib height and duct hydraulicdiameter the friction factor and the average Nusselt number are predicted inreasonable agreement with experimental data. For large such ratios the predictiveability is worse. This finding agrees with that in [19].

Figure 3 shows numerical and experimental non-dimensional surfacetemperature distributions on a triangular iron fin. This problem constitutes acoupled conduction-convection problem, where heat conduction takes place inthe solid fin material and convective cooling (or heating) occurs in the boundarylayers around the fin surface. The reference velocities given in Fig. 3 are theapproach velocities of the fluid. Both laminar and turbulent cases wereinvestigated. For the turbulence modeling, a one-equation model was applied.The agreement between experiments and numerical simulations is satisfactory.Further details can be found in [20].

Figure 4 depicts computed Nusselt numbers (compared to those of a straightduct) for helical square ducts in laminar flow. The Reynolds number based onmean velocity and duct hydraulic diameter is used on the horizontal axis becausethe Dean number (proper for curved ducts) is not defined for a straight duct. Theeffects of curvature (e, dimensionless with respect to duct hydraulic diameter)and torsion (r|, dimensionless with respect to duct hydraulic diameter) as well asReynolds and Prandtl numbers are provided. The geometry sketches show theduct layout for various curvature and torsion values. The figures show that theNusselt number is only slightly affected by the torsion as long as it is small.



However, for a high Prandtl number fluid, even a small torsion has a significanteffect. As the torsion becomes large the Nusselt number approaches that of astraight duct. Also noticeable in Fig. 4 is that for a straight twisted duct (e = 0, r)= 1), the Nusselt number is insensitive to variations in Reynolds and Prandtlnumbers. Further details can be found in [21].

NIL,

350

300 -

250 -

200 -

150

100 -

50 -

0. b . c

Figure 2: Local heat transfer coefficient (Nusselt number) distribution on aribbed wall and the associated flow pattern. Re ~ 20200, rib height to hydraulic

diameter ~ 0.056.



0,8

0,6

0,4

0,2 I

Q 8 Uref=10m/s laminar0 O UreW20 m/s laminar7 7 Uref=20m/s turbulent* * Uref-38m/s turbulent

Figure 3: Dimensionless surface temperature distributions for an iron fin. Curvesare calculated, symbols are measured values.

10 100 500 10Re

100 500Re

Figure 4: Computed average Nusselt numbers for helical square ducts.



7 Concluding remarks

This paper gave a brief introduction to computational conductive and convectiveheat transfer. Most of the key issues were introduced and references to modernliterature was provided. Although computational heat transfer has reached acertain level and can be significantly helpful in many engineering and industrialapplications, still comprehensive research is needed in, e.g., turbulence modeling,handling complex geometries, modeling two-phase convective flow and heattransfer etc.

To illustrate the success of computational heat transfer, results of a fewresearch projects were utilized.

Acknowledgement

For several years financial support for projects on development and applicationof numerical modeling and simulation of convective and conductive heat transferhas been received from the former National Swedish Board for Industrial andTechnical Development (NUTEK) and the National Swedish Energy Authority(STEM). This support is kindly acknowledged.

References

[1] Launder, B.E. On the computation of convective heat transfer in complexturbulent flows. ASMEJ. Heat Transfer, Vol. 110, pp. 1112-1128, 1988.[2] Wilcox, D.C. Turbulence Modeling for CFD, DCW Industries Inc., LaCanada, USA, 1993.

[3] Amano, R.S. Turbulent heat transfer in a corrugated wall channel (Chapter 5).Computer Simulations in Compact Heat Exchangers, eds. B. Sunden and M.Faghri, Computational Mechanics Publications, UK, 1998.

[4] Smith, G.D. Numerical solution of partial differential equations, OxfordUniversity Press, UK, 1978.

[5] Patankar, S.V. Numerical heat transfer and fluid flow, Me Graw-Hill BookCompany, 1980.

[6] Versteeg, H. and Malalasekera, W. An Introduction to Computational FluidDynamics - the Finite Volume Method, Longman Sc. & Tech., 1995.

[7] Fletcher, C.A.J. Computational techniques for fluid dynamics, Vol. I,Springer, Berlin, 1991.

[8] Reddy, J.N. and Gartling, O.K. The finite element method in heat transferand fluid dynamics, CRC Press, Boca Raton, FL, 1994.

[9] Masson, C, Saabas, H.J. and Baliga, R.B. Co-located equal-order control-volume finite element method for two-dimensional axisymmetric incompressibleflow. Int. J. Numer. Meth. Fluids, Vol. 18, pp. 1-26, 1994.



[10] Power, H. and Wrobel, L. Boundary integral methods in fluid mechanics,WIT Press (Computational Mechanics Publications), UK, 1995.

[11] Rhie, C.M. and Chow, W.L. Numerical study of the turbulent flow past anairfoil with trailing edge separation. AIAA J., Vol. 21, pp. 1525-1532, 1983.

[12] Patankar, S.V. and Spalding, D.B. A calculation procedure for heat, massand momentum transfer in three-dimensional parabolic flows. Int. J. Heat MassTransfer, Vol. 15, pp. 1787-11805, 1972.

[13] Anderson, D.A., Tannehill, J.C. and Pletcher, R.H. Computational fluidmechanics and heat transfer, Hemisphere Publ. Corp., 1984.

[14] Issa, R.I. Solution of the implicity discretised fluid flow equations byoperator-splitting. /. Comput. Phys., Vol. 62, pp. 40-65, 1986.

[15] Farhanieh, B., Davidson, L. and Sunden, B. Employment of second-momentclosure for calculation of turbulent recirculating flows in complex geometrieswith colocated variable arrangement. Int. J. Num. Meth. Fluids, Vol. 16, pp.525-544, 1993.

[16] Ferziger, J.H. and Peric, M. Computational methods for fluid dynamics,Springer-Verlag, 1996.

[17] Abdon, A. and Sunden, B. Investigation of a turbulence model for wallcooling of combustion chambers. ASME paper 98-GT-540, 1998.

[18] Sunden, B., Enhancement of convective heat transfer in rib-roughenedrectangular ducts. /. Enhanced Heat Transfer, Vol. 6, pp. 89-103, 1999.

[19] Saidi, A. and Sunden, B., Calculation of convective heat transfer in square-sectioned gas turbine blade cooling channels. ASME Paper 98-GT-204, 1998.

[20] Sunden, B., Kardos, K. and Torkelsson, S., Mechanistic investigation of theperformance of a triangular fin (Chapter 3). Recent Advances in Analysis of HeatTransfer for Fin Type Surfaces, eds,. B. Sunden and P.J. Heggs, WIT Press,Southampton, UK, 1999.

[21] Bolinder, J. and Sunden, B. Numerical prediction of laminar flow and forcedconvective heat transfer in a helical square duct with a finite pitch. Int. J. HeatMass Transfer, Vol. 39, No 15, pp. 3101-3115, 1996.


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