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POD reduced-order model for steady natural convection based on a body-fitted coordinate
International Communications in Heat and Mass Transfer 68 (2015) 104–113
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POD reduced-order model for steady natural convection based on a body-fitted coordinate☆ Dongxu Han a, Bo Yu a,⁎, Jingjing Chen a, Yi Wang a, Ye Wang b a National Engineering Laboratory for Pipeline Safety, Beijing Key Laboratory of Urban Oil and Gas Distribution Technology, China University of Petroleum, Beijing 102249, People's Republic of China b Key Laboratory of Railway Vehicle Thermal Engineering, Ministry of Education of China, Lanzhou Jiaotong University, Lanzhou 7370070, People's Republic of China
a r t i c l e
i n f o
Available online 4 September 2015 Keywords: Proper orthogonal decomposition Reduced-order model Geometry parameter Natural convection Body-fitted coordinate
a b s t r a c t In this article, a POD reduced-order model for steady-state natural convection based on a body-fitted coordinate system is established. The velocity basis functions and temperature basis functions are generated, respectively. Based on the basis functions, the governing equations for velocity spectrum coefficient and temperature spectrum coefficient are deduced, which are coupled pluralistic quadratic nonlinear equations and solved iteratively by the Newton–Raphson method. Compared with the existing natural convection POD reduced-order models, the major advantage of the proposed POD model is its capability to calculate natural convections in differentshape domains having identical geometry characters. Two typical examples are given to show the model implementation procedure as well as to illustrate its good performance in terms of accuracy and robustness. It is found that even though the geometries and the physical conditions of the test cases differ greatly from those of the sampling cases, the reduced-order model can acquire accurate results efficiently. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Proper orthogonal decomposition (POD) can extract characteristic information (basis functions) from large amount of data, and the original data can be reconstructed with just a few basis functions,which makes POD a powerful tool for model reduction. POD reduced-order models can depict the physical problems accurately and efficiently for engineering application. In the field of fluid mechanics and thermal engineering, some POD reduced-order models have been developed and some related studies are briefly reviewed below. Lumely [1] first introduced POD technique for turbulent analysis. Deane et al. [2] established a POD-Galerkin reduced-order model for transition flow. Ravindran [3,4] applied a POD reduced-order model to the real-time control in the flow passing an airfoil as well as the flow separation over a forward facing step. My-Ha [5] employed POD method for optimizing parameters for underwater bubble explosions to generate an expected free surface shape. Foglema et al. [6] analyzed the flow field in the internal-combustion engines. More studies on fluid flow can be referred to Ref. [7–12]. Ding et al. [13] applied a Galerkin-POD reduced-order model to fast calculate the heat transfer and fluid flow in a tube-fin heat exchanger; Du et al. [14] achieved the fast thermal calculation of an air-cooled
☆ Communicated by W.J. Minkowycz ⁎ Corresponding author. E-mail address: [email protected] (B. Yu).
http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.024 0735-1933/© 2015 Elsevier Ltd. All rights reserved.
condenser of power generating unit using a POD reduced-order model. Joshi et al. [15,16] employed this technique to the fast thermal calculation of an electronic cabinet and data centers. Few studies have been carried out on the Galerkin-POD reducedorder model for natural convection. Guns et al. established a POD reduced-order model in Ref. [17] and applied it to predict the natural convection in a vertical channel in Ref. [18]. In addition, Park and Lee [19] applied a POD reduced-order model on real-time control of natural convection. The above-mentioned studies on POD reduced-order models for natural convection are all for fixed domain. However, fast calculations of natural convection in domains with different shapes having identical geometry characters are always required in engineering. For example, heated crude oil pipelines have different diameters and buried depths. After shutdowns, fast oil temperature drop (natural convection problem) calculations in the pipes are required to provide useful information to guide the crude oil pipeline operation. The objective of this study is to develop a POD reduced-order model to predict natural convection in different-shape domains having identical geometry characters. Previously, our research group has developed POD reduced-order models based on body-fitted coordinate (BFC) for fast heat conduction calculation [20–22] and fluid flow calculation without heat transfer [23] in domains with different shapes having identical geometry characters. In this article, we extend the technique to steady-state natural convection. The layout of this paper is as follows: the methodology of the BFC based POD reduced-order model for natural convection is introduced
D. Han et al. / International Communications in Heat and Mass Transfer 68 (2015) 104–113
Nomenclature Roman symbols a Thermal diffusivity (m2/s) Spectrum coefficient for the kth velocity basis function ak Spectrum coefficient for the kth temperature basis bk function ! ! ! ds dη i −dξ j Relative error of velocity calculated by reduced-order ev model Relative error of dimensionless temperature calculated eΘ by reduced-order model g Gravity (m/s2) Gr Grashof number h Eccentricity between the inner and the outer semicircles (m) H Dimensionless eccentricity between the inner and the outer semi-circles J Jacobi factor l Length of the side in a parallelogram cavity (m) The total number of the grid points Nt Serial number of the test cases Ntest p Dimensionless pressure Pr Prandtl number Radius of outer semi-circle (m) r1 Radius of inner semi-circle (m) r2 Dimensionless radius of outer semi-circle R1 Dimensionless radius of inner semi-circle R2 T Dimensional temperature (∘C) Higher temperature, lower temperature (∘C) Th, Tl u, v Cartesian components of dimensionless velocity ! ! ! u u i þv j U, V Contravariant components of velocity x, y Cartesian coordinate Greek symbols α, β, γ Variables related with grids Cartesian components of basis functions, corresponding ϕu, ϕv to u, v ! ! ! ϕ ϕu i þ ϕv j Eigenvalue related to the nth basis function λn ν Kinematic viscosity (m2/s) Accumulative energy contribution of the first n basis ϖn function θ The angle of parallelogram cavity l Θ Dimensionless temperature, Θ ¼ TT−T h −T l ϑ Temperature basis functions Energy contribution of the nth basis function ςn ξ, η Body-fitted coordinate Contravariant components of basis functions, correψu, ψv sponding to U, V ! ! ! ψ ψu i þ ψv j (,) Hilbert inner product ∇ Hamilton operator ! ! ⋄ Author defined operator ⋄ ¼ ∂n∂ ξ i þ ∂n∂η j Subscript ()ξ, ()η
∂ðÞ ∂ðÞ ; ∂ξ ∂η
105
2. Methodology In this section, the governing equations for steady-state natural convection in a body-fitted coordinate are first introduced. Subsequently, the principle of basis functions calculation is given briefly. Finally, the derivation of the BFC based POD reduced-order model is presented in details. 2.1. Governing equation for steady-state natural convection A two-dimensional incompressible steady-state natural convection problem, in a physical domain Ωp, is considered herein, which is governed by Navier–Stokes equations and the energy equation. Based on Boussinesq assumption, the 2-D natural convection on Cartesian coordinate in dimensionless form with the gravity force on y axis can be depicted by Eqs. (1)–(4). ∂u ∂v þ ¼0 ∂x ∂y
ð1Þ
∂ðuuÞ ∂ðvuÞ ∂p ∂ ∂u ∂ ∂u þ þ ¼− þ ∂x ∂y ∂x ∂x ∂x ∂y ∂y
ð2Þ
∂ðuvÞ ∂ðvvÞ ∂p ∂ ∂v ∂ ∂v þ þ GrΘ þ ¼− þ ∂x ∂y ∂y ∂x ∂x ∂y ∂y
ð3Þ
∂ðuΘÞ ∂ðvΘÞ 1 ∂ ∂Θ ∂ ∂Θ þ þ ¼ Pr ∂x ∂x ∂x ∂y ∂y ∂y
ð4Þ
! where u is the dimensionless velocity, Θ stands for the dimensionless l temperature and Θ ¼ TT−T , Gr denotes Grashof number and Gr ¼ gβΔTl ν2 h −T l
3
(the β here stand for expansion parameter), and Pr represents Prandtl number and Pr ¼ νa . By body-fitted coordinate transformation, Ωp on the physical plane is mapped to Ωc on calculation plane. Correspondingly, the governing equations Eqs. (1)–(4) are transformed to Eqs. (5)–(8), respectively. ∂U ∂V þ ¼0 ∂ξ ∂η
ð5Þ
∂ 1 ∂ðUuÞ ∂ðVuÞ þ ¼ − P ξ yη −P η yξ þ αuξ −βuη ∂ξ ∂η ∂ξ J ∂ 1 γuη −βuξ þ ∂η J
ð6Þ
∂ 1 ∂ðUvÞ ∂ðVvÞ þ ¼ − −P ξ xη þ P η xξ þ αvξ −βvη ∂ξ ∂η ∂ξ J ∂ 1 γvη −βvξ þ JGrΘ þ ∂η J
ð7Þ
∂ðUΘÞ ∂ðVΘÞ 1 ∂ 1 1 ∂ 1 þ ¼ αΘξ −βΘη þ γΘη −βΘξ Pr ∂ξ J Pr ∂η J ∂ξ ∂η
ð8Þ
where α ¼ x2η þ y2η ; β ¼ xξ xη þ yξ yη ; γ ¼ x2ξ þ y2ξ ; J ¼ xξ yη −xη yξ Contravariant components of velocity
in Section 2; two examples are presented to illustrate the good performance of the proposed model in Section 3; and some conclusions are given in the Section 4.
