Identification of forced convection in pulsating flow at a backward facing step with a stationary cylinder subjected to nanofluid

International Communications in Heat and Mass Transfer 45 (2013) 111–121

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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt

Identification of forced convection in pulsating flow at a backward facing step with a stationary cylinder subjected to nanofluid☆ Fatih Selimefendigil a,⁎, Hakan F. Öztop b a b

Department of Mechanical Engineering, Celal Bayar University, Manisa, Turkey Department of Mechanical Engineering, Technology Faculty, Fırat University, Elazığ, Turkey

a r t i c l e

i n f o

Available online 30 April 2013 Keywords: Backward facing step Pulsating flow System identification

a b s t r a c t In the present study, the application of the system identification method for forecasting the thermal performance of forced pulsating flow at a backward facing step with a stationary cylinder subjected to nanofluid is presented. The governing equations are solved with a finite volume based code. The effects of various parameter frequencies (0.25 Hz–8 Hz), Reynolds number (50–200), nanoparticle volume fraction (0.00–0.06) on the fluid flow and heat transfer characteristics are numerically studied. Nonlinear system identification toolbox of Matlab is utilized to obtain nonlinear dynamic models of data sets corresponding to different nanoparticle volume fractions at frequencies of 1, 4 and 8 Hz. It is observed that heat transfer is enhanced with increasing the frequency of the oscillation, nanoparticle volume fraction and Reynolds number. The level of the nonlinearity (distortion from a pure sinusoid) decreases with increasing ϕ and with decreasing Reynolds number. It is also shown that nonlinear dynamic models obtained from system identification toolbox could produce thermal output (length averaged Nusselt number) as close to as output from a high fidelity CFD simulation. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction A vast amount of literature is dedicated to the flow separation and its subsequent reattachment due to its practical importance in many engineering applications such as flow around airfoils, buildings, combustors and collectors of power systems. The flow over a backward facing or forward facing step is a benchmark problem where flow separation and reattachment occur. Theoretical, analytical and experimental studies of backward or forward facing step flow have been presented by many researchers. A comprehensive review is presented by Abu-Mulaweh [1] for laminar mixed convection over vertical, horizontal and inclined backward- and forward-facing steps studied in the open literature. Nie and Armaly [2] have reported the wall temperature distributions, Nusselt number and the friction coefficient for the laminar 3D flow adjacent to backward-facing step in a rectangular duct. Barkley et al. [3] have numerically studied the 3D linear stability analysis of flow over a backward-facing step for Reynolds numbers between 450 and 1050. Iwai et al. [4] have numerically investigated the 3D flow over a backward facing step at low Reynolds number for various duct aspect ratios. They reported that the maximum Nusselt number did not appear on the centerline, but near the two side walls in every case. Experimental studies have also been conducted for the flow over a backward facing or forward facing step. ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail addresses: [email protected] (F. Selimefendigil), [email protected] (H.F. Öztop). 0735-1933/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.04.016

Terhaar et al. [5] have experimentally studied the unsteady heat transfer downstream over a backward-facing step in the laminar flow regime for pulsating flow conditions at Reynolds number of 300 (based on the hydraulic diameter of the channel). They showed that the Nusselt number increases up to a certain Strouhal number and then degrades as the pulsation frequency increases. Armaly et al. [6] have reported the velocity distribution and reattachment length using Laser-Doppler measurement for flow downstream of a backward facing step in a two-dimensional channel. Their results showed separation length varies with Reynolds number and various flow regimes are characterized by variations of the separation length. Stuer et al. [7] have experimentally studied the separation ahead of a forward facing step under laminar flow conditions using the hydrogen bubble technique. Abu-Mulaweh [8] has reported the measurements of heat transfer and flow over a vertical forward-facing step for turbulent mixed convection using two-component LDV and a cold wire anemometer. He reported that when the step height increases the turbulence intensity of the streamwise and transverse velocity fluctuations and the intensity of temperature fluctuations downstream of the forward-facing step increase. Heat transfer and fluid flow characteristics over a backward or forward facing step in a channel with the insertion of obstacles have received less attention in the literature. Yilmaz and Oztop [9] have numerically investigated the turbulent forced convection over double forward facing step. They put two adiabatic steps with different lengths and heights. They studied the effects of step heights, step lengths and Reynolds numbers on heat transfer and fluid flow. They reported that the second step can be used to control the fluid flow and heat transfer characteristics. Oztop et al. [10] have numerically

