J Braz. Soc. Mech. Sci. Eng. DOI 10.1007/s40430-013-0031-0
TECHNICAL PAPER
Numerical analysis and identification of mixed convection in pulsating flow in a square cavity with two ventilation ports in the presence of a heating block Fatih Selimefendigil
Received: 28 February 2012 / Accepted: 10 March 2013 The Brazilian Society of Mechanical Sciences and Engineering 2013
Abstract In this study, a square cavity with two ventilation ports in the presence of a isothermal heating block placed in the middle of the cavity is numerically analyzed for the mixed convection case in pulsating flow for a range of Richardson numbers (Ri = 1, 10, 100) and at Reynolds number of 300. At the inlet of the ventilation port, pulsating velocity is imposed for Strouhal numbers of 0.1, 0.5, 1, 2.5, 5 and velocity amplitude ratio of 0.3, 0.6 and 0.9. The effect of the pulsation frequency, amplitude and Richardson number on the heat transfer and fluid flow characteristics is numerically analyzed. The results are presented in terms of streamlines, isotherm plots and averaged Nusselt number plots. A neural network-based identification approach is utilized for the low amplitude of pulsating velocity to obtain an input–output model for the heat transfer for a range of pulsating frequencies. Keywords Mixed convection heat transfer Pulsating flow System identification
1 Introduction Mixed convection heat transfer is important for various engineering applications. Design of the heat exchangers, nuclear reactors, solar collectors, cooling of electronic equipments and food industry may be considered as some
Technical Editor: Hora´cio Vielmo. F. Selimefendigil (&) Department of Mechanical Engineering, Celal Bayar University, Manisa, Turkey e-mail:
[email protected]
of them where one has to improve the thermal performance of those systems. Heat transfer and flow field characteristics have been investigated for the mixed convection case by many researchers. Dogan et al. [1] have investigated the mixed convection heat transfer in a horizontal channel with heat sources placed on the top and bottom experimentally for different aspect ratios and for various Reynolds and Grashof numbers. Aminossadati and Ghasemi [2] have numerically studied the mixed convection where a heat source is placed in the cavity of a horizontal channel. The effect of the heat source location and aspect ratio on the thermal transport of the flow have been analyzed for different Richardson numbers. Tmartnhad et al. [3] have analyzed the mixed convection in a trapezoidal cavity heated from the below. They have studied two different configurations where the applied jets are horizontal and vertical at the inlet, respectively. In the first case, they reported a critical Reynolds number where the forced convection mode is dominant. They also proposed correlations between Nusselt number and Reynolds number in both cases. Ozsunar et al. [4] have numerically studied the mixed convection heat transfer in rectangular channels for a range of Reynolds number in the laminar range and at Prandtl number 0.7. The effect of the Grashof number, inclination angle and Reynolds number on the heat transfer enhancements is analyzed. They reported that thermal instability is delayed for higher Reynolds number and move upstream with the increase of Grashof number. Khanafer et al. [5] have numerically analyzed the mixed convection over a backward-facing step for laminar pulsating flow. Their results show that the average Nusselt number increases with an increase in Reynolds and Grashof numbers and decrease with the increase in the pulsating frequency. Boutina and Bessaih [6] have numerically
123
J Braz. Soc. Mech. Sci. Eng.
