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Progress in Computational Fluid Dynamics, Vol. 9, Nos. 3/4/5, 2009 231

Copyright © 2009 Inderscience Enterprises Ltd.

An inverse analysis for the estimation of boundary heat flux in a circular pipe employing the proper orthogonal decomposition

Peng Ding and Wen-Quan Tao* State Key Laboratory of Multiphase Flow and Heat Transfer, School of Energy and Power Engineering, Xi’an Jiaotong University, 28 Xian Ning Road, Xi’an Shaanxi, 710049, PR China Fax: 0086-29-82669106 E-mail: [email protected] E-mail: [email protected] *Corresponding author

Abstract: An inverse algorithm based on the proper orthogonal decomposition technique is developed in this paper. The performance of the present inverse algorithm is examined by an inverse forced convection problem of identifying the unknown space-dependent heat flux at the outer boundary of a circular pipe. The inverse problem is resolved in a functional optimisation way by the Fletcher-Reeves conjugate gradient method. The effects of the convection, the location of the thermocouples and the measurement error on the performance of the inverse algorithm are studied thoroughly. It was shown that the present POD-based inverse algorithm is very accurate as well as efficient. A large time saving is also found by using the present inverse algorithm.

Keywords: proper orthogonal decomposition; reduced order model; inverse convection problem; functional optimisation.

Reference to this paper should be made as follows: Ding, P. and Tao, W-Q. (2009) ‘An inverse analysis for the estimation of boundary heat flux in a circular pipe employing the proper orthogonal decomposition’, Progress in Computational Fluid Dynamics, Vol. 9, Nos. 3/4/5, pp.231–246.

Biographical notes: Peng Ding graduated from the Xi’an University of Architecture and Technology, in 2005, for his Master Degree and continued his study at Xi’an Jiaotong University on the CFD and NHT. Now, he works on the application of the proper orthogonal decomposition in the fluid flow and heat transfer area.

Wen-Quan Tao is a Professor of Power and Energy Engineering School at Xi’an Jiaotong University of China. His research interests include enhanced heat transfer, numerical heat transfer and micro scale heat transfer, and solar energy: science and engineering. Currently, he is the member of Chinese Academy of Science, Associate Editor of International Journal of Heat and Mass Transfer and International Communications in Heat and Mass Transfer and Vice Chairman of the Chinese Engineering Thermophysics Association.

1 Introduction

Inverse analysis is very valuable when the direct measurements of data are impossible or the measuring process is very expensive. For example, the determination of heat transfer coefficients and the heat loads acting on the outer surface of re-entry vehicle, the estimation of unknown thermophysical properties of unknown materials, the prediction of the glass ribbon temperature in the float glass process, the determination of contact resistance and the damage detection technique in the structure fields and so on.

It is well known that the solution of inverse problems is more difficult than direct problems due to their ill-posed nature, i.e., small random error in the measurement data may deteriorate the solution significantly (Hadamard, 1923). A lot of studies have been conducted over the last few decades to improve the stability of the inverse algorithm. These techniques fall into two main categories, namely, the function specification method developed by Beck and Blackwell (1985) and the iterative regularisation method (also referred to as conjugate gradient method) pioneered by Alifanov (1994) where the regularisation is inherently built

232 P. Ding and W-Q. Tao

in the iterative procedure. The success of Beck and Blackwell (1985) method depends greatly on the choice of the future time parameter, whereas the iterative regularisation method of Alifanov (1994) appears to be one of the most efficient and universal approaches for the construction of stable algorithms for solving inverse problems.

Inverse Heat Conduction Problems (IHCP) have been studied by many scientists over the past decades of years. A good review of the IHCP can be found in the book of Alifanov (1994). Huang et al. (1998) used a boundary-element-method-based inverse algorithm, utilising the iterative regularisation method to solve the IHCP of estimating the unknown transient boundary heat flux in a multi-dimensional domain with arbitrary geometry. An IHCP was solved using Alifanov’s iterative regularisation method to estimate the time-varying heat transfer coefficient of forced convection flow boiling over the outer surface of a heated tube by Su and Hewitt (2004). The estimation was based on the transient temperature measurements taken by a thermocouple on the inner surface of the circular tube on which the flow boiling occurs. Effects of the time scales of the heat transfer coefficient variation, the measurement error and the data acquisition rates were investigated. Loulou and Scott (2006) used heat flux measurement rather than temperature measurement in the object function to estimate the time-dependent blood perfusion and the thermal conductance between the probe and the tissue. The minimisation procedure was achieved by using conjugate gradient method and adjoint equations. In the work of Huang and Tsai (2000), the local time dependence of surface heat transfer coefficients for plate finned-tube heat exchangers were estimated in a three-dimensional IHCP and the code developed has the ability to communicate with the commercial CFD code CFX4.4 by means of data transportation.

Deng et al. (2005) dealt with the estimation of the heat flux distribution generated by a flame gun base on the temperature measurements in a cylindrical workpiece. A first calculation was performed by solving the inverse heat problem using the conjugate gradient method. Then, the results were used in a second calculation implementing an artificial neural network structure to set-up correlations between the temperature of the workpiece and the heat flux generated by the flame gun. Girault and Petit (2005) proposed a method for solving non-linear IHCP using reduced order model. The ROM was identified through a specific procedure based on the numerical optimisation technique to minimise the discrepancy between the responses of the CFD model and ROM when a specific input signal was used. Then, the ROM was used to solve the inverse problem through a function specification method.