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studied in the literature. One of the objectives of the present study is to explore the effects of oscillation frequency imposed at the inlet, nanoparticle volume fraction and Reynolds number on the fluid flow and heat transfer characteristics. Another aim of the present study is to obtain nonlinear dynamic models using input–output data set from the high fidelity CFD computations corresponding to different nanoparticle volume fractions at different forcing frequencies using the system identification toolbox of Matlab. The details about the system identification procedures are detailed in the subsequent sections.

Nomenclature h k H n Nu p Pr Re St T u, v x, y

local heat transfer coefficient, (W/m 2K) thermal conductivity, (W/m⋅K) step height, (m) unit normal vector local Nusselt number, hH/k pressure, (Pa) Prandtl number, αν Reynolds number, uνH Strouhal number, ufH temperature, (K) x–y velocity components, (m/s) Cartesian coordinates, (m) 0

2. Numerical simulation

0

2.1. Problem description

Greek characters α thermal diffusivity, (m 2/s) θ non-dimensional temperature, TT−T −T ν kinematic viscosity, (m 2/s) ωH ρ density of the fluid, (kg/m 3) velocity of cylinder, 2u ϕ nanoparticle volume fraction c

h

c

0

Subscripts c cold h hot

studied the turbulent forced convection of double forward facing step with obstacles. They put rectangular obstacles before each step and investigated the effects of step height, obstacle aspect ratio and Reynolds number on fluid flow and heat transfer. They reported that heat transfer rate increases when the obstacle aspect ratio increases. Kumar and Dhiman [11] have numerically studied the heat transfer enhancement in laminar forced convection flow over a backward facing step with the insertion of an adiabatic circular cylinder. They considered different cross-stream positions of the circular cylinder for the Reynolds number between 1 and 200. They obtained heat transfer enhancement up to 155% compared to no-cylinder case. In heat transfer applications, nano-sized particles (average particle size less than 100 nm) are added in the base fluid such as water or ethylene glycol to obtain better thermal properties compared to base flow. Nanofluids have improved heat transfer characteristics with little pressure drop as compared to base fluids (Oztop and Abu-Nada [12]). Houshang et al. [13] have studied the mixed convection in a vented cavity subjected to an external copper–water nanofluid. The effects of inlet and outlet port locations for a range of Reynolds number, Richardson number and particle volume fractions. Kamyar et al. [14] have revised the literature related to computational fluid dynamics application to nanofluids. The literature review showed that nanofluids lead to improvement in the heat transfer performance, which is in a good agreement with experimental works. Modifications could be made to achieve more accurate results from numerical processes. Shahi et al. [15] have numerically investigated the convective cooling in a square vented cavity and partially heated from below utilizing nanofluids. They showed that increase in solid concentration leads to increase in the average Nusselt number at the heat source surface and decrease in the average bulk temperature. To the best of authors' knowledge a study of laminar forced convection in pulsating flow over a backward facing step in a channel with a stationary cylinder subjected to nanofluid has never been

A schematic description of the physical problem considered in this study is shown in Fig. 1. A channel with a backward facing step is considered. The step size of backward facing step is H and channel height is 2H. At the inlet of the channel, a parabolic velocity with a sinusoidal time dependent part ðu ¼ u ð1 þ 0:75sinð2πft ÞÞÞ and a uniform temperature (T = 300K) are imposed. The downstream length starting from the edge of the step to the exit of the channel is 25H to ensure that the recirculation length downstream of the step is independent of the computational domain. The downstream bottom surface of the backward facing step is maintained at T = 300K, while the other walls of the channel are assumed to be adiabatic. An adiabatic stationary cylinder with diameter (D = H) is mounted at the location (xc,yc) = (H,H) where the coordinate system is positioned at the step on the bottom wall of the channel. The continuity, momentum and energy equations can be expressed for two dimensional, incompressible, laminar and steady case as in the following, ∂u ∂v þ ¼ 0; ∂x ∂y