U ¼ uyη −vxη
ð9Þ
V ¼ vxξ −uyξ
ð10Þ
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D. Han et al. / International Communications in Heat and Mass Transfer 68 (2015) 104–113
For the sake of convenience, Eqs. (5)–(8) are written in weak form respectively as follows: ! ∇ U ¼0
ð11Þ
! ! ∇ U u ¼ − g 1 ∇ p þ ∇ ð⋄uÞ
ð12Þ
! ! ∇ U v ¼ − g 2 ∇ p þ ∇ ð⋄vÞ þ JGrΘ
ð13Þ
and (18), and vice versa. This means in principle only one kind of velocity basis functions needs to be generated by the sampling matrix and the other can be obtained by simple algebraic calculations. However, only the contravariant component of velocity in the sampling matrix can ensure the obtained velocity basis function satisfy the continuity equation on any domain with different shapes, which is a precondition of the derivation of POD reduced-order model; see details in Ref. [23]. 2.3. Establishment of the reduced-order model
! 1 ∇ ð⋄ΘÞ ∇ UΘ ¼ Pr
ð14Þ
! ∂! ∂ where ∇ is Hamilton operator, ∇ ¼ ∂ξ i þ ∂η j ; ⋄ is an operator (similar
(1) Reduced-order model of the momentum equation M M N ! ! ! Substituting u ¼ ∑ ak ϕ k ; U ¼ ∑ ak ψ k ; Θ ¼ ∑ bk ϑ k into k¼1
as Hamilton operator) introduced by the authors for concise expression, ! ! α f −β f γ f −β f ⋄ ¼ ∂n∂ ξ i þ ∂n∂ η j ; and for any f, ∂n∂ ξ ðf Þ ¼ ξ J η , ∂n∂η ðf Þ ¼ η J ξ and ! ! ! ! ! ! g 1 ¼ yη i −yξ j , g 2 ¼ −xη i þ xξ j .
k¼1
k¼1
Eqs. (11) and (12) and projecting them onto the space of velocity basis function ϕui and ϕvi , respectively, yields M X M X
M ! X ! ak al ∇ ψ k ϕul ; ϕui ¼ − g 1 ∇ p; ϕui þ ak ∇ ⋄ϕuk ; ϕui
k¼1 l¼1
2.2. Calculation of the basis functions
k¼1
ð19Þ
Only velocity basis functions are generated for pure fluid flow [23], but two kinds of basis functions, velocity basis functions and temperature basis functions, are created for natural convection problem by decomposing their corresponding sampling matrix shown in Fig. 1. Both Cartesian velocity components and contravariant components are included in the momentum equation in a body-fitted coordinate, and they can be expressed as the linear superposition of their corresponding basis functions as follows M ! X ! ak ϕ k u ¼
M X M X k¼1 l¼1
k¼1
M X M X k¼1 l¼1
¼− ð16Þ
þ
k¼1
! ! g 1 ∇ p; ϕui − g 2 ∇ p; ϕvi
M X
N h X i ak ∇ ⋄ϕuk ; ϕui þ ∇ ⋄ϕvk ; ϕvi þ JGr bk ϑ k k¼1
For the pressure term, ! ! − g 1 ∇ p; ϕui − g 2 ∇ p; ϕvi ¼ − pξ yη −pη yξ ; ϕui − pη xξ −pξ xη ; ϕvi ¼ pξ ; −yη ϕui þ pη ; yξ ϕui − pη ; xξ ϕvi þ pξ ; xη ϕvi ¼ − pξ ; ψui − pη ; ψvi
ð18Þ
Once the contravariant velocity basis functions are obtained, the Cartesian velocity basis functions can be simply calculated by Eqs. (17)
Velocity Sampling Matrix
h ! ! i ∇ ψ k ϕul ; ϕui þ ∇ ψ k ϕvl ; ϕvi
ð21Þ
ð17Þ
ϕvk ¼ ψuk yξ þ ψvk yη =J
ak al
k¼1
! ! where ϕ k and ψ k are the Cartesian and contravariant velocity basis functions, respectively.Substitute Eqs. (15) and (16) into Eqs. (9) and (10), we can get
k¼1
Sum Eqs. (19) and (20), we obtain
k¼1
ϕuk ¼ ψuk xξ þ ψvk xη =J
Temperature Sampling Matrix
U1 U2 U3
T1 T2
...
.... T3
UNL V1 V2 V3
t
s1
...
si
...
sN
s1
...
si
...
sN
.
Tj
...
TN -1
VNL
TN
t
t
t
Any sampling vector si for velocity
ð20Þ
M N X X þ ak ∇ ⋄ϕvk ; ϕvi þ JGr bk ϑ k
ð15Þ
M ! ! X ak ψ k U ¼
! ! ak al ∇ ψ k ϕvl ; ϕvi ¼ − g 2 ∇ p; ϕvi
Any sampling vector si for temperature
Fig. 1. Sketch map of sampling matrices.
ð22Þ
D. Han et al. / International Communications in Heat and Mass Transfer 68 (2015) 104–113
l
As the contravariant velocity basis functions satisfy the continuity equation, namely, (ψui )ξ + (ψvi )η = 0, we get
Adiabatic Th
107
! ! − g 1 ∇ p; ϕui − g 2 ∇ p; ϕvi ¼ − ∮ Pψui dη−Pψvi dξ ð24Þ For natural convection in the enclosures, the velocity at the boundary is always zero in each sampling case, and the values of velocity basis functions at the boundary are also zero. Then the right side of the Eq. (24) diminishes and Eq.(24) becomes ! ! ð25Þ − g 1 ∇ p; ϕui − g 2 ∇ p; ϕvi ¼ 0
Tl l
Adiabatic
For the diffusion term,
Fig. 2. Natural convection in a parallelogram cavity.
M X
By applying Greene's theorem, Eq. (22) can be rewritten as follows: ! ! − g 1 ∇ p; ϕui − g 2 ∇ p; ϕvi ð23Þ ¼ P; ψui ξ þ ψvi η −∮ Pψui dη−Pψvi dξ
ak ∇ ⋄ϕuk ; ϕui þ ∇ ⋄ϕvk ; ϕvi
k¼1
M ! X
¼ ∮ ϕui ⋄u þ ϕvi ⋄v d s − ak ⋄ϕuk ; ∇ϕui þ ⋄ϕvk ; ∇ϕvi
! ! ! where d s ¼ dη i −dξ j
ð26Þ
k¼1
Table 1 Energy contributions of the first 6 basis function for Example 1. n Basis functions of velocity
Basis functions of temperature
λn ςn ϖn λn ςn ϖn
1
2
3
4
5
6
5.04 × 104 94.419 94.419 3.10 × 105 94.916 94.916
2.57 × 103 4.814 99.233 1.53 × 104 4.699 99.615
2.76 × 102 0.516 99.750 9.40 × 102 0.288 99.903
84.18 0.158 99.907 2.13 × 102 0.065 99.968
25.63 0.048 99.955 65.74 0.020 99.988
15.01 0.028 99.983 18.64 0.006 99.994
1st
5th
10 th
15th Fig. 3. Contours of ψuk for Example 1.
