abstract - wit press...rections (navier-stokes equations), the energy conservation equation...

Utilization of the Algor package in solution of

the inverse heat convection problems

Ireneusz Szczygiel

Institute of Thermal Technology

Technical University of Silesia

44-101 Gliwice, Konarskiego 22, Poland

Abstract

This paper presents the method of application of the commercial

package Algor in the inverse heat convection problems solutions. Due

to the Algor functions it has to be supplemented by additional mod-

ules which allows us to solve full inverse problem. Constructed hy-

brid program make it possible to solve various formulations of inverse

problems. Two of them are dealt with here. These are evaluatingboundary heat flux from the internal temperature measurements (in-

verse formulation of energy equation), and estimation of inlet veloc-

ity profile from internal velocity measurements (inverse formulation

of momentum equation).

1 Introduction

Inverse problems are formally defined as an estimation of the bound-ary quantities from the measurements at interior locations of bodyunder consideration. Due to the nature of such a problems they areoften ill posed and require special solution techniques. Inverse prob-lems have been dynamically developed in last twenty years. Solvinginverse problems are much more difficult then direct ones. One of thereasons is that they are ill posed, which manifests itself by the sen-sitivity of solutions to the input data errors, thus a little inaccuracyof data (which usually come from measurements) can result in large

Transactions on Engineering Sciences vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-3533

426 Advanced Computational Methods in Heat Transfer

racies of results. There is a general opinion that getting results fromaccurate input data does not allow us to conclude, that the employedmethod is appropriate for problem under consideration. The finalcriterion of the robustness of the method should be the sensitivityto the data errors. There is also one important difficulty in solvinginverse problems - weakening of signal and information noise. If welocate measurement sensors far from the boundary of our interest wewill get unreasonable results. Thus, proper location of sensors shouldbe a result of an introductory analysis. In heat transfer researche,the majority of work on inverse problems is devoted to heat conduc-tion. This problem has been thoroughly discussed, including solutionmethods as well as parameters values which guarantee the stability ofthe solution. However, the inverse heat convection problems are lessfamiliar. To the author's knowledge no algorithm for a solution offull inverse heat convection problem, i.e. containing both energy andmomentum equations has been published yet. In this paper the trialalgorithm of such a problem is considered. The first formulation ofinverse problem on convection heat transfer was introduced by Kellerand Cebbeci [1]. The inverse problem studied by them was concernedwith determination of the spatial free stream velocity variation for aprescribed local skin friction coefficient or a prescribed displacementthickness in compressible boundary layers flow. In this study how-ever, the calculated local skin friction coefficient and displacementthickness from the solution of direct problem were utilized as an in-put to the inverse problem, so the stability of the inverse problemwas not investigated. In other papers devoted to inverse problems ofheat convection, authors assumed both velocity and pressure fieldswere known and only the inverse formulation of energy equation wastaken under scrutiny. Examples of such approaches can be found inthe works of Huang and Ozisik [2] or Moutsoglou [3]. The problemconsidered herein is the estimation of boundary quantities (boundaryvelocity, boundary heat flux or temperature) from the temperatureand velocity measurements at interior locations of a fluid. The sen-sitivity of solution to the measurements errors is also investigated.The problems studied in this paper are characterized by:

• two dimensional, Cartesian geometry,

* flow described by the Navier Stokes equation,

# incompressible, isoparametric flow,

* laminar, steady state flow.


Advanced Computational Methods in Heat Transfer 427

Most inverse problem solutions can be divided into two subprocesses:

• solution of the direct problem formulation,

• stabilization of results.

The both steps are presented in the paper.

2 Direct heat convection problem

Convective heat transfer is described by the set of partial differentialequations, which consist of the momentum equation for each of di-rections (Navier-Stokes equations), the energy conservation equation(Fourier- Kirchoff equation) as well as of the continuity equation.In particular, mathematical description of convective heat transfer(for the assumptions described in the introduction) can be written asfollows:

• x momentum equation

dx dy ^ p dx

+v— ( —^ ) +v— ( —^ ) + Sx (1)ox \ dx J dy \ dy /

• y momentum equation

dx dy ^ pdyd f duy\ d /duy\

4-Z/TT- I ~ r ; I ~h Z/TT" I "% 1 + Sy (2)dx \ dx J dy \ dy J *

• continuity equation

dux duy~x *" ~ ~~ = 0 (3)ox dy

• energy conservation equation

, d (dT\ , d (dT\= *^-hr +*5- «~ox \ ox ) dy \oy J



u=0

Figure 1: Boundary conditions for fluid flow problem

In these equations p stands for the mass density of the fluid, p denotes

pressure, v stands for kinematic viscosity coefficient, u% and Uy are

the components of velocity vector in the x and y directions, respec-

tively, Sx and Sy stand for the source terms, T means temperature,

c specific heat, k is an heat conductivity coefficient and q^ stands

for the internal heat sources generation rate. This set of equation

should be also supplemented by the boundary conditions (as well as

by the initial conditions in transient problems). The proper selec-

tion of boundary conditions is essential problem, because of the fact,

that convergence of method depends on it. In this paper, bound-

ary conditions for momentum equation were assumed as shown in

the figure 1. Determination of the velocity and pressure fields is the

most time consuming part of solution of the full inverse problem.

This procedure is realized in several nested loops which makes the

computations very expensive. There are available several commercial

numerical packages, which allow to solve fluid flow problems. Oneof them is Algor, which is able to calculate velocity arid pressure

fields with high speed and accuracy. It was optimized with respect

to calculations time. After defining geometry, boundary conditions

and fluid physical parameters (i.e. viscosity, density etc) it allows to

determine velocity and pressure field. The computational procedure

can be invoked both from graphical user interface and from external

shell which makes possible application in so-called hybrid program.Unfortunatelly, Algor does not allow to solve energy equation (eq. 4).

This type of equation appears both in direct heat convection prob-

lem formulation (determination of temperature field) and in inverseformulation (determination of sensitivity coefficients). Due to this,self writeen module was inserted to the hybrid program.

For the solution of the energy type equation the Control Volume

formulation of Finite Element Method was used. Due to the natureof the problem some difficulties appear during mathematical model-

ing. One of them results from the non symmetric character of flow

- information is shifted in the flow direction, what can cause loss of



convergence of method. One way to avoid this unpleasant propriety

of fluid flow equations is to use curved interpolation functions. The

size of curvation should depend on the local Reynolds number in the

finite element. Such a technique is called up-winding.

After CVFEM discretization equation (4) can be written as fol-

lows:

CT = 0 (5)

Equation (5) constitutes a linear system of equation which can be

i easily solved. Full description of CVFEM method can be found in

[7]-The whole procedure of direct problem solution can be realized

in following steps:

1. solution of velocity and pressure fields (Algor module)

2. solution of temperature field (CVFEM module)

3. optional jump to 1. with corrected temperature dependent

physical fluid parameters.

3 Inverse heat convection problem

When the heat convection is considered, there are a lot of possible

formulations of inverse problem. In this paper two of them are stud-

ied:

• Estimation of the boundary velocity values from the velocity

measurements inside the region,

• Estimation of the boundary heat flux from the temperature

measurements inside the fluid.

For simplicity lets name the first case inverse momentum equation

formulation and the second one - inverse energy equation formulation.Because of the fact, that inverse momentum equation formulation is

much more difficult than inverse formulation of energy equation, the

estimation of heat flux is discussed first.There a lot of methods of solution of inverse problems. In the

studies presented herein combination of sensitivity coefficients method

and regularization method was used. In the inverse formulation of

momentum equation however, the fundamental reconstruction of the

sensitivity coefficients method was necessary.



3.1 Inverse formulation of the energy equation

As it was mentioned before, sensitivity coefficients method combined

with the regularization method was used for inverse energy equation

solution. The equation describing the energy balance in convective

heat transfer is linear one (in order to temperature) so it was possible

to employ sensitivity coefficients method in its pure form.

Sensitivity coefficients are defined as first derivative of tempera-

ture with respect to the boundary heat flux:

In the linear cases sensitivity coefficient is not a function of calcu-

lated variable, i.e. in this case it doesn't depend on . It also has

one positive propriety - it shows the regions which are the best for

measurements sensors locations. When the Z coefficient is relatively

large, it means high sensitivity of interior temperature to the fluxes

changes. On the other hand, when the values of the Z coefficient are

relatively small, our chances for reasonable results are rather poor.

Exemplary distribution of sensitivity coefficients for the laminar flowin the parallel channel is shown in the Figure 2.

The sensitivity coefficients can be derived from the equation of the

same structure like the problem governing equation. Let's notice, that

after differentiating energy equation (4) with respect to the boundary

heat flux , one can arrive at the equation:

»<„,*+ «*+ =*

Because of the same structure of equations (4) and (7) the same

computer code can be used for solution.