ð1Þ

! ∂u ∂u ∂u 1 ∂p ∂2 u ∂2 u þ ; þu þv ¼− þ νnf ρnf ∂x ∂t ∂x ∂y ∂x2 ∂y2

ð2Þ

∂v ∂v ∂v 1 ∂p ∂2 v ∂2 v þ þu þv ¼− þ ν nf ρnf ∂y ∂t ∂x ∂y ∂x2 ∂y2 ! ∂T ∂T ∂T ∂2 T ∂2 T þ þu þv ¼ α nf ; ∂t ∂x ∂y ∂x2 ∂y2

! ð3Þ

ð4Þ

where u, v, T and p represent the two velocity components, temperature and pressure, respectively. ρ, ν and α denote the density, kinematic viscosity and thermal diffusivity, respectively. The viscous dissipation is assumed negligible in the energy equation. The boundary conditions for the considered problem in dimensional form can be expressed as: • At the inlet, velocity is unidirectional and temperature is uniform, ð u ¼ u0 ð1 þ sinð2πft ÞÞ; v ¼ 0; T ¼ 300K Þ: • At the bottom wall, downstream of the step, temperature is constant, (T = 310K). • At the channel in the x-direction are  exit, gradients of all variables  set to zero,

∂u ∂x

¼ 0;

∂v ∂x

¼ 0;

∂T ∂x

¼0 .

• On the channel walls (except the downstream of the step), adiabatic wall with no-slip boundary conditions are assumed,   u ¼ 0; v ¼ 0;

∂T ∂n

¼0 .

• On the cylindersurface, no slip with adiabatic wall boundary con dition is used,

u ¼ 0;

v ¼ 0;

∂T ∂n

¼0 .

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113

Fig. 1. Geometry with boundary conditions.

The relevant physical nondimensional numbers are Reynolds number (Re) and nondimensional frequency Strouhal number (St). Local Nusselt number is defined as

Nux;t ¼

  hx;t L ∂θ ¼− : k ∂n S

ð5Þ

where hx,t represent the local heat transfer coefficient. Spatial averaged Nusselt number is obtained after integrating the local Nusselt number along the bottom wall downstream of the step as Nut ¼

1 L ∫ Nu dx: L 0 x;t

ð6Þ

Time and spatial averaged Nusselt number are obtained after integrating spatial averaged Nusselt number along the bottom wall downstream of the step for one period of the oscillation τ as 1 τ Nu ¼ ∫0 Nut dt: τ

ð7Þ

2.1.1. Thermo-physical properties of the Al2O3–water nanofluid The working fluid is water with Al2O3 nanoparticles. In the present study, single phase or homogeneous model is used. The single phase model assumes that the liquid phase and nanoparticles are in thermal equilibrium and move of same velocity. The effective thermophysical properties of nanofluids are defined by using the following formulas. The effective density of a nanofluid can be defined based on the classical two phase mixture as ρnf ¼ ð1−ϕÞρbf þ ϕρp :

ð8Þ

Under the thermal equilibrium conditions, the specific heat of nanofluid is given   Cp

nf

    ¼ ð1−ϕÞ C p þ ϕ Cp bf

ð9Þ

p

Table 1 Grid sensitivity study for at Reynolds number of 100 and ϕ = 0.00. Nusselt numbers are the length averaged values along the bottom wall downstream of the step. Grid size

Nusselt number

24,668 39,588 52,174 65,594

2.78 2.77 2.74 2.73

where the subscripts bf, nf and p denote the base fluid, nanofluid and nanoparticle volume fraction, respectively. Dynamic viscosity of the nanofluid is computed using the correlation obtained from the least squares curve fitting of the experimental data as Wang et al. [16]   2 μ nf ¼ μ bf 123ϕ þ 7:3ϕ þ 1 :