The studies in the inverse convection heat transfer problems are more recent than in the IHCP. Huang and Ozisik (1992) used a combination of conjugate gradient method and modified conjugate gradient method to solve the inverse problem of determining the spacewise variation of an unknown wall flux for laminar flow inside a parallel

plate duct. Bokar and Ozisik (1995) utilised the same method to estimate the timewise variation of inlet temperature of a thermally developing, hydrodynamically developed laminar flow between parallel plates by utilising transient temperature measurements from a single thermocouple located downstream of the entrance. In the work of Colaco and Orlande (2001), the conjugate gradient method was used for the simultaneous identification of two unknown boundary heat fluxes in an irregularly shaped channel with laminar flow. The inverse convection problems in turbulent channel flow were also studied in recent years (Chen et al., 2006; Li and Yan, 2003; Su and Neto, 2001).

All the works about the inverse convection heat transfer problems mentioned earlier mainly studied the effects of measurement error and the position of the thermocouples on the performance of the inverse algorithm at a single specified Reynolds number; the effects of the convection on the performance of the inverse algorithm have not been studied yet.

As we know, it is very computationally expensive to solve the inverse problems in a straightforward way since it needs to solve the governing equations repeatedly. Applying these CFD-models-based inverse algorithm into the online estimation is still a challenging work. A method that has received growing attention recently is the Proper Orthogonal Decomposition (POD) technique, which has been used widely in the area of data analysis, data compression and development of reduced order models for the distributed parameter systems. The greatest attraction of the POD technique comes from its energy-optimality characteristic, i.e., among all possible decompositions of a physical field, with the same number of modes, POD modes will, on average, contain the most energy. In the context of heat transfer, few papers have considered the application of the POD technique. Park and Cho (1996a) applied the POD to a non-linear heat conduction problem. The resulting empirical eigenfunctions were used as the basis functions of Galerkin procedure to obtain the reduced order model. It was shown that the reduced order model has the same accuracy as the CFD type model; a drastic reduction in computation time was also reported. In another work of Park and Cho (1996b), the POD technique was used to obtain the reduced order model of a flow reactor. Bendersky and Christofides (2000) developed a reduced order model for two representative transport-reaction processes utilising the POD technique and then the reduced order model was used to optimise the performance of the reactors.

In the present study, an inverse algorithm based on the reduced order model employing the conjugate gradient method was developed. The effects of convection on the stability and accuracy of the inverse algorithm were analysed in terms of Reynolds number. It will be shown that, Reynolds number has a significant effect on the performance of the inverse algorithm. This paper presents a detailed derivation of the sets of sensitivity and adjoint equations used in the inverse algorithm first.

An inverse analysis for the estimation of boundary heat flux in a circular pipe 233

Then, numerical results about the effects of measurement error, the position of the thermocouples at different Reynolds number will be presented next. Finally, the paper gave some brief conclusions.

2 Physical problem We consider a thermally developing, hydrodynamically developed laminar forced convection of the water through a circular pipe with the outer wall subjected to a spacewise varying heat flux q(x). Fluid enters the pipe at a uniform temperature of T0. Figure 1 presents a schematic view of the present pipe flow system. Note, in this study, all the thermocouples used to measure the temperature of fluid are placed along the streamwise direction at the same radial position. By assuming axisymmetry of the problem and neglecting axial conduction and viscous dissipation, the governing equation in dimensional form for this problem is given by

1( )pT TC u r rx r r r

ρ η∂ ∂ ∂ = ∂ ∂ ∂ (1)

with the boundary conditions

0

0r

Tr =

∂ =∂

(2)

( )or r

T q xr

η=

∂ =∂

(3)

0(0, )T x T= (4)

where ρ is the density, Cp is the capacity, η is the thermal conductivity. ro denotes the radius of the pipe and T0 is the inlet temperature that takes a constant value of 300 K.

Figure 1 Schematic view of the pipe system and position of the measuring points. The velocity profile is fully developed

The fully developed velocity profile of a fluid in circular pipes is given by

2 2avg( ) 2.0 (1 / )ou r u r r= − (5)

where uavg is the average velocity. In this work, we take ro = 0.01 m, ρ = 1000.0 kg/m3, Cp = 4183.0 J/(Kg K), η = 0.599 W/(m K), uavg is assigned with different values so as to investigate the influences of the Reynolds number on the performance of the inverse algorithm. The length of pipe

l is fixed for this investigation at a value of 2.0 m. A 2000 × 100 uniform grid system was used to discretise the space domain, which is sufficient for the present problem.