studied the laminar mixed convection in an inclined channel containing two heat sources. The effects of the Reynolds number, spacing between the heat sources and the inclination angle on the heat transfer and flow field have been analyzed. They reported that an increase in the Reynolds number and spacing between the heat sources enhances the heat transfer. They also reported an optimum inclination angle for the maximum heat transfer rate. Sourtiji et al. [7] have investigated the flow field and heat transfer characteristics in a square cavity with two ventilation ports for the mixed convection case. A pulsating velocity is imposed at the inlet port for a range of Strouhal numbers for Reynolds numbers between 10 and 500. They reported that optimum Strouhal number is between 0.5 and 1 for the best thermal performance and minimum pressure drop. Angirasa [8] has analyzed the mixed convection heat transfer in a vented enclosure. The vertical wall of the enclosure is kept at constant temperature and the ventilation ports are at the top and bottom of the enclosure. The effect of the buoyancy and forced flow is numerically analyzed for a fixed value of Grashof number and changing Reynolds number. They reported that heat transfer is enhanced with increasing Reynolds number, and the relative strengths of the two flow mechanisms determined the flow field and heat transfer characteristics of the cavity. Rahman et al. [9] have studied the steady mixed convection heat transfer in a rectangular cavity with a heat-conducting solid cylinder placed at the center with ventilation ports for Richardson numbers from 0 to 5, cavity aspect ratio from 0.5 to 2 and at Reynolds number of 100. The right vertical wall is kept at constant temperature. They reported that the average Nusselt number at the heated surface is the highest for the lowest value of cavity aspect ratio and the lowest value of the temperature at the cylinder is obtained for the highest values of the cavity aspect ratio. In the present work, a square cavity with two ventilation ports in the presence of a isothermal heating block placed in the middle of the cavity is numerically analyzed for the mixed convection case in pulsating flow (changing the pulsating amplitude and frequency at the inlet port) for a range of Richardson numbers (Ri = 1, 10, 100) at Reynolds number of 300. The effect of the various parameters on the fluid flow and heat transfer characteristics is numerically analyzed. The results are presented in terms of streamlines, isotherms, and averaged Nusselt number for different parameters. Finally, a neural network-based identification approach is utilized for the low amplitude of pulsating velocity to obtain an input–output model for the heat transfer for a range of pulsating frequencies. To the best of authors’ knowledge, a study for identification of laminar pulsating mixed convection in a cavity in the presence of a heating block has never been reported in the literature.
123
2 Numerical simulation 2.1 Problem description, governing equations, physical parameters and solution method A schematic description of the problem is shown in Fig. 1. A square enclosure with two ventilation ports is considered. The inlet and outlet ports are placed at the middle of the left and right vertical cavity walls. The width of the ports and height of the isothermal block are L/4 and L/5, respectively. All walls of the cavity are considered as adiabatic. At the inlet port, a sinusoidal velocity with different amplitudes and frequencies is imposed. Working fluid is air with a Prandtl number of Pr = 0.71. The flow is assumed to be two dimensional, Newtonian, incompressible and in the laminar flow regime. The physical properties are assumed to be temperature-independent except for the density in the buoyancy force according to Boussinesq approximation. For a two-dimensional, incompressible, laminar and unsteady case, the continuity, momentum and energy equations can be expressed as follows: ou ov þ ¼ 0; ox oy
ð1Þ
2 ou ou ou 1 op o u o2 u þu þv ¼ þm þ ; ð2Þ ot ox oy q ox ox2 oy2 2 ov ov ov 1 op o v o2 v þu þv ¼ þm þ þ gbðT Tc Þ; ot ox oy q oy ox2 oy2 ð3Þ
oT oT oT o2 T o2 T þu þv ¼a þ ; ot ox oy ox2 oy2
ð4Þ
Fig. 1 Geometry and the boundary conditions for the cavity with two ventilation ports and an isothermal block placed at the middle (kept at temperature Th). Air has a pulsating velocity u0 þ Au0 sinðxtÞ and temperature Tc at the inlet port. All walls of the enclosure are adiabatic
J Braz. Soc. Mech. Sci. Eng.
where u, v, T and p represent the two velocity components, temperature and pressure, respectively. q, m, b, g and a denote the density, kinematic viscosity, volumetric expansion coefficient, gravity and thermal diffusivity, respectively. The relevant physical parameters are Reynolds number, u0 L Re ¼ ; m
ð5Þ
where u0 denote the velocity imposed at the inlet and Richardson number Ri ¼
Gr ; Re2
ð6Þ
where Gr represent the Grashof number Gr ¼
gbðTh Tc ÞL3 m2
ð7Þ
and Strouhal number, St ¼
fL : u0
ð8Þ
The boundary conditions for the considered problem can be expressed as: •
•
•
•
At the inlet, air has a sinusoidal velocity uðtÞ ¼ u0 þ Au0 sinð2pftÞ for velocity amplitude A and forcing frequency f. Air temperature is Tc at the inlet port. All walls of the isothermal heating block are kept at temperature Th with no-slip boundary conditions (u = v = 0). Outflow boundary condition is imposed at the outlet port where gradients of all variables in the x-direction ov oT are set to zero, ou ox ¼ 0; ox ¼ 0; ox ¼ 0: For the cavity walls, adiabatic wall (for horizontal oT walls, oT oy ¼ 0 and for vertical walls, ox ¼ 0) with no-slip boundary conditions (u = v = 0) is imposed.