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D. Han et al. / International Communications in Heat and Mass Transfer 68 (2015) 104–113
Substituting Eqs. (25) and (26) into Eq. (21), we get M X M X k¼1 l¼1
M N X X ak al Dkli − ak H ki −Si − bk F ki ¼ 0 i ¼ 1; 2…M ð27Þ k¼1
k¼1
where ! ! Dkli ¼ ∇ ψ k ϕul ; ϕui þ ψ k ∇ ϕvl ; ϕvi ;
H ki ¼ − ⋄ϕuk ; ∇ϕui þ ⋄ϕvk ; ∇ϕvi ; F ki ¼ JGrϑk ; ϕvi ; ! Si ¼ ∮ ϕui ⋄u þ ϕvi ⋄v d s :
The combination of Eqs. (27) and (28) gives the POD reduced-order model for steady-state natural convection based on BFC. It consists of two series of pluralistic quadratic nonlinear equations for ak and bk. These equations are coupled and solved by a Newton–Raphson method in this study. Once ak and bk are obtained, the velocity field and temperature field can be constructed by the summation of the product of spectral coefficients and corresponding basis functions. It should be noted that even though the above-mentioned theory is in 2-D, it can be easily extended to 3-D. 3. Results and discussions
project the equation onto the space of temperature basis functions. The reduced-order model of energy equation can be obtained after deductions,
In this section, two examples are designed to test the performance of the proposed model to fast calculate natural convections in different domains with different physical conditions. Example 1 and Example 2 are involved with the steady-state natural convection of a fluid with Pr = 1 in the parallelogram cavity and the eccentric semi-annulus, respectively. Example 2 is more complex than Example 1 for its shape depends on more parameters.
M X N X
3.1. Natural convection in a parallelogram cavity (Example 1)
(2) Reduced-order model of the energy equation M N ! ! Similarly, substitute U ¼ ∑ ak ψ k ; Θ ¼ ∑ bk ϑ k into Eq. (14) and k¼1
k¼1 l¼1
k¼1
N X ak bl DEkli − bk H Eki −SEi ¼ 0 i ¼ 1; 2…N k¼1
where ! 1 DEkli ¼ ∇ ψ k ϑl ; ϑ i ; H Eki ¼ − ð⋄ϑ k ; ∇ϑi Þ; Pr 1 ! SEi ¼ ∮ ϑ i ⋄Θ d s : Pr
ð28Þ
The natural convection in parallelogram cavities with different inclination angle θ depicted in Fig. 2 is studied. All the walls remain stationary while the left and the right ones are kept at a higher temperature Th and lower temperature Tl, respectively. The top and the bottom boundaries are adiabatic. The implementation procedure of the proposed POD reduced-order model is given below.
1st
5 th
10th
15th Fig. 4. Contours of ϑk for Example 1.
D. Han et al. / International Communications in Heat and Mass Transfer 68 (2015) 104–113
N
N
l¼1 Nt
l¼1 Nt
l¼1
l¼1
t t ! ! ! ev ¼ N1t ∑ j u POD ðlÞ− u FVM ðlÞj= ∑ j u FVM ðlÞj
,
3.0
0.21
2.5
0.18
Temperature Velocity
eΘ (%)
0.15
2.0
0.12
1.5
0.09
ev (%)
The first step is sampling. A finite volume method ( FVM) is used to get the sampling matrix on a mesh system with 128 × 128 grids. The sampling conditions are as follows: every 5 values are chosen evenly in each interval, i.e., 102 to 103, 103 to 104, 104 to 105, and 105 to 106 for the Gr; θ ∈ {60∘, 75∘, 90∘}. By permutation of the above-mentioned parameter values, the total sampling cases with different shapes and boundary conditions are 60. The velocity and temperature fields of the sampling cases are obtained by FVM and are employed to construct the sampling matrices. Subsequently, the basis function is obtained by decomposing the sampling matrix with a “snapshot” method [24]. The eigenvalues, energy contribution, and accumulative energy of the first 6 velocity basis functions as well as temperature basis function are given in Table. 1. As it is shown in Table. 1, the first 6 basis functions have more than 99.9% of the total energy for both velocity and temperature. Four typical contours of the velocity and temperature basis functions are given in the Figs. 3 and 4, respectively. Generally speaking, the first 6 basis functions can capture the main characters of the natural convection since they occupy most of energy, but they are not enough to get an accurate result in this example. The 7th–15th basis functions have a very small amount of energy, but they can capture the necessary information in local places. To illustrate this, 6 and 15 basis functions are employed by the reduced-order model, respectively, to calculate one typical case (θ = 90∘, Gr = 1.0 × 105). Moreover, contours of u velocity obtained by reduced-order model are compared with the ones of FVM in Fig. 5. It is shown that even the first 6 basis functions can capture more than 99.9% of the total energy, to reach higher accuracy, the first 15 velocity and temperature basis functions are needed in this example. Finally, by permutation of the parameters of Grashof number and inclination angle, which are chosen from Gr ∈ {1.0 × 102, 3.0 × 102, 5.0 × 103, 7.0 × 104, 3.0 × 105, 1.1 × 106} and θ ∈ {50∘, 73∘, 80∘, 90∘}, 24 cases different from the sampling cases are obtained to test the accuracy and robustness of the proposed reduced-order model. In order to quantitatively evaluate the accuracy of the proposed model, the errors of velocity and temperature are defined as
109
1.0
0.06 0.5 0.03 0.0 0
4
8
12
16
20
24
Ntest Fig. 6. Relative error of velocity and temperature by POD in Example 1.
that the proposed model can predict the natural convection in the skew cavities with different shapes accurately, which shows its robustness and its advantages compared with the existing reduced-order models. The velocity contours and temperature contours of four typical cases obtained by both FVM and POD methods are compared in Fig. 7. Those cases are (a) θ = 50∘, Gr = 1.1 × 106; (b) θ = 73∘, Gr = 1.1 × 105; (c) θ = 80∘, Gr = 5.0 × 103; and (d) θ = 90∘, Gr = 1.0 × 102. The relative velocity errors are 1.78%, 1.31%, 0.39%, and 0.67%, respectively, and the relative temperatures errors 0.17%, 0.10%, 0.05%, and 0.04%, respectively. It is shown clearly from the figures that the agreements between the contours by FVM and POD methods are perfect for both velocity and temperature for all the four cases whose shape and Grashof number are quite different. The contour comparisons have vividly shown the degree of accuracy of the proposed POD reduced-order model. However, the accuracy of the established reduced-order model will decrease when the value of θ or Gr goes further and further away from the sampling intervals, and such kind of phenomenon is suffered by most of the POD reduced-order models.
eΘ ¼ N1t ∑ jΘPOD ðlÞ−Θ FVM ðlÞj= ∑ jΘ FVM ðlÞj
3.2. Natural convection in an eccentric semi-annulus (Example 2)
where the subscripts “FVM” and “POD” mean the results calculated by POD reduced-order model and FVM respectively. Fig. 6 shows the errors of the 24 test cases, in which the maximum errors for velocity and temperature are 2.06% and 0.20%, respectively. As the error is relative small, it indicates the proposed model can obtain an accurate solution in engineering view. In addition, it is clearly seen
The physical domain of Example 2 is shown in Fig. 8, and its shape depends on several geometry parameters, i.e., r1, r2, h. The established reduced-order model is applied to solve the natural convection in the domains with the fixed outer semi-circle radius r1, the changing inner semi-circle r2, and the changing eccentricity h.
(a) 6 basis functions
(b)15 basis functions
Fig. 5. u velocity contours calculated by POD ROM with 6 and 15 basis functions and their comparisons with FVM results in the case θ = 90∘, Gr = 1.0 × 105.
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D. Han et al. / International Communications in Heat and Mass Transfer 68 (2015) 104–113
(a)
(b)
(c)
(d) Fig. 7. Comparisons of the contours by FVM and POD methods in Example 1.