The foregoing equation is discretised by the finite volume method of Patankar (1980). We consider the control volume as shown in Figure 2. Integration of Equation (1) over the control volume gives

( ) d d d de n e n

pw s w s

T TC u r r r x r r xx r r

ρ η∂ ∂ ∂ = ∂ ∂ ∂ ∫ ∫ ∫ ∫ (6)

( )( ) d dn e

p e ws wn s

T TC u r T T r r r r xr r

ρ η η∂ ∂ − = − ∂ ∂ ∫ ∫ (7)

if we discretise the right hand side using a central differencing scheme and a first order differencing scheme to the convective terms on the left hand side, Equation (7) may be written as

( )( ) N P P Sp P W n s

PN PS

T T T TC u r T T r r r r x

r rρ η η

δ δ − −

− ∆ = − ∆

(8)

now we can write Equation (8) in the familiar standard form as:

P P N N S Sa T a T a T b= + + (9)

where 0

P S N Pa a a a= + + (10)

0 ( )Pa u r r rρ= ∆ (11)

,n sN S

PN PS

r x r xa ar rδ δ

Γ ∆ Γ ∆= = (12)

0 , / .P W pb a T Cη= Γ = (13)

Figure 2 Control volume used for the discretisation of governing equation

In Equations (8)–(12), ∆ means the increments between the grid interfaces, whereas δ means the increments between the grid lines. As the governing equation, Equation (1), is a


linear parabolic equation, a spatially marching procedure starting from the inlet to the outlet line by line can be used. Along each line, a Gaussian elimination method was adopted to solve the algebraic equation, Equation (9).

The code developed here was validated by reproducing solutions for some benchmark problems. The direct problem was solved for the following two classical problems: thermally developing, hydrodynamically developed laminar forced convection heat transfer in a circular pipe with a constant heat flux boundary condition and thermally developing, hydrodynamically developed laminar forced convection heat transfer in a circular pipe with a constant temperature boundary condition. The Nusselt number for the fully developed region in a circular pipe with a constant heat flux boundary condition using the developed code is 4.34, which agrees excellently with the Nusselt number value of 4.36 mentioned by Shah and London (1978) and Kays and Crawford (1980). The Nusselt number for the fully developed region in a circular pipe with a constant temperature boundary condition using the developed code is 3.651, which agrees excellently with the Nusselt number value of 3.647.

3 Fundamentals of the POD In this section, we introduce the basic idea of the POD technique, further detail of the POD technique can be found in Berkooz et al. (1993), Holmes et al. (1996) and Sirovich (1987). The basic procedure to implement the POD technique is to construct the snapshot matrix first, estimate the auto-covariance matrix, then solve the corresponding eigenvalue problem.

Let us store the solutions of the system under consideration in a rectangular L by N rectangular matrix F and each column vector Fi ( )x , i = 1, 2, …, N is a sample of solutions of the physical problem under consideration at a specific dependent parameter t. In transient problems, the dependent parameter t may denote the time variable, whereas in steady problems it may represent the Rayleigh number, Reynolds number, or such. The POD procedure aims at extracting a sequence of typical structures 1 ( ) k N

kxφ = among the snapshots matrix F and these eigenfunctions satisfy the following orthogonality condition given by

( ) 1( ), ( )

0i j

ij

i jx x

i jφ φ δ

== = ≠

(14)

where the parentheses (⋅,⋅) mean the standard Euclidean inner product. Any snapshots in the matrix F can be reconstructed accurately with a very small truncation degree of M with ( )xΦ as its basis function as follows:

1

( ) ( ) 1, ,M

i kk

kx x i N M Nα φ

=

= = <<∑ …iF (15)

where ikα is the spectral coefficient. According to

POD theory, the truncation error of the reconstruction

formula (15) is the least one among all possible sets of orthogonal basis functions. This is equivalent to find a function ( )xΦ that maximises the normalised average projection of ( )i xF onto ( )xΦ

2( ( ), ( ))

( ( ), ( ))

x F x

x x

φγ

φ φ= (16)

where the brackets ,⋅ ⋅ mean the ensemble average of the snapshots. Equation (16) can be reformulated as

21

1 ( ( ), ( ))

( ( ), ( ))

Nii

F x xN

x x

φγ

φ φ=

=∑

(17)

( )1

1 ( ) ( ) ( )d ( )d.

( ), ( )

Ni ii

F x F x x x x xN

x x

φ φ

φ φ=Ω Ω

′ ′ ′=

∑∫ ∫ (18)

This maximisation problem can be reduced to the following integral eigenvalue problem:

1

1 ( ) ( ) ( )d ( ).N

i ii

F x F x x x xN

φ λφΩ

=

′ ′ ′ ′=∑∫ (19)

Generally speaking, two methods can be found in the literatures to solve the integral eigenvalue problem of Equation (19), i.e., the classical POD method and the snapshots POD method. In the classical POD method, it is prohibited to evaluate the auto-covariance matrix when the sampling of data contains a large number of components. This often happens when the sampling of data comes from the numerical simulation method because the number of sampling points is equal to the number of the grid points. Whereas the snapshots POD method successfully reduces the computation task to a much more tractable eigenvalue problem with a size of N equal to the number of snapshots. The snapshots POD method is based on the theory that the snapshot vectors ( )i xF and the POD eigenfunctions

( )xΦ span the same linear space and these spatial eigenfunctions are represented as a linear combination of the snapshots

1

( ) ( )N

k kn n

n

x F xφ σ=

=∑ (20)

the eigenvectors σ can be easily obtained by solving another integral eigenvalue problem using a standard numerical technique

( ) ( ) 1, ,n nn n Nσ λ σ= = …A (21)

where the N by N auto-covariance matrix A is calculated as

,1 ( ) ( ) d 1, , , 1, , .i j i jA F x F x x i N j NN Ω

′ ′ ′= = =∫ … … (22)

At last, Equation (20) is used to resolve the empirical eigenfunctions ( )xΦ . According to the POD theory, the magnitude of eigenvalue λk provides a measure of the amount of energy captured by the corresponding eigenfunction ( )k xΦ , whereas the energy measures the


contribution of each eigenfunction to the overall system dynamics. So in practice, we often order the eigenfunctions

( )k xΦ by the magnitude of their corresponding eigenvalues λk, i.e., λ1 > λ2 > ⋅⋅⋅ > λN.