Local Nusselt number is defined as hx;t L oh ¼ Nux;t ¼ : k on S
0
1 Nut ¼ @ Lb 0 þ@
ZLb
1
1 þ@ Lb
Nux;t dxA
0
1 Lb
0
right
ZLb 0
1
0
ZLb
Nux;t dxA
0
1 Nuy;t dyA þ@ Lb top
1
left
ZLb 0
1 Nuy;t dyA
bottom
ð10Þ where Lb denotes the length of the walls of the body which is Lb = L/5. Time-spatial averaged Nusselt number is obtained for one period of oscillation as Z 1 T Num ¼ Nut dt ð11Þ T 0 Equations (1–4) along with the boundary and initial conditions are solved with FLUENT 6.1 solver (a general purpose finite volume solver). The body-adapted mesh consists of 22,000 triangular elements. The grid distribution of the computational domain and mesh distribution in the vicinity of the heating block is shown in Fig. 2. The mesh is finer near the walls to resolve the high gradients in the thermal and hydrodynamic boundary layer. Mesh independence of the solutions has been confirmed. A second order accurate spatial discretization is used. Second order upwind schemes are applied in terms
ð9Þ
Where hx, t represents the local heat transfer coefficient and k denotes the thermal conductivity of air. h is the nonc dimensional temperature which is defined as h ¼ TTT :n h Tc and S denote the surface normal component and heated part of the bottom and left vertical side walls of the cavity, respectively. Spatial averaged Nusselt number is obtained after integrating the local Nusselt number along the walls of the isothermal block as
Fig. 2 Grid distribution of the computational domain and in the vicinity of the heating block
123
J Braz. Soc. Mech. Sci. Eng.
of discretising the momentum and the energy equations. PISO algorithm is used for pressure velocity coupling. The global convergence for the continuity, momentum and energy residuals are set to 10-4, 10-5 and 10-5, respectively. 2.2 CFD results In the present study, Richardson number is varied between 1 and 100 (Ri ¼ 1; 10; 100) and Reynolds number is kept at Ri=1
Ri=10
20 20
15
15
10
35
40
45
50
55
10
60
35
40
45
50
55
60
7
7.5
8
Ri=100 A=0.3
Nut
25
A=0.6 A=0.9
20
15
35
40
45
50
55
60
tu /L 0
(a) St=0.1 Ri=1
Ri=10
18
16
16
14
14
12
5
5.5
6
6.5
7
7.5
8
5
5.5
6
6.5
Ri=100 28 A=0.3 27
Nu
t
A=0.6
26
A=0.9
25 5
5.5
6
6.5
7
7.5
8
tu /L 0
(b) St=1 Ri=1
Ri=10
17
14
16
13.5
15
13
300. In this case, only the Grashof number changes. One can also consider other Reynolds number in the laminar range of fluid flow, but for the unsteady calculation with different pulsating amplitudes and frequencies, this would require a large number of parameters to characterize the system. Therefore, in the considered problem Reynolds number is taken as 300 which is in the laminar range. Calculated spatial averaged Nusselt number plots are shown for Richardson numbers of 1, 10 and 100 in Fig. 3 at Strouhal numbers of 0.1, 1 and 5, respectively. In these plots, vertical axis denotes spatial averaged Nusselt number in pulsating flow and horizontal axis denotes the nondimensional time. A close inspection of these plots reveals that, with increasing the Strouhal number for Ri = 100, the mean is shifted upwards at velocity amplitude of 0.9. It is also observed that with increasing the Richardson number, distortions from a pure sinusoid (nonlinearity with respect to sinusoidal forcing) are seen at high amplitude of pulsating velocity. The system reaches the periodic state after 4–5 periods and the simulations are performed for 10–12 periods. Streamline and isotherm plots for Richardson number of 1, 10 and 100 at velocity amplitude 0.9 and Strouhal number of 0.5 are shown in Figs. 5 and 6, respectively. The time instances within half a period when the isotherms and streamlines are