All the boundaries are stationary with higher and lower temperature in the inner arc wall and outer arc wall, respectively. In addition, r1 is chosen to be the character length. Thus, R1 = r1/r = 1, R2 = r2/r1, H = h/r1, Gr = gβΔT(r1)3/ν2 ( the β here stand for expansion parameter). The sampling parameters are as follows: R2 ∈ {0.134, 0.267, 0.400}, H ∈ {− 0.2, 0.0, 0.2}, and 10 values are chosen evenly in the interval [1.0 × 102, 1.0 × 104]. Ninety sampling cases are obtained by permuting the above-mentioned parameters. The basis function is obtained by decomposing the sampling matrix. The obtained eigenvalues, energy contribution and the accumulative energy contribution of the first 6 velocity basis functions as well as
temperature basis functions are given in Table. 2. The first 20 velocity and temperature basis functions are utilized by the reduced-order model for the prediction. The contours of the 1st and the 20th velocity basis function are shown in Fig. 9, and the contours of the 1st and the 20th temperature basis function are shown in Fig. 10. By permutation of the values of the parameters R2 ∈ {0.067, 0.200, 0.333, 0.467}, H ∈ {0, 0.1, − 0.1, 0.233, − 0.233}, and Gr ∈ {500, 3000, 7000, 9000}, 80 test cases are obtained to test the accuracy and robustness of the proposed reduced-order model. The relative error of the 80 test cases can be seen in Fig. 11, which reveals that the errors of the reduced-order model are small. The velocity
D. Han et al. / International Communications in Heat and Mass Transfer 68 (2015) 104–113
contours and temperature contours of four typical cases obtained by both FVM and POD methods are compared in Fig. 12. These are (a) R2 = 0.067, H = 0, Gr = 9000; (b) R2 = 0.2, H = 0.233, Gr = 500; (c) R2 = 0.333, H = ‐ 0.233, Gr = 3000; and (d) R2 = 0.333, H = ‐ 0.233, Gr = 3000, respectively. The relative velocity errors of the four test cases are 1.71%, 0.43%, 0.33%, and 3.02%, respectively, and the relative temperature errors of the four test cases are 1.22%, 0.06%, 0.07%, and 0.18%, respectively. As it can be seen in Fig. 12, even though the geometries and physical fields of the four test cases are totally different from each other as well as the geometries and fields of the sampling cases, the established reduced-order model can also acquire the accurate results for all of them. This indicates the good robustness of the established reducedorder model. Statistically, the mean calculation speeds of POD reduced-order model are about 100 and 110 times faster than FVM on Example 1 and Example 2, respectively.
Adiabatic r1 O1 O2
h
Tl
Th
r2
111
Adiabatic Fig. 8. Natural convection in an eccentric semi-annulus.
Table 2 Energy contributions of the first 6 basis function for Example 2. n Basis function of velocity
Basis function of temperature
λn ςn ϖn λn ςn ϖn
1
2
3
4
5
6
1.11 × 104 98.118 98.118 1.89 × 105 97.904 97.904
1.31 × 102 1.161 99.279 3.41 × 103 1.764 99.668
2.76 × 102 0.547 99.826 4.64 × 102 0.240 99.908
61.96 0.124 99.950 1.21 × 102 0.062 99.970
14.05 0.022 99.972 46.77 0.024 99.994
2.50 0.019 99.992 4.41 0.002 99.997
1st
20 th Fig. 9. Contours of ψuk for Example 2.
1st
20th Fig. 10. Contours of ϑk for Example 2.
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D. Han et al. / International Communications in Heat and Mass Transfer 68 (2015) 104–113
4
1.5
3
Temperature Velocity
0.9
2
0.6
1
0.3
0
ev(%)
eΘ (%)
1.2
-1
0.0 0
20
40
60
80
Ntest Fig. 11. Relative error of velocity and temperature calculated by reduced-order model in Example 2.
Through the above two examples, the good performance of the proposed POD reduced-order model on accuracy, efficiency and robustness have been illustrated. 4. Conclusions In this paper, by Galerkin projection, the POD reduced-order model based on BFC for steady-state natural convection is established for the first time. By the established reduced-order model, the fast calculations of natural convection in different-shape domains having identical geometry characters can be achieved. In addition, the established reduced-order model has good degree of accuracy and robustness.
(a)
(c)
Two examples are given in this paper to illustrate the good performance of the established reduced-order model. It is found that the relative errors of velocity calculation are less than 2.06% and 3.02%, respectively; the relative errors of temperature calculation are less than 0.20% and 1.29%, respectively. In addition, the calculation speed of the established reduced-order model is no less than 100 times faster than that of FVM in the two given examples. Acknowledgments The study was supported by the National Science Foundation of China (No.51325603, No. 51134006, and No. 51476073). References [1] J.L. Lumley, The structure of inhomogeneous turbulent flows, Atmos. Turbul. Radio Wave Propag. (1967) 166–178. [2] A.E. Deane, I.G. Kevrekidis, G.E. Karniadakis, et al., Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders, Phys. Fluids A 3 (10) (1991) 2337–2354. [3] S.S. Ravindran, Reduced-order controllers for control of flow past an airfoil, Int. J. Numer. Methods Fluids 50 (5) (2006) 531–554. [4] S.S. Ravindran, Control of flow separation over a forward-facing step by model reduction, Comput. Methods Appl. Mech. Eng. 191 (41) (2002) 4599–4617. [5] D. My-Ha, K.M. Lim, B.C. Khoo, et al., Real-time optimization using proper orthogonal decomposition: free surface shape prediction due to underwater bubble dynamics, Comput. Fluids 36 (3) (2007) 499–512. [6] M. Fogleman, J.L. Lumley, D. Rempfer, et al., Application of the proper orthogonal decomposition to datasets of internal combustion engine flows, J. Turbul. 5 (23) (2004) 1–3. [7] B. Podvin, A proper-orthogonal-decomposition-based model for the wall layer of a turbulent channel flow, Phys. Fluids 21 (1) (2009). [8] B. Galletti, C.H. Bruneau, L. Zannetti, et al., Low-order modelling of laminar flow regimes past a confined square cylinder, J. Fluid Mech. 503 (2004) 161–170. [9] M. Couplet, C. Basdevant, P. Sagaut, Calibrated reduced-order POD-Galerkin system for fluid flow modeling, J. Comput. Phys. 207 (1) (2005) 192–220. [10] M. Bergmann, C.H. Bruneau, A. Iollo, Enablers for robust POD models, J. Comput. Phys. 228 (2) (2009) 516–538.
(b)
(d) Fig. 12. Comparisons of the contours by FVM and POD methods in Example 2.
D. Han et al. / International Communications in Heat and Mass Transfer 68 (2015) 104–113 [11] H.M. Park, D.H. Cho, Low dimensional modeling of flow reactors, Int. J. Heat Mass Transf. 39 (16) (1996) 3311–3323. [12] M.V. Tabib, J.B. Joshi, Analysis of dominant flow structures and their flow dynamics in chemical process equipment using snapshot proper orthogonal decomposition technique, Chem. Eng. Sci. 63 (14) (2008) 3695–3715. [13] P. Ding, W.Q. Tao, Reduced order modeling of fluid flow and heat transfer in tube-fin heat exchanger, J. China Univ. Petrol. 35 (2) (2011) 137–140. [14] X. Du, H. Hu, Y. Shen, et al., Reduced order analysis of flow and heat transfer for aircooled condenser of power generating unit, Appl. Therm. Eng. 51 (1) (2013) 383–392. [15] Q. Nie, Y. Joshi, Reduced order modeling and experimental validation of steady turbulent convection in connected domains, Int. J. Heat Mass Transf. 51 (25) (2008) 6063–6076. [16] R. Ghosh, Y. Joshi, Error estimation in POD-based dynamic reduced-order thermal modeling of data centers, Int. J. Heat Mass Transf. 57 (2) (2013) 698–707. [17] H. Gunes, R.A. Sahan, A. Liakopoulos, Low-dimensional representation of transitional buoyancy-driven flow in a vertical channel with discrete heaters, Asme Publ. Htd. 303 (1995) 125–138.