4 Construction of the POD eigenfunctions It is very clear that the POD eigenfunctions can be no better than the information contained in the snapshots since it comes from the POD of the snapshots matrix. Therefore, the construction of the snapshots matrix is the most important step in the POD procedure. A good snapshot set must represent the dynamic characteristic of the system under consideration and encompass the admissible solution space of the governing equation for different kinds of boundary heat flux functions q(x) and different Reynolds numbers in the laminar range.

In this paper, we tried to develop a reduced order model that can be used in the whole laminar regime. The snapshot matrix of the system was obtained in the following way. The governing equation was first solved with a constant heat flux function at four Reynolds numbers of 500, 1000, 1500 and 2000, respectively. Then, we recorded the temperature fields along the vertical grid lines in the x-direction at certain intervals. One may take snapshots more frequently at the forepart of the tube since the temperature field varies drastically at the beginning of the pipe. The snapshot matrix was at last constructed by these 4000 snapshots. It is expected that, by assembling snapshots at different Reynolds numbers into one snapshots matrix, a more universal reduced order model which encompasses the whole laminar regime can be obtained with the eigenfunctions extracting from these combination snapshots as its basis. When the POD technique was applied to the snapshot matrix, we obtained a total of 4000 eigenfunctions in the order of the magnitude of their corresponding eigenvalues. The first ten eigenfunctions and their corresponding eigenvalues are shown in Figure 3. An overview of Figure 3 shows that the magnitude of the eigenvalue decreases drastically from the first eigenvalue of 10.72 to the tenth eigenvalue of 4.96 × 10–9. This phenomenon gives a clear demonstration about the energy optimality characteristic of the POD eigenfunctions since the magnitude of eigenvalues λk provides a measure of the amount of energy captured by their corresponding eigenfunctions ( )k xφ . Figure 3 also says that the eigenfunctions with large eigenvalue take the shape of large scale smooth structures, whereas the eigenfunctions with large index number have a tendency to include more and more small scale structures and those small scale structures represent the structures not captured by the eigenfunctions of large eigenvalue, such as the velocity boundary layer or temperature boundary layer.

Figure 3 The most dominant ten eigenfunctions and their corresponding eigenvalues obtained from the POD procedure. (a)–(e) represent the first, second, third, fourth and fifth pair of eigenfunctions, respectively (continues on next page)

(a)

(b)

(c)

(d)


Figure 3 The most dominant ten eigenfunctions and their corresponding eigenvalues obtained from the POD procedure. (a)–(e) represent the first, second, third, fourth and fifth pair of eigenfunctions, respectively (continued)

(e)

5 Development of the reduced order model In this section, the basic procedure to develop the reduced order model is introduced. Let us expand the temperature fields as a linear combination of the POD eigenfunctions as follows:

1( , ) ( ) ( ) 1, ,

Mk

i k ik

T x r x r i N M Nα φ=

= = <<∑ … (23)

where Φk(r) is the kth eigenfunction vector, αk(xi) is the corresponding spectral coefficient and M is truncation degree that denotes the total number of POD eigenfunctions used in the reconstruction formula. The reduced order model is obtained by a Galerkin projection of the governing equation onto the POD eigenfunctions. The residual of the governing equation may be written as

1Res ( )pT TC u r rx r r r

ρ η∂ ∂ ∂ = − ∂ ∂ ∂ (24)

by requiring the residual to be orthogonal to each of the eigenfunctions

0Res ( ) d 0or i r r rφ =∫ (25)

we obtain

1 1

d ( )( ) (1) ( )

d

M Mj i i

ij ij j ij j

xM q x H x

xα

φ α= =

= −∑ ∑ (26)

with the following initial condition

01

0

( ) d( 0) 1, ,

( ) ( ) d

o

o

r iin

i r i i

r T r rx i M

r r r r

φα

φ φ= = =∫

∫… (27)

where 1

0( ) ( ) ( ) di j

ij pM C u r r r r rρ φ φ= ∫ (28)

0

( ) ( ) d .oi jr

ijr rH r r

r rφ φη ∂ ∂=∂ ∂∫ (29)

The performance of the reduced order model can be easily examined by comparing its solutions with the solutions obtained by the finite volume method at different Reynolds number. Figure 4 presents three different shapes of heat flux function q(x) that we considered in the validation process and the formulae of q(x) are as follows

Example 1

q(x) = 6000 tanh(1.0 xπ/l) 0 ≤ x ≤ l. (30)

Example 2

q(x) = (6000 – 11000x/l) 0 ≤ x ≤ l/2

q(x) = (500 + 11000(x/l–1/2))l/2 ≤ x ≤ l. (31)

Example 3

q(x) = 2000 sin(6xπ/l) + 40000 ≤ x ≤ l. (32)

Figure 4 Various shapes of heat flux functions q(x) used to examine the performance of the reduced order model. (a)–(c) correspond to examples 1–3, respectively (continues on next page)

(a)

(b)


Figure 4 Various shapes of heat flux functions q(x) used to examine the performance of the reduced order model. (a)–(c) correspond to examples 1–3, respectively (continued)

(c)

Generally speaking, the accuracy of the reduced order model increases with the increase of truncation degree M. It is well known that the truncation degree M has an optimal value beyond which further increase in the number of eigenfunctions in the series does not improve or even deteriorate the results. In this study, the optimal value of truncation degree is found to be ten, thus we are going to use the truncation degree M = 10 in the remaining parts of the paper.