illustrated for three Richardson numbers of 1, 10 and 100 are shown in Fig. 4. The vertical lines on the figure show that the same time instances are taken for Richardson number of 1 (top), 10 (middle) and 100 (bottom). The contribution of the right, top, left and bottom part of the isothermal block on the heat transfer is also seen in Fig. 4. Maximum Nusselt number is obtained for the left part at Ri = 1, 10 and for the right part at Ri = 100. For all the Richardson numbers considered, minimum contribution is obtained for the top part of heating block. At Ri = 1, two vortices are formed on the top and the bottom of the left wall at the beginning of the half period (refer to Fig. 4). At that time instance maximum and Ri=1 40
1
1.1
1.2
1.3
1.4
1.5
20
1
1.1
1.2
1.3
1.4
0
1.5
20.5
21
21.5
22
22.5
23
23.5
24
22.5
23
23.5
24
Ri=10 40
Ri=100 26
Nut
A=0.3 25.8
A=0.6
25.6
A=0.9
20 0
20.5
21
21.5
22
Ri=100
60
left right top bottom total
Nu
t
25.4 1
1.1
1.2
1.3
1.4
1.5
t u0 / L
(c) St=5
Fig. 3 Spatial averaged Nusselt number for Richardson number of 1, 10 and 100 and velocity amplitudes of 0.3, 0.6 and 0.9 for Strouhal numbers of 0.1, 1 and 5, respectively
123
40 20 0
20.5
21
21.5
22
22.5
23
23.5
24
u0 t / L
Fig. 4 Time instances with markers within a half period at Richardson number of 1,10 and 100 for Strouhal number of 0.5 and velocity amplitude of 0.9
J Braz. Soc. Mech. Sci. Eng. 1.1 A=0.3
1.09
A=0.6
1.08
A=0.9
Num / Nu0
1.07 1.06 1.05 1.04 1.03 1.02 1.01
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
St
(a) Ri=1 1.16 A=0.3
1.14
A=0.6 1.12 A=0.9
Fig. 5 Streamlines taken for time instances according to (Fig. 4) at Richardson number of 1 (first row), 10 (second row) and 100 (third row)
Nu / Nu m 0
1.1 1.08 1.06 1.04 1.02 1 0.98
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
St
(b) Ri=10 1.04 1.02
Nu / Nu m 0
1 0.98
A=0.3
0.96
A=0.6
0.94
A=0.9
0.92 0.9 0.88 0.86
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
St
(c) Ri=100
Fig. 7 Spatial and time averaged Nusselt number divided by the Nusselt number in non-pulsating case versus Strouhal number for velocity amplitudes of 0.3, 0.6 and 0.9 at Richardson number of 0.1, 1, 10 and 100, respectively Fig. 6 Isotherms taken for time instances according to (Fig. 4) at Richardson number of 1 (first row), 10 (second row) and 100 (third row)
minimum Nusselt numbers are obtained at the left and right part of the heating block, respectively. When the Nusselt number increases for the right part of the heating block, two more cells are formed on that part of the heating block and grow in size and strength as Nusselt number becomes maximum at the right of the heating block. At Ri = 1, maximum and minimum temperature gradients are obtained at the left and right part of the heating block, respectively. As the time increases within a half period, temperature gradient and boundary layer thickness along
the right part of the heating block increase and along the top–bottom part decrease. At Ri = 10, at the beginning of the half cycle, two weak recirculation patterns are observed at the top and bottom of the left wall of the cavity and the vortex at the left bottom increases in size and then disappears at the end of the cycle. When the Nusselt number becomes maximum at the left side of the heating block, two cells are formed on the lefttop and bottom-right corner of the heating block and the latter one then disappears at the end of the cycle. At Ri = 10, isotherm plots show that maximum temperature gradient is at the left part of the heating block at the beginning of the cycle. At the top, boundary layer thickness
123
J Braz. Soc. Mech. Sci. Eng.