In the next step, temperature should be written as a functionof Z. Utilizing the fact, that temperature is a continuum function

of boundary heat flux, one can expand temperature field around an

arbitrary, but known value of q* into the Taylor series:

+ ... (8)

9=9*



Figure 2: Exemplary values of sensitivity coefficients across parallel

channel

For the linear problems higher order terms are equal to zero. After

utilizing definition of sensitivity coefficients (6) equation (8) can be

written as:

3This equation allows us to estimate j values if we know Ti values.

In order to achieve better stability of the solution it's better to

employ greater number of measured temperatures than estimated

heat fluxes. In such a case the least squares method should be em-

ployed to the solution of problem. In other words, the minimum of

the following expression should be determined:

KA = V^ \Yk — T(xk u q)] (10)

where K stands for the number of measurements and Y& means values

of temperature measurements.After discretization, equation (10) can be rewritten in matrix

form:Z*ZQ = Z*(Y - T*) + Z*ZQ* (11)

where Z stands for the matrix of sensitivity coefficients, superscript t

means transposition, Q - the vector of estimated boundary fluxes, Y



exemplary location ofmeasurement points

Figure 3: Definition of inverse problem

- the vector of measured temperatures, T* and Q* mean the vectors

of arbitrary, but known values of temperature and heat flux, respec-

tively. In many cases expression (11) is still posed, so special tech-

niques of solution should be employed. One of them is regularization

method. In principles, regularization method consist of determining

the minimum of functional:

L(f) = \\Af-vs\$ + i\\f\& (12)

where 7 > 0 stands for the regularization parameter, V and F meansmathematical spaces, and

Af = v (13)

is the mathematical description of ill posed problem, v$ are the in-

put data containing errors. There are several methods of choosingregularization parameter. They are precisely described in (Beck et

al.,1985) in detail.After employing regularization method equation (11) arrives at:

Z*(Y - T*) 4- Z ZQ* (14)

where I stands for the identity matrix. Above expression allows to

determine unknown vector Q.

3.2 Inverse formulation of the momentum equation

The inverse problem of the fluid flow problem is defined as an esti-

mation of boundary velocity from the internal velocity measurements

(see Fig. 3)The set of equations which describes the fluid flow problem (l)-=-(3)

is nonlinear. Due to this nonlinearity sensitivity coefficients method



can not be employed to the problem straightway. In other words, the

set of equations should be at first linearized somehow.

Let us assume, the the -component of boundary velocity is of

our interest and the /-component is equal to zero. So

%% = .&(%zo) (15)

and

where subscript xQ stands for the boundary value.

Interior velocity is a function of boundary velocity (15), so it can

be expanded around arbitrary, but known values of u^:

dux

The above expression is not exact because of nonlinearity of function

(15). In that we introduced two velocity fields: desired u(ux,Uy) and

auxiliary u*(u*,it*) which also represents starting values for the iter-

ative procedure. If the iterative procedure is convergent, the velocity

u* tends to the desired u.

After defining sensitivity coefficients

(18)

expression (17) can be written as:

%% = ?4 + Z* (tw - <o) (19)

or

(20)

Introducing (20) to the Navier Stokes equation one can get

p dx V 3*2 "

where v stands for the kinematic viscosity coefficient.

Assuming, that the iterative procedure is convergent we may omit

certain terms in eq. (21) which yields:

+ Uy—^ = — -- + v\ —f + —j- 22" dy pdx * ^



This equation, after supplementation by similar one for the y di-

rection creates system of equations which allows to obtain u field

distribution from distribution of u* distribution. The ommited terms

in equation (22) can be written in the following form:

4- 7/ - 4- A?/ - 4-^ ^ 7 / 0 I *-*"Z o Ir\ I * 7/ r\ I *—»"*Z r\ ' '— "y ^az ^ ay ax oy

= yf ^ r + # l (23)

where underline term tends to zero when iterative procedure is con-

vergent (i.e. An^o — > 0). After neglecting this term the residual of

equation (23) can be expressed as:

Since u^ /(UXQ) and u* /(w o) we can differentiate eq. (24) with

respect to UXQ which yields:

* z c %4- -— = --- + --- (25)

This equation, which allows us to determine sensitivity coefficients

field, has the same structure as the energy equation, so the samecomputer code can be used for its solution. Finally, there is the

following set of equations to be solved:

(26)j-~ — —dy \ ox*

o (27)

1 dp%-%— = --•%- +^ ay pax

(29)ux uy p uy \ ux~ uy~ i

and

— I - = 0 (30)



The problem is fully linearized now.