For the convenience of comparison, we define the relative error as follows:

ROM FVM

1 FVM

( , ) ( , )1

( , )

J

j

T i j T i jE M

T i j=

−=∑ (33)

where M1 means the total number of grid points in the radial direction, the subscript ROM means the solutions obtained from the reduced order model and the subscript FVM denotes the solutions obtained from the finite volume method.

Figure 5 shows the computational results obtained by the reduced order model with a truncation degree of M = 10 for the case of example 1 at four different Reynolds numbers. The computational results obtained from the FVM are also shown in Figure 5 for comparison. The solid lines indicate the solutions from finite volume method and the dashed lines indicate the solutions from reduced order model. Figure 5(a) presents the evolution of temperature at three radial positions of r = 0.2r0, r = 0.5r0 and r = 0.8r0 for the case of Re = 500. It reveals that the solutions from the ROM and the solutions from the FVM are nearly the same. Figure 5(b)–(d) show the solutions at Re = 850, 1500 and 2000, respectively. It is obvious that the agreements are also very well. For example 1, the maximum value of relative error E takes a value of 0.01% and the minimum value of relative error E takes a value of 0.005%. Figures 6 and 7 show the computation results obtained by the reduced order model for the case of examples 2 and 3, respectively.

Figure 5 Temperature distribution obtained with the FVM as well as the ROM of order ten at r = 0.2r0, r = 0.5r0, r = 0.8r0 for the example 1: (a) corresponds to the solutions at Reynolds = 500; (b) corresponds to the solutions at Reynolds = 850; (c) corresponds to the solutions at Reynolds = 1500 and (d) corresponds to the solutions at Reynolds = 2000

(a)

(b)

(c)

(d)


Figure 6 Temperature distribution obtained with the FVM as well as the ROM of order ten at r = 0.2r0, r = 0.5r0, r = 0.8r0 for the example 2: (a) corresponds to the solutions at Reynolds = 400; (b) corresponds to the solutions at Reynolds = 800; (c) corresponds to the solutions at Reynolds = 1200 and (d) corresponds to the solutions at Reynolds = 1600 (continues on next column)

(a)

(b)

(c)

Figure 6 Temperature distribution obtained with the FVM as well as the ROM of order ten at r = 0.2r0, r = 0.5r0, r = 0.8r0 for the example 2: (a) corresponds to the solutions at Reynolds = 400; (b) corresponds to the solutions at Reynolds = 800; (c) corresponds to the solutions at Reynolds = 1200 and (d) corresponds to the solutions at Reynolds = 1600 (continued)

(d)

Figure 7 Temperature distribution obtained with the FVM as well as the ROM of order ten at r = 0.2r0; r = 0.5r0; r = 0.8r0 for the example 3: (a) corresponds to the solutions at Reynolds = 300; (b) corresponds to the solutions at Reynolds = 900; (c) corresponds to the solutions at Reynolds = 1600 and (d) corresponds to the solutions at Reynolds = 1900 (continues on next page)

(a)

(b)


Figure 7 Temperature distribution obtained with the FVM as well as the ROM of order ten at r = 0.2r0; r = 0.5r0; r = 0.8r0 for the example 3: (a) corresponds to the solutions at Reynolds = 300; (b) corresponds to the solutions at Reynolds = 900; (c) corresponds to the solutions at Reynolds = 1600 and (d) corresponds to the solutions at Reynolds = 1900 (continued)

(c)

(d)

All results as shown in Figures 5–7 give an excellent demonstration that the reduced order model constructed with the POD eigenfunctions can produce temperature fields as accurate as the original governing equation. Moreover, the solutions are obtained from the reduced order model with only ten degrees of freedom, but the original partial differential equation has a freedom degree of 2000 × 100; thus, a drastic reduction in computation time can be expected, which will facilitate the implementation of the inverse procedure greatly. In the sequel, the inverse procedure in this investigation is based on the reduced order model developed in this section.

6 Inverse problem For the inverse problem, we are going to identify the unknown wall heat flux q(x) by using the temperature measurements taken from within the pipe flow. Let the temperature measurements taken at some appropriate locations within the flow be denoted by *T and let

T denote the solutions of the direct problem at the thermocouples position, that is, the temperature corresponding to a particular value of heat flux function q(x). In the idealised situation without measurement error, the unknown heat flux q(x) can be resolved by requiring a exact equivalency between the measured temperature *T and the calculated temperature T. But in practice, the inverse problem should be solved in a least square way owing to the ill-posed nature of the inverse problem. Then, the inverse problem treated in this paper was defined as follows

To find a heat flux q(x) which minimises the object function J defined as

[ ]2

mea mea mea meamea 1

2

mea mea mea meamea 1 1

( , ) *( , )

( ) ( ) *( , )

Nm

Nm Mk

kk

J T x r T x r

x r T x rα φ

=

= =

= −

= −

∑

∑ ∑ (34)

where Nm represents the total number of measuring points, xmea and rmea represent the stream-wise coordinate and the radial coordinate of the measuring points, respectively. The minimisation of object function J is achieved by the Fletcher-Reeves conjugate gradient method in this paper. The essence of the Fletcher-Reeves conjugate gradient method is to decide a suitable descent direction and a suitable step size in the descent direction for the minimisation of object function J. The descent direction and the step size may be obtained, respectively, from the solutions of two auxiliary problems known as the sensitivity problem and adjoint problem. The derivation of sensitivity problem and adjoint problem in the CFD model type can be found in the references. In this section, we will present the detail of the derivation of the adjoint and sensitivity equations in the reduced order model form.