is minimum and distorted from right to the left during the half cycle. At Ri = 100, buoyancy effects become important and more cells are formed. At the beginning of the cycle, a cell is formed at the left-bottom of the cavity and moves towards the right. At the time when the maximum Nusselt number is obtained at the right part of the heating block, a strong recirculation pattern is seen at the left-bottom corner of the heating block and two weak cells are formed at the right part of the heating block. Isotherm plots at Ri=100 show that minimum temperature gradient is achieved at the top part of the heating block. Time-spatial averaged values of the Nusselt numbers for pulsating flow divided by non-pulsating case (heat transfer enhancement-HTE) are shown in Fig. 7 for a range of Strouhal number at Richardson numbers of 1, 10 and 100. At Ri = 1, HTE is 8.5 % for amplitude of 0.9 and Strouhal number of 0.5, and 4 % for velocity amplitude of 0.6. HTE is less than 2 % for velocity amplitude of 0.3 and Strouhal number does not have an effect on the heat transfer enhancement. At Ri = 10, maximum HTE is obtained at Strouhal number of 0.1 for all velocity amplitudes and generally decreases with increasing the Strouhal number. At Ri = 100, for Strouhal number of 0.1, HTE is minimum for A = 0.9 and increases with increasing the pulsating velocity amplitude. At Ri = 100, maximum HTE is obtained for Strouhal number of 1 for all the amplitudes considered.
Identification methods can generally be classified as parametric and non-parametric. In the parametric approaches, the system is described with differential/difference equations, and the aim is to find the parameters of this mathematical description. Well known non-parametric representations are the Impulse Response (in the time domain) and the Frequency Response (in the frequency domain). In the identification, a model structure is selected and the number of past inputs and outputs is specified. Identification methods have the following procedures in common Ljung; So¨derstro¨m and Stoica ; Sjo¨berg et al. [10, 11, 13]: •
• •
•
2.3 System identification System identification can be used to construct dynamic models from the input–output data sets that may be obtained from an experimental test rig or numerical simulation Ljung; So¨derstro¨m and Stoica [10, 11]. For the CFD model of the present configuration, obtaining a linear transfer function (heat transfer response of the isothermal block to small velocity perturbations at the inlet port for a frequency range) may be challenging due to the complex interactions in the hydrodynamic and thermal boundary layers along the isothermal block. Linear transfer function of the CFD model is computed from an FFT of the response with single sinusoid excitations at the inlet port for different frequencies. On the other hand, with the linear identification method based on correlation, one can obtain the same transfer function using a broadband excitation signal with less computational effort. It is advantageous to know some of the physical parameters of the system a priori, e.g. the order of the maximum lag, when constructing dynamic models from identification. This information can also be obtained from the identification as in a black-box approach Yuen et al. [12].
123
•
An appropriate choice of the input signal: The system is excited with a proper signal for the excitation of all relevant modes of interest. Generally, broadband forcing, chirp signals or pseudo random binary sequences, which have white noise characteristics, are used to excite the system for a wide range of frequencies. Model structure selection: Equation error or output error model structures are used. Selection of the number of past inputs and outputs used in the model structure (the system ‘‘memory’’: A priori information about the maximum time lag of the system is helpful. Depending on the maximum frequencies of interest and the time lag of the system under consideration, the number of regressors is specified. An algorithm to minimize the cost function: The difference between responses of the time series data generated from numerical simulation or experiment and identification is minimized. Marquardt-Levenberg algorithm, Gauss–Newton methods or other nonlinear optimization (genetic algorithms, particle swarm optimization) methods are used. Model validation: The identified model is tested against signals which have not been used in the estimation. In a broadband forcing, half of the data is used for the fit (minimization of the cost function) while the other half is used for validation.
3 Neural network identification For linear and nonlinear systems, black box identification can be used if little or no priori information about the complex physics is available Sjo¨berg et al.; Juditsky et al. [13, 14]. Neural network identification methods belong to the class of parametric nonlinear black box identification procedures [13, 15, 16, 17]. They are promising to identify any linear model or nonlinearity up to a specified degree of accuracy (universal function approximators). For the present case of mixed convection, complex interactions take place in the thermal and velocity
J Braz. Soc. Mech. Sci. Eng.