ð10Þ

Thermal conductivity of the nanofluid is obtained using the well known Hamilton and Crosser model (Hamilton and Crosser [17]) as,   2 knf ¼ kbf 4:97ϕ þ 2:72ϕ þ 1 :

ð11Þ

Considering the local thermal equilibrium, and assuming nanofluid behaves as a conventional single-phase fluid and its properties evaluated as a function of those of both constituents as given above in Eqs. (1)–(4) along with the boundary conditions are solved with Fluent (a general purpose finite volume solver [18]). The convective terms in the momentum and energy equations are solved using QUICK scheme and PISO algorithm is used for velocity–pressure coupling. The system of algebraic equations is solved with Gauss-Siedel point by point iterative method and algebraic multi-grid method. The convergence criteria for continuity, momentum and energy equations are set to 10−4, 10−5 and 10 −6, respectively. The unstructured body-adapted mesh of appropriate size consists of only triangular elements. The computational domain is divided into 52,174 triangular elements. The mesh is finer near the walls to resolve the high gradients in the thermal and hydrodynamic boundary layer and in the vicinity of the step for the recirculation region downstream of the step. Mesh independence study is also carried out to obtain an optimal grid distribution with accurate results and minimal computational time. Four different grid sizes are tested and the convergence in the length-averaged Nusselt number (along the bottom wall downstream Table 2 Reported values for the reattachment lengths XR at Reynolds number 100 (expansion ratio of 2).

Acharya et al. [19] Lin et al. [20] Dyne et al. [21] El-Refaee et al. [22] Cochran et al. [23] Present

XR/S

Error (%)

4.97 4.91 4.89 4.77 5.32 5.12

−2.93 −4.1 −4.49 −6.83 3.9 0

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where A, B, C, D, and F are polynomials in terms of z, u and Q are the input and output of the system, and v is the error term. z is a shift operator namely,

c

zQ ðt Þ ¼ Q ðt þ 1Þ:

Here, Q(t + 1) is a shorthand notation for Q(t + Δt), for the time step Δt. This operator simply shifts one step ahead value of the input or output to the current time. The past inputs (u(t − 1),…,u(t − Nu)) and outputs (Q(t − 1),…,Q(t − NQ)) are called the regressors. Depending on the polynomials used, different model structures appear [24]. FIR (Finite Impulse Response):

b

a

Fig. 2. Three points during the acceleration phase of a period.

A ¼ C ¼ D ¼ F ¼ 1:

of the step) is checked. The results at Reynolds number of 100 and ϕ = 0.00 are tabulated in Table 1. From this table, grid size of 52,174 is decided to be fine enough to resolve the flow and thermal field. The numerical code is first checked against the benchmarked results of backward facing step reported in the literature [19–23]. Table 2 shows the results of the reattachment length divided by step height at Reynolds number of 100 for expansion ratio of 2. Minimum deviation for the percentage in the error is obtained for the results of Acharya et al. [19] which is − 2.93%. The agreement between the other sources is less than 5%, only − 6.83% error is obtained for the results of El-Refaee et al. [22].

This is the simplest model structure to be considered. The past inputs are used as regressors. The structure results in a linear least square problem for minimizing the cost function (e.g. Euclidean norm of the residual between the actual and the estimated output). It requires many regressors and the convergence rate is slow. ARX (Auto Regressive with eXogenous input): C ¼ D ¼ F ¼ 1: This model structure uses the past inputs and past outputs as regressors. This again results in linear least square description where the cost function needs to be minimized. ARMAX (Auto Regressive Moving Average with eXogenous input): D = F = 1. OE (Output Error): A = C = D = 1. In the identification, a model structure is selected and the number of past inputs and outputs is specified. A criterion to minimize the difference between the actual output and output from identification is specified in order to get the parameters of the model structures. In an equation error/output error type modeling approach, this criterion results in a linear/nonlinear least square fit. 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 [24–26]:

2.2. System identification System identification methods can be utilized to construct dynamic models from the observation of input–output data sets that may be obtained from an experimental test rig or numerical simulation [24,25]. 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 nonparametric representations are the Impulse Response (in the time domain) and the Frequency Response (in the frequency domain). The input–output representation of a Linear Time Invariant, system in polynomial form is expressed as [24]: BðzÞ C ðzÞ uðt Þ þ vðt Þ; F ðzÞ DðzÞ

• 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 pseudorandom

ð12Þ

Re=50

3

4

Nu

3.5

2

3 2.5

1.5 1 0

Re=100

4.5

2.5

Nu

2 1

2

3

1.5 0

4

1

2

time (s)

3

4

time (s)

Re=200

10 8

Nu

AðzÞQ ðt Þ ¼

ð13Þ

f=0.25Hz f=1Hz

6

f=8Hz

4 2 0

1

2

3

4

time (s) Fig. 3. Time evolution of the length-averaged Nusselt number along the bottom wall for different Reynolds numbers and frequencies at ϕ = 0.02.

F. Selimefendigil, H.F. Öztop / International Communications in Heat and Mass Transfer 45 (2013) 111–121

(a) a

(b) b

f= 0.25Hz

(e) b

f= 1Hz

(h) b

f= 8Hz

115

(c) c

(d) a

(f) c

(g) a

(i) c Fig. 4. Streamlines for different frequencies at three time instances during the acceleration phase of a period (Fig. 3) at Re = 100 and ϕ = 0.002.

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.

(a) a

• 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

(b) b

f=0.25Hz

(e) b

f=1Hz

(h) b

f=8Hz

(c) c

(d) a

(f) c

(g) a

(i) c Fig. 5. Temperature contours for different frequencies at three time instances during the acceleration phase of a period (Fig. 3) at Re = 100 and ϕ = 0.002.

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f=0.25 Hz

f=1 Hz 20

Nux

Nux

20

10

0

0

10

10

0

20

0

x/H

10

20

x/H

f=8 Hz a

Nux

20

b 10

0

c

0

10

20

x/H Fig. 6. Local Nusselt number distribution along the bottom wall downstream of the step for different frequencies at three time instances during the acceleration phase of a period at Re = 100 and ϕ = 0.002.

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. 2.2.1. Nonlinear identification problem Input–output modeling of the nonlinear systems is generally categorized as nonparametric – functional series expansion (Volterra, Wiener series expansion) – and parametric (differential/difference equation models, neural network models, polynomial models) [27,28]. Volterra series is the extension of the impulse response of the linear system to the nonlinear case [29,30]. These are the Taylor series expansion applied to functionals. In Hammerstein and Wiener identification methods, a linear dynamic block is connected to a static nonlinear input and/or output block structure [31,28]. Neural network is a black-box identification method which uses expansion functions through the units (layers) to model the nonlinear input–output relation [32,33,26]. There also exist local linear models which use fuzzy based algorithms that could be utilized as nonlinear identification tools [34,35]. The nonlinear dynamic fit is generally expressed as,     Q ðt Þ ¼ F Q ðt−1Þ; …; Q t−NQ ; uðt Þ; …; uðt−Nu Þ :

dynamic model from the input–output data set corresponding to different parameter sets. 3.1. CFD results 3.1.1. Effects of frequency Length-averaged Nusselt number along the bottom wall downstream of the step versus time is shown in Fig. 3 for different frequencies at Re = 50, 100 and 200. The number of cycles to reach steady state increases with increasing frequencies and increasing Reynolds number. This could be attributed to the increase in the adaptation time of the system to pulsating velocities with increasing frequencies. Another observation is that with increasing the Reynolds number the level of nonlinearity (distortion from a pure sinusoid) increases. This is due to the flow reversal of the heated fluid particles during the cycle. Streamline plots showing the flow patterns at Reynolds number of 100 and ϕ = 0.02 are shown in Fig. 4 for different frequencies during the acceleration phase of a period when the steady state periodic oscillations are reached. The points of the acceleration phase of the period are shown in Fig. 2. From the streamline plots, it is seen that with increasing the frequency, the flow structure will be different behind the cylinder and near the bottom wall downstream of the step at point a (the point where peak value is achieved). At f = 1 Hz, two counter rotating vortices are formed behind the cylinder and at f = 8 Hz, the intensity of these vortices increases. At the highest frequency of the interest, several cell centers are also observed downstream of the step. Isotherm plots are shown in Fig. 5 for different frequencies. The steepest temperature gradients are