6.1 Sensitivity problem

Let us define ∆α as the variation of spectral coefficient resulting from the change of the unknown heat flux q(x) in the amount of ε∆q, i.e.,

( , , ) ( , , )T x r q q T x r qα ε∆ = + ∆ − (35)

we define the sensitivity function of the spectral coefficient α as the directional derivative of α at q in the direction of ∆q, i.e.,

0ˆ limQD

ε

αα αε∆ →

∆= = (36)

the reduced order model of the sensitivity function can be obtained in the following way. The direct problem given by Equations (25)–(28) are written first for (q + ε∆q) and then subtract Equations (25)–(28) from it and the limiting process defined by Equation (36) is applied. Then, we obtain the following formula


1 1

ˆd ( )ˆ( ) ( ) ( )

d

M Mj i i

ij o ij j ij j

xM q x r H x

xα

φ α= =

= ∆ −∑ ∑ (37)

and the boundary condition is written as

1ˆ ( 0) 0 1, , .i x i Mα = = = … (38)

6.2 Adjoint problem and gradient equation

The gradient of object function J with respect to the heat flux q ∇Jq can be obtained from the solution of the adjoint problem. The adjoint problem is obtained by introducing the Lagrange multiplier ( )xα into the direct problem and by writing the modified object function as follows:

2

mea mea mea mea01

01 1

1

( ) ( ) *( , ) d

d ( )( )

d

( ) (1) ( ) d

Ml kk

k

M Ml j kk kj

k j

Mk

kj j kj

J x r T x r x

xx M

x

q x H x x

α φ

αα

φ α

=

= =

=

= −

−

− +

∑∫

∑ ∑∫

∑ (39)

where xmea and rmea are the axial and radial coordinate of the measuring points, respectively.

Following the same procedure as the derivation of the sensitivity equation, the directional derivative of the object function J in the direction of ∆q can be written as

]

mea mea01

mea mea mea mea1

01 1

1

2 ( ) ( )

*( , ) ( ) ( ) d

d ( )( )

d

( ) (1) ( ) d

Ml kq k

k

Mk

kk

M Ml j kk kj

k j

Mk

kj j kj

D J x r

T x r x r x

xx M

x

q x H x x

α φ

δα φ

δαα

φ δα

∆=

=

= =

=

=

−

−

− ∆ +

∑∫

∑

∑ ∑∫

∑ (40)

by employing interchanging the summation indices in the summation at the right-hand side of Equation (40), using the initial and boundary conditions of the sensitivity problem and also requiring that the coefficients of δαk(x) vanish, then we obtain the reduced order model for the adjoint function ( )xα

1 1

mea mea mea mea1

mea

d ( )( ) 2

d

( ) ( ) *( , )

( )

M Mj k

kj kj j kj j

Mk

kk

k

xM H x

x

x r T x r

r

αα

α φ

φ

= =

=

= −

−

∑ ∑

∑ (41)

with the starting condition at x = l

( ) 0 1, ,i l i Mα = = … (42)

finally, the following term is left

01

( ) (1) dMl k

q kk

D J x q xα φ∆=

= ∆∑∫ (43)

the directional derivative of object function J in the direction of ∆q can be expressed in another way, i.e., projection gradient ∇Jq onto the direction ∆q, we have

0

( , )

d .

q q

l

q

D J J q

J q x

∆ = ∇ ∆

= ∇ ∆∫ (44)

A comparison between Equations (43) and (44) gives

1( ) (1).

Mk

q kk

J xα φ=

∇ =∑ (45)

It should be noted that the adjoint problem involving the starting condition at the end of the pipe instead of the usual inlet condition. To solve the adjoint problem, we often utilise the following independent variable transformation

* .x l x= − (46)

6.3 Fletcher-Reeves conjugate gradient method

In this investigation, the Fletcher-Reeves version is used to minimise the object function J.

The iterative procedure of the Fletcher-Reeves conjugate gradient method to identify the unknown heat flux q(x) is given by

1( ) ( ) ( ) 0, 1, 2,n n n nq x q x P x nβ+ = + = … (47)

where βn is the search step size from the nth iterative to the (n + 1)th and Pn(x) is the search direction expressed by

1( ) ( ) ( )n n n nqP x J x P xγ −= ∇ + (48)

in the Fletcher-Reeves conjugate gradient method, the conjugate coefficient γn is determined from

2

00

1 2

0

( ) d, 0.

( ) d

L nqn

L nq

J x

J xγ γ

−

∇= =

∇

∫∫

(49)

The search step size βn is determined by

[ ]mea mea mea mea mea mea0

2mea mea0

mea mea mea mea10

mea mea102

mea mea10

ˆ( , ) * ( , ) ( , ) d

ˆ ( , ) d

( ) ( ) *( , ).