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[18] H. Gunes, A. Liakopoulos, R.A. Sahan, Low-dimensional description of oscillatory thermal convection: the small Prandtl number limit, Theor. Comput. Fluid Dyn. 9 (1) (1997) 1–16. [19] H.M. Park, W.J. Lee, Feedback control of natural convection, Comput. Methods Appl. Mech. Eng. 191 (8) (2001) 1013–1028. [20] B. Yu, G. Yu, Z. Cao, et al., Fast calculation of the soil temperature field around a buried oil pipeline using a body-fitted coordinates-based POD-Galerkin reducedorder model, Numer. Heat Transfer, Part A 63 (10) (2013) 776–794. [21] D. Han, B. Yu, G. Yu, et al., Study on a BFC-based POD-Galerkin ROM for the steadystate heat transfer problem, Int. J. Heat Mass Transf. 69 (2014) 1–5. [22] D. Han, B. Yu, X. Zhang, Study on a BFC-based POD-Galerkin reduced-order model for the unsteady-state variable-property heat transfer problem, Numer. Heat Transfer, Part B: Fundam. 65 (3) (2014) 256–281. [23] D. Han, B. Yu, Y. Wang, et al., POD reduced-order model for steady laminar flow based on the body-fitted coordinate, Commun. Comput. Phys. (2015) (Submitted for publication). [24] L. Sirovich, Turbulence and the dynamics of coherent structures. Part I–III, Q. Appl. Math. 45 (3) (1987) 561–590.
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
POD reduced-order model for steady natural convection based on a body-fitted coordinate☆ Dongxu Han a, Bo Yu a,⁎, Jingjing Chen a, Yi Wang a, Ye Wang b a National Engineering Laboratory for Pipeline Safety, Beijing Key Laboratory of Urban Oil and Gas Distribution Technology, China University of Petroleum, Beijing 102249, People's Republic of China b Key Laboratory of Railway Vehicle Thermal Engineering, Ministry of Education of China, Lanzhou Jiaotong University, Lanzhou 7370070, People's Republic of China
a r t i c l e
i n f o
Available online 4 September 2015 Keywords: Proper orthogonal decomposition Reduced-order model Geometry parameter Natural convection Body-fitted coordinate
a b s t r a c t In this article, a POD reduced-order model for steady-state natural convection based on a body-fitted coordinate system is established. The velocity basis functions and temperature basis functions are generated, respectively. Based on the basis functions, the governing equations for velocity spectrum coefficient and temperature spectrum coefficient are deduced, which are coupled pluralistic quadratic nonlinear equations and solved iteratively by the Newton–Raphson method. Compared with the existing natural convection POD reduced-order models, the major advantage of the proposed POD model is its capability to calculate natural convections in differentshape domains having identical geometry characters. Two typical examples are given to show the model implementation procedure as well as to illustrate its good performance in terms of accuracy and robustness. It is found that even though the geometries and the physical conditions of the test cases differ greatly from those of the sampling cases, the reduced-order model can acquire accurate results efficiently. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Proper orthogonal decomposition (POD) can extract characteristic information (basis functions) from large amount of data, and the original data can be reconstructed with just a few basis functions,which makes POD a powerful tool for model reduction. POD reduced-order models can depict the physical problems accurately and efficiently for engineering application. In the field of fluid mechanics and thermal engineering, some POD reduced-order models have been developed and some related studies are briefly reviewed below. Lumely [1] first introduced POD technique for turbulent analysis. Deane et al. [2] established a POD-Galerkin reduced-order model for transition flow. Ravindran [3,4] applied a POD reduced-order model to the real-time control in the flow passing an airfoil as well as the flow separation over a forward facing step. My-Ha [5] employed POD method for optimizing parameters for underwater bubble explosions to generate an expected free surface shape. Foglema et al. [6] analyzed the flow field in the internal-combustion engines. More studies on fluid flow can be referred to Ref. [7–12]. Ding et al. [13] applied a Galerkin-POD reduced-order model to fast calculate the heat transfer and fluid flow in a tube-fin heat exchanger; Du et al. [14] achieved the fast thermal calculation of an air-cooled
☆ Communicated by W.J. Minkowycz ⁎ Corresponding author. E-mail address: [email protected] (B. Yu).
http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.024 0735-1933/© 2015 Elsevier Ltd. All rights reserved.
condenser of power generating unit using a POD reduced-order model. Joshi et al. [15,16] employed this technique to the fast thermal calculation of an electronic cabinet and data centers. Few studies have been carried out on the Galerkin-POD reducedorder model for natural convection. Guns et al. established a POD reduced-order model in Ref. [17] and applied it to predict the natural convection in a vertical channel in Ref. [18]. In addition, Park and Lee [19] applied a POD reduced-order model on real-time control of natural convection. The above-mentioned studies on POD reduced-order models for natural convection are all for fixed domain. However, fast calculations of natural convection in domains with different shapes having identical geometry characters are always required in engineering. For example, heated crude oil pipelines have different diameters and buried depths. After shutdowns, fast oil temperature drop (natural convection problem) calculations in the pipes are required to provide useful information to guide the crude oil pipeline operation. The objective of this study is to develop a POD reduced-order model to predict natural convection in different-shape domains having identical geometry characters. Previously, our research group has developed POD reduced-order models based on body-fitted coordinate (BFC) for fast heat conduction calculation [20–22] and fluid flow calculation without heat transfer [23] in domains with different shapes having identical geometry characters. In this article, we extend the technique to steady-state natural convection. The layout of this paper is as follows: the methodology of the BFC based POD reduced-order model for natural convection is introduced
D. Han et al. / International Communications in Heat and Mass Transfer 68 (2015) 104–113