ð14Þ

8

The function F is linear with respect to its arguments (regressors, Q(t–i), u(t–j)) for a linear system. In linear identification, it is not so difficult to obtain a dynamic model using one of the existing model structures and appropriate choice of the excitation signal. For a nonlinear system, the form of the function F is not known a priori. This function is approximated using expansion functions and polynomials. Nonlinear extensions of the linear model structures are named as NFIR, NARX and NARMAX, and NOE [36].

7

3. Results and discussion In the following, streamlines, isotherms and averaged Nusselt numbers calculated from CFD will be presented for different parameter sets and nonlinear system identification will be used to construct a

Re=50 Re=100

6

Nu

Re=200

5 4 3 2 10

0

f Fig. 7. Time-spatial averaged Nusselt number along the bottom wall versus frequency for different Reynolds numbers at ϕ = 0.002.

F. Selimefendigil, H.F. Öztop / International Communications in Heat and Mass Transfer 45 (2013) 111–121

Re=50

3.5

117

Re=100

5 4.5

Nu

Nu

3 2.5

4 3.5 3

2

2.5 0

0.5

1

1.5

2

0

0.5

time (s)

1

1.5

2

time (s)

Re=200

10

φ=0

Nu

8

φ=0.02 φ=0.04

6

φ=0.06

4 2

0

0.5

1

1.5

2

time (s) Fig. 8. Time evolution of the length-averaged Nusselt number along the bottom wall for different Reynolds numbers and nanoparticle volume fractions at f = 2 Hz.

seen at the reattachment points of the vortices along the bottom wall. With increasing the frequency of oscillation, more fluctuating of the isotherms from bottom wall towards the upper wall is observed. Local Nusselt number distribution along the bottom wall downstream of the step for various frequencies at Re = 100 and ϕ = 0.02 is depicted in Fig. 6 for the points according to Fig. 2. The horizontal axis denotes the distance from the edge of the step to the exit normalized by the step height. In these plots, a peak in the Nusselt number is seen which corresponds to a location very close to the reattachment point where steep temperature gradient occurs. The peak value increases with phase approaching from minimum to the maximum.

(a) a

With increasing the frequency of the oscillation, several local peaks are observed downstream of the step. Time-spatial averaged Nusselt number versus frequency is shown in Fig. 7 for different Reynolds numbers. With increasing the frequency, heat transfer is enhanced and this is more effective at Re = 200. 3.1.2. Effects of nanoparticle volume fraction Fig. 8 shows the time evolution of the length-averaged Nusselt number along the bottom wall downstream of the step at frequency of 2 Hz and Reynolds number of 100 for different ϕ values. Increasing the ϕ value increases the peak value of the Nusselt number for all Re

(b) b

= 0%

(e) b

= 2%

(h) b

= 6%

(c) c

(d) a

(f) c

(g) a

(i) c Fig. 9. Streamlines for different nanoparticle volume fractions at three time instances during the acceleration phase of a period (Fig. 3) at Re = 100 and f = 2 Hz.

118

F. Selimefendigil, H.F. Öztop / International Communications in Heat and Mass Transfer 45 (2013) 111–121

(a) a

(b) b

= 0%

(e) b

= 2%

(h) b

= 6%

(c) c

(d) a

(f) c

(g) a

(i) c Fig. 10. Isotherms for different nanoparticle volume fractions at three time instances during the acceleration phase of a period (Fig. 3) at Re = 100 and f = 2 Hz.

show that there is almost a linear increase in the heat transfer enhancement with increasing ϕ due to the fact that a higher value of the particle volume fraction results in a higher thermal conductivity of the nanofluid and higher energy which accelerates the fluid.