In order to increase the stability of solution the number of measure-

ment sensors might be greater then the number of estimated veloc-

ities, and the regularization method can be employed as well. Thus

equation (27) after discretization can written in matrix notation inthe following way:

Z^ZxUxo = Z%(Y - U;) + Z ZxU o (31)

and after employing regularization method:

= Z%(Y - U%) + Z%ZxU;o (32)

where I stands for the identity matrix. The full iterative processconsist of the following steps:

1. Assumption of fields Ux and Uy\

2. Solution of set of equation in order to calculate U* and U*

vectors;

3. Evaluation of sensitivity coefficients matrix;

4. Calculation of unknown boundary velocities;

5. Calculation of remaining internal velocities;

6. Back to 2. if convergence is not achieved.

Steps 1 and 2 are realized in Algor package, as well as 3,4 and 5 are

terminated in own CVFEM module. The block diagram of the whole

computational procedure is shown in the figure 4.

4 Numerical examples

In order to estimate accuracy of the described techniques we have

solved several examples. As a measurement values the results of di-

rect problem solution disturbed by a random error were employed.Such a technique is called numerical experiment. The inverse formu-lation of energy equation in parallel channel (see Fig. 6) has been

examined. On the upper wall we have assumed linear distributionof heat flux while bottom boundary was isolated. Results of calcu-

lations are shown in figure 5, where solid lines denotes the accurate



Assumption of initialvelocities

Solution of directfluid flow problem

Algor

Evaluation of sensitivitycoefficients matrix

Estimation of unknownboundary velocities

CVFEM

Fortran module

Convergence testpassed

STOP

Figure 4: The block diagram of computational procedure

value of heat flux, and stars denote results of inverse procedure. Inboth figures measured values (numerical experiment) were disturbed

by maximum random error 5%. Figure (a) present result with the

least square method employed, while figure (b) shows results withadditional regularization. Momentum equation has been solved for

the same geometric arrangement. To obtain results we have used

numerical experiment technique assuming irregular ux inlet velocity

component profile. Results are shown in figure 6. Similarly to the

previous case solid lines denotes real velocity profile and stars stand

for inverse problem solution. After disturbing input error by a ran-

dom error (maximum value 10%) we have calculated inlet velocity

profile from the inverse formulation with and without regularizationmethod. Results are shown in the figure 7.

5 Conclusions

The combination of sensitivity coefficients and regularization method

was used in solution of inverse formulation of momentum and energy

equations. Momentum equation requires special techniques of solu-

tion when the sensitivity method is to be employed while this method



.2 ooo

Figure 5: Geometry of inverse problem

top wall of channel top wall of channel

Figure 6: Results for inverse formulation of energy equation without

(left) and with (right) regularization

can be introduced to the inverse energy equation formulation in itspure form. As it can be noticed in the figures, regularization method

is strongly recommended in inverse energy equation. The influence of

regularization in inverse momentum equation formulation is of minor

importance. Utilization of the Algor package decreased considerablytime of computations.

References

[1] Cebbeci T. et al.: Turbulent Boundary Layers With AssignedWall Shear, Computers and Fluids, vol. 3, 1975 37-49

[2] Huang C.A., Ozisik M.N.: Inverse Problem of Determining Un-

known Wall Heat Flux In Laminar Flow Through a Parallel Plate


* ,mnputalion2i

Figure 7: Results tor inverse formulation of momentum equation with

(left) and without (left) and with (right) regularizatton

Duct. Numerical Heat Transfer, vol. 21, 192. G5-70

|3l Moutsoglou A.: Inverse Convection Problem, ASME J. HeatTransfer vol. Ill, 1989, 37-43

14] Patankar S.V. Numerical Heat Transfer and Fluid Flow. MeGrill-Hill Co. New York 1980

[5] Baliga B.R. et al.: Solution of Some Two Dimensional Incorn-pressible Fluid Flow and Heat Transfer Problems, Using a Con-

trol Volume Finite Element Method, Numerical Heat Transfer,

vol.6, 1983, 263-282

[6] Beck J.V. et al.: Inverse Heat Conduction: 111 Posed Problems,

Wiley Intersc., New York 1985

[7] Minkowycz W.J et al.: Handbook of Numerical Heat Transfer,

J. Willey and Sons, New York 1988


abstract - wit press...rections (navier-stokes equations), the energy conservation equation...

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