Nomenclature Roman symbols a Thermal diffusivity (m2/s) Spectrum coefficient for the kth velocity basis function ak Spectrum coefficient for the kth temperature basis bk function ! ! ! ds dη i −dξ j Relative error of velocity calculated by reduced-order ev model Relative error of dimensionless temperature calculated eΘ by reduced-order model g Gravity (m/s2) Gr Grashof number h Eccentricity between the inner and the outer semicircles (m) H Dimensionless eccentricity between the inner and the outer semi-circles J Jacobi factor l Length of the side in a parallelogram cavity (m) The total number of the grid points Nt Serial number of the test cases Ntest p Dimensionless pressure Pr Prandtl number Radius of outer semi-circle (m) r1 Radius of inner semi-circle (m) r2 Dimensionless radius of outer semi-circle R1 Dimensionless radius of inner semi-circle R2 T Dimensional temperature (∘C) Higher temperature, lower temperature (∘C) Th, Tl u, v Cartesian components of dimensionless velocity ! ! ! u u i þv j U, V Contravariant components of velocity x, y Cartesian coordinate Greek symbols α, β, γ Variables related with grids Cartesian components of basis functions, corresponding ϕu, ϕv to u, v ! ! ! ϕ ϕu i þ ϕv j Eigenvalue related to the nth basis function λn ν Kinematic viscosity (m2/s) Accumulative energy contribution of the first n basis ϖn function θ The angle of parallelogram cavity l Θ Dimensionless temperature, Θ ¼ TT−T h −T l ϑ Temperature basis functions Energy contribution of the nth basis function ςn ξ, η Body-fitted coordinate Contravariant components of basis functions, correψu, ψv sponding to U, V ! ! ! ψ ψu i þ ψv j (,) Hilbert inner product ∇ Hamilton operator ! ! ⋄ Author defined operator ⋄ ¼ ∂n∂ ξ i þ ∂n∂η j Subscript ()ξ, ()η
∂ðÞ ∂ðÞ ; ∂ξ ∂η
105
2. Methodology In this section, the governing equations for steady-state natural convection in a body-fitted coordinate are first introduced. Subsequently, the principle of basis functions calculation is given briefly. Finally, the derivation of the BFC based POD reduced-order model is presented in details. 2.1. Governing equation for steady-state natural convection A two-dimensional incompressible steady-state natural convection problem, in a physical domain Ωp, is considered herein, which is governed by Navier–Stokes equations and the energy equation. Based on Boussinesq assumption, the 2-D natural convection on Cartesian coordinate in dimensionless form with the gravity force on y axis can be depicted by Eqs. (1)–(4). ∂u ∂v þ ¼0 ∂x ∂y
ð1Þ
∂ðuuÞ ∂ðvuÞ ∂p ∂ ∂u ∂ ∂u þ þ ¼− þ ∂x ∂y ∂x ∂x ∂x ∂y ∂y
ð2Þ
∂ðuvÞ ∂ðvvÞ ∂p ∂ ∂v ∂ ∂v þ þ GrΘ þ ¼− þ ∂x ∂y ∂y ∂x ∂x ∂y ∂y
ð3Þ
∂ðuΘÞ ∂ðvΘÞ 1 ∂ ∂Θ ∂ ∂Θ þ þ ¼ Pr ∂x ∂x ∂x ∂y ∂y ∂y
ð4Þ
! where u is the dimensionless velocity, Θ stands for the dimensionless l temperature and Θ ¼ TT−T , Gr denotes Grashof number and Gr ¼ gβΔTl ν2 h −T l
3
(the β here stand for expansion parameter), and Pr represents Prandtl number and Pr ¼ νa . By body-fitted coordinate transformation, Ωp on the physical plane is mapped to Ωc on calculation plane. Correspondingly, the governing equations Eqs. (1)–(4) are transformed to Eqs. (5)–(8), respectively. ∂U ∂V þ ¼0 ∂ξ ∂η
ð5Þ
∂ 1 ∂ðUuÞ ∂ðVuÞ þ ¼ − P ξ yη −P η yξ þ αuξ −βuη ∂ξ ∂η ∂ξ J ∂ 1 γuη −βuξ þ ∂η J
ð6Þ
∂ 1 ∂ðUvÞ ∂ðVvÞ þ ¼ − −P ξ xη þ P η xξ þ αvξ −βvη ∂ξ ∂η ∂ξ J ∂ 1 γvη −βvξ þ JGrΘ þ ∂η J
ð7Þ
∂ðUΘÞ ∂ðVΘÞ 1 ∂ 1 1 ∂ 1 þ ¼ αΘξ −βΘη þ γΘη −βΘξ Pr ∂ξ J Pr ∂η J ∂ξ ∂η
ð8Þ
where α ¼ x2η þ y2η ; β ¼ xξ xη þ yξ yη ; γ ¼ x2ξ þ y2ξ ; J ¼ xξ yη −xη yξ Contravariant components of velocity
in Section 2; two examples are presented to illustrate the good performance of the proposed model in Section 3; and some conclusions are given in the Section 4.
U ¼ uyη −vxη
ð9Þ
V ¼ vxξ −uyξ
ð10Þ
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For the sake of convenience, Eqs. (5)–(8) are written in weak form respectively as follows: ! ∇ U ¼0
ð11Þ
! ! ∇ U u ¼ − g 1 ∇ p þ ∇ ð⋄uÞ
ð12Þ
! ! ∇ U v ¼ − g 2 ∇ p þ ∇ ð⋄vÞ þ JGrΘ
ð13Þ
and (18), and vice versa. This means in principle only one kind of velocity basis functions needs to be generated by the sampling matrix and the other can be obtained by simple algebraic calculations. However, only the contravariant component of velocity in the sampling matrix can ensure the obtained velocity basis function satisfy the continuity equation on any domain with different shapes, which is a precondition of the derivation of POD reduced-order model; see details in Ref. [23]. 2.3. Establishment of the reduced-order model
! 1 ∇ ð⋄ΘÞ ∇ UΘ ¼ Pr
ð14Þ
! ∂! ∂ where ∇ is Hamilton operator, ∇ ¼ ∂ξ i þ ∂η j ; ⋄ is an operator (similar
(1) Reduced-order model of the momentum equation M M N ! ! ! Substituting u ¼ ∑ ak ϕ k ; U ¼ ∑ ak ψ k ; Θ ¼ ∑ bk ϑ k into k¼1
as Hamilton operator) introduced by the authors for concise expression, ! ! α f −β f γ f −β f ⋄ ¼ ∂n∂ ξ i þ ∂n∂ η j ; and for any f, ∂n∂ ξ ðf Þ ¼ ξ J η , ∂n∂η ðf Þ ¼ η J ξ and ! ! ! ! ! ! g 1 ¼ yη i −yξ j , g 2 ¼ −xη i þ xξ j .
k¼1
k¼1
Eqs. (11) and (12) and projecting them onto the space of velocity basis function ϕui and ϕvi , respectively, yields M X M X
M ! X ! ak al ∇ ψ k ϕul ; ϕui ¼ − g 1 ∇ p; ϕui þ ak ∇ ⋄ϕuk ; ϕui
k¼1 l¼1
2.2. Calculation of the basis functions
k¼1
ð19Þ
Only velocity basis functions are generated for pure fluid flow [23], but two kinds of basis functions, velocity basis functions and temperature basis functions, are created for natural convection problem by decomposing their corresponding sampling matrix shown in Fig. 1. Both Cartesian velocity components and contravariant components are included in the momentum equation in a body-fitted coordinate, and they can be expressed as the linear superposition of their corresponding basis functions as follows M ! X ! ak ϕ k u ¼
M X M X k¼1 l¼1
k¼1
M X M X k¼1 l¼1
¼− ð16Þ
þ
k¼1
! ! g 1 ∇ p; ϕui − g 2 ∇ p; ϕvi
M X
N h X i ak ∇ ⋄ϕuk ; ϕui þ ∇ ⋄ϕvk ; ϕvi þ JGr bk ϑ k k¼1
For the pressure term, ! ! − g 1 ∇ p; ϕui − g 2 ∇ p; ϕvi ¼ − pξ yη −pη yξ ; ϕui − pη xξ −pξ xη ; ϕvi ¼ pξ ; −yη ϕui þ pη ; yξ ϕui − pη ; xξ ϕvi þ pξ ; xη ϕvi ¼ − pξ ; ψui − pη ; ψvi
ð18Þ
Once the contravariant velocity basis functions are obtained, the Cartesian velocity basis functions can be simply calculated by Eqs. (17)
Velocity Sampling Matrix
h ! ! i ∇ ψ k ϕul ; ϕui þ ∇ ψ k ϕvl ; ϕvi
ð21Þ
ð17Þ
ϕvk ¼ ψuk yξ þ ψvk yη =J
ak al
k¼1
! ! where ϕ k and ψ k are the Cartesian and contravariant velocity basis functions, respectively.Substitute Eqs. (15) and (16) into Eqs. (9) and (10), we can get
k¼1
Sum Eqs. (19) and (20), we obtain
k¼1
ϕuk ¼ ψuk xξ þ ψvk xη =J
Temperature Sampling Matrix
U1 U2 U3
T1 T2
...
.... T3
UNL V1 V2 V3
t
s1
...
si
...
sN
s1
...
si
...
sN
.
Tj
...
TN -1
VNL
TN
t
t
t
Any sampling vector si for velocity
ð20Þ
M N X X þ ak ∇ ⋄ϕvk ; ϕvi þ JGr bk ϑ k
ð15Þ
M ! ! X ak ψ k U ¼
! ! ak al ∇ ψ k ϕvl ; ϕvi ¼ − g 2 ∇ p; ϕvi
Any sampling vector si for temperature
Fig. 1. Sketch map of sampling matrices.
ð22Þ
D. Han et al. / International Communications in Heat and Mass Transfer 68 (2015) 104–113
l
As the contravariant velocity basis functions satisfy the continuity equation, namely, (ψui )ξ + (ψvi )η = 0, we get
Adiabatic Th
107
! ! − g 1 ∇ p; ϕui − g 2 ∇ p; ϕvi ¼ − ∮ Pψui dη−Pψvi dξ ð24Þ For natural convection in the enclosures, the velocity at the boundary is always zero in each sampling case, and the values of velocity basis functions at the boundary are also zero. Then the right side of the Eq. (24) diminishes and Eq.(24) becomes ! ! ð25Þ − g 1 ∇ p; ϕui − g 2 ∇ p; ϕvi ¼ 0
Tl l
Adiabatic
For the diffusion term,
Fig. 2. Natural convection in a parallelogram cavity.