3.1 Extraction of kernels in terms of neural network weights
Fig. 8 1 hidden layer feed-forward neural network structure along with regressors with tangent hyperbolic activation function f and the linear function F at the output layer, M is the number of units and L is the memory length of the regressors
As mentioned before, a wide class of linear nonlinear systems can be represented in Volterra series form for the input–output relation (extension of the Taylor series approximation for functionals Boyd and Chua; Billings [19, 20]. For an input–output data set (u(t), y(t)) with memory length L, this relation will be written for a linear model in discrete form as yðtÞ ¼ h0 þ
L X
h1 ðsÞuðt sÞ:
ð15Þ
s¼1
boundary layer along the isothermal heating block. It is possible to obtain a dynamic model (input–output relation) that is valid for a range of frequencies with black box identification from measured input and output time series data aas described by Selimefendigil et al. [18]. Network structures can be classified as feed-forward or recurrent Narendra and Parthasarathy; Pha [15, 16]. In a recurrent network structure, the computed outputs from the network will be fed as the input to the layers. In this study, a feed-forward network structure is used and only the past inputs as the input to the neural network (NFIR model) are utilized. The network has only 1 hidden layer and tangent hyperbolic as the activation function. A schematic of the neural network topology along with the regressors used as the input is shown in Fig. 8. Let u be the set of regressors with memory length of L, u as the input, y as the output / ¼ ½1 uðt 1Þ. . .uðt LÞ;
ð12Þ
and ZN be the set of the input–output data (training set) up to time N. Then the identification problem will be formulated as the minimization of the error between the CFD model output and the output from neural network as VN ðhÞ ¼
N 1X
N
½yCFD ðtÞ yNeuralNet ðtÞ2 :
ð13Þ
t¼1
This function will be minimized by some nonlinear iterative search algorithms and we use LevenbergMarquardt technique to find the minimum of the function and hence the weights of neural networks which are denoted by h. The output from the neural network will be written in terms of the weights of the network as ! M L X X yNeuralNet ðtÞ ¼ Wj f wjl /l þ wj0 þ W0 ; ð14Þ j¼1
Different approaches exist in the literature to extract the kernels. A correlation-based analysis with broadband forcing has been developed by Schetzen [21]. Wray and Green [22] have developed a strategy to get an improved accuracy of the approximation in comparison to Toeplitz matrix inversion proposed by Korenberg et al. [23]. In this study, we use the approach proposed by Wray and Green [22]. The idea is to expand the neural network approximation output as in Eq. (14) for the tangent hyperbolic function around the bias term. Taylor series approximation of the tangent hyperbolic function around zero can be written as Wray and Green [22] tanhðxÞ ¼
ð1Þnþ1
n¼1
Bn ð24n 22n Þx2n1 ; ð2nÞ!
ð16Þ
where Bn is the nth order Bernoulli number and is defined as Bn ¼
1 2ð2nÞ! X 1 : 2n s ð2pÞ s¼1 2n
ð17Þ
Expanding the activation function in Eq. (14), the neural network output will be written as yNeuralNet ðtÞ ¼ "P1
M X
Wj
j¼1
k¼1 ð1Þ
kþ1
P 2k1 # Bk ð24k 22k Þ Ll¼1 wjl /l þ wj0 þ W0 : ð2kÞ!
ð18Þ Combining this representation with Volterra series of first order in the form as in Eq. (15), we will express the kernels in terms of the weights of the neural network. The zeroth order kernel: h0 ¼
M X
Wj
j¼1
"P1
l¼1
where w and W0 s are the weights of the neural network for the input and output to the hidden layer and f is the tangent hyperbolic activation function.
1 X
k¼1 ð1Þ
kþ1
# Bk ð24k 22k ÞCð2k 1; 0Þw2k1 j0 þ W0 : ð2kÞ!
ð19Þ
123
J Braz. Soc. Mech. Sci. Eng. 0.05
St=0.1
CFD Sys ident
St=0.5
Nu’ / Nu
0
0.1
0
0
−0.05
5
10
15
0
20
−0.1 10
tU /L 0
20
30
40
50
2
4
6
8
10
−3
x 10
0.05
St=2.5
St=5
−3
x 10
5
CFD
Nu ’ / Nu0
0.01 0 0
0
−0.05
1
1.5
2
2.5
3
3.5
4
20
20.2
20.4
20.6
20.8
Sys ident
5
0
21
−0.01
Fig. 9 Nusselt number response after a chirp signal excitation calculated with CFD and estimated from system identification, bottom-left low frequency region, bottom-right high frequency region
1
1.2
1.4
1.6
1.8
2
0.85
0.9
0.95
1
1.05
1.1
tU0 / L
Fig. 10 Nusselt number outputs calculated with CFD and obtained from system identification after a single sinusoid harmonic excitation at low amplitude for Strouhal numbers of 0.1, 0.5, 2.5 and 5, respectively
The first order kernels: M X
"P1 Wj
k¼1 ð1Þ
kþ1
j¼1
Bk ð24k 22k ÞCð2k 1; 1Þwja w2k2 j0 ð2kÞ!