numbers. At Re = 50, the shape of the oscillation is preserved but at Re = 100 and Re = 200, the level of the nonlinearity (distortion from a pure sinusoid) decreases with increasing ϕ values. Figs. 9 and 10 show the streamlines and isotherms for three points (Fig. 2) during the acceleration phase of a period for different nanoparticle volume fractions at Re = 100 and f = 2 Hz. The shape of the flow patterns and isotherms generally do not change with changing ϕ values, but the strength changes. Local Nusselt number distribution and length averaged Nusselt versus nanoparticle volume fraction are shown in Figs. 11 and 12. Length averaged Nusselt number plots in Fig. 12

3.2. System identification results The input data set corresponds to velocity forcing at the inlet of backward facing problem for different nanoparticle volume fractions while the output data set corresponds to length averaged Nusselt

φ =0.00

30

φ =0.02

30 20

Nux

Nux

20

10

10 0

0

10

0

20

0

10

x/S

φ =0.06

30

Nux

20

x/S

20

a b

10

c 0

0

10

20

x/S Fig. 11. Local Nusselt number distribution along the bottom wall downstream of the step for volume fractions at three time instances during the acceleration phase of a period at Re = 100 and f = 2 Hz.

F. Selimefendigil, H.F. Öztop / International Communications in Heat and Mass Transfer 45 (2013) 111–121

8

where NQ = 4, and NU = 4 represent the number of delays in the output and input, respectively. The function F1Hz is composed of a linear part and a nonlinear part which is a wavenet structure with 24 units. It is observed that adding higher order terms in the set of regressors (quadratic, cubic and fourth order terms of delayed inputs and outputs) increases the accuracy in the estimation. The model structure and delays for data set at 4 Hz are given below as

7 Re=50

Num

6

Re=100 Re=200

5 4

h Q ðt Þ ¼ F 4Hz Q ðt−1Þ; Q ðt−1Þ2 ; …     ; Q t−NQ ; Q t−NQ 2

3 2

0

0.01

0.02

0.03

0.04

0.05

0.06

φ

where NQ = 10, and NU = 10. The function F4Hz is composed of a linear part and a nonlinear part which is a wavenet structure with 16 units. The model structure and delays for data set at 8 Hz are given below as

Table 3 Data set for different frequencies and number of samples for each data set. f = 1 Hz

f = 4 Hz

f = 8 Hz

Experiment

Samples

Samples

Samples

Φ Φ Φ Φ

1297 1201 1201 1211

321 389 391 281

181 260 181 159

0.00 0.02 0.04 0.06

h   Q ðt Þ ¼ F 8Hz Q ðt−1Þ; …; Q t−NQ ;

h Q ðt Þ ¼ F 1Hz Q ðt−1Þ; Q ðt−1Þ2 ; Q ðt−1Þ3 ; Q ðt−1Þ4 ; …    2  3  4 ; Q t−NQ ; Q t−NQ ; Q t−NQ ; Q t−NQ ;

where NQ = 22, NU = 22 and 0 ≤ i,j ≤ 22. The function F8Hz is composed of a linear part and a nonlinear part which is a wavenet structure with 4 units. 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:

3

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3  2 u u N u∑k¼1 NuðkÞ−NuðkÞpred 7 6 t 7  100: Fit ¼ 6 41− 5 N ∑k¼1 NuðkÞ2 2

4

φ=0, f=1Hz, Fit=84.97%

φ=0.02, f=1Hz, Fit=91.03% 4.5

4.5

Nu

Nu

4 3.5

3.5

3 2.5

9

10

11

12

2.5

13

7

8

Time

4.5

4.5

4

4

Nu

Nu

5

3.5 3

3 2.5

9

Time

10

Sys ident CFD

3.5

2.5

8

10

φ=0.06, f=1Hz, Fit=93.25%

5

7

9

Time

φ=0.04, f=1Hz, Fit=90.44%

ð18Þ

Fig. 13 shows the fit between the CFD output and predicted model output at 1 Hz. Fig. 14 shows the low and uncorrelated residuals which is considered as one of the criteria for model determination and validation. The model fits for different frequencies are tabulated in Table 4. Except for one case (f = 8 Hz, ϕ = 0), the model fits are