M X
By applying Greene's theorem, Eq. (22) can be rewritten as follows: ! ! − g 1 ∇ p; ϕui − g 2 ∇ p; ϕvi ð23Þ ¼ P; ψui ξ þ ψvi η −∮ Pψui dη−Pψvi dξ
ak ∇ ⋄ϕuk ; ϕui þ ∇ ⋄ϕvk ; ϕvi
k¼1
M ! X
¼ ∮ ϕui ⋄u þ ϕvi ⋄v d s − ak ⋄ϕuk ; ∇ϕui þ ⋄ϕvk ; ∇ϕvi
! ! ! where d s ¼ dη i −dξ j
ð26Þ
k¼1
Table 1 Energy contributions of the first 6 basis function for Example 1. n Basis functions of velocity
Basis functions of temperature
λn ςn ϖn λn ςn ϖn
1
2
3
4
5
6
5.04 × 104 94.419 94.419 3.10 × 105 94.916 94.916
2.57 × 103 4.814 99.233 1.53 × 104 4.699 99.615
2.76 × 102 0.516 99.750 9.40 × 102 0.288 99.903
84.18 0.158 99.907 2.13 × 102 0.065 99.968
25.63 0.048 99.955 65.74 0.020 99.988
15.01 0.028 99.983 18.64 0.006 99.994
1st
5th
10 th
15th Fig. 3. Contours of ψuk for Example 1.
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D. Han et al. / International Communications in Heat and Mass Transfer 68 (2015) 104–113
Substituting Eqs. (25) and (26) into Eq. (21), we get M X M X k¼1 l¼1
M N X X ak al Dkli − ak H ki −Si − bk F ki ¼ 0 i ¼ 1; 2…M ð27Þ k¼1
k¼1
where ! ! Dkli ¼ ∇ ψ k ϕul ; ϕui þ ψ k ∇ ϕvl ; ϕvi ;
H ki ¼ − ⋄ϕuk ; ∇ϕui þ ⋄ϕvk ; ∇ϕvi ; F ki ¼ JGrϑk ; ϕvi ; ! Si ¼ ∮ ϕui ⋄u þ ϕvi ⋄v d s :
The combination of Eqs. (27) and (28) gives the POD reduced-order model for steady-state natural convection based on BFC. It consists of two series of pluralistic quadratic nonlinear equations for ak and bk. These equations are coupled and solved by a Newton–Raphson method in this study. Once ak and bk are obtained, the velocity field and temperature field can be constructed by the summation of the product of spectral coefficients and corresponding basis functions. It should be noted that even though the above-mentioned theory is in 2-D, it can be easily extended to 3-D. 3. Results and discussions
project the equation onto the space of temperature basis functions. The reduced-order model of energy equation can be obtained after deductions,
In this section, two examples are designed to test the performance of the proposed model to fast calculate natural convections in different domains with different physical conditions. Example 1 and Example 2 are involved with the steady-state natural convection of a fluid with Pr = 1 in the parallelogram cavity and the eccentric semi-annulus, respectively. Example 2 is more complex than Example 1 for its shape depends on more parameters.
M X N X
3.1. Natural convection in a parallelogram cavity (Example 1)
(2) Reduced-order model of the energy equation M N ! ! Similarly, substitute U ¼ ∑ ak ψ k ; Θ ¼ ∑ bk ϑ k into Eq. (14) and k¼1
k¼1 l¼1
k¼1
N X ak bl DEkli − bk H Eki −SEi ¼ 0 i ¼ 1; 2…N k¼1
where ! 1 DEkli ¼ ∇ ψ k ϑl ; ϑ i ; H Eki ¼ − ð⋄ϑ k ; ∇ϑi Þ; Pr 1 ! SEi ¼ ∮ ϑ i ⋄Θ d s : Pr
ð28Þ
The natural convection in parallelogram cavities with different inclination angle θ depicted in Fig. 2 is studied. All the walls remain stationary while the left and the right ones are kept at a higher temperature Th and lower temperature Tl, respectively. The top and the bottom boundaries are adiabatic. The implementation procedure of the proposed POD reduced-order model is given below.
1st
5 th
10th
15th Fig. 4. Contours of ϑk for Example 1.
D. Han et al. / International Communications in Heat and Mass Transfer 68 (2015) 104–113
N
N
l¼1 Nt
l¼1 Nt
l¼1
l¼1
t t ! ! ! ev ¼ N1t ∑ j u POD ðlÞ− u FVM ðlÞj= ∑ j u FVM ðlÞj
,
3.0
0.21
2.5
0.18
Temperature Velocity
eΘ (%)
0.15
2.0
0.12
1.5
0.09
ev (%)
The first step is sampling. A finite volume method ( FVM) is used to get the sampling matrix on a mesh system with 128 × 128 grids. The sampling conditions are as follows: every 5 values are chosen evenly in each interval, i.e., 102 to 103, 103 to 104, 104 to 105, and 105 to 106 for the Gr; θ ∈ {60∘, 75∘, 90∘}. By permutation of the above-mentioned parameter values, the total sampling cases with different shapes and boundary conditions are 60. The velocity and temperature fields of the sampling cases are obtained by FVM and are employed to construct the sampling matrices. Subsequently, the basis function is obtained by decomposing the sampling matrix with a “snapshot” method [24]. The eigenvalues, energy contribution, and accumulative energy of the first 6 velocity basis functions as well as temperature basis function are given in Table. 1. As it is shown in Table. 1, the first 6 basis functions have more than 99.9% of the total energy for both velocity and temperature. Four typical contours of the velocity and temperature basis functions are given in the Figs. 3 and 4, respectively. Generally speaking, the first 6 basis functions can capture the main characters of the natural convection since they occupy most of energy, but they are not enough to get an accurate result in this example. The 7th–15th basis functions have a very small amount of energy, but they can capture the necessary information in local places. To illustrate this, 6 and 15 basis functions are employed by the reduced-order model, respectively, to calculate one typical case (θ = 90∘, Gr = 1.0 × 105). Moreover, contours of u velocity obtained by reduced-order model are compared with the ones of FVM in Fig. 5. It is shown that even the first 6 basis functions can capture more than 99.9% of the total energy, to reach higher accuracy, the first 15 velocity and temperature basis functions are needed in this example. Finally, by permutation of the parameters of Grashof number and inclination angle, which are chosen from Gr ∈ {1.0 × 102, 3.0 × 102, 5.0 × 103, 7.0 × 104, 3.0 × 105, 1.1 × 106} and θ ∈ {50∘, 73∘, 80∘, 90∘}, 24 cases different from the sampling cases are obtained to test the accuracy and robustness of the proposed reduced-order model. In order to quantitatively evaluate the accuracy of the proposed model, the errors of velocity and temperature are defined as
109
1.0
0.06 0.5 0.03 0.0 0
4
8
12
16
20
24
Ntest Fig. 6. Relative error of velocity and temperature by POD in Example 1.
that the proposed model can predict the natural convection in the skew cavities with different shapes accurately, which shows its robustness and its advantages compared with the existing reduced-order models. The velocity contours and temperature contours of four typical cases obtained by both FVM and POD methods are compared in Fig. 7. Those cases are (a) θ = 50∘, Gr = 1.1 × 106; (b) θ = 73∘, Gr = 1.1 × 105; (c) θ = 80∘, Gr = 5.0 × 103; and (d) θ = 90∘, Gr = 1.0 × 102. The relative velocity errors are 1.78%, 1.31%, 0.39%, and 0.67%, respectively, and the relative temperatures errors 0.17%, 0.10%, 0.05%, and 0.04%, respectively. It is shown clearly from the figures that the agreements between the contours by FVM and POD methods are perfect for both velocity and temperature for all the four cases whose shape and Grashof number are quite different. The contour comparisons have vividly shown the degree of accuracy of the proposed POD reduced-order model. However, the accuracy of the established reduced-order model will decrease when the value of θ or Gr goes further and further away from the sampling intervals, and such kind of phenomenon is suffered by most of the POD reduced-order models.
eΘ ¼ N1t ∑ jΘPOD ðlÞ−Θ FVM ðlÞj= ∑ jΘ FVM ðlÞj
3.2. Natural convection in an eccentric semi-annulus (Example 2)
where the subscripts “FVM” and “POD” mean the results calculated by POD reduced-order model and FVM respectively. Fig. 6 shows the errors of the 24 test cases, in which the maximum errors for velocity and temperature are 2.06% and 0.20%, respectively. As the error is relative small, it indicates the proposed model can obtain an accurate solution in engineering view. In addition, it is clearly seen
The physical domain of Example 2 is shown in Fig. 8, and its shape depends on several geometry parameters, i.e., r1, r2, h. The established reduced-order model is applied to solve the natural convection in the domains with the fixed outer semi-circle radius r1, the changing inner semi-circle r2, and the changing eccentricity h.