ˆ ( ) ( ) d

ˆ ( ) ( ) d

L

nL

L M kkk

L M kkk

L M kkk

T x r T x r T x r x

T x r x

x r T x r

x r x

x r x

β

α φ

α φ

α φ

=

=

=

−=

− =

∫∫

∑∫∑∫∑∫

(50)

The computational algorithm to identify the unknown heat flux q(x) can be arranged in the following manner


• solve the direct problem with an assumed initial heat flux

• solve the adjoint problem

• compute the gradient ∇jq from Equation (45)

• compute the conjugate coefficient γn from Equation (49)

• compute the search direction p from Equation (48)

• solve the sensitivity problem by set q P∆ = in Equation (37)

• compute the search step size βn from Equation (50)

• update the unknown heat flux q from Equation (47)

• repeat the above step until convergence criterion satisfied.

7 Results and discussion In practice, the temperature measurements always contain some degrees of measurement error. As real experimental data were not available, in this study, we generate the simulated measurement data by adding the random error to the exact temperature T. The simulated measurement data can be expressed in the following dimensional way as

*T T ϖσ= + (51)

where T denotes the exact temperature which is the solution of the direct problem, σ is the standard deviation of the measurement error (in °C), which takes values of 0.0, 0.08 and 0.15 in this investigation. ϖ is a random number with normal distribution lying in the range of –2.567 < ϖ < 2.567 generated by a random number generator.

Two kinds of convergence criterions were used in this paper for case of σ = 0.0 and σ ≠ 0.0, respectively. When σ = 0.0, in theory, the object function J may converge to zero. In practice, it is feasible to stop the iteration after a specified number since the decrease of the error will become very slow as iteration goes on. For the case of σ ≠ 0.0, the object function J cannot converge to zero as a certain level of noise exists in the measurements. The iteration must be stopped at an optimum iteration number. When the iteration is stopped before this optimum number, we cannot obtain a fully convergent solution. In contrast, the solutions will be contaminated by high-frequency components of the noise contained in the measurement data. The discrepancy-principle-based convergence criterion has been adopted by many other works for the case of σ ≠ 0.0 where the iteration is stopped when

J ε< (52)

here

21 .2

Nmε σ= (53)

According to our experience, Equation (53) can only provide an estimation of the optimal iteration number roughly. So in practice computation, we perform computation using the convergence criterion Equation (53) first to estimate an optimal iteration number, then we perform more computations with iteration going beyond or below this number to determine a most likely solution by watching the smoothness of the predicted heat flux profiles.

The accuracy and robust of the inverse algorithm can be easily determined since the exact solutions were known already. Let us define ERR as

estimated exact

exact

ERR = 1 100q q

LqΩ

−×∑ (54)

where L1 means the total number of grid points in the stream-wise direction, the subscript exact means the exact heat flux function and the subscript estimated denotes the heat flux profile obtained by the inverse algorithm.

7.1 Effects of Reynolds number and thermocouple position

Figure 8 shows the estimated heat flux function by the inverse algorithm for example 1 at Reynolds number of 200 with errorless measurement data. Figure 8(a)–(c) correspond to the computational results when the thermocouples are placed at r = 0.3r0, r = 0.4r0 and r = 0.85r0, respectively. It is easy to find from Figure 8 that the accuracy of the estimation become more accurate if we take the sensors closer to the boundary. The relative error ERRs for these three cases are 14.12, 1.69 and 0.93%, respectively. The computational result for the case of r = 0.3r0 takes the largest relative error, but the result is still reasonable. If we consider a measurement position farther from the boundary wall than r = 0.3r0, the inverse algorithm diverges.

Figure 8 The estimated heat flux function of example 1 for the case of Re = 200 with errorless data: (a) measurement points at r = 0.3r0; σ = 0.0; (b) measurement points at r = 0.4r0; σ = 0.0 and (c) measurement points at r = 0.85r0; σ = 0.0 (continues on next page)

(a)


Figure 8 The estimated heat flux function of example 1 for the case of Re = 200 with errorless data: (a) measurement points at r = 0.3r0; σ = 0.0; (b) measurement points at r = 0.4r0; σ = 0.0 and (c) measurement points at r = 0.85r0; σ = 0.0 (continued)

(b)

(c)

As we know, the accuracy of the inverse algorithm largely depends on the sensitivity of the measurements to the unknown quantity. For Re = 200, the convection of fluid in the pipe is relatively low; consequently, the boundary layer is thicker; any variation occurred in the boundary wall can be felt by the sensors placed at a relatively far places. As the thermal sensors moved closer to the boundary wall, the sensitivity at the measurement points also increase; then we get a more accurate result as shown in Figure 8.

Figure 9 plots the estimated heat flux from the inverse algorithm when the Reynolds number takes the value of 600. Comparing the results of Figure 9 with those of Figure 8, we find that the sensitivity at the same measurement points decrease as the Reynolds number increases. Reasonable results can be obtained only when the thermal sensors moved to r = 0.5r0 for case of Re = 600. The estimation error ERR decreases significantly from a value of 11.74% to 1.29% when the thermocouples are moved from r = 0.5r0 to r = 0.7r0 because the sensitivity is greater for sensors placed at r = 0.7r0 than that for sensors placed at r = 0.5r0.