ð15Þ

uðt Þ; uðt Þ ; uðt Þ ; uðt Þ ; … i 2 ; uðt−Nu Þ; uðt−N u Þ ; uðt−Nu Þ3 ; uðt−Nu Þ4

ð17Þ

uðt Þ; …; uðt−Nu Þ; i ; uðt−iÞuðt−jÞÞ

number along the bottom wall downstream of the step. The number of samples considered in the experiments for frequencies of 1 Hz, 4 Hz and 8 Hz are tabulated in Table 3. The Matlab identification toolbox has been used to select the model structure and delays, fit and validate the proposed models. The model structure and delays for data set at 1 Hz are given below as

2

ð16Þ

2 uðt Þ; uðt Þ ; … i 2 ; uðt−Nu Þ; uðt−Nu Þ

Fig. 12. Time-spatial averaged Nusselt number distribution along the bottom wall versus volume fraction for different Reynolds numbers at f = 2 Hz.

= = = =

119

7

8

9

10

Time

Fig. 13. CFD outputs and predicted outputs with system identification versus time at frequency of 1 Hz for different nanoparticle volume fractions.

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F. Selimefendigil, H.F. Öztop / International Communications in Heat and Mass Transfer 45 (2013) 111–121

Autocorrelation of residuals for output y1 (Nu)

0.1 0.05 0 −0.05 −0.1 −20

−15

−10

−5

0

5

10

15

20

Cross corr for input u1 (=V) and output y1 (Nu) resids 0.05 0 −0.05 −20

−15

−10

−5

0

5

10

15

20

Samples Fig. 14. Residual autocorrelation and cross-correlation.

around 85%. The residuals for frequencies of 4 and 8 Hz are also low and uncorrelated (Fig. 14). 4. Conclusions Numerical study and identification of forced pulsating flow at a backward facing step with a stationary cylinder subjected to nanofluid are presented. Following results are achieved: • The number of cycles to reach steady state increases with increasing frequencies and increasing Reynolds number. • Increasing the Reynolds number increases the level of nonlinearity (distortion from a pure sinusoid). This is due to the flow reversal of the heated fluid particles during the cycle. • Pulsation frequency changes the flow structure and temperature counters behind the cylinder and near the bottom wall downstream of the step especially at Re = 200. • The steepest temperature gradients are seen at the reattachment points of the vortices along the bottom wall. With increasing the frequency of oscillation, more fluctuating of the isotherms from bottom wall towards the upper wall is observed. • With increasing the frequency, heat transfer is enhanced and this is more effective at Re = 200. • The shape of the flow patterns and isotherms generally does not change with changing ϕ values, but the strength changes. • Increasing the ϕ value increases the peak value of the Nusselt number for all Re numbers, but at Re = 100 and Re = 200, the level of the nonlinearity (distortion from a pure sinusoid) decreases with increasing ϕ values. • Length averaged Nusselt number plots show that there is a linear increase in the heat transfer enhancement with increasing ϕ due to the fact that a higher value of the particle volume fraction results in a higher thermal conductivity of the nanofluid and higher energy which accelerates the fluid. Table 4 Results—fit between the CFD output and identified model output for each of the data set. Φ f (Hz)

0

0.02

0.04

0.06

1 4 8

84.97 84.56 77.96

91.03 93.79 91.08

90.44 93.24 84.46

93.25 93.42 92.79

• Nonlinear system identification toolbox of Matlab is utilized to obtain nonlinear dynamic models of data sets corresponding to different nanoparticle volume fractions at frequencies of 1, 4 and 8 Hz. • The model fits for different frequencies show that except for one case (f = 8 Hz, ϕ = 0), the model fits are around 85%. The residuals for frequencies are also low and uncorrelated. • It is shown that nonlinear dynamic models obtained from system identification toolbox could produce length averaged Nusselt number as close to as output from a high fidelity CFD computation.

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