(a) 6 basis functions
(b)15 basis functions
Fig. 5. u velocity contours calculated by POD ROM with 6 and 15 basis functions and their comparisons with FVM results in the case θ = 90∘, Gr = 1.0 × 105.
110
D. Han et al. / International Communications in Heat and Mass Transfer 68 (2015) 104–113
(a)
(b)
(c)
(d) Fig. 7. Comparisons of the contours by FVM and POD methods in Example 1.
All the boundaries are stationary with higher and lower temperature in the inner arc wall and outer arc wall, respectively. In addition, r1 is chosen to be the character length. Thus, R1 = r1/r = 1, R2 = r2/r1, H = h/r1, Gr = gβΔT(r1)3/ν2 ( the β here stand for expansion parameter). The sampling parameters are as follows: R2 ∈ {0.134, 0.267, 0.400}, H ∈ {− 0.2, 0.0, 0.2}, and 10 values are chosen evenly in the interval [1.0 × 102, 1.0 × 104]. Ninety sampling cases are obtained by permuting the above-mentioned parameters. The basis function is obtained by decomposing the sampling matrix. The obtained eigenvalues, energy contribution and the accumulative energy contribution of the first 6 velocity basis functions as well as
temperature basis functions are given in Table. 2. The first 20 velocity and temperature basis functions are utilized by the reduced-order model for the prediction. The contours of the 1st and the 20th velocity basis function are shown in Fig. 9, and the contours of the 1st and the 20th temperature basis function are shown in Fig. 10. By permutation of the values of the parameters R2 ∈ {0.067, 0.200, 0.333, 0.467}, H ∈ {0, 0.1, − 0.1, 0.233, − 0.233}, and Gr ∈ {500, 3000, 7000, 9000}, 80 test cases are obtained to test the accuracy and robustness of the proposed reduced-order model. The relative error of the 80 test cases can be seen in Fig. 11, which reveals that the errors of the reduced-order model are small. The velocity
D. Han et al. / International Communications in Heat and Mass Transfer 68 (2015) 104–113
contours and temperature contours of four typical cases obtained by both FVM and POD methods are compared in Fig. 12. These are (a) R2 = 0.067, H = 0, Gr = 9000; (b) R2 = 0.2, H = 0.233, Gr = 500; (c) R2 = 0.333, H = ‐ 0.233, Gr = 3000; and (d) R2 = 0.333, H = ‐ 0.233, Gr = 3000, respectively. The relative velocity errors of the four test cases are 1.71%, 0.43%, 0.33%, and 3.02%, respectively, and the relative temperature errors of the four test cases are 1.22%, 0.06%, 0.07%, and 0.18%, respectively. As it can be seen in Fig. 12, even though the geometries and physical fields of the four test cases are totally different from each other as well as the geometries and fields of the sampling cases, the established reduced-order model can also acquire the accurate results for all of them. This indicates the good robustness of the established reducedorder model. Statistically, the mean calculation speeds of POD reduced-order model are about 100 and 110 times faster than FVM on Example 1 and Example 2, respectively.
Adiabatic r1 O1 O2
h
Tl
Th
r2
111
Adiabatic Fig. 8. Natural convection in an eccentric semi-annulus.
Table 2 Energy contributions of the first 6 basis function for Example 2. n Basis function of velocity
Basis function of temperature
λn ςn ϖn λn ςn ϖn
1
2
3
4
5
6
1.11 × 104 98.118 98.118 1.89 × 105 97.904 97.904
1.31 × 102 1.161 99.279 3.41 × 103 1.764 99.668
2.76 × 102 0.547 99.826 4.64 × 102 0.240 99.908
61.96 0.124 99.950 1.21 × 102 0.062 99.970
14.05 0.022 99.972 46.77 0.024 99.994
2.50 0.019 99.992 4.41 0.002 99.997
1st
20 th Fig. 9. Contours of ψuk for Example 2.
1st
20th Fig. 10. Contours of ϑk for Example 2.
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D. Han et al. / International Communications in Heat and Mass Transfer 68 (2015) 104–113
4
1.5
3
Temperature Velocity
0.9
2
0.6
1
0.3
0
ev(%)
eΘ (%)
1.2
-1
0.0 0
20
40
60
80
Ntest Fig. 11. Relative error of velocity and temperature calculated by reduced-order model in Example 2.
Through the above two examples, the good performance of the proposed POD reduced-order model on accuracy, efficiency and robustness have been illustrated. 4. Conclusions In this paper, by Galerkin projection, the POD reduced-order model based on BFC for steady-state natural convection is established for the first time. By the established reduced-order model, the fast calculations of natural convection in different-shape domains having identical geometry characters can be achieved. In addition, the established reduced-order model has good degree of accuracy and robustness.
(a)
(c)
Two examples are given in this paper to illustrate the good performance of the established reduced-order model. It is found that the relative errors of velocity calculation are less than 2.06% and 3.02%, respectively; the relative errors of temperature calculation are less than 0.20% and 1.29%, respectively. In addition, the calculation speed of the established reduced-order model is no less than 100 times faster than that of FVM in the two given examples. Acknowledgments The study was supported by the National Science Foundation of China (No.51325603, No. 51134006, and No. 51476073). References [1] J.L. Lumley, The structure of inhomogeneous turbulent flows, Atmos. Turbul. Radio Wave Propag. (1967) 166–178. [2] A.E. Deane, I.G. Kevrekidis, G.E. Karniadakis, et al., Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders, Phys. Fluids A 3 (10) (1991) 2337–2354. [3] S.S. Ravindran, Reduced-order controllers for control of flow past an airfoil, Int. J. Numer. Methods Fluids 50 (5) (2006) 531–554. [4] S.S. Ravindran, Control of flow separation over a forward-facing step by model reduction, Comput. Methods Appl. Mech. Eng. 191 (41) (2002) 4599–4617. [5] D. My-Ha, K.M. Lim, B.C. Khoo, et al., Real-time optimization using proper orthogonal decomposition: free surface shape prediction due to underwater bubble dynamics, Comput. Fluids 36 (3) (2007) 499–512. [6] M. Fogleman, J.L. Lumley, D. Rempfer, et al., Application of the proper orthogonal decomposition to datasets of internal combustion engine flows, J. Turbul. 5 (23) (2004) 1–3. [7] B. Podvin, A proper-orthogonal-decomposition-based model for the wall layer of a turbulent channel flow, Phys. Fluids 21 (1) (2009). [8] B. Galletti, C.H. Bruneau, L. Zannetti, et al., Low-order modelling of laminar flow regimes past a confined square cylinder, J. Fluid Mech. 503 (2004) 161–170. [9] M. Couplet, C. Basdevant, P. Sagaut, Calibrated reduced-order POD-Galerkin system for fluid flow modeling, J. Comput. Phys. 207 (1) (2005) 192–220. [10] M. Bergmann, C.H. Bruneau, A. Iollo, Enablers for robust POD models, J. Comput. Phys. 228 (2) (2009) 516–538.
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(d) Fig. 12. Comparisons of the contours by FVM and POD methods in Example 2.
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