#
1
Gain
h1 ðaÞ ¼
ð20Þ
H1 ðxÞ ¼
L X
h1 ðkÞeixkDt :
ð21Þ
0
0
1
0
1
2
3
4
5
2
3
4
5
200
Phase [Deg]
Once the kernels of first order are calculated, these will then be extended into frequency domain to obtain the transfer function. Transfer function will be computed from z transform of the first order kernel as,
0.5
0
−200
Strouhal Number
k¼1
3.2 Identification results
Fig. 11 Gain and phase of the transfer function calculated with combining neural network weights with Volterra kernel
As the input to the system at the inlet ventilation, a chirp signal with Str ranges from 0.087 to 5.26 with the perturbation amplitude A = 0.1 is used at Richardson number of 10. The chirp signal has linearly varying frequency component over time and is defined as x x max min uðtÞ ¼ u þ A u sin xmax k kDt ; M ð22Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 uPN u k¼1 ðNuðkÞ NuðkÞpred Þ2 4 5 100: Fit ¼ 1 t PN 2 NuðkÞ k¼1
with maximum and minimum frequencies of xmin, xmax and amplitude A. M; Dt denote the number of the total samples used and time step, respectively. The memory length of the regressors is 200 and the neural network is a structure with one hidden layer and 5 neurons, composed of tangent hyperbolic functions and a linear output layer. A measure of the quality of identification is the match between the output Nupred predicted by the dynamic model and the actual output (Nusselt number) Nu computed with CFD:
123
2
ð23Þ
The fits for the linear approximations with the neural network approximation to the CFD output is 99 and is shown in Fig. 9 for the low and high frequency regions. Validation is performed for single sinusoid forcing. A comparison with the single sinusoidal response from CFD along with the response predicted from the dynamic model is shown in Fig. 10 for Str ¼ 0:1; 1; 2:5; 5: The fits between the CFD and dynamic model are 95, 97, 94 and 91 for Strouhal numbers of 0.1,1, 2.5 and 5, respectively. The overall agreement between the model output and CFD output is adequate for the considered range of frequencies. Gain and phase of the transfer function which is calculated from the above detailed method (combining the neural network weights with Volterra kernel of first order) are shown in Fig. 11. The response in the gain decreases as the Strouhal number increases. The phase first increases
J Braz. Soc. Mech. Sci. Eng.
around Strouhal number of 2, then changes its sign and decreases with increasing frequency.
4 Discussions and conclusion In this study, a square cavity with two ventilation ports in the presence of a isothermal heating block placed in the middle of the cavity is numerically analyzed for the mixed convection case in pulsating flow (changing the pulsating amplitude and frequency at the inlet port) for a range of Richardson numbers (Ri = 1, 10, 100) at Reynolds number of 300. Following results are obtained: • •
• •
•
•
• •
•
With increasing the Strouhal number for Ri = 100, the mean is shifted upwards at velocity amplitude of 0.9 With increasing the Richardson number, distortions from a pure sinusoid (nonlinearity with respect to sinusoidal forcing) is seen at high amplitude of pulsating velocity. The system reaches the periodic state after 4–5 periods. At Ri = 1, HTE is 8.5 % for amplitude of 0.9 and Strouhal number of 0.5, and 4 % for velocity amplitude of 0.6. HTE is less than 2 % for velocity amplitude of 0.3 and Strouhal number does not have an effect on the heat transfer enhancement. At Ri = 10, maximum HTE is obtained at Strouhal number of 0.1 for all velocity amplitudes and generally decreases with increasing the Strouhal number At Ri = 100, for Strouhal number of 0.1, HTE is minimum for A = 0.9 and increases with increasing the pulsating velocity amplitude. At Ri = 100, maximum HTE is obtained for Strouhal number of 1 for all the amplitudes considered. A dynamic model for the input–output model (inputvelocity forcing, output-Nusselt number at the surface of the heating block) is obtained using a neural network-based identification method. This model is valid for the range of the frequencies considered. The response in the gain decreases as the Strouhal number increases. The phase first increases around Strouhal number of 2, then changes its sign and decreases with increasing frequency.