The computational results for the case of Re = 1600 are shown in Figure 10. The thermal sensors should be moved closer to the boundary wall further to obtain reasonable results as shown in Figure 10(a). Comparisons between Figures 9 and 10 also say that at the same measurement points, the increase of the convection will decrease the accuracy of the estimation significantly. For example, the relative error ERR increases from 1.29% to 4.01% when the Reynolds number increases from 600 to 1600.

Figure 9 The estimated heat flux function of example 1 for the case of Re = 600 with errorless data: (a) measurement points at r = 0.5r0; σ = 0.0 and (b) measurement points at r = 0.7r0; σ = 0.0

(a)

(b)

Figure 10 The estimated heat flux function of example 1 for case of Re = 1600 with errorless data: (a) measurement points at r = 0.7r0; σ = 0.0 and (b) measurement points at r = 0.9r0; σ = 0.0 (continues on next page)

(a)


Figure 10 The estimated heat flux function of example 1 for case of Re = 1600 with errorless data: (a) measurement points at r = 0.7r0; σ = 0.0 and (b) measurement points at r = 0.9r0; σ = 0.0 (continued)

(b)

The estimated heat flux function identified by the inverse algorithm for example 2 are shown in Figures 11–14. The same conclusions with that of example 1 can be drawn from Figures 11–14.

Figure 11 The estimated heat flux function of example 2 for the case of Re = 200 with errorless data: (a) measurement points at r = 0.4r0; σ = 0.0; (b) measurement points at r = 0.6r0; σ = 0.0 and (c) measurement points at r = 0.8r0, σ = 0.0 (continues on next column)

(a)

(b)

Figure 11 The estimated heat flux function of example 2 for the case of Re = 200 with errorless data: (a) measurement points at r = 0.4r0; σ = 0.0; (b) measurement points at r = 0.6r0; σ = 0.0 and (c) measurement points at r = 0.8r0, σ = 0.0 (continued)

(c)

Figure 12 The estimated heat flux function of example 2 for the case of Re = 1600 with errorless data: (a) measurement points at r = 0.8r0; σ = 0.0 and (b) measurement points at r = 0.95r0; σ = 0.0

(a)

(b)


Figure 13 The estimated heat flux function of example 3 for the case of Re = 200 with σ = 0.08: (a) measurement points at r = 0.7r0; σ = 0.08 and (b) measurement points at r = 0.9r0; σ = 0.08

(a)

(b)

Figure 14 The estimated heat flux function of example 3 for the case of Re = 200 with σ = 0.15: (a) measurement points at r = 0.8r0; σ = 0.15 and (b) measurement points at r = 0.9r0; σ = 0.15 (continues on next column)

(a)

Figure 14 The estimated heat flux function of example 3 for the case of Re = 200 with σ = 0.15: (a) measurement points at r = 0.8r0; σ = 0.15 and (b) measurement points at r = 0.9r0; σ = 0.15 (continued)

(b)

7.2 Effects of measurement error

The effect of the measurement error on the accuracy of the estimation is investigated in this section. The computational results for example 3 are shown in Figures 13–15. The discrepancy-principle-based convergence criterion Equations (52) and (53) was used to stop the iteration. Figures 13 and 14 are the estimated heat flux function for example 3 at Reynolds number of 200 when the noise level are σ = 0.08 and σ = 0.15, respectively. As expected, increasing the measurement error decreases the accuracy of the estimation. Figure 15 shows the estimated results for example 3 with a noise level of 0.15 at Reynolds number of 1600. The accuracy of the estimation was greatly improved by moving the sensors from r = 0.85r0 to r = 0.95r0. The relative errors for these two cases are 6.61% and 2.14%, respectively.

It is interesting to compare the CPU time required for the POD-based algorithm with that of the CFD-model-based algorithm. It takes 6.5 S for the CFD-based algorithm to iterate 1 step, whereas it takes only 0.078 S to iterate 1 step.

Figure 15 The estimated heat flux function of example 3 for the case of Re = 600 with σ = 0.15: (a) measurement points at r = 0.85r0; σ = 0.15 and (b) measurement points at r = 0.95r0; σ = 0.15 (continues on next page)

(a)


Figure 15 The estimated heat flux function of example 3 for the case of Re = 600 with σ = 0.15: (a) measurement points at r = 0.85r0; σ = 0.15 and (b) measurement points at r = 0.95r0; σ = 0.15 (continued)

(b)

8 Concluding remarks An inverse algorithm based on the POD technique is developed in this paper. The inverse problem is resolved by minimising the object function, which represents the least square norm between the measurements and its corresponding calculated values. The Fletcher-Reeves conjugate gradient method is used to minimise the object function. The reduced order model of the physical problem is obtained by a Galerkin projection of the governing equation onto the POD eigenfunctions.

It is found that by assembling snapshots at different Reynolds numbers into one snapshots matrix, a more universal reduced order model, which encompasses the whole laminar regime can be obtained with the eigenfunctions extracting from these combination snapshots as its basis.

The performance of the present inverse algorithm is examined by an inverse forced convection problem of identifying the unknown space-dependent heat flux at the outer boundary of a circular pipe. The effects of the convection, the number of thermocouples, the location of the thermocouples and the measurement error on the performance of the inverse algorithm are studied thoroughly.

The sensitivity at the measurement points is greatly decreased as the Reynolds number increases. The accuracy of the inverse algorithm can be improved greatly by moving the measurement points closer to the unknown heat flux. It also brings significant computational time savings compared with the CFD-mode-based algorithm.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 50476046, No.50636050).

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