References 1. Dogan A, Sivrioglu M, Baskaya S (2006) Investigation of mixed convection heat transfer in a horizontal channel with discrete heat sources at the top and at the bottom. Int J Heat Mass Transf 49:2652–2662 2. Aminossadati SM, Ghasemi B (2009) A numerical study of mixed convection in a horizontal channel with a discrete heat source in an open cavity. Eur J Mech 28:590–598
3. Tmartnhad I, Alami M E, Najam M, Oubarra A (2008) Numerical investigation on mixed convection flow in a trapezoidal cavity heated from below. Energy Convers Manag 49:3205–3210 4. Ozsunar A, Baskaya S, Sivrioglu M (2001) Numerical analysis of grashof number, reynolds number and inclination effects on mixed convection heat transfer in rectangular channels. Int Commun Heat Mass Transf 28:985–994 5. Khanafer K, Al-Azmi B, Al-Shammari A, Pop I (2008) Mixed convection analysis of laminar pulsating flow and heat transfer over a backwardfacing step. Int J Heat Mass Transf 51:5785– 5793 6. Boutina L, Bessaih R (2011) Numerical simulation of mixed convection air-cooling of electronic components mounted in an inclined channel. Appl Therm Eng 31:2052–2062 7. Sourtiji E, Hosseinizadeh S, Gorji-Bandpy M, Ganji D (2011) Heat transfer enhancement of mixed convection in a square cavity with inlet and outlet ports due to oscillation of incoming flow. Int Commun Heat Mass Transf 8:806–814 8. Angirasa D (2000) Mixed convection in a vented enclosure with an isothermal vertical surface. Fluid Dyn Res 26:219–233 9. Rahman MM, Elias M, Alim M (2011) Mixed convection flow in a rectangular ventilated cavity with a heat conducting solid circular cylinder at the center. IJE Trans B Appl 24(1):93–105 10. Ljung L (1987) System identification: theory for the user. Prentice Hall, Englewood Cliffs 11. So¨derstro¨m T, Stoica P (1989) System identification, PrenticeHall International, Hemel Hempstead 12. Yuen SW, Gentemann AMG, Polifke W Investigation of the influence of boundary conditions on system identifiability using real time system modeling. In: 11th international congress on sound and vibration (ICSV11), IIAV, Saint-Petersburg, Russia, pp 3501–3508 13. Sjo¨berg J, Zhang Q, Ljung L, Benveniste A, Delyon B, Glorennec P, Hjalmarsson H, Juditsky A (1995) Nonlinear black-box modeling in system identification: a unified overview. Automatica 31(12):1691–1724 14. Juditsky A, Hjalmarsson H, Beneviste A, Delyon B, Ljung L, Sjoberg J, Zhang Q (1995) Nonlinear black-box models in system identification: mathematical foundations. Automatica 31:1725– 1750 15. Narendra K, Parthasarathy K (1990) Identification and control of dynamical systems using neural networks. IEEE Trans Neural Netw 1:4–27 16. Pha DT (1995) Neural networks for identification, prediction and control. Springer, Berlin 17. NØrgaard M, Ravn O, Poulsen N (2002) Nnsysid-toolbox for system identification with neural networks. mathematical and computer modelling of dynamical systems. Math Comput Model Dyn Syst 8(1):1–20 18. Selimefendigil F, Fo¨ller S, Polifke W (2012) Nonlinear identification of the unsteady heat transfer of a cylinder in pulsating crossflow. Comput Fluids 53:1–14 19. Boyd S, Chua LO (1985) Fading memory and the problem of approximating nonlinear operators with volterra series. IEEE Trans Circuits Syst 32:1150–1161 20. Billings S (1980) Identification of nonlinear systems a survey. In: Proceedings of IEEE 127(6):272–285 21. Schetzen M (1965) Measurement of the kernels of a non-linear system of finite order. Int J Control 1(3):251–263 22. Wray J, Green GGR (1994) Calculation of the volterra kernels of non-linear dynamic systems using an artificial neural network. Biol Cybern 71:187–195 23. Korenberg MJ, Bruder SB, McLlroy PJ (1998) Exact orthogonal kernel estimation from finite data records: extending wiener’s identification of nonlinear systems. Ann Biomed Eng 16:201–214
123
